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YEN-JIE LEE: And
welcome back, everybody,
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00:00:25,270 --> 00:00:31,960
to this fun class, 8.03.
10
00:00:31,960 --> 00:00:35,150
Let's get started.
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00:00:35,150 --> 00:00:37,960
So the first thing
which we will do
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00:00:37,960 --> 00:00:41,650
is to review a bit what
we have learned last time.
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00:00:41,650 --> 00:00:44,440
And then we'll go
to the next level
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00:00:44,440 --> 00:00:48,780
to study coupled oscillators.
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00:00:48,780 --> 00:00:50,440
OK.
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00:00:50,440 --> 00:00:55,550
Last time, we had learned a lot
on damped driven oscillators.
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00:00:55,550 --> 00:00:59,710
So as far as the course
we've been going,
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00:00:59,710 --> 00:01:04,510
actually, we only
study a single object,
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00:01:04,510 --> 00:01:09,040
and then we introduce
more and more force
20
00:01:09,040 --> 00:01:10,560
acting on this object.
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00:01:10,560 --> 00:01:14,710
We introduce damping
force, we introduce
22
00:01:14,710 --> 00:01:17,300
a driving force last time.
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00:01:17,300 --> 00:01:24,330
And we see that
the system becomes
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00:01:24,330 --> 00:01:28,720
more and more difficult to
understand because of the added
25
00:01:28,720 --> 00:01:30,150
component.
26
00:01:30,150 --> 00:01:33,930
But after the class
last time, I hope
27
00:01:33,930 --> 00:01:38,980
I convinced you that we can
understand driven oscillators.
28
00:01:38,980 --> 00:01:42,920
And there are two very important
things we learned last time.
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00:01:42,920 --> 00:01:46,610
The first one is the
transient behavior,
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00:01:46,610 --> 00:01:48,970
which is actually
a superposition
31
00:01:48,970 --> 00:01:53,560
of the homogeneous solution
and the steady state solution.
32
00:01:53,560 --> 00:01:55,130
OK.
33
00:01:55,130 --> 00:01:59,060
One very good news is that
if you are patient enough,
34
00:01:59,060 --> 00:02:03,530
you shake the
system continuously,
35
00:02:03,530 --> 00:02:08,030
and if you wait long enough,
then the homogeneous solution
36
00:02:08,030 --> 00:02:11,009
contribution goes away.
37
00:02:11,009 --> 00:02:16,200
And what is actually left over
is the steady state solution,
38
00:02:16,200 --> 00:02:20,690
which is actually much simpler
than what we saw beforehand.
39
00:02:20,690 --> 00:02:24,170
It's actually just
harmonic oscillation
40
00:02:24,170 --> 00:02:26,600
at driving frequency.
41
00:02:26,600 --> 00:02:32,060
Also, I hope that we also have
learned a very interesting
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00:02:32,060 --> 00:02:34,190
phenomenon, which is resonance.
43
00:02:34,190 --> 00:02:39,710
When the driving frequency is
close to the natural frequency
44
00:02:39,710 --> 00:02:46,280
of the system, then the
system apparently likes it.
45
00:02:46,280 --> 00:02:51,350
Then it would respond
with larger amplitude
46
00:02:51,350 --> 00:02:55,790
and oscillating up and
down at driving frequency.
47
00:02:55,790 --> 00:02:59,510
So that, we call it resonance.
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00:02:59,510 --> 00:03:01,250
This is the equation
of motion, which
49
00:03:01,250 --> 00:03:03,140
we have learned last time.
50
00:03:03,140 --> 00:03:07,280
You can see is theta double
dot plus Gamma theta dot,
51
00:03:07,280 --> 00:03:10,010
is a contribution
from the drag force,
52
00:03:10,010 --> 00:03:14,780
and omega 0 squared
theta is the contribution
53
00:03:14,780 --> 00:03:17,270
from the so-called spring force.
54
00:03:17,270 --> 00:03:23,810
And finally, that is equal to
f0 cosine omega d t, the driving
55
00:03:23,810 --> 00:03:24,950
force.
56
00:03:24,950 --> 00:03:27,170
And as we mentioned
in the beginning,
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00:03:27,170 --> 00:03:34,110
if we prepare this system
and under-damp the situation,
58
00:03:34,110 --> 00:03:38,820
then the full solution
is a superposition
59
00:03:38,820 --> 00:03:41,130
of the steady state
solution, which
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00:03:41,130 --> 00:03:46,050
is the left-hand side, the
red thing I'm pointing to,
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00:03:46,050 --> 00:03:48,600
this steady state solution.
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00:03:48,600 --> 00:03:54,390
There's no free parameter in
the steady state solution.
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00:03:54,390 --> 00:03:57,290
So A, the amplitude, is
determined by omega d.
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00:04:00,260 --> 00:04:03,820
Delta, which is the phase is
also determined by omega d.
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00:04:03,820 --> 00:04:06,940
There's no free parameter.
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00:04:06,940 --> 00:04:08,180
OK.
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00:04:08,180 --> 00:04:13,180
And in order to make the
solution a full solution,
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00:04:13,180 --> 00:04:16,630
we actually have to add in
this homogeneous solution
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00:04:16,630 --> 00:04:18,310
back into this again.
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00:04:18,310 --> 00:04:21,339
And basically, you
have B and alpha,
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00:04:21,339 --> 00:04:23,320
those are the free
parameters, which
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00:04:23,320 --> 00:04:28,780
can be determined by the
given initial conditions.
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00:04:28,780 --> 00:04:29,590
OK.
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00:04:29,590 --> 00:04:32,650
So if we go ahead and
plot some of the examples
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00:04:32,650 --> 00:04:36,910
as a function of
time, so the y-axis
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00:04:36,910 --> 00:04:38,860
is actually the amplitude.
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00:04:38,860 --> 00:04:41,290
And the x-axis is time.
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00:04:41,290 --> 00:04:43,570
And what is actually
plotted here
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00:04:43,570 --> 00:04:46,840
is a combination or
the superposition
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00:04:46,840 --> 00:04:51,640
of the steady state solution
and the homogeneous solution.
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00:04:51,640 --> 00:04:55,690
And you can see that the
individual components are also
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00:04:55,690 --> 00:04:57,610
shown in this slide.
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00:04:57,610 --> 00:05:01,960
You can see the red thing
oscillating up and down
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00:05:01,960 --> 00:05:06,540
harmonically, is steady
state solution contribution.
85
00:05:06,540 --> 00:05:10,870
And also, you have
the blue curve,
86
00:05:10,870 --> 00:05:14,180
which is decaying away
as a function of time.
87
00:05:14,180 --> 00:05:17,990
And you can see that if you
add these two curves together,
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00:05:17,990 --> 00:05:20,740
you get something
rather complicated.
89
00:05:20,740 --> 00:05:25,720
You will get some kind of
motion like, do, do, do, do.
90
00:05:25,720 --> 00:05:30,710
Then the homogeneous
solution actually dies out.
91
00:05:30,710 --> 00:05:34,960
Then what is actually left
over is just the steady state
92
00:05:34,960 --> 00:05:38,030
solution, harmonic oscillation.
93
00:05:38,030 --> 00:05:42,320
And in this case, omega d
is actually 10 times larger
94
00:05:42,320 --> 00:05:45,290
than the natural frequency.
95
00:05:45,290 --> 00:05:48,890
And there's another example
which is also very interesting.
96
00:05:48,890 --> 00:05:57,470
It's that if I make the
omega d closer to omega 0--
97
00:05:57,470 --> 00:06:00,740
OK, in this case it's actually
omega d equal to 2 times omega
98
00:06:00,740 --> 00:06:04,070
0, then you can produce
some kind of a motion, which
99
00:06:04,070 --> 00:06:04,910
is like this.
100
00:06:04,910 --> 00:06:06,920
So you have the oscillation.
101
00:06:06,920 --> 00:06:09,080
And they stayed
there for a while,
102
00:06:09,080 --> 00:06:11,440
then goes back, and
oscillates down,
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00:06:11,440 --> 00:06:14,930
and stay there then goes back.
104
00:06:14,930 --> 00:06:17,300
OK, as you can see on there.
105
00:06:17,300 --> 00:06:22,250
The homogeneous solution part
and steady state solution part
106
00:06:22,250 --> 00:06:27,170
work together and produce
this kind of strange behavior.
107
00:06:27,170 --> 00:06:27,980
OK.
108
00:06:27,980 --> 00:06:30,860
And that's just another example.
109
00:06:30,860 --> 00:06:33,010
And if you wait
long enough, again
110
00:06:33,010 --> 00:06:37,400
what is actually left over
is the steady state solution.
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00:06:37,400 --> 00:06:38,760
OK.
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00:06:38,760 --> 00:06:41,440
So what are we
going to do today?
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00:06:41,440 --> 00:06:45,660
So today, we are
going to investigate
114
00:06:45,660 --> 00:06:52,260
what will happen if we try to
put together multiple objects
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00:06:52,260 --> 00:06:55,170
and also allow them
to talk to each other.
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00:06:55,170 --> 00:06:55,770
OK.
117
00:06:55,770 --> 00:06:58,740
So if we have two objects, and
they don't talk to each other,
118
00:06:58,740 --> 00:07:01,050
then they are still
like a single object.
119
00:07:01,050 --> 00:07:04,800
They are still like simple
harmonic motion on their own.
120
00:07:04,800 --> 00:07:08,130
But if you allow them
to talk to each other,
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00:07:08,130 --> 00:07:11,820
this is the so-called
coupled oscillator, then
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00:07:11,820 --> 00:07:15,130
interesting thing happen.
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00:07:15,130 --> 00:07:23,250
So in general, coupled systems
are super, super complicated.
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00:07:23,250 --> 00:07:24,420
OK.
125
00:07:24,420 --> 00:07:27,660
So let me give you
one example here.
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00:07:27,660 --> 00:07:33,690
This is actually
two pendulums that
127
00:07:33,690 --> 00:07:36,720
are a coupled to each
other, they are actually
128
00:07:36,720 --> 00:07:39,950
connected to each
other, one pendulum,
129
00:07:39,950 --> 00:07:42,270
the second one is here, OK.
130
00:07:42,270 --> 00:07:48,270
And for example, I can actually
give it an initial velocity
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00:07:48,270 --> 00:07:49,695
and see what is going to happen.
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00:07:52,680 --> 00:07:56,130
You can see that the
resulting motion--
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00:07:56,130 --> 00:07:57,150
OK.
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00:07:57,150 --> 00:08:00,980
Remember, we are just talking
about two pendulums that
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00:08:00,980 --> 00:08:02,620
are connected to each other.
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00:08:02,620 --> 00:08:08,530
The resulting motion
is super complicated.
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00:08:08,530 --> 00:08:11,200
This is one of my
favorite demonstrations.
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00:08:11,200 --> 00:08:17,291
You can actually stare at
this machine the whole time.
139
00:08:17,291 --> 00:08:18,790
And you can see
that, huh, sometimes
140
00:08:18,790 --> 00:08:20,370
it does this rotation.
141
00:08:20,370 --> 00:08:22,190
Sometimes it doesn't do that.
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00:08:22,190 --> 00:08:25,750
And it's almost like
a living creature.
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00:08:28,840 --> 00:08:31,310
So we are going to
solve this system.
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00:08:31,310 --> 00:08:33,606
No, probably not, he knows.
145
00:08:33,606 --> 00:08:36,190
[LAUGHTER]
146
00:08:36,190 --> 00:08:41,080
But as I mentioned before,
you can always write down
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00:08:41,080 --> 00:08:43,409
the equation of motion.
148
00:08:43,409 --> 00:08:47,350
And you can solve
it by computer.
149
00:08:47,350 --> 00:08:49,560
Maybe some of the
course 6 people
150
00:08:49,560 --> 00:08:52,980
can actually try and write the
program to solve this thing
151
00:08:52,980 --> 00:08:56,820
and to simulate what
is going to happen.
152
00:08:56,820 --> 00:09:02,730
So let's take a look at this
complicated motion again.
153
00:09:02,730 --> 00:09:06,480
So you can see that the good
news is that there are only two
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00:09:06,480 --> 00:09:07,470
objects.
155
00:09:07,470 --> 00:09:10,260
And you can see--
156
00:09:10,260 --> 00:09:14,280
look at the green,
sorry, the orange dot.
157
00:09:14,280 --> 00:09:21,460
The orange dot is always
moving along a semi circle.
158
00:09:21,460 --> 00:09:27,680
But if you focus
on the yellow dot,
159
00:09:27,680 --> 00:09:30,940
the yellow is doing all
kinds of different things.
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00:09:30,940 --> 00:09:35,640
It's very hard to predict
what is going to happen.
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00:09:35,640 --> 00:09:40,130
So what I want to say is,
those are interesting examples
162
00:09:40,130 --> 00:09:41,960
of coupled systems.
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00:09:41,960 --> 00:09:45,800
But they are actually far
more complicated than what
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00:09:45,800 --> 00:09:51,050
we thought, because they
are not smooth oscillation
165
00:09:51,050 --> 00:09:54,260
around equilibrium position.
166
00:09:54,260 --> 00:09:56,920
So you can see that now
if I stop this machine
167
00:09:56,920 --> 00:10:02,940
and just perturb it slightly,
giving it a small angle
168
00:10:02,940 --> 00:10:08,370
displacement, then you
can see that the motion is
169
00:10:08,370 --> 00:10:12,560
much more easier to understand.
170
00:10:12,560 --> 00:10:14,030
You see.
171
00:10:14,030 --> 00:10:17,530
You even get one of these
questions in your p-set.
172
00:10:17,530 --> 00:10:18,795
OK, that's good news.
173
00:10:21,390 --> 00:10:25,920
So our job today
is to understand
174
00:10:25,920 --> 00:10:30,150
what is going to happen to
those coupled oscillators.
175
00:10:30,150 --> 00:10:34,560
Let me give you a few
examples before we start
176
00:10:34,560 --> 00:10:38,130
to work on a specific question.
177
00:10:38,130 --> 00:10:43,170
The second example I would
like to show you is a saw
178
00:10:43,170 --> 00:10:47,220
and you actually connect
it to two - actually
179
00:10:47,220 --> 00:10:50,970
a ruler, a metal ruler,
which is connected
180
00:10:50,970 --> 00:10:54,750
to two massive objects.
181
00:10:54,750 --> 00:11:00,090
Now I can actually give
it the initial velocity
182
00:11:00,090 --> 00:11:02,470
and see what happens.
183
00:11:02,470 --> 00:11:04,560
And you can see
that they do talk
184
00:11:04,560 --> 00:11:09,490
to each other through this
ruler, this metal ruler.
185
00:11:09,490 --> 00:11:10,020
Can you see?
186
00:11:10,020 --> 00:11:11,820
I hope you can see.
187
00:11:11,820 --> 00:11:13,530
It's a bit small.
188
00:11:13,530 --> 00:11:16,630
But it's really interesting
that you can see-- originally,
189
00:11:16,630 --> 00:11:20,440
I just introduced
some displacement
190
00:11:20,440 --> 00:11:23,100
in the left-hand side mass.
191
00:11:23,100 --> 00:11:27,270
And the left-had side
thing start to move or so
192
00:11:27,270 --> 00:11:30,260
after a while.
193
00:11:30,260 --> 00:11:36,690
There are two more examples
which I would like to give you,
194
00:11:36,690 --> 00:11:38,920
introduce to you.
195
00:11:38,920 --> 00:11:42,390
There are two
kinds of pendulums,
196
00:11:42,390 --> 00:11:44,380
which I prepared here.
197
00:11:44,380 --> 00:11:49,050
The first one is there are two
pendulums that are connected
198
00:11:49,050 --> 00:11:52,530
to each other by a spring.
199
00:11:52,530 --> 00:11:57,850
And if I try to
introduce displacement,
200
00:11:57,850 --> 00:12:03,340
I move both masses slightly and
see what is going to happen.
201
00:12:03,340 --> 00:12:07,400
And we see that the motion
is still complicated.
202
00:12:07,400 --> 00:12:12,250
Although, if you stare at
one objects, it looks more
203
00:12:12,250 --> 00:12:16,870
like harmonic oscillation,
but not quite.
204
00:12:16,870 --> 00:12:19,360
For example, this
guy is slowing down,
205
00:12:19,360 --> 00:12:21,680
and this is actually
moving faster.
206
00:12:21,680 --> 00:12:26,500
And now the right hand side
guy is actually moving faster.
207
00:12:26,500 --> 00:12:29,590
Motion seems to be completed.
208
00:12:29,590 --> 00:12:32,810
Also, you can look at this one.
209
00:12:32,810 --> 00:12:34,900
Those are the two pendulums.
210
00:12:34,900 --> 00:12:39,670
They are connected to each
other by this rod here.
211
00:12:39,670 --> 00:12:42,340
And of course, you
can displace the mass
212
00:12:42,340 --> 00:12:44,964
from the equilibrium position.
213
00:12:44,964 --> 00:12:45,880
I'm not going to hit--
214
00:12:45,880 --> 00:12:47,080
not hitting each other.
215
00:12:47,080 --> 00:12:49,730
So you can displace the
masses from each other.
216
00:12:49,730 --> 00:12:54,040
And you can see that they
do complicated things
217
00:12:54,040 --> 00:12:55,640
as a function the time.
218
00:12:55,640 --> 00:12:58,280
How are we going
to understand this?
219
00:12:58,280 --> 00:13:02,890
And I hope that by the
end of this lecture
220
00:13:02,890 --> 00:13:06,940
you are convinced that you can
as you solve this really easy,
221
00:13:06,940 --> 00:13:10,390
following a fixed procedure.
222
00:13:10,390 --> 00:13:15,130
In those examples, we
have two objects that
223
00:13:15,130 --> 00:13:17,120
are connected to each other.
224
00:13:17,120 --> 00:13:19,450
And therefore, they
talk to each other
225
00:13:19,450 --> 00:13:23,800
and produce coupled motion.
226
00:13:23,800 --> 00:13:27,650
Those are a couple
oscillator examples.
227
00:13:27,650 --> 00:13:31,000
There's another very
interesting example, which
228
00:13:31,000 --> 00:13:36,370
is called Wilberforce pendulum.
229
00:13:36,370 --> 00:13:39,220
So this is actually a pendulum.
230
00:13:39,220 --> 00:13:42,820
You can rotate like this.
231
00:13:42,820 --> 00:13:44,900
And it can also
move up and down.
232
00:13:44,900 --> 00:13:48,310
It's connected to a spring.
233
00:13:48,310 --> 00:13:51,200
The interesting thing
is that if I just
234
00:13:51,200 --> 00:13:56,270
start with some
rotation, you can
235
00:13:56,270 --> 00:14:01,910
see that it starts to also
oscillate up and down.
236
00:14:01,910 --> 00:14:02,590
You see?
237
00:14:02,590 --> 00:14:08,330
So initially I just
introduced a rotation.
238
00:14:08,330 --> 00:14:10,120
Now it's actually
fully rotating.
239
00:14:10,120 --> 00:14:13,150
And now it starts
to move up and down.
240
00:14:13,150 --> 00:14:18,600
And you can see that the
energy stored in the pendulum
241
00:14:18,600 --> 00:14:27,260
is going back and forth between
the gravitational potential,
242
00:14:27,260 --> 00:14:31,360
between the potential
of the spring,
243
00:14:31,360 --> 00:14:38,920
and also between the kinetic
energy of up and down motion
244
00:14:38,920 --> 00:14:40,320
and the rotation.
245
00:14:40,320 --> 00:14:46,840
They're actually doing all
those transitions all the time.
246
00:14:46,840 --> 00:14:50,280
So you can see--
247
00:14:50,280 --> 00:14:53,830
so initially it's just rotating.
248
00:14:53,830 --> 00:14:56,440
And then it starts
to move up and down.
249
00:14:56,440 --> 00:14:58,840
And this one is
also very similar.
250
00:14:58,840 --> 00:15:02,110
But now the mass is
much more displaced.
251
00:15:02,110 --> 00:15:07,340
And if I try to rotate this
system without introducing
252
00:15:07,340 --> 00:15:10,360
a horizontal direction
displacement,
253
00:15:10,360 --> 00:15:17,170
it still does is up and down
motion, like a simple spring
254
00:15:17,170 --> 00:15:18,620
mass system.
255
00:15:18,620 --> 00:15:23,481
So what causes this
kind of motion?
256
00:15:23,481 --> 00:15:29,590
That is because when we move
this pendulum up and down,
257
00:15:29,590 --> 00:15:34,180
we also slightly
unwind the spring.
258
00:15:34,180 --> 00:15:39,070
That can generate some
kind of torque to this mass
259
00:15:39,070 --> 00:15:44,900
and produce rotational behavior.
260
00:15:44,900 --> 00:15:47,755
And you can see
that this is just
261
00:15:47,755 --> 00:15:49,480
involving one single object.
262
00:15:49,480 --> 00:15:53,650
But there's a coupling
between the rotation
263
00:15:53,650 --> 00:15:56,890
and the horizontal
direction motion.
264
00:15:56,890 --> 00:16:04,240
So that's also special
kind of coupled oscillator.
265
00:16:04,240 --> 00:16:10,580
So after all this,
before we get started,
266
00:16:10,580 --> 00:16:13,150
I would like to say that
what we are going to do
267
00:16:13,150 --> 00:16:17,800
is to assume all those
things are ideal,
268
00:16:17,800 --> 00:16:21,670
without them being forced,
without a driving force.
269
00:16:21,670 --> 00:16:24,410
We may introduce that
later in the class.
270
00:16:24,410 --> 00:16:28,870
But for simplicity, we'll
just stay with this idea case,
271
00:16:28,870 --> 00:16:31,390
before the mass becomes
super complicated
272
00:16:31,390 --> 00:16:34,430
to solve it in front of you.
273
00:16:34,430 --> 00:16:43,020
And also, we can see that
all those complicated motion
274
00:16:43,020 --> 00:16:43,955
are just illusion.
275
00:16:46,590 --> 00:16:49,900
Actually, the reality
is that all of those
276
00:16:49,900 --> 00:16:54,250
are just superposition
of harmonic motions.
277
00:16:54,250 --> 00:16:57,680
You will see that by
the end of this class.
278
00:16:57,680 --> 00:16:59,961
So that is really amazing.
279
00:16:59,961 --> 00:17:00,460
OK.
280
00:17:00,460 --> 00:17:02,870
Let's immediately get started.
281
00:17:02,870 --> 00:17:06,609
So let's take a
look at this system
282
00:17:06,609 --> 00:17:08,980
together and see
if we can actually
283
00:17:08,980 --> 00:17:12,670
figure out the motion
of this system together.
284
00:17:12,670 --> 00:17:17,109
So I have a system with
three little masses.
285
00:17:17,109 --> 00:17:20,609
So there are three little
masses in this system.
286
00:17:20,609 --> 00:17:25,240
They are connected to
each other by spring.
287
00:17:25,240 --> 00:17:29,590
Those springs are highly
idealized, the springs.
288
00:17:29,590 --> 00:17:32,320
And they have spring constant k.
289
00:17:32,320 --> 00:17:35,950
And the natural length's l0.
290
00:17:35,950 --> 00:17:40,120
And they are placed on Earth.
291
00:17:40,120 --> 00:17:44,800
And I carefully
design the lab so
292
00:17:44,800 --> 00:17:50,150
that there's no friction
between the desk and all
293
00:17:50,150 --> 00:17:52,450
those little masses.
294
00:17:52,450 --> 00:17:56,370
So once you get started
and look at this system,
295
00:17:56,370 --> 00:17:58,540
you can imagine
that there can be
296
00:17:58,540 --> 00:18:02,860
all kinds of different
complicated motions.
297
00:18:02,860 --> 00:18:06,950
You can actually, for
example, just move this mass
298
00:18:06,950 --> 00:18:12,220
and put the other two on hold.
299
00:18:12,220 --> 00:18:14,770
And they can
oscillate like crazy.
300
00:18:14,770 --> 00:18:20,540
They can do very
similar kind of motion.
301
00:18:20,540 --> 00:18:23,950
There are many,
many possibilities.
302
00:18:23,950 --> 00:18:30,100
But if you stare at
this system long enough,
303
00:18:30,100 --> 00:18:35,560
you will be able to identify
special kinds of motion which
304
00:18:35,560 --> 00:18:39,670
are easier to understand.
305
00:18:39,670 --> 00:18:42,490
So what I would like
to introduce to you
306
00:18:42,490 --> 00:18:45,190
is a special kind
of motion which
307
00:18:45,190 --> 00:18:51,720
you can identify from the
symmetry of this system.
308
00:18:51,720 --> 00:18:53,930
That is your
so-called normal mode.
309
00:19:01,310 --> 00:19:04,970
So what is a normal mode,
a special kind of motion
310
00:19:04,970 --> 00:19:07,700
we are trying to identify?
311
00:19:07,700 --> 00:19:10,430
That is actually
the kind of motion
312
00:19:10,430 --> 00:19:23,100
which every part
of the system is
313
00:19:23,100 --> 00:19:40,975
oscillating at the same
frequency and the same phase.
314
00:19:49,770 --> 00:19:52,700
So that is your
so-called normal mode,
315
00:19:52,700 --> 00:19:56,420
and is a special kind
of motion, which I would
316
00:19:56,420 --> 00:20:00,620
like you to identify with me.
317
00:20:00,620 --> 00:20:06,170
And we would later realize that
those special kinds of motions,
318
00:20:06,170 --> 00:20:09,830
which are easier to
understand, actually
319
00:20:09,830 --> 00:20:14,660
helps us to understand the
general motion of the system.
320
00:20:14,660 --> 00:20:19,820
You will realize that the most
general motion of the system
321
00:20:19,820 --> 00:20:26,860
is just a superposition of all
the identified normal modes.
322
00:20:26,860 --> 00:20:28,580
And then we are
done, because we have
323
00:20:28,580 --> 00:20:31,300
a general solution already.
324
00:20:31,300 --> 00:20:34,100
So that's very good news.
325
00:20:34,100 --> 00:20:38,020
That tells us that
we can understand
326
00:20:38,020 --> 00:20:43,190
the system systematically,
and step by step.
327
00:20:43,190 --> 00:20:47,710
And then we can write the
general motion of the system
328
00:20:47,710 --> 00:20:51,700
as a superposition of
all the normal modes.
329
00:20:51,700 --> 00:20:54,440
So let's get started.
330
00:20:54,440 --> 00:21:00,040
So can you guess what are
the possible normal modes
331
00:21:00,040 --> 00:21:02,330
of this system?
332
00:21:02,330 --> 00:21:05,230
So that means each
part of the system
333
00:21:05,230 --> 00:21:09,670
is oscillating at the same
frequency and the same phase.
334
00:21:09,670 --> 00:21:15,970
Can anybody and any one of
you guess what can happen,
335
00:21:15,970 --> 00:21:18,550
each part of this
is an oscillating
336
00:21:18,550 --> 00:21:21,390
at the same frequency?
337
00:21:21,390 --> 00:21:22,845
Yeah?
338
00:21:22,845 --> 00:21:23,870
AUDIENCE: If the two masses
on that side are displaced
339
00:21:23,870 --> 00:21:25,328
the same amount
and then they're --
340
00:21:29,150 --> 00:21:30,610
YEN-JIE LEE: Very good.
341
00:21:30,610 --> 00:21:36,460
So he was saying that now
I displace the right hand
342
00:21:36,460 --> 00:21:40,600
side two masses all together
by a fixed amount, and also
343
00:21:40,600 --> 00:21:47,000
the left hand side, right, by
a fixed amount and then let go.
344
00:21:47,000 --> 00:21:49,011
So that's what
you're saying, right?
345
00:21:49,011 --> 00:21:49,510
OK.
346
00:21:49,510 --> 00:21:56,500
So the first mode we have
identified is like this.
347
00:21:56,500 --> 00:22:03,490
So you have left hand side
mass displace by delta x.
348
00:22:03,490 --> 00:22:07,670
And the right hand
side two masses
349
00:22:07,670 --> 00:22:09,965
are also displaced by delta x.
350
00:22:15,440 --> 00:22:20,590
So basically you hold
this three little masses
351
00:22:20,590 --> 00:22:25,530
and stretch it by the same--
352
00:22:25,530 --> 00:22:29,470
introduce the same amplitude to
all those three little masses,
353
00:22:29,470 --> 00:22:31,000
and let go.
354
00:22:31,000 --> 00:22:34,830
So that is actually
one possible mode.
355
00:22:34,830 --> 00:22:41,090
And if we do this,
then basically
356
00:22:41,090 --> 00:22:43,930
what you are going to see
is that this is actually
357
00:22:43,930 --> 00:22:50,130
roughly equal to this system.
358
00:22:52,920 --> 00:22:57,630
They're connected to each
other by two springs.
359
00:22:57,630 --> 00:23:02,250
And the right hand side part
of the system, both masses
360
00:23:02,250 --> 00:23:06,150
are oscillating at the same
amplitude and the same phase.
361
00:23:06,150 --> 00:23:14,040
They look like as if they are
just single mass with mass
362
00:23:14,040 --> 00:23:15,130
equal to 2m.
363
00:23:17,830 --> 00:23:22,490
And if you introduce a
displacement of delta x,
364
00:23:22,490 --> 00:23:25,510
then what is going to
happen is that if I
365
00:23:25,510 --> 00:23:31,210
take a look at the mass, left
hand side mass, and the force
366
00:23:31,210 --> 00:23:34,300
acting on this
mass, the force will
367
00:23:34,300 --> 00:23:42,580
be equal to minus
2k times 2 delta x,
368
00:23:42,580 --> 00:23:45,520
because that's the
amount of stretch you
369
00:23:45,520 --> 00:23:48,970
introduce to the spring.
370
00:23:48,970 --> 00:23:52,765
And that will give
you minus 4k delta x.
371
00:23:57,390 --> 00:24:00,300
And we have already solved
this kind of problem
372
00:24:00,300 --> 00:24:02,310
in the first lecture.
373
00:24:02,310 --> 00:24:05,430
So therefore you can
immediately identify
374
00:24:05,430 --> 00:24:10,690
omega, in this case,
omega a squared will
375
00:24:10,690 --> 00:24:16,500
be equal to 4k divided by 2m.
376
00:24:16,500 --> 00:24:20,880
This is actually the
effective spring constant,
377
00:24:20,880 --> 00:24:24,210
and this is actually the mass.
378
00:24:24,210 --> 00:24:31,440
So that is actually the
frequency of mode A.
379
00:24:31,440 --> 00:24:38,020
Can you identify a second kind
of motion which does that?
380
00:24:38,020 --> 00:24:39,990
So in this case, what
is going to happen
381
00:24:39,990 --> 00:24:42,600
is that the three masses will--
382
00:24:42,600 --> 00:24:45,120
OK, one, two, and three.
383
00:24:45,120 --> 00:24:50,560
The three masses will
oscillate as a function of time
384
00:24:50,560 --> 00:25:01,610
like this with angular frequency
of square root of 4k over 2m.
385
00:25:01,610 --> 00:25:05,586
What is actually a
second possible motion?
386
00:25:05,586 --> 00:25:07,410
Yes?
387
00:25:07,410 --> 00:25:10,570
AUDIENCE: All masses being
stretched [INAUDIBLE]
388
00:25:10,570 --> 00:25:11,641
YEN-JIE LEE: Compressed.
389
00:25:11,641 --> 00:25:14,082
AUDIENCE: Compressed the same--
390
00:25:14,082 --> 00:25:15,040
YEN-JIE LEE: Very good.
391
00:25:18,100 --> 00:25:24,910
I'm very lucky that I'm in front
of such a smart crowd today.
392
00:25:24,910 --> 00:25:28,330
And we have
successfully identified
393
00:25:28,330 --> 00:25:34,060
the second mode, mode B.
So what is going to happen
394
00:25:34,060 --> 00:25:41,470
is that the left hand
side mass is not moving.
395
00:25:41,470 --> 00:25:49,400
And you compress the
upper one slightly
396
00:25:49,400 --> 00:25:56,570
and you stretch the lower
one, the lower little mass
397
00:25:56,570 --> 00:25:59,700
to the opposite direction.
398
00:25:59,700 --> 00:26:03,140
The displacement is delta
x, and the displacement
399
00:26:03,140 --> 00:26:07,210
of the second mass is delta x.
400
00:26:07,210 --> 00:26:09,980
So what is going to happen?
401
00:26:09,980 --> 00:26:13,220
What is going to happen
is that the left hand side
402
00:26:13,220 --> 00:26:19,400
mass will not move at all
because the force, the spring
403
00:26:19,400 --> 00:26:25,490
force, acting on this
mass is going to cancel.
404
00:26:25,490 --> 00:26:27,650
And apparently, these
two little masses
405
00:26:27,650 --> 00:26:31,370
are going to be doing
harmonic motion.
406
00:26:34,160 --> 00:26:38,136
Since this left hand
side mass is not moving,
407
00:26:38,136 --> 00:26:47,900
it's as if this is a wall and
this were a single spring, k,
408
00:26:47,900 --> 00:26:51,130
that's connected
to a little mass.
409
00:26:51,130 --> 00:26:54,265
And it got displaced by delta x.
410
00:26:56,890 --> 00:27:02,690
So what will happen is that this
mass will experience a spring
411
00:27:02,690 --> 00:27:09,700
force, which is F equal
to minus k delta x.
412
00:27:09,700 --> 00:27:12,860
Therefore, we can
immediately identify
413
00:27:12,860 --> 00:27:17,803
omega b squared will
be equal to k over m.
414
00:27:20,380 --> 00:27:27,350
So you can see that we have
identified two kinds of modes,
415
00:27:27,350 --> 00:27:30,760
which every part of the
system is oscillating
416
00:27:30,760 --> 00:27:36,040
at the same frequency
and the same phase.
417
00:27:36,040 --> 00:27:38,100
Everybody agree?
418
00:27:38,100 --> 00:27:41,440
No not everybody agree.
419
00:27:41,440 --> 00:27:45,870
Look at this guy this
guy is not moving.
420
00:27:45,870 --> 00:27:46,830
How could this be?
421
00:27:46,830 --> 00:27:51,340
This is not the normal mode.
422
00:27:51,340 --> 00:27:51,840
Isn't it?
423
00:27:54,640 --> 00:27:55,320
OK.
424
00:27:55,320 --> 00:27:57,510
I hope that will
wake you up a bit.
425
00:28:00,030 --> 00:28:02,220
I can be very tricky here.
426
00:28:02,220 --> 00:28:08,100
I can say that this mass
is also oscillating,
427
00:28:08,100 --> 00:28:12,065
but with what amplitude?
428
00:28:12,065 --> 00:28:12,690
AUDIENCE: Zero.
429
00:28:12,690 --> 00:28:13,856
YEN-JIE LEE: Zero amplitude.
430
00:28:13,856 --> 00:28:18,870
Right So the conclusion is
that, aha, everybody is actually
431
00:28:18,870 --> 00:28:21,250
oscillating at the
same frequency,
432
00:28:21,250 --> 00:28:23,380
but these guy with
zero amplitude.
433
00:28:25,545 --> 00:28:27,920
AUDIENCE: Are they oscillating
at the same phase as well?
434
00:28:27,920 --> 00:28:29,130
YEN-JIE LEE: Yeah.
435
00:28:29,130 --> 00:28:32,400
Oh very good question.
436
00:28:32,400 --> 00:28:34,770
Another objection I receive.
437
00:28:34,770 --> 00:28:38,130
So life is hard for me today.
438
00:28:38,130 --> 00:28:39,300
Hey.
439
00:28:39,300 --> 00:28:43,530
This guy is oscillating
out of phase.
440
00:28:43,530 --> 00:28:45,660
These two guys are out of phase.
441
00:28:45,660 --> 00:28:50,310
But I can argue that the
amplitude of the first mass
442
00:28:50,310 --> 00:28:56,760
is actually has a minus sign
compared to the second mass.
443
00:28:56,760 --> 00:29:00,750
Then they are again in phase.
444
00:29:03,970 --> 00:29:05,440
So very good.
445
00:29:05,440 --> 00:29:07,540
I like those questions.
446
00:29:07,540 --> 00:29:13,220
And I hope I have convinced you
that everybody is oscillating,
447
00:29:13,220 --> 00:29:17,000
although you cannot see it,
because the amplitude is small,
448
00:29:17,000 --> 00:29:18,130
is zero.
449
00:29:18,130 --> 00:29:23,990
And they are all oscillating
at the same phase.
450
00:29:23,990 --> 00:29:24,872
Yes.
451
00:29:24,872 --> 00:29:26,689
AUDIENCE: How come
there's only one mass?
452
00:29:26,689 --> 00:29:28,230
YEN-JIE LEE: Oh,
the right hand side?
453
00:29:28,230 --> 00:29:28,874
AUDIENCE: Yeah.
454
00:29:28,874 --> 00:29:29,790
YEN-JIE LEE: Oh, yeah.
455
00:29:29,790 --> 00:29:33,780
That is because the left
hand side mass, the 2m one,
456
00:29:33,780 --> 00:29:35,490
is actually not moving.
457
00:29:35,490 --> 00:29:37,980
Because they are
two spring forces,
458
00:29:37,980 --> 00:29:39,800
one is actually
pushing the mass,
459
00:29:39,800 --> 00:29:42,330
the other one's
pulling the mass.
460
00:29:42,330 --> 00:29:45,620
And they cancel perfectly.
461
00:29:45,620 --> 00:29:52,140
Therefore, it's as if
those two guys are not--
462
00:29:52,140 --> 00:29:54,480
they don't find each other.
463
00:29:54,480 --> 00:29:57,510
And then it's like,
they are just tools
464
00:29:57,510 --> 00:30:01,890
mass connected to a wall along.
465
00:30:01,890 --> 00:30:05,950
And then you can now identify
what is the frequency.
466
00:30:05,950 --> 00:30:06,450
OK.
467
00:30:06,450 --> 00:30:08,760
Very good.
468
00:30:08,760 --> 00:30:15,630
So we make the made a lot of the
progress from the discussion.
469
00:30:15,630 --> 00:30:21,220
And now I would like
to ask you for help.
470
00:30:21,220 --> 00:30:24,797
What is the third oscillation?
471
00:30:24,797 --> 00:30:25,731
Yes?
472
00:30:25,731 --> 00:30:28,540
AUDIENCE: There's no
third normal mode.
473
00:30:28,540 --> 00:30:30,365
YEN-JIE LEE: There's
no third normal mode.
474
00:30:30,365 --> 00:30:31,948
AUDIENCE: There's
no third normal mode
475
00:30:31,948 --> 00:30:34,906
because there are
restricted to one dimension.
476
00:30:34,906 --> 00:30:37,371
I can not imagine another
mode that would not
477
00:30:37,371 --> 00:30:39,965
displace the central mass.
478
00:30:39,965 --> 00:30:41,130
YEN-JIE LEE: Very good.
479
00:30:41,130 --> 00:30:42,680
That's very good.
480
00:30:42,680 --> 00:30:46,100
On the other hand,
you can also say,
481
00:30:46,100 --> 00:30:48,960
I also take the
center mass motion
482
00:30:48,960 --> 00:30:50,410
as one of the normal mode.
483
00:30:50,410 --> 00:30:54,260
I think that's also
fair to do that.
484
00:30:54,260 --> 00:30:55,640
Very good observation.
485
00:30:55,640 --> 00:31:00,770
You can see that the whole
can move simultaneously.
486
00:31:06,120 --> 00:31:10,400
I can also argue that
they are oscillating
487
00:31:10,400 --> 00:31:12,710
at the same frequency
and the same phase,
488
00:31:12,710 --> 00:31:15,200
because they are
all moving together.
489
00:31:21,560 --> 00:31:25,630
So these are the 2m
connected to mass one.
490
00:31:31,160 --> 00:31:35,690
All of them are moving
in the same direction.
491
00:31:35,690 --> 00:31:38,350
So now I can
calculate the force.
492
00:31:38,350 --> 00:31:40,100
What is the force?
493
00:31:40,100 --> 00:31:43,050
F is 0.
494
00:31:43,050 --> 00:31:49,490
Therefore, omega c is 0.
495
00:31:49,490 --> 00:31:53,630
So you can the small
limit of omega.
496
00:31:53,630 --> 00:31:59,120
So of course, I can pretend
that those mass are connected
497
00:31:59,120 --> 00:32:05,000
to a really, really small spring
to the wall with is a spring
498
00:32:05,000 --> 00:32:06,350
constant k'.
499
00:32:06,350 --> 00:32:08,600
And I have k' goes to zero.
500
00:32:08,600 --> 00:32:11,450
And they are actually
going to oscillate
501
00:32:11,450 --> 00:32:16,880
with omega c goes to zero.
502
00:32:16,880 --> 00:32:20,640
So in this case,
the amplitude is
503
00:32:20,640 --> 00:32:27,240
going to increase forever,
because you have A sin omega c
504
00:32:27,240 --> 00:32:28,890
t.
505
00:32:28,890 --> 00:32:33,260
And this roughly A omega c t.
506
00:32:33,260 --> 00:32:37,620
And this is just vt.
507
00:32:37,620 --> 00:32:41,970
So what I want to argue
is that this is actually
508
00:32:41,970 --> 00:32:47,640
also oscillation, but with
angular frequency zero.
509
00:32:47,640 --> 00:32:58,470
And the general motion can
be in written as vt times c,
510
00:32:58,470 --> 00:33:02,240
for example, some constant.
511
00:33:02,240 --> 00:33:05,770
Any questions?
512
00:33:05,770 --> 00:33:09,310
So what I'm going to
do next may amaze you.
513
00:33:11,890 --> 00:33:12,550
Very good.
514
00:33:12,550 --> 00:33:18,790
So we have identified three
different kinds of modes.
515
00:33:18,790 --> 00:33:30,301
We have mode A, which is with
omega a squared equal to omega
516
00:33:30,301 --> 00:33:30,800
a squared.
517
00:33:30,800 --> 00:33:32,740
Where is omega a squared.
518
00:33:32,740 --> 00:33:33,610
There.
519
00:33:33,610 --> 00:33:36,410
It's 4k over 2m.
520
00:33:36,410 --> 00:33:40,870
And also, the
motion is like this.
521
00:33:40,870 --> 00:33:51,720
x1 equal t A cosine
omega a t plus phi a.
522
00:33:51,720 --> 00:34:00,420
x2 is equal to minus A, because
they have different sine.
523
00:34:00,420 --> 00:34:05,550
So if the motion is in the
left hand side direction,
524
00:34:05,550 --> 00:34:07,530
then the two masses
are oscillating
525
00:34:07,530 --> 00:34:09,420
in the opposite direction.
526
00:34:09,420 --> 00:34:14,719
So therefore, I get a minus
sign in front of A. Cosine omega
527
00:34:14,719 --> 00:34:17,310
a t plus phi a.
528
00:34:19,889 --> 00:34:28,590
x3 will be also equal to minus
A cosine omega a t plus phi a.
529
00:34:28,590 --> 00:34:32,730
Of course, I need to define
what this x1, x2, x3.
530
00:34:32,730 --> 00:34:37,770
That's why most of you
got super confused.
531
00:34:37,770 --> 00:34:43,690
So the x1, what
I mean is that is
532
00:34:43,690 --> 00:34:50,719
that the displacement of
the mass 2m, I call it x1.
533
00:34:50,719 --> 00:34:56,590
The displacement of the upper
mass, the upper little mass,
534
00:34:56,590 --> 00:34:59,110
I call it x2.
535
00:34:59,110 --> 00:35:05,410
And finally, the displacement
of the third mass, I call it x3.
536
00:35:05,410 --> 00:35:09,160
Therefore, you can
see that mode A,
537
00:35:09,160 --> 00:35:12,340
you have this kind of motion.
538
00:35:12,340 --> 00:35:17,500
The amplitude of the first
mass is A. Therefore,
539
00:35:17,500 --> 00:35:21,320
if I define that to be A,
then the second and third one,
540
00:35:21,320 --> 00:35:24,220
or the amplitude will
be defined as minus A.
541
00:35:24,220 --> 00:35:27,640
And you can see that all
of them are oscillating
542
00:35:27,640 --> 00:35:31,090
at fixed angular
frequency, omega
543
00:35:31,090 --> 00:35:37,156
a, omega a, omega a; and also
fixed phase, phi a, phi a, phi
544
00:35:37,156 --> 00:35:37,655
a.
545
00:35:42,260 --> 00:35:44,570
Of course, we can
also write down
546
00:35:44,570 --> 00:35:49,210
what we get for
mode B. For mode B,
547
00:35:49,210 --> 00:35:53,780
the left hand side mass
is not moving, stay put.
548
00:35:53,780 --> 00:35:56,930
And the other two
masses are oscillating
549
00:35:56,930 --> 00:36:00,470
at the frequency of omega b.
550
00:36:00,470 --> 00:36:06,680
And amplitude, they
differ by a minus sign.
551
00:36:06,680 --> 00:36:08,580
OK.
552
00:36:08,580 --> 00:36:12,820
Omega b squared is
equal to k over m
553
00:36:12,820 --> 00:36:18,320
from that logical argument.
554
00:36:18,320 --> 00:36:26,495
And then we get x1 equal
to 0 times cosine omega
555
00:36:26,495 --> 00:36:31,020
b t plus phi b.
556
00:36:31,020 --> 00:36:40,460
x2, I get B cosine
omega b t plus phi b.
557
00:36:40,460 --> 00:36:49,329
x3, I get minus B cosine
omega b t plus phi b.
558
00:36:49,329 --> 00:36:50,120
Any questions here?
559
00:36:53,120 --> 00:36:58,480
Finally, mode C.
All the mass, x1
560
00:36:58,480 --> 00:37:06,460
is equal to x2 is equal to
x3, is equal to C plus vt.
561
00:37:10,450 --> 00:37:17,470
So you can see that we have
identified three modes, mode
562
00:37:17,470 --> 00:37:25,660
A, mode B, and the mode C.
And there are three angular
563
00:37:25,660 --> 00:37:31,840
frequencies which we identified
for all of those normal modes,
564
00:37:31,840 --> 00:37:35,950
omega a, omega b, and omega c.
565
00:37:35,950 --> 00:37:38,380
And you can see that
we also identified
566
00:37:38,380 --> 00:37:40,920
how many free parameters.
567
00:37:40,920 --> 00:37:51,515
One free parameter, two,
three, four, five, and six.
568
00:37:54,470 --> 00:37:58,220
If you careful, you
write down the equation
569
00:37:58,220 --> 00:38:04,850
of motion of this system,
you will have three
570
00:38:04,850 --> 00:38:08,900
coupled differential equations.
571
00:38:08,900 --> 00:38:12,470
And those are second order
differential equations.
572
00:38:12,470 --> 00:38:17,570
If you have three variables,
three second order differential
573
00:38:17,570 --> 00:38:19,310
equations.
574
00:38:19,310 --> 00:38:23,660
If you manage it magically,
with the help from a computer
575
00:38:23,660 --> 00:38:30,800
or from math department
people, how many free parameter
576
00:38:30,800 --> 00:38:35,420
would you expect in
you a general solution?
577
00:38:35,420 --> 00:38:38,680
Can anybody tell
me well how many?
578
00:38:38,680 --> 00:38:44,131
I have three second order
differential equations.
579
00:38:44,131 --> 00:38:44,630
Yes?
580
00:38:44,630 --> 00:38:45,670
AUDIENCE: 6?
581
00:38:45,670 --> 00:38:48,180
YEN-JIE LEE: 6.
582
00:38:48,180 --> 00:38:53,220
So look at what we have
done we identified already
583
00:38:53,220 --> 00:38:56,190
1, 2, 3, three normal modes.
584
00:38:56,190 --> 00:39:03,030
By there are 1, 2, 3, 4,
5 6, six free parameters.
585
00:39:03,030 --> 00:39:05,985
That tells me I am done.
586
00:39:09,640 --> 00:39:10,280
I'm done.
587
00:39:10,280 --> 00:39:15,140
Because what is the
general solution?
588
00:39:15,140 --> 00:39:20,960
The general solution is just a
superposition of mode A mode B
589
00:39:20,960 --> 00:39:27,620
and mode C. You have six
free meters to be determined
590
00:39:27,620 --> 00:39:31,790
by six initial conditions,
which I would like--
591
00:39:31,790 --> 00:39:36,650
I have to tell you what are
those initial conditions.
592
00:39:36,650 --> 00:39:40,730
So isn't this amazing to you?
593
00:39:40,730 --> 00:39:44,390
I didn't even solve the
differential equation,
594
00:39:44,390 --> 00:39:48,050
and I already get the solution.
595
00:39:48,050 --> 00:39:50,540
And you can see another
lesson we learned
596
00:39:50,540 --> 00:39:53,750
from here is that,
oh no, you can
597
00:39:53,750 --> 00:39:56,180
imagine that the
motion of the system
598
00:39:56,180 --> 00:39:58,310
can be super complicated.
599
00:39:58,310 --> 00:40:02,180
This whole thing can do
this, all the crazy things
600
00:40:02,180 --> 00:40:04,190
are all displaced,
and the center of mass
601
00:40:04,190 --> 00:40:06,860
can move, as you said.
602
00:40:06,860 --> 00:40:12,830
But the result is actually
very easy to understand.
603
00:40:12,830 --> 00:40:17,360
It's just three kinds of motion,
the displacement, and two kinds
604
00:40:17,360 --> 00:40:21,410
of simple harmonic motion.
605
00:40:21,410 --> 00:40:22,760
We add them together.
606
00:40:22,760 --> 00:40:27,590
And then you get the general
description of that system.
607
00:40:27,590 --> 00:40:31,010
So everything is so nice.
608
00:40:31,010 --> 00:40:34,690
We understand the
motion of that system.
609
00:40:34,690 --> 00:40:41,180
But in general,
life is very hard.
610
00:40:41,180 --> 00:40:46,010
For example, now I do
something crazy here.
611
00:40:46,010 --> 00:40:51,070
I change this to 3.
612
00:40:51,070 --> 00:40:53,350
What are the normal modes?
613
00:40:53,350 --> 00:40:55,550
Can anybody tell me?
614
00:40:55,550 --> 00:40:59,050
It becomes very, very
difficult, because there's
615
00:40:59,050 --> 00:41:03,370
no general symmetry of
that kind of system.
616
00:41:03,370 --> 00:41:05,890
So we are in trouble.
617
00:41:05,890 --> 00:41:09,250
One of the modes
maybe still there,
618
00:41:09,250 --> 00:41:13,770
which is actually mode
B. But the other modes
619
00:41:13,770 --> 00:41:16,900
are harder to actually guess.
620
00:41:16,900 --> 00:41:22,790
So you can see that that already
brings you a lot of trouble.
621
00:41:22,790 --> 00:41:29,330
And you can see that I can now
couple not just two objects,
622
00:41:29,330 --> 00:41:32,750
I can couple three objects,
four objects, five objects,
623
00:41:32,750 --> 00:41:33,920
10 objects.
624
00:41:33,920 --> 00:41:37,020
Maybe I will put that in your
p set and see what happens.
625
00:41:37,020 --> 00:41:39,980
And you can see
that this becomes
626
00:41:39,980 --> 00:41:42,590
very difficult to manage.
627
00:41:42,590 --> 00:41:47,030
So what I'm going to do in
the rest of this lecture
628
00:41:47,030 --> 00:41:49,760
is to introduce
you a method which
629
00:41:49,760 --> 00:41:54,800
you can follow in general
to solve the question
630
00:41:54,800 --> 00:41:59,120
and get the normal mode
frequencies and normal modes.
631
00:41:59,120 --> 00:42:01,950
So we will take a
four minute break.
632
00:42:01,950 --> 00:42:04,940
And we come back at 12:20.
633
00:42:04,940 --> 00:42:07,160
So if you have any
questions, let me know.
634
00:42:12,790 --> 00:42:17,380
What we are going to do
in the following exercise
635
00:42:17,380 --> 00:42:22,750
is to try to understand a
general strategy to solve
636
00:42:22,750 --> 00:42:26,980
the normal mode frequencies
and the normal mode amplitudes,
637
00:42:26,980 --> 00:42:30,640
so that you can apply this
technique to all kinds
638
00:42:30,640 --> 00:42:32,320
of different systems.
639
00:42:32,320 --> 00:42:36,590
So what I am going to do today
is to take these three mass
640
00:42:36,590 --> 00:42:41,595
system, and of course as
usual, I try to define what
641
00:42:41,595 --> 00:42:43,600
is this coordinate system?
642
00:42:43,600 --> 00:42:45,520
The coordinate system
I'm going to use
643
00:42:45,520 --> 00:42:51,790
is x1 and x2 an x3 describing
the displacement of the mass
644
00:42:51,790 --> 00:42:53,750
from the equilibrium position.
645
00:42:53,750 --> 00:42:55,420
And the equilibrium
means that there's
646
00:42:55,420 --> 00:42:57,970
no stretch on the spring.
647
00:42:57,970 --> 00:43:00,310
The string is unstretched.
648
00:43:00,310 --> 00:43:05,780
It's at their own
natural length, l0.
649
00:43:05,780 --> 00:43:12,910
So once I define that, I can
do a force diagram analysis.
650
00:43:12,910 --> 00:43:17,140
So that starts from
the left hand side mass
651
00:43:17,140 --> 00:43:19,870
with mass equal to 2m.
652
00:43:19,870 --> 00:43:23,350
I can write down the equation
of motion, 2m x1 double dot.
653
00:43:26,290 --> 00:43:37,730
And this is going to be equal
to k x2 minus x1 plus k x3
654
00:43:37,730 --> 00:43:38,230
minus x1.
655
00:43:40,870 --> 00:43:45,850
So there are two spring
forces acting on this mass,
656
00:43:45,850 --> 00:43:47,140
the left hand side mass.
657
00:43:47,140 --> 00:43:49,990
The first one is
the upper spring.
658
00:43:49,990 --> 00:43:55,720
The second one is coming
from the lower one.
659
00:43:55,720 --> 00:43:58,340
And you can see
that both of them
660
00:43:58,340 --> 00:44:00,940
are proportional to
spring constant k,
661
00:44:00,940 --> 00:44:05,650
and also proportional to
the relative displacement.
662
00:44:05,650 --> 00:44:09,610
And you can see that the two
relative displacement, which
663
00:44:09,610 --> 00:44:14,130
is the amount of
stretch to the spring,
664
00:44:14,130 --> 00:44:20,540
is actually x2 minus
x1, and the x3 minus x1.
665
00:44:20,540 --> 00:44:22,720
Am I going too fast?
666
00:44:22,720 --> 00:44:23,220
OK.
667
00:44:23,220 --> 00:44:24,410
Everybody's following.
668
00:44:24,410 --> 00:44:26,420
And you can actually
check the sign.
669
00:44:26,420 --> 00:44:27,620
So you may not be sure.
670
00:44:27,620 --> 00:44:30,470
Maybe this is your x1 minus x2.
671
00:44:30,470 --> 00:44:33,200
But you can check that,
because if you increase
672
00:44:33,200 --> 00:44:36,200
x1, what is going to happen?
673
00:44:36,200 --> 00:44:38,780
This term will
become more negative.
674
00:44:38,780 --> 00:44:42,820
More negative in this
coordinate system
675
00:44:42,820 --> 00:44:45,410
is pointing to the
left hand side.
676
00:44:45,410 --> 00:44:46,670
So that makes sense.
677
00:44:46,670 --> 00:44:49,890
Because if I move this
mass to the right side,
678
00:44:49,890 --> 00:44:53,380
then I am compressing
the springs.
679
00:44:53,380 --> 00:44:55,520
Therefore, they are
pushing it back.
680
00:44:55,520 --> 00:45:02,210
Therefore, this is actually
the correct sign, x2 minus x1.
681
00:45:02,210 --> 00:45:07,130
The same thing also applies
to the second spring force.
682
00:45:07,130 --> 00:45:10,655
So that's a way I double
check if I make a mistake.
683
00:45:13,220 --> 00:45:16,450
Now, this is actually the
first equations of motion.
684
00:45:16,450 --> 00:45:20,300
And I can now also
work on a second mass.
685
00:45:20,300 --> 00:45:23,540
Now I focus on a
mass number two.
686
00:45:23,540 --> 00:45:26,020
The displacement is x2.
687
00:45:26,020 --> 00:45:28,760
Therefore the left hand
side of Newton's Law
688
00:45:28,760 --> 00:45:32,780
is m x2 double dot.
689
00:45:32,780 --> 00:45:36,920
And that is equal
to the spring force.
690
00:45:36,920 --> 00:45:42,650
The spring force, there's
only one spring force
691
00:45:42,650 --> 00:45:44,510
acting on the mass.
692
00:45:44,510 --> 00:45:48,790
Therefore, what I am going
to get is k x1 minus x2.
693
00:45:53,850 --> 00:45:55,060
Everybody's following?
694
00:45:57,830 --> 00:46:02,730
You can actually check
the sign carefully, also.
695
00:46:02,730 --> 00:46:05,780
And finally, I have the
third mass, very similar
696
00:46:05,780 --> 00:46:07,850
to mass number two.
697
00:46:07,850 --> 00:46:10,380
I can write down the
equation of motion,
698
00:46:10,380 --> 00:46:16,120
which should k x1 minus x3.
699
00:46:16,120 --> 00:46:21,740
So that is my coupled second
order differential equations.
700
00:46:21,740 --> 00:46:23,060
There are three equations.
701
00:46:23,060 --> 00:46:27,260
And all of them are
second order equations.
702
00:46:27,260 --> 00:46:30,050
So this looks a bit messy.
703
00:46:30,050 --> 00:46:32,810
So what I'm going
to do Is no magic.
704
00:46:32,810 --> 00:46:37,550
I'm just collecting all
the terms belonging to x1,
705
00:46:37,550 --> 00:46:41,480
and put them together, all
the terms belonging to x2,
706
00:46:41,480 --> 00:46:45,470
putting all together, and
just rearranging things.
707
00:46:45,470 --> 00:46:47,300
So no magic.
708
00:46:47,300 --> 00:46:50,340
So I copied this
thing, left hand side.
709
00:46:50,340 --> 00:46:52,790
2m x1 double dot.
710
00:46:52,790 --> 00:47:00,260
And the dot will be equal
to minus 2k x1 plus,
711
00:47:00,260 --> 00:47:05,930
I collect all the times
related to x2, plus k x2,
712
00:47:05,930 --> 00:47:13,070
there's only one term
here, then plus k x3.
713
00:47:13,070 --> 00:47:17,065
I'm just trying to
organize my question.
714
00:47:17,065 --> 00:47:19,190
So you can see that I
collect all the terms related
715
00:47:19,190 --> 00:47:21,800
to x1 to here.
716
00:47:21,800 --> 00:47:24,560
Minus k, minus k,
I get minus 2k.
717
00:47:24,560 --> 00:47:28,850
And the plus k for
x2, plus k for the x3.
718
00:47:28,850 --> 00:47:32,480
And I can also do that
for m x2 double dot.
719
00:47:32,480 --> 00:47:41,930
That will be equal to k x1
minus k x2 plus zero x3,
720
00:47:41,930 --> 00:47:44,900
just for completeness.
721
00:47:44,900 --> 00:47:50,207
I can also do the same thing
for the third mass, m x3 double
722
00:47:50,207 --> 00:47:50,940
dot.
723
00:47:50,940 --> 00:47:58,920
This is equal to k x1 plus 0 x2.
724
00:47:58,920 --> 00:48:04,880
There's no dependence on
x2, because x1 and x2--
725
00:48:04,880 --> 00:48:09,440
x3 and x2 are not talking
to each either directly.
726
00:48:09,440 --> 00:48:13,880
Finally, I have the third,
which is minus k x3.
727
00:48:19,720 --> 00:48:27,400
Now our job is to solve
those coupled equations.
728
00:48:27,400 --> 00:48:29,720
Of course, you have
the freedom, if you
729
00:48:29,720 --> 00:48:33,130
know how to solve
it yourself, you
730
00:48:33,130 --> 00:48:35,570
can already go
ahead and solve it.
731
00:48:35,570 --> 00:48:37,540
But what I am
going to do here is
732
00:48:37,540 --> 00:48:41,560
to introduce technique,
which can be useful for you
733
00:48:41,560 --> 00:48:44,750
and make it easier to follow.
734
00:48:44,750 --> 00:48:47,020
It's a fixed procedure.
735
00:48:47,020 --> 00:48:49,880
So what I can do
is the following.
736
00:48:49,880 --> 00:48:56,450
I can write everything
in them form of a matrix.
737
00:48:56,450 --> 00:49:00,480
How many of you heard the
matrix for the first time?
738
00:49:00,480 --> 00:49:03,550
1, 2, 3, 4.
739
00:49:03,550 --> 00:49:04,050
OK.
740
00:49:04,050 --> 00:49:05,470
Only four.
741
00:49:05,470 --> 00:49:09,490
But if you are not being
familiar with matrix,
742
00:49:09,490 --> 00:49:11,160
let me know, and I can help you.
743
00:49:11,160 --> 00:49:12,370
Let the TA know.
744
00:49:12,370 --> 00:49:17,860
And also, there's a section
in the textbooks, which
745
00:49:17,860 --> 00:49:21,790
I posted on announcement,
which is actually very
746
00:49:21,790 --> 00:49:24,260
helpful to understand matrices.
747
00:49:24,260 --> 00:49:26,680
But sorry to these
four students,
748
00:49:26,680 --> 00:49:28,000
we are going to use that.
749
00:49:28,000 --> 00:49:32,500
And maybe, you already learn
how it works from here.
750
00:49:32,500 --> 00:49:36,280
So one trick which we
will use in this class
751
00:49:36,280 --> 00:49:41,500
is to convert everything
into matrix format.
752
00:49:41,500 --> 00:49:44,260
What I am going to do
is to write everything
753
00:49:44,260 --> 00:49:49,270
in terms of M,
capital X, capital M,
754
00:49:49,270 --> 00:49:56,620
capital X double dot equal
to minus capital K capital X.
755
00:49:56,620 --> 00:50:01,450
Capital M, capital
X, and capital K,
756
00:50:01,450 --> 00:50:05,800
those are all matrices.
757
00:50:05,800 --> 00:50:09,130
Because I write this
thing I really carefully,
758
00:50:09,130 --> 00:50:12,640
therefore we can already
immediately identify
759
00:50:12,640 --> 00:50:18,010
what would be M, capital
M and capital X and a K.
760
00:50:18,010 --> 00:50:28,210
So I can write down immediately
will be equal to 2m, 0, 0, 0,
761
00:50:28,210 --> 00:50:35,190
m, 0, 0, 0, m.
762
00:50:35,190 --> 00:50:39,750
Because there's only one in each
line, you only have one term.
763
00:50:39,750 --> 00:50:44,250
X1 double dot, x2 double
dot, x3 double dot.
764
00:50:44,250 --> 00:50:46,290
And also, you can
write down what
765
00:50:46,290 --> 00:50:51,030
will be the X. This
is actually a vector.
766
00:50:51,030 --> 00:50:56,040
X will be equal to x1, x2, x3.
767
00:50:59,680 --> 00:51:02,310
Finally, you have the K?
768
00:51:02,310 --> 00:51:05,860
How do I read off K?
769
00:51:05,860 --> 00:51:08,010
Careful, there's
a minus sign here,
770
00:51:08,010 --> 00:51:13,510
because I would like to make
this matrix equation as if it's
771
00:51:13,510 --> 00:51:16,740
describing a simple
harmonic motion of a one
772
00:51:16,740 --> 00:51:18,080
dimensional system.
773
00:51:18,080 --> 00:51:20,940
So it looks the same, but they
are different because those
774
00:51:20,940 --> 00:51:22,810
are matrices.
775
00:51:22,810 --> 00:51:25,710
But therefore, I
have in my convention
776
00:51:25,710 --> 00:51:27,780
I have this minus sign there.
777
00:51:27,780 --> 00:51:32,300
Therefore, when you read off
the K, you have to be careful.
778
00:51:32,300 --> 00:51:35,124
So what is K?
779
00:51:35,124 --> 00:51:40,950
K is equal to 2k.
780
00:51:40,950 --> 00:51:44,190
You have the minus 2k
here in front of x1.
781
00:51:44,190 --> 00:51:47,520
But because I have a minus
sign there, therefore
782
00:51:47,520 --> 00:51:50,610
this one is actually taken out.
783
00:51:50,610 --> 00:51:53,580
So we have 2k there.
784
00:51:53,580 --> 00:52:04,570
Then you have minus k,
minus k, minus k, k, 0.
785
00:52:04,570 --> 00:52:08,370
Minus k, plus k, 0.
786
00:52:08,370 --> 00:52:14,220
And finally, you can
also finish the last row.
787
00:52:14,220 --> 00:52:17,618
You get minus k, 0, k.
788
00:52:21,960 --> 00:52:23,740
K becomes minus k.
789
00:52:23,740 --> 00:52:27,560
Minus k becomes k.
790
00:52:27,560 --> 00:52:32,930
So we have read off all
those matrices successfully.
791
00:52:32,930 --> 00:52:38,060
So you may ask,
what do they mean?
792
00:52:38,060 --> 00:52:40,040
Do they get the meaning?
793
00:52:40,040 --> 00:52:45,350
M, K, X, what those?
794
00:52:45,350 --> 00:52:52,370
M, capital M matrix, tells
you the mass distribution
795
00:52:52,370 --> 00:52:54,270
inside the system.
796
00:52:54,270 --> 00:52:58,390
So that's the meaning
of this matrix.
797
00:52:58,390 --> 00:53:02,180
X is actually
vector, which tells
798
00:53:02,180 --> 00:53:08,800
the position of individual
components in the system.
799
00:53:08,800 --> 00:53:11,140
Finally, what is K?
800
00:53:11,140 --> 00:53:16,890
K is telling you how each
component in the system
801
00:53:16,890 --> 00:53:21,420
talks to the other components.
802
00:53:21,420 --> 00:53:25,470
So K is telling you
the communication
803
00:53:25,470 --> 00:53:29,550
inside the system.
804
00:53:29,550 --> 00:53:33,180
So now we understand a
bit what is going on.
805
00:53:33,180 --> 00:53:38,040
And as usual, I will go
to the complex notation.
806
00:53:45,790 --> 00:53:53,310
So I have xj, the small
xj are the position
807
00:53:53,310 --> 00:53:57,310
of the mass, x1, x2, and x3.
808
00:53:57,310 --> 00:54:04,540
xj will be real
part of small zj.
809
00:54:04,540 --> 00:54:07,710
Small xj equal to
real part of zj.
810
00:54:07,710 --> 00:54:14,370
Therefore, I can now write
everything in terms of matrices
811
00:54:14,370 --> 00:54:15,150
again.
812
00:54:15,150 --> 00:54:20,708
So now I can write
the solution to be Z,
813
00:54:20,708 --> 00:54:30,410
the capitol z is a matrix,
exponential i omega t plus phi.
814
00:54:30,410 --> 00:54:33,320
This is the guess
the solution I have.
815
00:54:33,320 --> 00:54:37,640
A1, A2, and A3.
816
00:54:37,640 --> 00:54:41,130
Those are the
amplitudes, amplitude A
817
00:54:41,130 --> 00:54:45,620
of the first mass, amplitude
of the second mass, amplitude
818
00:54:45,620 --> 00:54:52,020
of the third mass,
in their normal mode.
819
00:54:52,020 --> 00:55:00,300
And all of those are oscillating
at the same frequency, omega,
820
00:55:00,300 --> 00:55:03,720
and the same phase, phi.
821
00:55:03,720 --> 00:55:08,580
Does that tell you something
which we learned before?
822
00:55:08,580 --> 00:55:12,150
Oh, that's the definition
of the normal mode.
823
00:55:12,150 --> 00:55:15,150
I'm using the definition
of the normal mode.
824
00:55:15,150 --> 00:55:17,580
Every part of the
system is oscillating
825
00:55:17,580 --> 00:55:20,520
at the same frequency
in the same phase.
826
00:55:20,520 --> 00:55:25,590
And we use that to
construct my solution.
827
00:55:25,590 --> 00:55:29,940
The complex version is
exponential i omega t plus phi,
828
00:55:29,940 --> 00:55:32,000
oscillating at the
same frequency,
829
00:55:32,000 --> 00:55:34,680
oscillating at the same phase.
830
00:55:34,680 --> 00:55:37,730
And those are the amplitude,
which I will solve later.
831
00:55:37,730 --> 00:55:41,280
OK, any questions?
832
00:55:41,280 --> 00:55:44,100
I hope I'm not going too fast.
833
00:55:44,100 --> 00:55:48,060
If everybody can
follow, now I can
834
00:55:48,060 --> 00:55:54,540
go ahead and solve the
equation in the matrix format.
835
00:55:54,540 --> 00:55:59,550
So now I go to the
complex notation.
836
00:55:59,550 --> 00:56:06,510
So the equation M X double
dot equal to minus KX
837
00:56:06,510 --> 00:56:12,810
becomes M Z double
dot equal to minus KZ.
838
00:56:17,990 --> 00:56:22,530
And also, I can immediately
get the Z double dot will
839
00:56:22,530 --> 00:56:27,920
be equal to minus
omega squared Z,
840
00:56:27,920 --> 00:56:31,800
because each time I
do a differentiation,
841
00:56:31,800 --> 00:56:37,470
I get i omega out of the
exponential function.
842
00:56:37,470 --> 00:56:39,460
And I cannot kill that
exponential function,
843
00:56:39,460 --> 00:56:40,950
so it's still there.
844
00:56:40,950 --> 00:56:43,440
Therefore, I get
minus omega squared
845
00:56:43,440 --> 00:56:48,360
in front of Z. I hope
that doesn't surprise you.
846
00:56:48,360 --> 00:56:50,880
So that's very nice
and very good news.
847
00:56:50,880 --> 00:56:55,010
That means I can replace this
Z double prime with minus omega
848
00:56:55,010 --> 00:57:01,050
squared Z. Then I get
minus M omega squared Z.
849
00:57:01,050 --> 00:57:04,200
And this is equal to minus KZ.
850
00:57:04,200 --> 00:57:06,600
And I can cancel the minus sign.
851
00:57:06,600 --> 00:57:08,230
That becomes
something like this.
852
00:57:12,930 --> 00:57:20,790
So, I can now cancel the
exponential i omega t plus phi,
853
00:57:20,790 --> 00:57:23,430
because I have Z in
the left hand side.
854
00:57:23,430 --> 00:57:28,270
And exponential i omega t
plus phi is just a number.
855
00:57:28,270 --> 00:57:30,780
So therefore, I can cancel it.
856
00:57:30,780 --> 00:57:34,420
So what is going to
happen if I do that?
857
00:57:34,420 --> 00:57:37,740
Basically, what
I'm going get is I
858
00:57:37,740 --> 00:57:44,670
get M omega squared
A equal to K A.
859
00:57:44,670 --> 00:57:48,330
I'm trying to go extremely
slowly, because this
860
00:57:48,330 --> 00:57:52,200
is the first time we
go through matrices.
861
00:57:52,200 --> 00:57:55,020
So now you have this expression.
862
00:57:55,020 --> 00:58:00,410
Left hand is a matrix, M, times
some constant, omega squared.
863
00:58:00,410 --> 00:58:02,910
I can actually get omega
squared in front of it,
864
00:58:02,910 --> 00:58:06,060
because this is
actually just a number.
865
00:58:06,060 --> 00:58:12,480
A is just a vector, which is
A1, A2, A3, also a matrix.
866
00:58:12,480 --> 00:58:17,340
K is actually how the individual
components talks to the others.
867
00:58:17,340 --> 00:58:20,490
So that's there, times A.
868
00:58:20,490 --> 00:58:23,910
Now I would like
to move everything
869
00:58:23,910 --> 00:58:26,640
to the right hand side,
all the matrices in front A
870
00:58:26,640 --> 00:58:27,840
to the right hand side.
871
00:58:27,840 --> 00:58:32,760
Then I multiply both
sides by M minus 1.
872
00:58:32,760 --> 00:58:39,100
So I multiply M minus 1
to the whole equation.
873
00:58:39,100 --> 00:58:43,110
M minus 1, what is M minus 1?
874
00:58:43,110 --> 00:58:46,710
The definition is
that the inverse of M
875
00:58:46,710 --> 00:58:48,940
is called M minus 1.
876
00:58:48,940 --> 00:58:57,240
M minus 1 times M is equal to
I, which is actually 1, 1, 1.
877
00:58:57,240 --> 00:59:02,670
Therefore, if I do this thing,
then I would get omega squared.
878
00:59:02,670 --> 00:59:09,970
M minus 1 times M
becomes I, unit matrix.
879
00:59:09,970 --> 00:59:17,280
And this is equal to M
minus 1 K A. And be careful,
880
00:59:17,280 --> 00:59:21,140
I multiply M minus
1, the inverse of M,
881
00:59:21,140 --> 00:59:23,060
from the left hand side.
882
00:59:23,060 --> 00:59:25,800
That matters.
883
00:59:25,800 --> 00:59:30,390
So now I can move
everything to the same side.
884
00:59:30,390 --> 00:59:33,970
I moved the left hand side
term to the right hand side.
885
00:59:33,970 --> 00:59:40,670
Therefore, I get M minus
1 K minus omega squared
886
00:59:40,670 --> 00:59:46,430
I. Those are all matrices.
887
00:59:46,430 --> 00:59:49,330
Times A, this is equal to 0.
888
00:59:54,400 --> 00:59:55,370
Any questions?
889
01:00:00,430 --> 01:00:02,950
So a lot of manipulation.
890
01:00:02,950 --> 01:00:07,880
But if you think about it,
and you are following me,
891
01:00:07,880 --> 01:00:11,840
you'll see that all
those steps are exactly
892
01:00:11,840 --> 01:00:17,140
identical to what we have been
doing for a single harmonic
893
01:00:17,140 --> 01:00:18,440
oscillator.
894
01:00:18,440 --> 01:00:20,000
Looks pretty familiar to you.
895
01:00:20,000 --> 01:00:24,708
But the difference is that now
we are dealing with matrices.
896
01:00:24,708 --> 01:00:25,910
AUDIENCE: What is A?
897
01:00:25,910 --> 01:00:30,870
YEN-JIE LEE: Oh, A. A
is actually this guy.
898
01:00:30,870 --> 01:00:33,620
I define this to be
A. And that means
899
01:00:33,620 --> 01:00:40,930
Z will be exponential i omega
t plus phi times A. I didn't
900
01:00:40,930 --> 01:00:42,360
actually write it explicitly.
901
01:00:42,360 --> 01:00:43,910
But that's what I mean.
902
01:00:46,500 --> 01:00:49,412
Any more questions?
903
01:00:49,412 --> 01:00:50,861
Yes?
904
01:00:50,861 --> 01:00:55,605
AUDIENCE: [INAUDIBLE]
is for [INAUDIBLE]??
905
01:00:55,605 --> 01:00:56,980
YEN-JIE LEE: Can
you repeat that?
906
01:00:56,980 --> 01:01:01,230
AUDIENCE: So this whole
process, this is mode A, right?
907
01:01:01,230 --> 01:01:01,980
YEN-JIE LEE: Yeah.
908
01:01:01,980 --> 01:01:05,970
So this whole process is
for, not really the for mode
909
01:01:05,970 --> 01:01:09,390
A. So that A may be confusing.
910
01:01:09,390 --> 01:01:12,080
But in general, if
I have a solution,
911
01:01:12,080 --> 01:01:16,830
and I assume that the amplitude
can be described by a matrix.
912
01:01:16,830 --> 01:01:17,820
So it's in general.
913
01:01:17,820 --> 01:01:19,630
And you'll see that
we can actually
914
01:01:19,630 --> 01:01:24,690
derive the angular frequency
of mode A, mode B, and mode C
915
01:01:24,690 --> 01:01:26,290
afterward.
916
01:01:26,290 --> 01:01:28,020
I hope that answers
your question.
917
01:01:28,020 --> 01:01:31,350
So you see that for in
general, what I have been doing
918
01:01:31,350 --> 01:01:34,320
is that now, all
those things are
919
01:01:34,320 --> 01:01:39,520
equivalent to the original
equation of motion.
920
01:01:39,520 --> 01:01:44,780
What I am doing is
purely cosmetic.
921
01:01:44,780 --> 01:01:47,810
You see, make it beautiful.
922
01:01:47,810 --> 01:01:52,850
So all those things, this
thing is exactly the equivalent
923
01:01:52,850 --> 01:01:56,190
to that thing, up there.
924
01:01:56,190 --> 01:01:59,300
Up to M X double dot
equal to minus K x.
925
01:01:59,300 --> 01:02:01,140
Cosmetics.
926
01:02:01,140 --> 01:02:02,560
Beautiful.
927
01:02:02,560 --> 01:02:06,570
Looks-- I like it.
928
01:02:06,570 --> 01:02:07,650
All right.
929
01:02:07,650 --> 01:02:10,170
Then what I have been
doing is that now
930
01:02:10,170 --> 01:02:15,210
I introduce using a
definition of normal mode.
931
01:02:15,210 --> 01:02:20,190
I guess the solution will
have this functional form.
932
01:02:20,190 --> 01:02:25,300
Z equals to exponential i
omega t plus phi, everybody
933
01:02:25,300 --> 01:02:28,830
oscillating at the same
frequency, the same phase.
934
01:02:28,830 --> 01:02:32,010
Frequency omega, phase phi.
935
01:02:32,010 --> 01:02:35,130
And everybody can have
different amplitude.
936
01:02:35,130 --> 01:02:39,240
You can see from this
example, normal modes,
937
01:02:39,240 --> 01:02:42,710
they can have
different amplitude.
938
01:02:42,710 --> 01:02:44,490
The amplitude is what?
939
01:02:44,490 --> 01:02:45,900
I don't know yet.
940
01:02:45,900 --> 01:02:48,150
But we will figure it out.
941
01:02:48,150 --> 01:02:50,520
Then that's my assumption.
942
01:02:50,520 --> 01:02:52,510
The definition of normal mode.
943
01:02:52,510 --> 01:02:56,220
And I plug in to the
equation of motion.
944
01:02:56,220 --> 01:02:59,790
Then this is what we
are doing to simplify
945
01:02:59,790 --> 01:03:01,350
the equation of motion.
946
01:03:01,350 --> 01:03:03,540
There's no magic here.
947
01:03:03,540 --> 01:03:09,670
If I plug in the definition on
normal mode to that equation,
948
01:03:09,670 --> 01:03:15,460
this is actually going to bring
you to this equation, matrix
949
01:03:15,460 --> 01:03:18,210
equation.
950
01:03:18,210 --> 01:03:24,000
So if you have learned matrices
before, you have something,
951
01:03:24,000 --> 01:03:30,930
some matrix, times Z.
This is equal to zero.
952
01:03:30,930 --> 01:03:33,660
A is not zero.
953
01:03:33,660 --> 01:03:34,600
I hope.
954
01:03:34,600 --> 01:03:37,780
If it's zero, then the
whole system is not moving.
955
01:03:37,780 --> 01:03:39,780
Then it's not fun.
956
01:03:42,520 --> 01:03:47,040
So if A is not zero, then
this thing should be--
957
01:03:47,040 --> 01:03:52,290
this thing times A should
make this equation equal to 0.
958
01:03:52,290 --> 01:03:56,980
So what is actually
the required condition?
959
01:03:56,980 --> 01:03:58,080
I get stuck,
960
01:03:58,080 --> 01:04:01,440
and of course again, my
friend from math department
961
01:04:01,440 --> 01:04:04,080
comes to save me.
962
01:04:04,080 --> 01:04:10,930
That means if this
thing has a solution,
963
01:04:10,930 --> 01:04:12,640
this equation has
a solution, that
964
01:04:12,640 --> 01:04:23,090
means that determinant of
M minus 1 K minus omega
965
01:04:23,090 --> 01:04:29,750
squared I has to be equal to 0.
966
01:04:29,750 --> 01:04:34,790
So that is the condition
for this equation
967
01:04:34,790 --> 01:04:41,030
to satisfy this
to be equal to 0.
968
01:04:41,030 --> 01:04:45,320
And just to make sure that I
don't know what is the angular
969
01:04:45,320 --> 01:04:46,820
frequency omega yet.
970
01:04:46,820 --> 01:04:49,990
I don't know what
is the phi yet.
971
01:04:49,990 --> 01:04:55,730
We can actually solve the
angular frequency, omega.
972
01:04:55,730 --> 01:05:01,070
So now, turn everything around.
973
01:05:01,070 --> 01:05:05,060
And basically now,
using this normal mode
974
01:05:05,060 --> 01:05:11,290
definition, and some
mathematical manipulation,
975
01:05:11,290 --> 01:05:16,440
the condition we need for this
equation to satisfy equal to 0,
976
01:05:16,440 --> 01:05:22,190
is determinant M minus 1
K minus omega squared I.
977
01:05:22,190 --> 01:05:25,930
I can write down M
minus 1 K minus omega
978
01:05:25,930 --> 01:05:32,870
squared I explicitly, just
to help you with mathematics.
979
01:05:36,910 --> 01:05:50,500
M minus 1 K is equal to 1 over
2m, 0, 0, 0, 1/m, 0, 0, 0, 1/m.
980
01:05:50,500 --> 01:05:56,230
It's just the inverse
matrix of the M matrix.
981
01:05:56,230 --> 01:06:07,090
Therefore, now I can write
down the explicit expression
982
01:06:07,090 --> 01:06:10,350
of M minus 1 K
minus omega squared
983
01:06:10,350 --> 01:06:23,730
I. This will be equal to k
over m minus omega squared,
984
01:06:23,730 --> 01:06:34,370
minus k over 2m, minus k
over 2, minus k over m,
985
01:06:34,370 --> 01:06:40,030
k over m minus omega squared, 0.
986
01:06:40,030 --> 01:06:42,600
I will write down all
the elements first.
987
01:06:42,600 --> 01:06:46,392
Then I will explain to you how
I arrived at the expression.
988
01:06:52,200 --> 01:06:53,150
OK.
989
01:06:53,150 --> 01:06:55,180
So this is M minus 1 k.
990
01:06:55,180 --> 01:06:58,790
The definition of
M minus 1 is that.
991
01:06:58,790 --> 01:07:02,280
And the definition of
K is in the upper right
992
01:07:02,280 --> 01:07:04,980
corner of the black board.
993
01:07:04,980 --> 01:07:08,620
Therefore, if you
multiply M minus 1 K,
994
01:07:08,620 --> 01:07:14,130
basically, the first
column will get--
995
01:07:18,240 --> 01:07:18,850
wait a second.
996
01:07:18,850 --> 01:07:20,100
Did I make a mistake?
997
01:07:24,670 --> 01:07:25,170
No.
998
01:07:25,170 --> 01:07:26,130
OK.
999
01:07:26,130 --> 01:07:32,440
So basically, what you
arrive at is k/m, k/m, k/m.
1000
01:07:36,430 --> 01:07:42,760
And also, the minus k
over 2m for the rest part
1001
01:07:42,760 --> 01:07:44,440
of the matrix.
1002
01:07:44,440 --> 01:07:48,235
And the minus omega
squared I will give you
1003
01:07:48,235 --> 01:07:51,145
the diagonal component.
1004
01:07:51,145 --> 01:07:52,227
Yes?
1005
01:07:52,227 --> 01:07:54,268
AUDIENCE: Why do you have
to take the determinant
1006
01:07:54,268 --> 01:07:56,287
and set it equal to
0 instead of just
1007
01:07:56,287 --> 01:07:59,230
setting that equal to zero?
1008
01:07:59,230 --> 01:08:02,740
AUDIENCE: This is a matrix.
1009
01:08:02,740 --> 01:08:04,150
So these are the matrix.
1010
01:08:04,150 --> 01:08:08,020
So a matrix times A
will be equal to zero.
1011
01:08:08,020 --> 01:08:12,670
The general condition
for that to be satisfied
1012
01:08:12,670 --> 01:08:14,440
is more general.
1013
01:08:14,440 --> 01:08:19,359
It's actually the determinant
of this matrix equal to zero.
1014
01:08:19,359 --> 01:08:27,790
Because this is actually
multiplied by some back to A.
1015
01:08:27,790 --> 01:08:31,430
So I think there are
mathematical manipulation.
1016
01:08:31,430 --> 01:08:33,270
Basically, you would
just collect the terms.
1017
01:08:33,270 --> 01:08:37,140
And then calculate
M minus 1 K first.
1018
01:08:37,140 --> 01:08:40,930
And the minus omega squared I
will give you all the diagonal
1019
01:08:40,930 --> 01:08:44,899
and terms have a minus
omega square there.
1020
01:08:44,899 --> 01:08:48,540
And that is actually the matrix.
1021
01:08:48,540 --> 01:08:52,290
And of course, I can
calculate the determinant.
1022
01:08:52,290 --> 01:08:54,479
So if I calculate
the determinant,
1023
01:08:54,479 --> 01:08:58,319
then basically I get this
times that times that.
1024
01:08:58,319 --> 01:09:07,950
So what you get is k over m
minus omega squared times k
1025
01:09:07,950 --> 01:09:15,430
over m minus omega squared times
k over m minus omega squared.
1026
01:09:15,430 --> 01:09:17,510
So these are all diagonal terms.
1027
01:09:17,510 --> 01:09:26,340
And the minus 1 over 2 k
squared over m squared, k
1028
01:09:26,340 --> 01:09:32,879
squared over m squared.
1029
01:09:35,770 --> 01:09:36,790
sorry.
1030
01:09:36,790 --> 01:09:40,600
Minus omega squared.
1031
01:09:40,600 --> 01:09:46,649
So that's this off diagonal
term, this times this times
1032
01:09:46,649 --> 01:09:47,310
that.
1033
01:09:47,310 --> 01:09:48,130
OK.
1034
01:09:48,130 --> 01:09:50,050
It will give you
the second term.
1035
01:09:50,050 --> 01:09:52,390
And the third one,
which survived
1036
01:09:52,390 --> 01:09:56,320
because of those zeros, many,
many terms are equal to 0.
1037
01:09:56,320 --> 01:10:01,590
And then the third term, which
is nonzero, is again minus 1
1038
01:10:01,590 --> 01:10:09,790
over 2k squared over m squared,
k over m minus omega squared.
1039
01:10:09,790 --> 01:10:13,270
And this is actually equal
to 0, because the determinant
1040
01:10:13,270 --> 01:10:17,380
of this matrix is equal to zero.
1041
01:10:17,380 --> 01:10:19,320
Everybody following?
1042
01:10:19,320 --> 01:10:20,320
A little bit of a mess.
1043
01:10:20,320 --> 01:10:23,760
Because I have been doing
something very challenging.
1044
01:10:23,760 --> 01:10:32,410
I'm solving a 3 by 3 matrix
problem in front of you right.
1045
01:10:32,410 --> 01:10:35,600
So the math can get
a bit complicated.
1046
01:10:35,600 --> 01:10:40,135
But next time, I think we are
going to go to a second order
1047
01:10:40,135 --> 01:10:42,070
one, 2 by 2 matrix.
1048
01:10:42,070 --> 01:10:45,040
And I think that will
be slightly easier.
1049
01:10:45,040 --> 01:10:47,650
But the general
approach is the same.
1050
01:10:47,650 --> 01:10:52,480
So basically, you calculate M
minus 1 K minus omega squared.
1051
01:10:52,480 --> 01:10:57,820
Then you get what is inside, all
the content inside this matrix.
1052
01:10:57,820 --> 01:11:00,640
Then you would calculate
the determinant.
1053
01:11:00,640 --> 01:11:05,800
And basically, you can
solve this equation.
1054
01:11:05,800 --> 01:11:11,250
Now I can define omega0
squared to be k/m.
1055
01:11:11,250 --> 01:11:15,210
And I can actually make this
expression much simpler.
1056
01:11:18,840 --> 01:11:20,970
Then basically,
what you are getting
1057
01:11:20,970 --> 01:11:25,660
is omega0 squared
minus omega squared
1058
01:11:25,660 --> 01:11:33,480
to the third minus 1/2 omega0
to the fourth, omega0 squared
1059
01:11:33,480 --> 01:11:35,650
minus omega squared.
1060
01:11:35,650 --> 01:11:42,930
Minus 1/2 omega0 to the
fourth, omega0 to the square,
1061
01:11:42,930 --> 01:11:43,980
minus omega squared.
1062
01:11:43,980 --> 01:11:47,340
And this is equal to 0.
1063
01:11:47,340 --> 01:11:51,684
And you can factor out
the common components.
1064
01:11:51,684 --> 01:11:53,100
Then basically,
what you are going
1065
01:11:53,100 --> 01:11:56,370
to get is, you can
write this thing
1066
01:11:56,370 --> 01:12:03,270
to be omega0 squared minus
omega squared, omega squared.
1067
01:12:03,270 --> 01:12:06,390
Because all of them
have omega squared.
1068
01:12:06,390 --> 01:12:12,430
And omega squared
minus 2 omega0 squared.
1069
01:12:12,430 --> 01:12:14,770
And that's equal to 0.
1070
01:12:14,770 --> 01:12:18,520
So I am skipping a lot of steps
from this one to that one.
1071
01:12:18,520 --> 01:12:22,690
But in general, you can solve
this third order equation.
1072
01:12:22,690 --> 01:12:29,530
And I can first combine
all those terms together.
1073
01:12:29,530 --> 01:12:33,050
And then I factor out
the common components.
1074
01:12:33,050 --> 01:12:35,500
Then basically, what you
are going to arrive at
1075
01:12:35,500 --> 01:12:38,560
is something like this.
1076
01:12:38,560 --> 01:12:39,890
A lot of math here.
1077
01:12:39,890 --> 01:12:43,240
But we are close to the end.
1078
01:12:43,240 --> 01:12:48,440
So you can see now what are the
possible solutions for omega.
1079
01:12:48,440 --> 01:12:53,860
That is the omega,
unknown angular frequency
1080
01:12:53,860 --> 01:12:56,380
we are trying to figure out.
1081
01:12:56,380 --> 01:13:00,460
You can see that there are
three possible omegas that can
1082
01:13:00,460 --> 01:13:03,220
make this equation equal to 0.
1083
01:13:03,220 --> 01:13:12,000
The first one is
omega equal to omega0.
1084
01:13:12,000 --> 01:13:19,140
The second one is
square root of omega 0,
1085
01:13:19,140 --> 01:13:21,450
coming from this
expression, that omega
1086
01:13:21,450 --> 01:13:24,360
squared minus 2
omega zero squared.
1087
01:13:24,360 --> 01:13:26,280
If omega equal to
square root 2 omega0,
1088
01:13:26,280 --> 01:13:28,110
this will be equal to zero.
1089
01:13:28,110 --> 01:13:31,180
And that will give you the
whole expression equal to 0.
1090
01:13:31,180 --> 01:13:33,670
Then finally, I take this term.
1091
01:13:33,670 --> 01:13:36,030
And then you will get zero.
1092
01:13:36,030 --> 01:13:40,170
Omega squared, if
omega is equal to 0,
1093
01:13:40,170 --> 01:13:43,770
then the whole expression is 0.
1094
01:13:43,770 --> 01:13:50,130
I have defined omega0 squared
to be equal to k over m.
1095
01:13:50,130 --> 01:13:54,060
Therefore, I can conclude
that omega squared
1096
01:13:54,060 --> 01:14:04,002
is equal to k over
m, 2k over m, and 0.
1097
01:14:06,960 --> 01:14:10,390
Look at what we have done,
a lot of mathematics.
1098
01:14:10,390 --> 01:14:15,240
But in the end, after you
solve the eigenvalue problem,
1099
01:14:15,240 --> 01:14:18,720
or the determinant
equal to zero problem,
1100
01:14:18,720 --> 01:14:20,940
you arrive at that
there are only
1101
01:14:20,940 --> 01:14:25,740
three possible
values of omega which
1102
01:14:25,740 --> 01:14:28,620
can make the
determinant of M minus 1
1103
01:14:28,620 --> 01:14:32,600
K minus omega
squared I equal to 0.
1104
01:14:32,600 --> 01:14:35,550
What are the three?
1105
01:14:35,550 --> 01:14:38,710
k/m, 2k/m, and 0.
1106
01:14:42,260 --> 01:14:45,150
If you look at this
value, then we'll say,
1107
01:14:45,150 --> 01:14:49,800
this is essentially what we
actually argued before, right?
1108
01:14:49,800 --> 01:14:55,000
Omega A squared is equal
to 4k over 2m is 2k over m.
1109
01:14:55,000 --> 01:14:55,930
Wow.
1110
01:14:55,930 --> 01:14:56,560
We got it.
1111
01:15:00,320 --> 01:15:03,210
The second one is,
we think about really
1112
01:15:03,210 --> 01:15:04,930
keep a straight
question in my head
1113
01:15:04,930 --> 01:15:07,460
and understand this system.
1114
01:15:07,460 --> 01:15:12,260
The second identified
normal more is having omega
1115
01:15:12,260 --> 01:15:14,090
squared be equal to k/m.
1116
01:15:14,090 --> 01:15:18,650
I got this also here magically,
after all those magics.
1117
01:15:18,650 --> 01:15:24,960
And finally, the third one,
the math also knows physics.
1118
01:15:24,960 --> 01:15:29,700
It also predicted that this is
one mode which have oscillation
1119
01:15:29,700 --> 01:15:32,430
frequency of 0.
1120
01:15:32,430 --> 01:15:34,924
Isn't that amazing to you?
1121
01:15:39,690 --> 01:15:43,900
But that also gives
us a sense of safety.
1122
01:15:43,900 --> 01:15:48,320
Because I can now add
10 pendulums, or 10
1123
01:15:48,320 --> 01:15:51,080
coupled system to
your homework, and you
1124
01:15:51,080 --> 01:15:52,210
will be able to solve it.
1125
01:15:55,350 --> 01:15:56,790
So very good.
1126
01:15:56,790 --> 01:16:00,530
This example seems
to be complicated.
1127
01:16:00,530 --> 01:16:03,570
But the what I want to say,
I have one minute left,
1128
01:16:03,570 --> 01:16:05,790
is that what we
have been doing is
1129
01:16:05,790 --> 01:16:11,370
to write the equation of
motion based on force diagram.
1130
01:16:11,370 --> 01:16:15,660
Then I convert that
to matrix format,
1131
01:16:15,660 --> 01:16:19,090
and X double dot
equal to minus KX.
1132
01:16:19,090 --> 01:16:21,270
Then I follow the
whole procedure,
1133
01:16:21,270 --> 01:16:24,110
solve the eigenvalue problem.
1134
01:16:24,110 --> 01:16:30,060
Then I will be able to figure
out what are the possible omega
1135
01:16:30,060 --> 01:16:37,555
values which can satisfy
this eigenvalue problem
1136
01:16:37,555 --> 01:16:38,920
or this determinant.
1137
01:16:38,920 --> 01:16:42,630
M minus 1 K minus omega
squared I equal to 0 problem.
1138
01:16:42,630 --> 01:16:46,130
And after solving
all those, you will
1139
01:16:46,130 --> 01:16:51,000
be able to solve the
corresponding so-called normal
1140
01:16:51,000 --> 01:16:52,530
mode frequencies.
1141
01:16:52,530 --> 01:16:53,760
You can solve it.
1142
01:16:53,760 --> 01:16:57,700
And of course, you can plug
those normal mode frequencies
1143
01:16:57,700 --> 01:17:01,170
back in, then you
will be able to drive
1144
01:17:01,170 --> 01:17:05,430
the relative amplitude,
A1, A2, and A3.
1145
01:17:05,430 --> 01:17:07,890
So what we have
we learned today?
1146
01:17:07,890 --> 01:17:11,670
We have learned how
to predict the motion
1147
01:17:11,670 --> 01:17:14,460
of coupled oscillators.
1148
01:17:14,460 --> 01:17:15,900
That's really cool.
1149
01:17:15,900 --> 01:17:18,540
And then next time,
we are going to learn
1150
01:17:18,540 --> 01:17:22,050
a special kind of motion in
coupled oscillators, which
1151
01:17:22,050 --> 01:17:23,850
is the big phenomena.
1152
01:17:23,850 --> 01:17:26,710
And also, what will
happen if I start to drive
1153
01:17:26,710 --> 01:17:28,110
the coupled oscillators?
1154
01:17:28,110 --> 01:17:31,830
So I will be here if
you have any questions
1155
01:17:31,830 --> 01:17:34,130
about the lecture.
1156
01:17:34,130 --> 01:17:36,080
Thank you very much.