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YEN-JIE LEE: And
welcome back, everybody,
00:00:25.270 --> 00:00:31.960
to this fun class, 8.03.
00:00:31.960 --> 00:00:35.150
Let's get started.
00:00:35.150 --> 00:00:37.960
So the first thing
which we will do
00:00:37.960 --> 00:00:41.650
is to review a bit what
we have learned last time.
00:00:41.650 --> 00:00:44.440
And then we'll go
to the next level
00:00:44.440 --> 00:00:48.780
to study coupled oscillators.
00:00:48.780 --> 00:00:50.440
OK.
00:00:50.440 --> 00:00:55.550
Last time, we had learned a lot
on damped driven oscillators.
00:00:55.550 --> 00:00:59.710
So as far as the course
we've been going,
00:00:59.710 --> 00:01:04.510
actually, we only
study a single object,
00:01:04.510 --> 00:01:09.040
and then we introduce
more and more force
00:01:09.040 --> 00:01:10.560
acting on this object.
00:01:10.560 --> 00:01:14.710
We introduce damping
force, we introduce
00:01:14.710 --> 00:01:17.300
a driving force last time.
00:01:17.300 --> 00:01:24.330
And we see that
the system becomes
00:01:24.330 --> 00:01:28.720
more and more difficult to
understand because of the added
00:01:28.720 --> 00:01:30.150
component.
00:01:30.150 --> 00:01:33.930
But after the class
last time, I hope
00:01:33.930 --> 00:01:38.980
I convinced you that we can
understand driven oscillators.
00:01:38.980 --> 00:01:42.920
And there are two very important
things we learned last time.
00:01:42.920 --> 00:01:46.610
The first one is the
transient behavior,
00:01:46.610 --> 00:01:48.970
which is actually
a superposition
00:01:48.970 --> 00:01:53.560
of the homogeneous solution
and the steady state solution.
00:01:53.560 --> 00:01:55.130
OK.
00:01:55.130 --> 00:01:59.060
One very good news is that
if you are patient enough,
00:01:59.060 --> 00:02:03.530
you shake the
system continuously,
00:02:03.530 --> 00:02:08.030
and if you wait long enough,
then the homogeneous solution
00:02:08.030 --> 00:02:11.009
contribution goes away.
00:02:11.009 --> 00:02:16.200
And what is actually left over
is the steady state solution,
00:02:16.200 --> 00:02:20.690
which is actually much simpler
than what we saw beforehand.
00:02:20.690 --> 00:02:24.170
It's actually just
harmonic oscillation
00:02:24.170 --> 00:02:26.600
at driving frequency.
00:02:26.600 --> 00:02:32.060
Also, I hope that we also have
learned a very interesting
00:02:32.060 --> 00:02:34.190
phenomenon, which is resonance.
00:02:34.190 --> 00:02:39.710
When the driving frequency is
close to the natural frequency
00:02:39.710 --> 00:02:46.280
of the system, then the
system apparently likes it.
00:02:46.280 --> 00:02:51.350
Then it would respond
with larger amplitude
00:02:51.350 --> 00:02:55.790
and oscillating up and
down at driving frequency.
00:02:55.790 --> 00:02:59.510
So that, we call it resonance.
00:02:59.510 --> 00:03:01.250
This is the equation
of motion, which
00:03:01.250 --> 00:03:03.140
we have learned last time.
00:03:03.140 --> 00:03:07.280
You can see is theta double
dot plus Gamma theta dot,
00:03:07.280 --> 00:03:10.010
is a contribution
from the drag force,
00:03:10.010 --> 00:03:14.780
and omega 0 squared
theta is the contribution
00:03:14.780 --> 00:03:17.270
from the so-called spring force.
00:03:17.270 --> 00:03:23.810
And finally, that is equal to
f0 cosine omega d t, the driving
00:03:23.810 --> 00:03:24.950
force.
00:03:24.950 --> 00:03:27.170
And as we mentioned
in the beginning,
00:03:27.170 --> 00:03:34.110
if we prepare this system
and under-damp the situation,
00:03:34.110 --> 00:03:38.820
then the full solution
is a superposition
00:03:38.820 --> 00:03:41.130
of the steady state
solution, which
00:03:41.130 --> 00:03:46.050
is the left-hand side, the
red thing I'm pointing to,
00:03:46.050 --> 00:03:48.600
this steady state solution.
00:03:48.600 --> 00:03:54.390
There's no free parameter in
the steady state solution.
00:03:54.390 --> 00:03:57.290
So A, the amplitude, is
determined by omega d.
00:04:00.260 --> 00:04:03.820
Delta, which is the phase is
also determined by omega d.
00:04:03.820 --> 00:04:06.940
There's no free parameter.
00:04:06.940 --> 00:04:08.180
OK.
00:04:08.180 --> 00:04:13.180
And in order to make the
solution a full solution,
00:04:13.180 --> 00:04:16.630
we actually have to add in
this homogeneous solution
00:04:16.630 --> 00:04:18.310
back into this again.
00:04:18.310 --> 00:04:21.339
And basically, you
have B and alpha,
00:04:21.339 --> 00:04:23.320
those are the free
parameters, which
00:04:23.320 --> 00:04:28.780
can be determined by the
given initial conditions.
00:04:28.780 --> 00:04:29.590
OK.
00:04:29.590 --> 00:04:32.650
So if we go ahead and
plot some of the examples
00:04:32.650 --> 00:04:36.910
as a function of
time, so the y-axis
00:04:36.910 --> 00:04:38.860
is actually the amplitude.
00:04:38.860 --> 00:04:41.290
And the x-axis is time.
00:04:41.290 --> 00:04:43.570
And what is actually
plotted here
00:04:43.570 --> 00:04:46.840
is a combination or
the superposition
00:04:46.840 --> 00:04:51.640
of the steady state solution
and the homogeneous solution.
00:04:51.640 --> 00:04:55.690
And you can see that the
individual components are also
00:04:55.690 --> 00:04:57.610
shown in this slide.
00:04:57.610 --> 00:05:01.960
You can see the red thing
oscillating up and down
00:05:01.960 --> 00:05:06.540
harmonically, is steady
state solution contribution.
00:05:06.540 --> 00:05:10.870
And also, you have
the blue curve,
00:05:10.870 --> 00:05:14.180
which is decaying away
as a function of time.
00:05:14.180 --> 00:05:17.990
And you can see that if you
add these two curves together,
00:05:17.990 --> 00:05:20.740
you get something
rather complicated.
00:05:20.740 --> 00:05:25.720
You will get some kind of
motion like, do, do, do, do.
00:05:25.720 --> 00:05:30.710
Then the homogeneous
solution actually dies out.
00:05:30.710 --> 00:05:34.960
Then what is actually left
over is just the steady state
00:05:34.960 --> 00:05:38.030
solution, harmonic oscillation.
00:05:38.030 --> 00:05:42.320
And in this case, omega d
is actually 10 times larger
00:05:42.320 --> 00:05:45.290
than the natural frequency.
00:05:45.290 --> 00:05:48.890
And there's another example
which is also very interesting.
00:05:48.890 --> 00:05:57.470
It's that if I make the
omega d closer to omega 0--
00:05:57.470 --> 00:06:00.740
OK, in this case it's actually
omega d equal to 2 times omega
00:06:00.740 --> 00:06:04.070
0, then you can produce
some kind of a motion, which
00:06:04.070 --> 00:06:04.910
is like this.
00:06:04.910 --> 00:06:06.920
So you have the oscillation.
00:06:06.920 --> 00:06:09.080
And they stayed
there for a while,
00:06:09.080 --> 00:06:11.440
then goes back, and
oscillates down,
00:06:11.440 --> 00:06:14.930
and stay there then goes back.
00:06:14.930 --> 00:06:17.300
OK, as you can see on there.
00:06:17.300 --> 00:06:22.250
The homogeneous solution part
and steady state solution part
00:06:22.250 --> 00:06:27.170
work together and produce
this kind of strange behavior.
00:06:27.170 --> 00:06:27.980
OK.
00:06:27.980 --> 00:06:30.860
And that's just another example.
00:06:30.860 --> 00:06:33.010
And if you wait
long enough, again
00:06:33.010 --> 00:06:37.400
what is actually left over
is the steady state solution.
00:06:37.400 --> 00:06:38.760
OK.
00:06:38.760 --> 00:06:41.440
So what are we
going to do today?
00:06:41.440 --> 00:06:45.660
So today, we are
going to investigate
00:06:45.660 --> 00:06:52.260
what will happen if we try to
put together multiple objects
00:06:52.260 --> 00:06:55.170
and also allow them
to talk to each other.
00:06:55.170 --> 00:06:55.770
OK.
00:06:55.770 --> 00:06:58.740
So if we have two objects, and
they don't talk to each other,
00:06:58.740 --> 00:07:01.050
then they are still
like a single object.
00:07:01.050 --> 00:07:04.800
They are still like simple
harmonic motion on their own.
00:07:04.800 --> 00:07:08.130
But if you allow them
to talk to each other,
00:07:08.130 --> 00:07:11.820
this is the so-called
coupled oscillator, then
00:07:11.820 --> 00:07:15.130
interesting thing happen.
00:07:15.130 --> 00:07:23.250
So in general, coupled systems
are super, super complicated.
00:07:23.250 --> 00:07:24.420
OK.
00:07:24.420 --> 00:07:27.660
So let me give you
one example here.
00:07:27.660 --> 00:07:33.690
This is actually
two pendulums that
00:07:33.690 --> 00:07:36.720
are a coupled to each
other, they are actually
00:07:36.720 --> 00:07:39.950
connected to each
other, one pendulum,
00:07:39.950 --> 00:07:42.270
the second one is here, OK.
00:07:42.270 --> 00:07:48.270
And for example, I can actually
give it an initial velocity
00:07:48.270 --> 00:07:49.695
and see what is going to happen.
00:07:52.680 --> 00:07:56.130
You can see that the
resulting motion--
00:07:56.130 --> 00:07:57.150
OK.
00:07:57.150 --> 00:08:00.980
Remember, we are just talking
about two pendulums that
00:08:00.980 --> 00:08:02.620
are connected to each other.
00:08:02.620 --> 00:08:08.530
The resulting motion
is super complicated.
00:08:08.530 --> 00:08:11.200
This is one of my
favorite demonstrations.
00:08:11.200 --> 00:08:17.291
You can actually stare at
this machine the whole time.
00:08:17.291 --> 00:08:18.790
And you can see
that, huh, sometimes
00:08:18.790 --> 00:08:20.370
it does this rotation.
00:08:20.370 --> 00:08:22.190
Sometimes it doesn't do that.
00:08:22.190 --> 00:08:25.750
And it's almost like
a living creature.
00:08:28.840 --> 00:08:31.310
So we are going to
solve this system.
00:08:31.310 --> 00:08:33.606
No, probably not, he knows.
00:08:33.606 --> 00:08:36.190
[LAUGHTER]
00:08:36.190 --> 00:08:41.080
But as I mentioned before,
you can always write down
00:08:41.080 --> 00:08:43.409
the equation of motion.
00:08:43.409 --> 00:08:47.350
And you can solve
it by computer.
00:08:47.350 --> 00:08:49.560
Maybe some of the
course 6 people
00:08:49.560 --> 00:08:52.980
can actually try and write the
program to solve this thing
00:08:52.980 --> 00:08:56.820
and to simulate what
is going to happen.
00:08:56.820 --> 00:09:02.730
So let's take a look at this
complicated motion again.
00:09:02.730 --> 00:09:06.480
So you can see that the good
news is that there are only two
00:09:06.480 --> 00:09:07.470
objects.
00:09:07.470 --> 00:09:10.260
And you can see--
00:09:10.260 --> 00:09:14.280
look at the green,
sorry, the orange dot.
00:09:14.280 --> 00:09:21.460
The orange dot is always
moving along a semi circle.
00:09:21.460 --> 00:09:27.680
But if you focus
on the yellow dot,
00:09:27.680 --> 00:09:30.940
the yellow is doing all
kinds of different things.
00:09:30.940 --> 00:09:35.640
It's very hard to predict
what is going to happen.
00:09:35.640 --> 00:09:40.130
So what I want to say is,
those are interesting examples
00:09:40.130 --> 00:09:41.960
of coupled systems.
00:09:41.960 --> 00:09:45.800
But they are actually far
more complicated than what
00:09:45.800 --> 00:09:51.050
we thought, because they
are not smooth oscillation
00:09:51.050 --> 00:09:54.260
around equilibrium position.
00:09:54.260 --> 00:09:56.920
So you can see that now
if I stop this machine
00:09:56.920 --> 00:10:02.940
and just perturb it slightly,
giving it a small angle
00:10:02.940 --> 00:10:08.370
displacement, then you
can see that the motion is
00:10:08.370 --> 00:10:12.560
much more easier to understand.
00:10:12.560 --> 00:10:14.030
You see.
00:10:14.030 --> 00:10:17.530
You even get one of these
questions in your p-set.
00:10:17.530 --> 00:10:18.795
OK, that's good news.
00:10:21.390 --> 00:10:25.920
So our job today
is to understand
00:10:25.920 --> 00:10:30.150
what is going to happen to
those coupled oscillators.
00:10:30.150 --> 00:10:34.560
Let me give you a few
examples before we start
00:10:34.560 --> 00:10:38.130
to work on a specific question.
00:10:38.130 --> 00:10:43.170
The second example I would
like to show you is a saw
00:10:43.170 --> 00:10:47.220
and you actually connect
it to two - actually
00:10:47.220 --> 00:10:50.970
a ruler, a metal ruler,
which is connected
00:10:50.970 --> 00:10:54.750
to two massive objects.
00:10:54.750 --> 00:11:00.090
Now I can actually give
it the initial velocity
00:11:00.090 --> 00:11:02.470
and see what happens.
00:11:02.470 --> 00:11:04.560
And you can see
that they do talk
00:11:04.560 --> 00:11:09.490
to each other through this
ruler, this metal ruler.
00:11:09.490 --> 00:11:10.020
Can you see?
00:11:10.020 --> 00:11:11.820
I hope you can see.
00:11:11.820 --> 00:11:13.530
It's a bit small.
00:11:13.530 --> 00:11:16.630
But it's really interesting
that you can see-- originally,
00:11:16.630 --> 00:11:20.440
I just introduced
some displacement
00:11:20.440 --> 00:11:23.100
in the left-hand side mass.
00:11:23.100 --> 00:11:27.270
And the left-had side
thing start to move or so
00:11:27.270 --> 00:11:30.260
after a while.
00:11:30.260 --> 00:11:36.690
There are two more examples
which I would like to give you,
00:11:36.690 --> 00:11:38.920
introduce to you.
00:11:38.920 --> 00:11:42.390
There are two
kinds of pendulums,
00:11:42.390 --> 00:11:44.380
which I prepared here.
00:11:44.380 --> 00:11:49.050
The first one is there are two
pendulums that are connected
00:11:49.050 --> 00:11:52.530
to each other by a spring.
00:11:52.530 --> 00:11:57.850
And if I try to
introduce displacement,
00:11:57.850 --> 00:12:03.340
I move both masses slightly and
see what is going to happen.
00:12:03.340 --> 00:12:07.400
And we see that the motion
is still complicated.
00:12:07.400 --> 00:12:12.250
Although, if you stare at
one objects, it looks more
00:12:12.250 --> 00:12:16.870
like harmonic oscillation,
but not quite.
00:12:16.870 --> 00:12:19.360
For example, this
guy is slowing down,
00:12:19.360 --> 00:12:21.680
and this is actually
moving faster.
00:12:21.680 --> 00:12:26.500
And now the right hand side
guy is actually moving faster.
00:12:26.500 --> 00:12:29.590
Motion seems to be completed.
00:12:29.590 --> 00:12:32.810
Also, you can look at this one.
00:12:32.810 --> 00:12:34.900
Those are the two pendulums.
00:12:34.900 --> 00:12:39.670
They are connected to each
other by this rod here.
00:12:39.670 --> 00:12:42.340
And of course, you
can displace the mass
00:12:42.340 --> 00:12:44.964
from the equilibrium position.
00:12:44.964 --> 00:12:45.880
I'm not going to hit--
00:12:45.880 --> 00:12:47.080
not hitting each other.
00:12:47.080 --> 00:12:49.730
So you can displace the
masses from each other.
00:12:49.730 --> 00:12:54.040
And you can see that they
do complicated things
00:12:54.040 --> 00:12:55.640
as a function the time.
00:12:55.640 --> 00:12:58.280
How are we going
to understand this?
00:12:58.280 --> 00:13:02.890
And I hope that by the
end of this lecture
00:13:02.890 --> 00:13:06.940
you are convinced that you can
as you solve this really easy,
00:13:06.940 --> 00:13:10.390
following a fixed procedure.
00:13:10.390 --> 00:13:15.130
In those examples, we
have two objects that
00:13:15.130 --> 00:13:17.120
are connected to each other.
00:13:17.120 --> 00:13:19.450
And therefore, they
talk to each other
00:13:19.450 --> 00:13:23.800
and produce coupled motion.
00:13:23.800 --> 00:13:27.650
Those are a couple
oscillator examples.
00:13:27.650 --> 00:13:31.000
There's another very
interesting example, which
00:13:31.000 --> 00:13:36.370
is called Wilberforce pendulum.
00:13:36.370 --> 00:13:39.220
So this is actually a pendulum.
00:13:39.220 --> 00:13:42.820
You can rotate like this.
00:13:42.820 --> 00:13:44.900
And it can also
move up and down.
00:13:44.900 --> 00:13:48.310
It's connected to a spring.
00:13:48.310 --> 00:13:51.200
The interesting thing
is that if I just
00:13:51.200 --> 00:13:56.270
start with some
rotation, you can
00:13:56.270 --> 00:14:01.910
see that it starts to also
oscillate up and down.
00:14:01.910 --> 00:14:02.590
You see?
00:14:02.590 --> 00:14:08.330
So initially I just
introduced a rotation.
00:14:08.330 --> 00:14:10.120
Now it's actually
fully rotating.
00:14:10.120 --> 00:14:13.150
And now it starts
to move up and down.
00:14:13.150 --> 00:14:18.600
And you can see that the
energy stored in the pendulum
00:14:18.600 --> 00:14:27.260
is going back and forth between
the gravitational potential,
00:14:27.260 --> 00:14:31.360
between the potential
of the spring,
00:14:31.360 --> 00:14:38.920
and also between the kinetic
energy of up and down motion
00:14:38.920 --> 00:14:40.320
and the rotation.
00:14:40.320 --> 00:14:46.840
They're actually doing all
those transitions all the time.
00:14:46.840 --> 00:14:50.280
So you can see--
00:14:50.280 --> 00:14:53.830
so initially it's just rotating.
00:14:53.830 --> 00:14:56.440
And then it starts
to move up and down.
00:14:56.440 --> 00:14:58.840
And this one is
also very similar.
00:14:58.840 --> 00:15:02.110
But now the mass is
much more displaced.
00:15:02.110 --> 00:15:07.340
And if I try to rotate this
system without introducing
00:15:07.340 --> 00:15:10.360
a horizontal direction
displacement,
00:15:10.360 --> 00:15:17.170
it still does is up and down
motion, like a simple spring
00:15:17.170 --> 00:15:18.620
mass system.
00:15:18.620 --> 00:15:23.481
So what causes this
kind of motion?
00:15:23.481 --> 00:15:29.590
That is because when we move
this pendulum up and down,
00:15:29.590 --> 00:15:34.180
we also slightly
unwind the spring.
00:15:34.180 --> 00:15:39.070
That can generate some
kind of torque to this mass
00:15:39.070 --> 00:15:44.900
and produce rotational behavior.
00:15:44.900 --> 00:15:47.755
And you can see
that this is just
00:15:47.755 --> 00:15:49.480
involving one single object.
00:15:49.480 --> 00:15:53.650
But there's a coupling
between the rotation
00:15:53.650 --> 00:15:56.890
and the horizontal
direction motion.
00:15:56.890 --> 00:16:04.240
So that's also special
kind of coupled oscillator.
00:16:04.240 --> 00:16:10.580
So after all this,
before we get started,
00:16:10.580 --> 00:16:13.150
I would like to say that
what we are going to do
00:16:13.150 --> 00:16:17.800
is to assume all those
things are ideal,
00:16:17.800 --> 00:16:21.670
without them being forced,
without a driving force.
00:16:21.670 --> 00:16:24.410
We may introduce that
later in the class.
00:16:24.410 --> 00:16:28.870
But for simplicity, we'll
just stay with this idea case,
00:16:28.870 --> 00:16:31.390
before the mass becomes
super complicated
00:16:31.390 --> 00:16:34.430
to solve it in front of you.
00:16:34.430 --> 00:16:43.020
And also, we can see that
all those complicated motion
00:16:43.020 --> 00:16:43.955
are just illusion.
00:16:46.590 --> 00:16:49.900
Actually, the reality
is that all of those
00:16:49.900 --> 00:16:54.250
are just superposition
of harmonic motions.
00:16:54.250 --> 00:16:57.680
You will see that by
the end of this class.
00:16:57.680 --> 00:16:59.961
So that is really amazing.
00:16:59.961 --> 00:17:00.460
OK.
00:17:00.460 --> 00:17:02.870
Let's immediately get started.
00:17:02.870 --> 00:17:06.609
So let's take a
look at this system
00:17:06.609 --> 00:17:08.980
together and see
if we can actually
00:17:08.980 --> 00:17:12.670
figure out the motion
of this system together.
00:17:12.670 --> 00:17:17.109
So I have a system with
three little masses.
00:17:17.109 --> 00:17:20.609
So there are three little
masses in this system.
00:17:20.609 --> 00:17:25.240
They are connected to
each other by spring.
00:17:25.240 --> 00:17:29.590
Those springs are highly
idealized, the springs.
00:17:29.590 --> 00:17:32.320
And they have spring constant k.
00:17:32.320 --> 00:17:35.950
And the natural length's l0.
00:17:35.950 --> 00:17:40.120
And they are placed on Earth.
00:17:40.120 --> 00:17:44.800
And I carefully
design the lab so
00:17:44.800 --> 00:17:50.150
that there's no friction
between the desk and all
00:17:50.150 --> 00:17:52.450
those little masses.
00:17:52.450 --> 00:17:56.370
So once you get started
and look at this system,
00:17:56.370 --> 00:17:58.540
you can imagine
that there can be
00:17:58.540 --> 00:18:02.860
all kinds of different
complicated motions.
00:18:02.860 --> 00:18:06.950
You can actually, for
example, just move this mass
00:18:06.950 --> 00:18:12.220
and put the other two on hold.
00:18:12.220 --> 00:18:14.770
And they can
oscillate like crazy.
00:18:14.770 --> 00:18:20.540
They can do very
similar kind of motion.
00:18:20.540 --> 00:18:23.950
There are many,
many possibilities.
00:18:23.950 --> 00:18:30.100
But if you stare at
this system long enough,
00:18:30.100 --> 00:18:35.560
you will be able to identify
special kinds of motion which
00:18:35.560 --> 00:18:39.670
are easier to understand.
00:18:39.670 --> 00:18:42.490
So what I would like
to introduce to you
00:18:42.490 --> 00:18:45.190
is a special kind
of motion which
00:18:45.190 --> 00:18:51.720
you can identify from the
symmetry of this system.
00:18:51.720 --> 00:18:53.930
That is your
so-called normal mode.
00:19:01.310 --> 00:19:04.970
So what is a normal mode,
a special kind of motion
00:19:04.970 --> 00:19:07.700
we are trying to identify?
00:19:07.700 --> 00:19:10.430
That is actually
the kind of motion
00:19:10.430 --> 00:19:23.100
which every part
of the system is
00:19:23.100 --> 00:19:40.975
oscillating at the same
frequency and the same phase.
00:19:49.770 --> 00:19:52.700
So that is your
so-called normal mode,
00:19:52.700 --> 00:19:56.420
and is a special kind
of motion, which I would
00:19:56.420 --> 00:20:00.620
like you to identify with me.
00:20:00.620 --> 00:20:06.170
And we would later realize that
those special kinds of motions,
00:20:06.170 --> 00:20:09.830
which are easier to
understand, actually
00:20:09.830 --> 00:20:14.660
helps us to understand the
general motion of the system.
00:20:14.660 --> 00:20:19.820
You will realize that the most
general motion of the system
00:20:19.820 --> 00:20:26.860
is just a superposition of all
the identified normal modes.
00:20:26.860 --> 00:20:28.580
And then we are
done, because we have
00:20:28.580 --> 00:20:31.300
a general solution already.
00:20:31.300 --> 00:20:34.100
So that's very good news.
00:20:34.100 --> 00:20:38.020
That tells us that
we can understand
00:20:38.020 --> 00:20:43.190
the system systematically,
and step by step.
00:20:43.190 --> 00:20:47.710
And then we can write the
general motion of the system
00:20:47.710 --> 00:20:51.700
as a superposition of
all the normal modes.
00:20:51.700 --> 00:20:54.440
So let's get started.
00:20:54.440 --> 00:21:00.040
So can you guess what are
the possible normal modes
00:21:00.040 --> 00:21:02.330
of this system?
00:21:02.330 --> 00:21:05.230
So that means each
part of the system
00:21:05.230 --> 00:21:09.670
is oscillating at the same
frequency and the same phase.
00:21:09.670 --> 00:21:15.970
Can anybody and any one of
you guess what can happen,
00:21:15.970 --> 00:21:18.550
each part of this
is an oscillating
00:21:18.550 --> 00:21:21.390
at the same frequency?
00:21:21.390 --> 00:21:22.845
Yeah?
00:21:22.845 --> 00:21:23.870
AUDIENCE: If the two masses
on that side are displaced
00:21:23.870 --> 00:21:25.328
the same amount
and then they're --
00:21:29.150 --> 00:21:30.610
YEN-JIE LEE: Very good.
00:21:30.610 --> 00:21:36.460
So he was saying that now
I displace the right hand
00:21:36.460 --> 00:21:40.600
side two masses all together
by a fixed amount, and also
00:21:40.600 --> 00:21:47.000
the left hand side, right, by
a fixed amount and then let go.
00:21:47.000 --> 00:21:49.011
So that's what
you're saying, right?
00:21:49.011 --> 00:21:49.510
OK.
00:21:49.510 --> 00:21:56.500
So the first mode we have
identified is like this.
00:21:56.500 --> 00:22:03.490
So you have left hand side
mass displace by delta x.
00:22:03.490 --> 00:22:07.670
And the right hand
side two masses
00:22:07.670 --> 00:22:09.965
are also displaced by delta x.
00:22:15.440 --> 00:22:20.590
So basically you hold
this three little masses
00:22:20.590 --> 00:22:25.530
and stretch it by the same--
00:22:25.530 --> 00:22:29.470
introduce the same amplitude to
all those three little masses,
00:22:29.470 --> 00:22:31.000
and let go.
00:22:31.000 --> 00:22:34.830
So that is actually
one possible mode.
00:22:34.830 --> 00:22:41.090
And if we do this,
then basically
00:22:41.090 --> 00:22:43.930
what you are going to see
is that this is actually
00:22:43.930 --> 00:22:50.130
roughly equal to this system.
00:22:52.920 --> 00:22:57.630
They're connected to each
other by two springs.
00:22:57.630 --> 00:23:02.250
And the right hand side part
of the system, both masses
00:23:02.250 --> 00:23:06.150
are oscillating at the same
amplitude and the same phase.
00:23:06.150 --> 00:23:14.040
They look like as if they are
just single mass with mass
00:23:14.040 --> 00:23:15.130
equal to 2m.
00:23:17.830 --> 00:23:22.490
And if you introduce a
displacement of delta x,
00:23:22.490 --> 00:23:25.510
then what is going to
happen is that if I
00:23:25.510 --> 00:23:31.210
take a look at the mass, left
hand side mass, and the force
00:23:31.210 --> 00:23:34.300
acting on this
mass, the force will
00:23:34.300 --> 00:23:42.580
be equal to minus
2k times 2 delta x,
00:23:42.580 --> 00:23:45.520
because that's the
amount of stretch you
00:23:45.520 --> 00:23:48.970
introduce to the spring.
00:23:48.970 --> 00:23:52.765
And that will give
you minus 4k delta x.
00:23:57.390 --> 00:24:00.300
And we have already solved
this kind of problem
00:24:00.300 --> 00:24:02.310
in the first lecture.
00:24:02.310 --> 00:24:05.430
So therefore you can
immediately identify
00:24:05.430 --> 00:24:10.690
omega, in this case,
omega a squared will
00:24:10.690 --> 00:24:16.500
be equal to 4k divided by 2m.
00:24:16.500 --> 00:24:20.880
This is actually the
effective spring constant,
00:24:20.880 --> 00:24:24.210
and this is actually the mass.
00:24:24.210 --> 00:24:31.440
So that is actually the
frequency of mode A.
00:24:31.440 --> 00:24:38.020
Can you identify a second kind
of motion which does that?
00:24:38.020 --> 00:24:39.990
So in this case, what
is going to happen
00:24:39.990 --> 00:24:42.600
is that the three masses will--
00:24:42.600 --> 00:24:45.120
OK, one, two, and three.
00:24:45.120 --> 00:24:50.560
The three masses will
oscillate as a function of time
00:24:50.560 --> 00:25:01.610
like this with angular frequency
of square root of 4k over 2m.
00:25:01.610 --> 00:25:05.586
What is actually a
second possible motion?
00:25:05.586 --> 00:25:07.410
Yes?
00:25:07.410 --> 00:25:10.570
AUDIENCE: All masses being
stretched [INAUDIBLE]
00:25:10.570 --> 00:25:11.641
YEN-JIE LEE: Compressed.
00:25:11.641 --> 00:25:14.082
AUDIENCE: Compressed the same--
00:25:14.082 --> 00:25:15.040
YEN-JIE LEE: Very good.
00:25:18.100 --> 00:25:24.910
I'm very lucky that I'm in front
of such a smart crowd today.
00:25:24.910 --> 00:25:28.330
And we have
successfully identified
00:25:28.330 --> 00:25:34.060
the second mode, mode B.
So what is going to happen
00:25:34.060 --> 00:25:41.470
is that the left hand
side mass is not moving.
00:25:41.470 --> 00:25:49.400
And you compress the
upper one slightly
00:25:49.400 --> 00:25:56.570
and you stretch the lower
one, the lower little mass
00:25:56.570 --> 00:25:59.700
to the opposite direction.
00:25:59.700 --> 00:26:03.140
The displacement is delta
x, and the displacement
00:26:03.140 --> 00:26:07.210
of the second mass is delta x.
00:26:07.210 --> 00:26:09.980
So what is going to happen?
00:26:09.980 --> 00:26:13.220
What is going to happen
is that the left hand side
00:26:13.220 --> 00:26:19.400
mass will not move at all
because the force, the spring
00:26:19.400 --> 00:26:25.490
force, acting on this
mass is going to cancel.
00:26:25.490 --> 00:26:27.650
And apparently, these
two little masses
00:26:27.650 --> 00:26:31.370
are going to be doing
harmonic motion.
00:26:34.160 --> 00:26:38.136
Since this left hand
side mass is not moving,
00:26:38.136 --> 00:26:47.900
it's as if this is a wall and
this were a single spring, k,
00:26:47.900 --> 00:26:51.130
that's connected
to a little mass.
00:26:51.130 --> 00:26:54.265
And it got displaced by delta x.
00:26:56.890 --> 00:27:02.690
So what will happen is that this
mass will experience a spring
00:27:02.690 --> 00:27:09.700
force, which is F equal
to minus k delta x.
00:27:09.700 --> 00:27:12.860
Therefore, we can
immediately identify
00:27:12.860 --> 00:27:17.803
omega b squared will
be equal to k over m.
00:27:20.380 --> 00:27:27.350
So you can see that we have
identified two kinds of modes,
00:27:27.350 --> 00:27:30.760
which every part of the
system is oscillating
00:27:30.760 --> 00:27:36.040
at the same frequency
and the same phase.
00:27:36.040 --> 00:27:38.100
Everybody agree?
00:27:38.100 --> 00:27:41.440
No not everybody agree.
00:27:41.440 --> 00:27:45.870
Look at this guy this
guy is not moving.
00:27:45.870 --> 00:27:46.830
How could this be?
00:27:46.830 --> 00:27:51.340
This is not the normal mode.
00:27:51.340 --> 00:27:51.840
Isn't it?
00:27:54.640 --> 00:27:55.320
OK.
00:27:55.320 --> 00:27:57.510
I hope that will
wake you up a bit.
00:28:00.030 --> 00:28:02.220
I can be very tricky here.
00:28:02.220 --> 00:28:08.100
I can say that this mass
is also oscillating,
00:28:08.100 --> 00:28:12.065
but with what amplitude?
00:28:12.065 --> 00:28:12.690
AUDIENCE: Zero.
00:28:12.690 --> 00:28:13.856
YEN-JIE LEE: Zero amplitude.
00:28:13.856 --> 00:28:18.870
Right So the conclusion is
that, aha, everybody is actually
00:28:18.870 --> 00:28:21.250
oscillating at the
same frequency,
00:28:21.250 --> 00:28:23.380
but these guy with
zero amplitude.
00:28:25.545 --> 00:28:27.920
AUDIENCE: Are they oscillating
at the same phase as well?
00:28:27.920 --> 00:28:29.130
YEN-JIE LEE: Yeah.
00:28:29.130 --> 00:28:32.400
Oh very good question.
00:28:32.400 --> 00:28:34.770
Another objection I receive.
00:28:34.770 --> 00:28:38.130
So life is hard for me today.
00:28:38.130 --> 00:28:39.300
Hey.
00:28:39.300 --> 00:28:43.530
This guy is oscillating
out of phase.
00:28:43.530 --> 00:28:45.660
These two guys are out of phase.
00:28:45.660 --> 00:28:50.310
But I can argue that the
amplitude of the first mass
00:28:50.310 --> 00:28:56.760
is actually has a minus sign
compared to the second mass.
00:28:56.760 --> 00:29:00.750
Then they are again in phase.
00:29:03.970 --> 00:29:05.440
So very good.
00:29:05.440 --> 00:29:07.540
I like those questions.
00:29:07.540 --> 00:29:13.220
And I hope I have convinced you
that everybody is oscillating,
00:29:13.220 --> 00:29:17.000
although you cannot see it,
because the amplitude is small,
00:29:17.000 --> 00:29:18.130
is zero.
00:29:18.130 --> 00:29:23.990
And they are all oscillating
at the same phase.
00:29:23.990 --> 00:29:24.872
Yes.
00:29:24.872 --> 00:29:26.689
AUDIENCE: How come
there's only one mass?
00:29:26.689 --> 00:29:28.230
YEN-JIE LEE: Oh,
the right hand side?
00:29:28.230 --> 00:29:28.874
AUDIENCE: Yeah.
00:29:28.874 --> 00:29:29.790
YEN-JIE LEE: Oh, yeah.
00:29:29.790 --> 00:29:33.780
That is because the left
hand side mass, the 2m one,
00:29:33.780 --> 00:29:35.490
is actually not moving.
00:29:35.490 --> 00:29:37.980
Because they are
two spring forces,
00:29:37.980 --> 00:29:39.800
one is actually
pushing the mass,
00:29:39.800 --> 00:29:42.330
the other one's
pulling the mass.
00:29:42.330 --> 00:29:45.620
And they cancel perfectly.
00:29:45.620 --> 00:29:52.140
Therefore, it's as if
those two guys are not--
00:29:52.140 --> 00:29:54.480
they don't find each other.
00:29:54.480 --> 00:29:57.510
And then it's like,
they are just tools
00:29:57.510 --> 00:30:01.890
mass connected to a wall along.
00:30:01.890 --> 00:30:05.950
And then you can now identify
what is the frequency.
00:30:05.950 --> 00:30:06.450
OK.
00:30:06.450 --> 00:30:08.760
Very good.
00:30:08.760 --> 00:30:15.630
So we make the made a lot of the
progress from the discussion.
00:30:15.630 --> 00:30:21.220
And now I would like
to ask you for help.
00:30:21.220 --> 00:30:24.797
What is the third oscillation?
00:30:24.797 --> 00:30:25.731
Yes?
00:30:25.731 --> 00:30:28.540
AUDIENCE: There's no
third normal mode.
00:30:28.540 --> 00:30:30.365
YEN-JIE LEE: There's
no third normal mode.
00:30:30.365 --> 00:30:31.948
AUDIENCE: There's
no third normal mode
00:30:31.948 --> 00:30:34.906
because there are
restricted to one dimension.
00:30:34.906 --> 00:30:37.371
I can not imagine another
mode that would not
00:30:37.371 --> 00:30:39.965
displace the central mass.
00:30:39.965 --> 00:30:41.130
YEN-JIE LEE: Very good.
00:30:41.130 --> 00:30:42.680
That's very good.
00:30:42.680 --> 00:30:46.100
On the other hand,
you can also say,
00:30:46.100 --> 00:30:48.960
I also take the
center mass motion
00:30:48.960 --> 00:30:50.410
as one of the normal mode.
00:30:50.410 --> 00:30:54.260
I think that's also
fair to do that.
00:30:54.260 --> 00:30:55.640
Very good observation.
00:30:55.640 --> 00:31:00.770
You can see that the whole
can move simultaneously.
00:31:06.120 --> 00:31:10.400
I can also argue that
they are oscillating
00:31:10.400 --> 00:31:12.710
at the same frequency
and the same phase,
00:31:12.710 --> 00:31:15.200
because they are
all moving together.
00:31:21.560 --> 00:31:25.630
So these are the 2m
connected to mass one.
00:31:31.160 --> 00:31:35.690
All of them are moving
in the same direction.
00:31:35.690 --> 00:31:38.350
So now I can
calculate the force.
00:31:38.350 --> 00:31:40.100
What is the force?
00:31:40.100 --> 00:31:43.050
F is 0.
00:31:43.050 --> 00:31:49.490
Therefore, omega c is 0.
00:31:49.490 --> 00:31:53.630
So you can the small
limit of omega.
00:31:53.630 --> 00:31:59.120
So of course, I can pretend
that those mass are connected
00:31:59.120 --> 00:32:05.000
to a really, really small spring
to the wall with is a spring
00:32:05.000 --> 00:32:06.350
constant k'.
00:32:06.350 --> 00:32:08.600
And I have k' goes to zero.
00:32:08.600 --> 00:32:11.450
And they are actually
going to oscillate
00:32:11.450 --> 00:32:16.880
with omega c goes to zero.
00:32:16.880 --> 00:32:20.640
So in this case,
the amplitude is
00:32:20.640 --> 00:32:27.240
going to increase forever,
because you have A sin omega c
00:32:27.240 --> 00:32:28.890
t.
00:32:28.890 --> 00:32:33.260
And this roughly A omega c t.
00:32:33.260 --> 00:32:37.620
And this is just vt.
00:32:37.620 --> 00:32:41.970
So what I want to argue
is that this is actually
00:32:41.970 --> 00:32:47.640
also oscillation, but with
angular frequency zero.
00:32:47.640 --> 00:32:58.470
And the general motion can
be in written as vt times c,
00:32:58.470 --> 00:33:02.240
for example, some constant.
00:33:02.240 --> 00:33:05.770
Any questions?
00:33:05.770 --> 00:33:09.310
So what I'm going to
do next may amaze you.
00:33:11.890 --> 00:33:12.550
Very good.
00:33:12.550 --> 00:33:18.790
So we have identified three
different kinds of modes.
00:33:18.790 --> 00:33:30.301
We have mode A, which is with
omega a squared equal to omega
00:33:30.301 --> 00:33:30.800
a squared.
00:33:30.800 --> 00:33:32.740
Where is omega a squared.
00:33:32.740 --> 00:33:33.610
There.
00:33:33.610 --> 00:33:36.410
It's 4k over 2m.
00:33:36.410 --> 00:33:40.870
And also, the
motion is like this.
00:33:40.870 --> 00:33:51.720
x1 equal t A cosine
omega a t plus phi a.
00:33:51.720 --> 00:34:00.420
x2 is equal to minus A, because
they have different sine.
00:34:00.420 --> 00:34:05.550
So if the motion is in the
left hand side direction,
00:34:05.550 --> 00:34:07.530
then the two masses
are oscillating
00:34:07.530 --> 00:34:09.420
in the opposite direction.
00:34:09.420 --> 00:34:14.719
So therefore, I get a minus
sign in front of A. Cosine omega
00:34:14.719 --> 00:34:17.310
a t plus phi a.
00:34:19.889 --> 00:34:28.590
x3 will be also equal to minus
A cosine omega a t plus phi a.
00:34:28.590 --> 00:34:32.730
Of course, I need to define
what this x1, x2, x3.
00:34:32.730 --> 00:34:37.770
That's why most of you
got super confused.
00:34:37.770 --> 00:34:43.690
So the x1, what
I mean is that is
00:34:43.690 --> 00:34:50.719
that the displacement of
the mass 2m, I call it x1.
00:34:50.719 --> 00:34:56.590
The displacement of the upper
mass, the upper little mass,
00:34:56.590 --> 00:34:59.110
I call it x2.
00:34:59.110 --> 00:35:05.410
And finally, the displacement
of the third mass, I call it x3.
00:35:05.410 --> 00:35:09.160
Therefore, you can
see that mode A,
00:35:09.160 --> 00:35:12.340
you have this kind of motion.
00:35:12.340 --> 00:35:17.500
The amplitude of the first
mass is A. Therefore,
00:35:17.500 --> 00:35:21.320
if I define that to be A,
then the second and third one,
00:35:21.320 --> 00:35:24.220
or the amplitude will
be defined as minus A.
00:35:24.220 --> 00:35:27.640
And you can see that all
of them are oscillating
00:35:27.640 --> 00:35:31.090
at fixed angular
frequency, omega
00:35:31.090 --> 00:35:37.156
a, omega a, omega a; and also
fixed phase, phi a, phi a, phi
00:35:37.156 --> 00:35:37.655
a.
00:35:42.260 --> 00:35:44.570
Of course, we can
also write down
00:35:44.570 --> 00:35:49.210
what we get for
mode B. For mode B,
00:35:49.210 --> 00:35:53.780
the left hand side mass
is not moving, stay put.
00:35:53.780 --> 00:35:56.930
And the other two
masses are oscillating
00:35:56.930 --> 00:36:00.470
at the frequency of omega b.
00:36:00.470 --> 00:36:06.680
And amplitude, they
differ by a minus sign.
00:36:06.680 --> 00:36:08.580
OK.
00:36:08.580 --> 00:36:12.820
Omega b squared is
equal to k over m
00:36:12.820 --> 00:36:18.320
from that logical argument.
00:36:18.320 --> 00:36:26.495
And then we get x1 equal
to 0 times cosine omega
00:36:26.495 --> 00:36:31.020
b t plus phi b.
00:36:31.020 --> 00:36:40.460
x2, I get B cosine
omega b t plus phi b.
00:36:40.460 --> 00:36:49.329
x3, I get minus B cosine
omega b t plus phi b.
00:36:49.329 --> 00:36:50.120
Any questions here?
00:36:53.120 --> 00:36:58.480
Finally, mode C.
All the mass, x1
00:36:58.480 --> 00:37:06.460
is equal to x2 is equal to
x3, is equal to C plus vt.
00:37:10.450 --> 00:37:17.470
So you can see that we have
identified three modes, mode
00:37:17.470 --> 00:37:25.660
A, mode B, and the mode C.
And there are three angular
00:37:25.660 --> 00:37:31.840
frequencies which we identified
for all of those normal modes,
00:37:31.840 --> 00:37:35.950
omega a, omega b, and omega c.
00:37:35.950 --> 00:37:38.380
And you can see that
we also identified
00:37:38.380 --> 00:37:40.920
how many free parameters.
00:37:40.920 --> 00:37:51.515
One free parameter, two,
three, four, five, and six.
00:37:54.470 --> 00:37:58.220
If you careful, you
write down the equation
00:37:58.220 --> 00:38:04.850
of motion of this system,
you will have three
00:38:04.850 --> 00:38:08.900
coupled differential equations.
00:38:08.900 --> 00:38:12.470
And those are second order
differential equations.
00:38:12.470 --> 00:38:17.570
If you have three variables,
three second order differential
00:38:17.570 --> 00:38:19.310
equations.
00:38:19.310 --> 00:38:23.660
If you manage it magically,
with the help from a computer
00:38:23.660 --> 00:38:30.800
or from math department
people, how many free parameter
00:38:30.800 --> 00:38:35.420
would you expect in
you a general solution?
00:38:35.420 --> 00:38:38.680
Can anybody tell
me well how many?
00:38:38.680 --> 00:38:44.131
I have three second order
differential equations.
00:38:44.131 --> 00:38:44.630
Yes?
00:38:44.630 --> 00:38:45.670
AUDIENCE: 6?
00:38:45.670 --> 00:38:48.180
YEN-JIE LEE: 6.
00:38:48.180 --> 00:38:53.220
So look at what we have
done we identified already
00:38:53.220 --> 00:38:56.190
1, 2, 3, three normal modes.
00:38:56.190 --> 00:39:03.030
By there are 1, 2, 3, 4,
5 6, six free parameters.
00:39:03.030 --> 00:39:05.985
That tells me I am done.
00:39:09.640 --> 00:39:10.280
I'm done.
00:39:10.280 --> 00:39:15.140
Because what is the
general solution?
00:39:15.140 --> 00:39:20.960
The general solution is just a
superposition of mode A mode B
00:39:20.960 --> 00:39:27.620
and mode C. You have six
free meters to be determined
00:39:27.620 --> 00:39:31.790
by six initial conditions,
which I would like--
00:39:31.790 --> 00:39:36.650
I have to tell you what are
those initial conditions.
00:39:36.650 --> 00:39:40.730
So isn't this amazing to you?
00:39:40.730 --> 00:39:44.390
I didn't even solve the
differential equation,
00:39:44.390 --> 00:39:48.050
and I already get the solution.
00:39:48.050 --> 00:39:50.540
And you can see another
lesson we learned
00:39:50.540 --> 00:39:53.750
from here is that,
oh no, you can
00:39:53.750 --> 00:39:56.180
imagine that the
motion of the system
00:39:56.180 --> 00:39:58.310
can be super complicated.
00:39:58.310 --> 00:40:02.180
This whole thing can do
this, all the crazy things
00:40:02.180 --> 00:40:04.190
are all displaced,
and the center of mass
00:40:04.190 --> 00:40:06.860
can move, as you said.
00:40:06.860 --> 00:40:12.830
But the result is actually
very easy to understand.
00:40:12.830 --> 00:40:17.360
It's just three kinds of motion,
the displacement, and two kinds
00:40:17.360 --> 00:40:21.410
of simple harmonic motion.
00:40:21.410 --> 00:40:22.760
We add them together.
00:40:22.760 --> 00:40:27.590
And then you get the general
description of that system.
00:40:27.590 --> 00:40:31.010
So everything is so nice.
00:40:31.010 --> 00:40:34.690
We understand the
motion of that system.
00:40:34.690 --> 00:40:41.180
But in general,
life is very hard.
00:40:41.180 --> 00:40:46.010
For example, now I do
something crazy here.
00:40:46.010 --> 00:40:51.070
I change this to 3.
00:40:51.070 --> 00:40:53.350
What are the normal modes?
00:40:53.350 --> 00:40:55.550
Can anybody tell me?
00:40:55.550 --> 00:40:59.050
It becomes very, very
difficult, because there's
00:40:59.050 --> 00:41:03.370
no general symmetry of
that kind of system.
00:41:03.370 --> 00:41:05.890
So we are in trouble.
00:41:05.890 --> 00:41:09.250
One of the modes
maybe still there,
00:41:09.250 --> 00:41:13.770
which is actually mode
B. But the other modes
00:41:13.770 --> 00:41:16.900
are harder to actually guess.
00:41:16.900 --> 00:41:22.790
So you can see that that already
brings you a lot of trouble.
00:41:22.790 --> 00:41:29.330
And you can see that I can now
couple not just two objects,
00:41:29.330 --> 00:41:32.750
I can couple three objects,
four objects, five objects,
00:41:32.750 --> 00:41:33.920
10 objects.
00:41:33.920 --> 00:41:37.020
Maybe I will put that in your
p set and see what happens.
00:41:37.020 --> 00:41:39.980
And you can see
that this becomes
00:41:39.980 --> 00:41:42.590
very difficult to manage.
00:41:42.590 --> 00:41:47.030
So what I'm going to do in
the rest of this lecture
00:41:47.030 --> 00:41:49.760
is to introduce
you a method which
00:41:49.760 --> 00:41:54.800
you can follow in general
to solve the question
00:41:54.800 --> 00:41:59.120
and get the normal mode
frequencies and normal modes.
00:41:59.120 --> 00:42:01.950
So we will take a
four minute break.
00:42:01.950 --> 00:42:04.940
And we come back at 12:20.
00:42:04.940 --> 00:42:07.160
So if you have any
questions, let me know.
00:42:12.790 --> 00:42:17.380
What we are going to do
in the following exercise
00:42:17.380 --> 00:42:22.750
is to try to understand a
general strategy to solve
00:42:22.750 --> 00:42:26.980
the normal mode frequencies
and the normal mode amplitudes,
00:42:26.980 --> 00:42:30.640
so that you can apply this
technique to all kinds
00:42:30.640 --> 00:42:32.320
of different systems.
00:42:32.320 --> 00:42:36.590
So what I am going to do today
is to take these three mass
00:42:36.590 --> 00:42:41.595
system, and of course as
usual, I try to define what
00:42:41.595 --> 00:42:43.600
is this coordinate system?
00:42:43.600 --> 00:42:45.520
The coordinate system
I'm going to use
00:42:45.520 --> 00:42:51.790
is x1 and x2 an x3 describing
the displacement of the mass
00:42:51.790 --> 00:42:53.750
from the equilibrium position.
00:42:53.750 --> 00:42:55.420
And the equilibrium
means that there's
00:42:55.420 --> 00:42:57.970
no stretch on the spring.
00:42:57.970 --> 00:43:00.310
The string is unstretched.
00:43:00.310 --> 00:43:05.780
It's at their own
natural length, l0.
00:43:05.780 --> 00:43:12.910
So once I define that, I can
do a force diagram analysis.
00:43:12.910 --> 00:43:17.140
So that starts from
the left hand side mass
00:43:17.140 --> 00:43:19.870
with mass equal to 2m.
00:43:19.870 --> 00:43:23.350
I can write down the equation
of motion, 2m x1 double dot.
00:43:26.290 --> 00:43:37.730
And this is going to be equal
to k x2 minus x1 plus k x3
00:43:37.730 --> 00:43:38.230
minus x1.
00:43:40.870 --> 00:43:45.850
So there are two spring
forces acting on this mass,
00:43:45.850 --> 00:43:47.140
the left hand side mass.
00:43:47.140 --> 00:43:49.990
The first one is
the upper spring.
00:43:49.990 --> 00:43:55.720
The second one is coming
from the lower one.
00:43:55.720 --> 00:43:58.340
And you can see
that both of them
00:43:58.340 --> 00:44:00.940
are proportional to
spring constant k,
00:44:00.940 --> 00:44:05.650
and also proportional to
the relative displacement.
00:44:05.650 --> 00:44:09.610
And you can see that the two
relative displacement, which
00:44:09.610 --> 00:44:14.130
is the amount of
stretch to the spring,
00:44:14.130 --> 00:44:20.540
is actually x2 minus
x1, and the x3 minus x1.
00:44:20.540 --> 00:44:22.720
Am I going too fast?
00:44:22.720 --> 00:44:23.220
OK.
00:44:23.220 --> 00:44:24.410
Everybody's following.
00:44:24.410 --> 00:44:26.420
And you can actually
check the sign.
00:44:26.420 --> 00:44:27.620
So you may not be sure.
00:44:27.620 --> 00:44:30.470
Maybe this is your x1 minus x2.
00:44:30.470 --> 00:44:33.200
But you can check that,
because if you increase
00:44:33.200 --> 00:44:36.200
x1, what is going to happen?
00:44:36.200 --> 00:44:38.780
This term will
become more negative.
00:44:38.780 --> 00:44:42.820
More negative in this
coordinate system
00:44:42.820 --> 00:44:45.410
is pointing to the
left hand side.
00:44:45.410 --> 00:44:46.670
So that makes sense.
00:44:46.670 --> 00:44:49.890
Because if I move this
mass to the right side,
00:44:49.890 --> 00:44:53.380
then I am compressing
the springs.
00:44:53.380 --> 00:44:55.520
Therefore, they are
pushing it back.
00:44:55.520 --> 00:45:02.210
Therefore, this is actually
the correct sign, x2 minus x1.
00:45:02.210 --> 00:45:07.130
The same thing also applies
to the second spring force.
00:45:07.130 --> 00:45:10.655
So that's a way I double
check if I make a mistake.
00:45:13.220 --> 00:45:16.450
Now, this is actually the
first equations of motion.
00:45:16.450 --> 00:45:20.300
And I can now also
work on a second mass.
00:45:20.300 --> 00:45:23.540
Now I focus on a
mass number two.
00:45:23.540 --> 00:45:26.020
The displacement is x2.
00:45:26.020 --> 00:45:28.760
Therefore the left hand
side of Newton's Law
00:45:28.760 --> 00:45:32.780
is m x2 double dot.
00:45:32.780 --> 00:45:36.920
And that is equal
to the spring force.
00:45:36.920 --> 00:45:42.650
The spring force, there's
only one spring force
00:45:42.650 --> 00:45:44.510
acting on the mass.
00:45:44.510 --> 00:45:48.790
Therefore, what I am going
to get is k x1 minus x2.
00:45:53.850 --> 00:45:55.060
Everybody's following?
00:45:57.830 --> 00:46:02.730
You can actually check
the sign carefully, also.
00:46:02.730 --> 00:46:05.780
And finally, I have the
third mass, very similar
00:46:05.780 --> 00:46:07.850
to mass number two.
00:46:07.850 --> 00:46:10.380
I can write down the
equation of motion,
00:46:10.380 --> 00:46:16.120
which should k x1 minus x3.
00:46:16.120 --> 00:46:21.740
So that is my coupled second
order differential equations.
00:46:21.740 --> 00:46:23.060
There are three equations.
00:46:23.060 --> 00:46:27.260
And all of them are
second order equations.
00:46:27.260 --> 00:46:30.050
So this looks a bit messy.
00:46:30.050 --> 00:46:32.810
So what I'm going
to do Is no magic.
00:46:32.810 --> 00:46:37.550
I'm just collecting all
the terms belonging to x1,
00:46:37.550 --> 00:46:41.480
and put them together, all
the terms belonging to x2,
00:46:41.480 --> 00:46:45.470
putting all together, and
just rearranging things.
00:46:45.470 --> 00:46:47.300
So no magic.
00:46:47.300 --> 00:46:50.340
So I copied this
thing, left hand side.
00:46:50.340 --> 00:46:52.790
2m x1 double dot.
00:46:52.790 --> 00:47:00.260
And the dot will be equal
to minus 2k x1 plus,
00:47:00.260 --> 00:47:05.930
I collect all the times
related to x2, plus k x2,
00:47:05.930 --> 00:47:13.070
there's only one term
here, then plus k x3.
00:47:13.070 --> 00:47:17.065
I'm just trying to
organize my question.
00:47:17.065 --> 00:47:19.190
So you can see that I
collect all the terms related
00:47:19.190 --> 00:47:21.800
to x1 to here.
00:47:21.800 --> 00:47:24.560
Minus k, minus k,
I get minus 2k.
00:47:24.560 --> 00:47:28.850
And the plus k for
x2, plus k for the x3.
00:47:28.850 --> 00:47:32.480
And I can also do that
for m x2 double dot.
00:47:32.480 --> 00:47:41.930
That will be equal to k x1
minus k x2 plus zero x3,
00:47:41.930 --> 00:47:44.900
just for completeness.
00:47:44.900 --> 00:47:50.207
I can also do the same thing
for the third mass, m x3 double
00:47:50.207 --> 00:47:50.940
dot.
00:47:50.940 --> 00:47:58.920
This is equal to k x1 plus 0 x2.
00:47:58.920 --> 00:48:04.880
There's no dependence on
x2, because x1 and x2--
00:48:04.880 --> 00:48:09.440
x3 and x2 are not talking
to each either directly.
00:48:09.440 --> 00:48:13.880
Finally, I have the third,
which is minus k x3.
00:48:19.720 --> 00:48:27.400
Now our job is to solve
those coupled equations.
00:48:27.400 --> 00:48:29.720
Of course, you have
the freedom, if you
00:48:29.720 --> 00:48:33.130
know how to solve
it yourself, you
00:48:33.130 --> 00:48:35.570
can already go
ahead and solve it.
00:48:35.570 --> 00:48:37.540
But what I am
going to do here is
00:48:37.540 --> 00:48:41.560
to introduce technique,
which can be useful for you
00:48:41.560 --> 00:48:44.750
and make it easier to follow.
00:48:44.750 --> 00:48:47.020
It's a fixed procedure.
00:48:47.020 --> 00:48:49.880
So what I can do
is the following.
00:48:49.880 --> 00:48:56.450
I can write everything
in them form of a matrix.
00:48:56.450 --> 00:49:00.480
How many of you heard the
matrix for the first time?
00:49:00.480 --> 00:49:03.550
1, 2, 3, 4.
00:49:03.550 --> 00:49:04.050
OK.
00:49:04.050 --> 00:49:05.470
Only four.
00:49:05.470 --> 00:49:09.490
But if you are not being
familiar with matrix,
00:49:09.490 --> 00:49:11.160
let me know, and I can help you.
00:49:11.160 --> 00:49:12.370
Let the TA know.
00:49:12.370 --> 00:49:17.860
And also, there's a section
in the textbooks, which
00:49:17.860 --> 00:49:21.790
I posted on announcement,
which is actually very
00:49:21.790 --> 00:49:24.260
helpful to understand matrices.
00:49:24.260 --> 00:49:26.680
But sorry to these
four students,
00:49:26.680 --> 00:49:28.000
we are going to use that.
00:49:28.000 --> 00:49:32.500
And maybe, you already learn
how it works from here.
00:49:32.500 --> 00:49:36.280
So one trick which we
will use in this class
00:49:36.280 --> 00:49:41.500
is to convert everything
into matrix format.
00:49:41.500 --> 00:49:44.260
What I am going to do
is to write everything
00:49:44.260 --> 00:49:49.270
in terms of M,
capital X, capital M,
00:49:49.270 --> 00:49:56.620
capital X double dot equal
to minus capital K capital X.
00:49:56.620 --> 00:50:01.450
Capital M, capital
X, and capital K,
00:50:01.450 --> 00:50:05.800
those are all matrices.
00:50:05.800 --> 00:50:09.130
Because I write this
thing I really carefully,
00:50:09.130 --> 00:50:12.640
therefore we can already
immediately identify
00:50:12.640 --> 00:50:18.010
what would be M, capital
M and capital X and a K.
00:50:18.010 --> 00:50:28.210
So I can write down immediately
will be equal to 2m, 0, 0, 0,
00:50:28.210 --> 00:50:35.190
m, 0, 0, 0, m.
00:50:35.190 --> 00:50:39.750
Because there's only one in each
line, you only have one term.
00:50:39.750 --> 00:50:44.250
X1 double dot, x2 double
dot, x3 double dot.
00:50:44.250 --> 00:50:46.290
And also, you can
write down what
00:50:46.290 --> 00:50:51.030
will be the X. This
is actually a vector.
00:50:51.030 --> 00:50:56.040
X will be equal to x1, x2, x3.
00:50:59.680 --> 00:51:02.310
Finally, you have the K?
00:51:02.310 --> 00:51:05.860
How do I read off K?
00:51:05.860 --> 00:51:08.010
Careful, there's
a minus sign here,
00:51:08.010 --> 00:51:13.510
because I would like to make
this matrix equation as if it's
00:51:13.510 --> 00:51:16.740
describing a simple
harmonic motion of a one
00:51:16.740 --> 00:51:18.080
dimensional system.
00:51:18.080 --> 00:51:20.940
So it looks the same, but they
are different because those
00:51:20.940 --> 00:51:22.810
are matrices.
00:51:22.810 --> 00:51:25.710
But therefore, I
have in my convention
00:51:25.710 --> 00:51:27.780
I have this minus sign there.
00:51:27.780 --> 00:51:32.300
Therefore, when you read off
the K, you have to be careful.
00:51:32.300 --> 00:51:35.124
So what is K?
00:51:35.124 --> 00:51:40.950
K is equal to 2k.
00:51:40.950 --> 00:51:44.190
You have the minus 2k
here in front of x1.
00:51:44.190 --> 00:51:47.520
But because I have a minus
sign there, therefore
00:51:47.520 --> 00:51:50.610
this one is actually taken out.
00:51:50.610 --> 00:51:53.580
So we have 2k there.
00:51:53.580 --> 00:52:04.570
Then you have minus k,
minus k, minus k, k, 0.
00:52:04.570 --> 00:52:08.370
Minus k, plus k, 0.
00:52:08.370 --> 00:52:14.220
And finally, you can
also finish the last row.
00:52:14.220 --> 00:52:17.618
You get minus k, 0, k.
00:52:21.960 --> 00:52:23.740
K becomes minus k.
00:52:23.740 --> 00:52:27.560
Minus k becomes k.
00:52:27.560 --> 00:52:32.930
So we have read off all
those matrices successfully.
00:52:32.930 --> 00:52:38.060
So you may ask,
what do they mean?
00:52:38.060 --> 00:52:40.040
Do they get the meaning?
00:52:40.040 --> 00:52:45.350
M, K, X, what those?
00:52:45.350 --> 00:52:52.370
M, capital M matrix, tells
you the mass distribution
00:52:52.370 --> 00:52:54.270
inside the system.
00:52:54.270 --> 00:52:58.390
So that's the meaning
of this matrix.
00:52:58.390 --> 00:53:02.180
X is actually
vector, which tells
00:53:02.180 --> 00:53:08.800
the position of individual
components in the system.
00:53:08.800 --> 00:53:11.140
Finally, what is K?
00:53:11.140 --> 00:53:16.890
K is telling you how each
component in the system
00:53:16.890 --> 00:53:21.420
talks to the other components.
00:53:21.420 --> 00:53:25.470
So K is telling you
the communication
00:53:25.470 --> 00:53:29.550
inside the system.
00:53:29.550 --> 00:53:33.180
So now we understand a
bit what is going on.
00:53:33.180 --> 00:53:38.040
And as usual, I will go
to the complex notation.
00:53:45.790 --> 00:53:53.310
So I have xj, the small
xj are the position
00:53:53.310 --> 00:53:57.310
of the mass, x1, x2, and x3.
00:53:57.310 --> 00:54:04.540
xj will be real
part of small zj.
00:54:04.540 --> 00:54:07.710
Small xj equal to
real part of zj.
00:54:07.710 --> 00:54:14.370
Therefore, I can now write
everything in terms of matrices
00:54:14.370 --> 00:54:15.150
again.
00:54:15.150 --> 00:54:20.708
So now I can write
the solution to be Z,
00:54:20.708 --> 00:54:30.410
the capitol z is a matrix,
exponential i omega t plus phi.
00:54:30.410 --> 00:54:33.320
This is the guess
the solution I have.
00:54:33.320 --> 00:54:37.640
A1, A2, and A3.
00:54:37.640 --> 00:54:41.130
Those are the
amplitudes, amplitude A
00:54:41.130 --> 00:54:45.620
of the first mass, amplitude
of the second mass, amplitude
00:54:45.620 --> 00:54:52.020
of the third mass,
in their normal mode.
00:54:52.020 --> 00:55:00.300
And all of those are oscillating
at the same frequency, omega,
00:55:00.300 --> 00:55:03.720
and the same phase, phi.
00:55:03.720 --> 00:55:08.580
Does that tell you something
which we learned before?
00:55:08.580 --> 00:55:12.150
Oh, that's the definition
of the normal mode.
00:55:12.150 --> 00:55:15.150
I'm using the definition
of the normal mode.
00:55:15.150 --> 00:55:17.580
Every part of the
system is oscillating
00:55:17.580 --> 00:55:20.520
at the same frequency
in the same phase.
00:55:20.520 --> 00:55:25.590
And we use that to
construct my solution.
00:55:25.590 --> 00:55:29.940
The complex version is
exponential i omega t plus phi,
00:55:29.940 --> 00:55:32.000
oscillating at the
same frequency,
00:55:32.000 --> 00:55:34.680
oscillating at the same phase.
00:55:34.680 --> 00:55:37.730
And those are the amplitude,
which I will solve later.
00:55:37.730 --> 00:55:41.280
OK, any questions?
00:55:41.280 --> 00:55:44.100
I hope I'm not going too fast.
00:55:44.100 --> 00:55:48.060
If everybody can
follow, now I can
00:55:48.060 --> 00:55:54.540
go ahead and solve the
equation in the matrix format.
00:55:54.540 --> 00:55:59.550
So now I go to the
complex notation.
00:55:59.550 --> 00:56:06.510
So the equation M X double
dot equal to minus KX
00:56:06.510 --> 00:56:12.810
becomes M Z double
dot equal to minus KZ.
00:56:17.990 --> 00:56:22.530
And also, I can immediately
get the Z double dot will
00:56:22.530 --> 00:56:27.920
be equal to minus
omega squared Z,
00:56:27.920 --> 00:56:31.800
because each time I
do a differentiation,
00:56:31.800 --> 00:56:37.470
I get i omega out of the
exponential function.
00:56:37.470 --> 00:56:39.460
And I cannot kill that
exponential function,
00:56:39.460 --> 00:56:40.950
so it's still there.
00:56:40.950 --> 00:56:43.440
Therefore, I get
minus omega squared
00:56:43.440 --> 00:56:48.360
in front of Z. I hope
that doesn't surprise you.
00:56:48.360 --> 00:56:50.880
So that's very nice
and very good news.
00:56:50.880 --> 00:56:55.010
That means I can replace this
Z double prime with minus omega
00:56:55.010 --> 00:57:01.050
squared Z. Then I get
minus M omega squared Z.
00:57:01.050 --> 00:57:04.200
And this is equal to minus KZ.
00:57:04.200 --> 00:57:06.600
And I can cancel the minus sign.
00:57:06.600 --> 00:57:08.230
That becomes
something like this.
00:57:12.930 --> 00:57:20.790
So, I can now cancel the
exponential i omega t plus phi,
00:57:20.790 --> 00:57:23.430
because I have Z in
the left hand side.
00:57:23.430 --> 00:57:28.270
And exponential i omega t
plus phi is just a number.
00:57:28.270 --> 00:57:30.780
So therefore, I can cancel it.
00:57:30.780 --> 00:57:34.420
So what is going to
happen if I do that?
00:57:34.420 --> 00:57:37.740
Basically, what
I'm going get is I
00:57:37.740 --> 00:57:44.670
get M omega squared
A equal to K A.
00:57:44.670 --> 00:57:48.330
I'm trying to go extremely
slowly, because this
00:57:48.330 --> 00:57:52.200
is the first time we
go through matrices.
00:57:52.200 --> 00:57:55.020
So now you have this expression.
00:57:55.020 --> 00:58:00.410
Left hand is a matrix, M, times
some constant, omega squared.
00:58:00.410 --> 00:58:02.910
I can actually get omega
squared in front of it,
00:58:02.910 --> 00:58:06.060
because this is
actually just a number.
00:58:06.060 --> 00:58:12.480
A is just a vector, which is
A1, A2, A3, also a matrix.
00:58:12.480 --> 00:58:17.340
K is actually how the individual
components talks to the others.
00:58:17.340 --> 00:58:20.490
So that's there, times A.
00:58:20.490 --> 00:58:23.910
Now I would like
to move everything
00:58:23.910 --> 00:58:26.640
to the right hand side,
all the matrices in front A
00:58:26.640 --> 00:58:27.840
to the right hand side.
00:58:27.840 --> 00:58:32.760
Then I multiply both
sides by M minus 1.
00:58:32.760 --> 00:58:39.100
So I multiply M minus 1
to the whole equation.
00:58:39.100 --> 00:58:43.110
M minus 1, what is M minus 1?
00:58:43.110 --> 00:58:46.710
The definition is
that the inverse of M
00:58:46.710 --> 00:58:48.940
is called M minus 1.
00:58:48.940 --> 00:58:57.240
M minus 1 times M is equal to
I, which is actually 1, 1, 1.
00:58:57.240 --> 00:59:02.670
Therefore, if I do this thing,
then I would get omega squared.
00:59:02.670 --> 00:59:09.970
M minus 1 times M
becomes I, unit matrix.
00:59:09.970 --> 00:59:17.280
And this is equal to M
minus 1 K A. And be careful,
00:59:17.280 --> 00:59:21.140
I multiply M minus
1, the inverse of M,
00:59:21.140 --> 00:59:23.060
from the left hand side.
00:59:23.060 --> 00:59:25.800
That matters.
00:59:25.800 --> 00:59:30.390
So now I can move
everything to the same side.
00:59:30.390 --> 00:59:33.970
I moved the left hand side
term to the right hand side.
00:59:33.970 --> 00:59:40.670
Therefore, I get M minus
1 K minus omega squared
00:59:40.670 --> 00:59:46.430
I. Those are all matrices.
00:59:46.430 --> 00:59:49.330
Times A, this is equal to 0.
00:59:54.400 --> 00:59:55.370
Any questions?
01:00:00.430 --> 01:00:02.950
So a lot of manipulation.
01:00:02.950 --> 01:00:07.880
But if you think about it,
and you are following me,
01:00:07.880 --> 01:00:11.840
you'll see that all
those steps are exactly
01:00:11.840 --> 01:00:17.140
identical to what we have been
doing for a single harmonic
01:00:17.140 --> 01:00:18.440
oscillator.
01:00:18.440 --> 01:00:20.000
Looks pretty familiar to you.
01:00:20.000 --> 01:00:24.708
But the difference is that now
we are dealing with matrices.
01:00:24.708 --> 01:00:25.910
AUDIENCE: What is A?
01:00:25.910 --> 01:00:30.870
YEN-JIE LEE: Oh, A. A
is actually this guy.
01:00:30.870 --> 01:00:33.620
I define this to be
A. And that means
01:00:33.620 --> 01:00:40.930
Z will be exponential i omega
t plus phi times A. I didn't
01:00:40.930 --> 01:00:42.360
actually write it explicitly.
01:00:42.360 --> 01:00:43.910
But that's what I mean.
01:00:46.500 --> 01:00:49.412
Any more questions?
01:00:49.412 --> 01:00:50.861
Yes?
01:00:50.861 --> 01:00:55.605
AUDIENCE: [INAUDIBLE]
is for [INAUDIBLE]??
01:00:55.605 --> 01:00:56.980
YEN-JIE LEE: Can
you repeat that?
01:00:56.980 --> 01:01:01.230
AUDIENCE: So this whole
process, this is mode A, right?
01:01:01.230 --> 01:01:01.980
YEN-JIE LEE: Yeah.
01:01:01.980 --> 01:01:05.970
So this whole process is
for, not really the for mode
01:01:05.970 --> 01:01:09.390
A. So that A may be confusing.
01:01:09.390 --> 01:01:12.080
But in general, if
I have a solution,
01:01:12.080 --> 01:01:16.830
and I assume that the amplitude
can be described by a matrix.
01:01:16.830 --> 01:01:17.820
So it's in general.
01:01:17.820 --> 01:01:19.630
And you'll see that
we can actually
01:01:19.630 --> 01:01:24.690
derive the angular frequency
of mode A, mode B, and mode C
01:01:24.690 --> 01:01:26.290
afterward.
01:01:26.290 --> 01:01:28.020
I hope that answers
your question.
01:01:28.020 --> 01:01:31.350
So you see that for in
general, what I have been doing
01:01:31.350 --> 01:01:34.320
is that now, all
those things are
01:01:34.320 --> 01:01:39.520
equivalent to the original
equation of motion.
01:01:39.520 --> 01:01:44.780
What I am doing is
purely cosmetic.
01:01:44.780 --> 01:01:47.810
You see, make it beautiful.
01:01:47.810 --> 01:01:52.850
So all those things, this
thing is exactly the equivalent
01:01:52.850 --> 01:01:56.190
to that thing, up there.
01:01:56.190 --> 01:01:59.300
Up to M X double dot
equal to minus K x.
01:01:59.300 --> 01:02:01.140
Cosmetics.
01:02:01.140 --> 01:02:02.560
Beautiful.
01:02:02.560 --> 01:02:06.570
Looks-- I like it.
01:02:06.570 --> 01:02:07.650
All right.
01:02:07.650 --> 01:02:10.170
Then what I have been
doing is that now
01:02:10.170 --> 01:02:15.210
I introduce using a
definition of normal mode.
01:02:15.210 --> 01:02:20.190
I guess the solution will
have this functional form.
01:02:20.190 --> 01:02:25.300
Z equals to exponential i
omega t plus phi, everybody
01:02:25.300 --> 01:02:28.830
oscillating at the same
frequency, the same phase.
01:02:28.830 --> 01:02:32.010
Frequency omega, phase phi.
01:02:32.010 --> 01:02:35.130
And everybody can have
different amplitude.
01:02:35.130 --> 01:02:39.240
You can see from this
example, normal modes,
01:02:39.240 --> 01:02:42.710
they can have
different amplitude.
01:02:42.710 --> 01:02:44.490
The amplitude is what?
01:02:44.490 --> 01:02:45.900
I don't know yet.
01:02:45.900 --> 01:02:48.150
But we will figure it out.
01:02:48.150 --> 01:02:50.520
Then that's my assumption.
01:02:50.520 --> 01:02:52.510
The definition of normal mode.
01:02:52.510 --> 01:02:56.220
And I plug in to the
equation of motion.
01:02:56.220 --> 01:02:59.790
Then this is what we
are doing to simplify
01:02:59.790 --> 01:03:01.350
the equation of motion.
01:03:01.350 --> 01:03:03.540
There's no magic here.
01:03:03.540 --> 01:03:09.670
If I plug in the definition on
normal mode to that equation,
01:03:09.670 --> 01:03:15.460
this is actually going to bring
you to this equation, matrix
01:03:15.460 --> 01:03:18.210
equation.
01:03:18.210 --> 01:03:24.000
So if you have learned matrices
before, you have something,
01:03:24.000 --> 01:03:30.930
some matrix, times Z.
This is equal to zero.
01:03:30.930 --> 01:03:33.660
A is not zero.
01:03:33.660 --> 01:03:34.600
I hope.
01:03:34.600 --> 01:03:37.780
If it's zero, then the
whole system is not moving.
01:03:37.780 --> 01:03:39.780
Then it's not fun.
01:03:42.520 --> 01:03:47.040
So if A is not zero, then
this thing should be--
01:03:47.040 --> 01:03:52.290
this thing times A should
make this equation equal to 0.
01:03:52.290 --> 01:03:56.980
So what is actually
the required condition?
01:03:56.980 --> 01:03:58.080
I get stuck,
01:03:58.080 --> 01:04:01.440
and of course again, my
friend from math department
01:04:01.440 --> 01:04:04.080
comes to save me.
01:04:04.080 --> 01:04:10.930
That means if this
thing has a solution,
01:04:10.930 --> 01:04:12.640
this equation has
a solution, that
01:04:12.640 --> 01:04:23.090
means that determinant of
M minus 1 K minus omega
01:04:23.090 --> 01:04:29.750
squared I has to be equal to 0.
01:04:29.750 --> 01:04:34.790
So that is the condition
for this equation
01:04:34.790 --> 01:04:41.030
to satisfy this
to be equal to 0.
01:04:41.030 --> 01:04:45.320
And just to make sure that I
don't know what is the angular
01:04:45.320 --> 01:04:46.820
frequency omega yet.
01:04:46.820 --> 01:04:49.990
I don't know what
is the phi yet.
01:04:49.990 --> 01:04:55.730
We can actually solve the
angular frequency, omega.
01:04:55.730 --> 01:05:01.070
So now, turn everything around.
01:05:01.070 --> 01:05:05.060
And basically now,
using this normal mode
01:05:05.060 --> 01:05:11.290
definition, and some
mathematical manipulation,
01:05:11.290 --> 01:05:16.440
the condition we need for this
equation to satisfy equal to 0,
01:05:16.440 --> 01:05:22.190
is determinant M minus 1
K minus omega squared I.
01:05:22.190 --> 01:05:25.930
I can write down M
minus 1 K minus omega
01:05:25.930 --> 01:05:32.870
squared I explicitly, just
to help you with mathematics.
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.
01:05:50.500 --> 01:05:56.230
It's just the inverse
matrix of the M matrix.
01:05:56.230 --> 01:06:07.090
Therefore, now I can write
down the explicit expression
01:06:07.090 --> 01:06:10.350
of M minus 1 K
minus omega squared
01:06:10.350 --> 01:06:23.730
I. This will be equal to k
over m minus omega squared,
01:06:23.730 --> 01:06:34.370
minus k over 2m, minus k
over 2, minus k over m,
01:06:34.370 --> 01:06:40.030
k over m minus omega squared, 0.
01:06:40.030 --> 01:06:42.600
I will write down all
the elements first.
01:06:42.600 --> 01:06:46.392
Then I will explain to you how
I arrived at the expression.
01:06:52.200 --> 01:06:53.150
OK.
01:06:53.150 --> 01:06:55.180
So this is M minus 1 k.
01:06:55.180 --> 01:06:58.790
The definition of
M minus 1 is that.
01:06:58.790 --> 01:07:02.280
And the definition of
K is in the upper right
01:07:02.280 --> 01:07:04.980
corner of the black board.
01:07:04.980 --> 01:07:08.620
Therefore, if you
multiply M minus 1 K,
01:07:08.620 --> 01:07:14.130
basically, the first
column will get--
01:07:18.240 --> 01:07:18.850
wait a second.
01:07:18.850 --> 01:07:20.100
Did I make a mistake?
01:07:24.670 --> 01:07:25.170
No.
01:07:25.170 --> 01:07:26.130
OK.
01:07:26.130 --> 01:07:32.440
So basically, what you
arrive at is k/m, k/m, k/m.
01:07:36.430 --> 01:07:42.760
And also, the minus k
over 2m for the rest part
01:07:42.760 --> 01:07:44.440
of the matrix.
01:07:44.440 --> 01:07:48.235
And the minus omega
squared I will give you
01:07:48.235 --> 01:07:51.145
the diagonal component.
01:07:51.145 --> 01:07:52.227
Yes?
01:07:52.227 --> 01:07:54.268
AUDIENCE: Why do you have
to take the determinant
01:07:54.268 --> 01:07:56.287
and set it equal to
0 instead of just
01:07:56.287 --> 01:07:59.230
setting that equal to zero?
01:07:59.230 --> 01:08:02.740
AUDIENCE: This is a matrix.
01:08:02.740 --> 01:08:04.150
So these are the matrix.
01:08:04.150 --> 01:08:08.020
So a matrix times A
will be equal to zero.
01:08:08.020 --> 01:08:12.670
The general condition
for that to be satisfied
01:08:12.670 --> 01:08:14.440
is more general.
01:08:14.440 --> 01:08:19.359
It's actually the determinant
of this matrix equal to zero.
01:08:19.359 --> 01:08:27.790
Because this is actually
multiplied by some back to A.
01:08:27.790 --> 01:08:31.430
So I think there are
mathematical manipulation.
01:08:31.430 --> 01:08:33.270
Basically, you would
just collect the terms.
01:08:33.270 --> 01:08:37.140
And then calculate
M minus 1 K first.
01:08:37.140 --> 01:08:40.930
And the minus omega squared I
will give you all the diagonal
01:08:40.930 --> 01:08:44.899
and terms have a minus
omega square there.
01:08:44.899 --> 01:08:48.540
And that is actually the matrix.
01:08:48.540 --> 01:08:52.290
And of course, I can
calculate the determinant.
01:08:52.290 --> 01:08:54.479
So if I calculate
the determinant,
01:08:54.479 --> 01:08:58.319
then basically I get this
times that times that.
01:08:58.319 --> 01:09:07.950
So what you get is k over m
minus omega squared times k
01:09:07.950 --> 01:09:15.430
over m minus omega squared times
k over m minus omega squared.
01:09:15.430 --> 01:09:17.510
So these are all diagonal terms.
01:09:17.510 --> 01:09:26.340
And the minus 1 over 2 k
squared over m squared, k
01:09:26.340 --> 01:09:32.879
squared over m squared.
01:09:35.770 --> 01:09:36.790
sorry.
01:09:36.790 --> 01:09:40.600
Minus omega squared.
01:09:40.600 --> 01:09:46.649
So that's this off diagonal
term, this times this times
01:09:46.649 --> 01:09:47.310
that.
01:09:47.310 --> 01:09:48.130
OK.
01:09:48.130 --> 01:09:50.050
It will give you
the second term.
01:09:50.050 --> 01:09:52.390
And the third one,
which survived
01:09:52.390 --> 01:09:56.320
because of those zeros, many,
many terms are equal to 0.
01:09:56.320 --> 01:10:01.590
And then the third term, which
is nonzero, is again minus 1
01:10:01.590 --> 01:10:09.790
over 2k squared over m squared,
k over m minus omega squared.
01:10:09.790 --> 01:10:13.270
And this is actually equal
to 0, because the determinant
01:10:13.270 --> 01:10:17.380
of this matrix is equal to zero.
01:10:17.380 --> 01:10:19.320
Everybody following?
01:10:19.320 --> 01:10:20.320
A little bit of a mess.
01:10:20.320 --> 01:10:23.760
Because I have been doing
something very challenging.
01:10:23.760 --> 01:10:32.410
I'm solving a 3 by 3 matrix
problem in front of you right.
01:10:32.410 --> 01:10:35.600
So the math can get
a bit complicated.
01:10:35.600 --> 01:10:40.135
But next time, I think we are
going to go to a second order
01:10:40.135 --> 01:10:42.070
one, 2 by 2 matrix.
01:10:42.070 --> 01:10:45.040
And I think that will
be slightly easier.
01:10:45.040 --> 01:10:47.650
But the general
approach is the same.
01:10:47.650 --> 01:10:52.480
So basically, you calculate M
minus 1 K minus omega squared.
01:10:52.480 --> 01:10:57.820
Then you get what is inside, all
the content inside this matrix.
01:10:57.820 --> 01:11:00.640
Then you would calculate
the determinant.
01:11:00.640 --> 01:11:05.800
And basically, you can
solve this equation.
01:11:05.800 --> 01:11:11.250
Now I can define omega0
squared to be k/m.
01:11:11.250 --> 01:11:15.210
And I can actually make this
expression much simpler.
01:11:18.840 --> 01:11:20.970
Then basically,
what you are getting
01:11:20.970 --> 01:11:25.660
is omega0 squared
minus omega squared
01:11:25.660 --> 01:11:33.480
to the third minus 1/2 omega0
to the fourth, omega0 squared
01:11:33.480 --> 01:11:35.650
minus omega squared.
01:11:35.650 --> 01:11:42.930
Minus 1/2 omega0 to the
fourth, omega0 to the square,
01:11:42.930 --> 01:11:43.980
minus omega squared.
01:11:43.980 --> 01:11:47.340
And this is equal to 0.
01:11:47.340 --> 01:11:51.684
And you can factor out
the common components.
01:11:51.684 --> 01:11:53.100
Then basically,
what you are going
01:11:53.100 --> 01:11:56.370
to get is, you can
write this thing
01:11:56.370 --> 01:12:03.270
to be omega0 squared minus
omega squared, omega squared.
01:12:03.270 --> 01:12:06.390
Because all of them
have omega squared.
01:12:06.390 --> 01:12:12.430
And omega squared
minus 2 omega0 squared.
01:12:12.430 --> 01:12:14.770
And that's equal to 0.
01:12:14.770 --> 01:12:18.520
So I am skipping a lot of steps
from this one to that one.
01:12:18.520 --> 01:12:22.690
But in general, you can solve
this third order equation.
01:12:22.690 --> 01:12:29.530
And I can first combine
all those terms together.
01:12:29.530 --> 01:12:33.050
And then I factor out
the common components.
01:12:33.050 --> 01:12:35.500
Then basically, what you
are going to arrive at
01:12:35.500 --> 01:12:38.560
is something like this.
01:12:38.560 --> 01:12:39.890
A lot of math here.
01:12:39.890 --> 01:12:43.240
But we are close to the end.
01:12:43.240 --> 01:12:48.440
So you can see now what are the
possible solutions for omega.
01:12:48.440 --> 01:12:53.860
That is the omega,
unknown angular frequency
01:12:53.860 --> 01:12:56.380
we are trying to figure out.
01:12:56.380 --> 01:13:00.460
You can see that there are
three possible omegas that can
01:13:00.460 --> 01:13:03.220
make this equation equal to 0.
01:13:03.220 --> 01:13:12.000
The first one is
omega equal to omega0.
01:13:12.000 --> 01:13:19.140
The second one is
square root of omega 0,
01:13:19.140 --> 01:13:21.450
coming from this
expression, that omega
01:13:21.450 --> 01:13:24.360
squared minus 2
omega zero squared.
01:13:24.360 --> 01:13:26.280
If omega equal to
square root 2 omega0,
01:13:26.280 --> 01:13:28.110
this will be equal to zero.
01:13:28.110 --> 01:13:31.180
And that will give you the
whole expression equal to 0.
01:13:31.180 --> 01:13:33.670
Then finally, I take this term.
01:13:33.670 --> 01:13:36.030
And then you will get zero.
01:13:36.030 --> 01:13:40.170
Omega squared, if
omega is equal to 0,
01:13:40.170 --> 01:13:43.770
then the whole expression is 0.
01:13:43.770 --> 01:13:50.130
I have defined omega0 squared
to be equal to k over m.
01:13:50.130 --> 01:13:54.060
Therefore, I can conclude
that omega squared
01:13:54.060 --> 01:14:04.002
is equal to k over
m, 2k over m, and 0.
01:14:06.960 --> 01:14:10.390
Look at what we have done,
a lot of mathematics.
01:14:10.390 --> 01:14:15.240
But in the end, after you
solve the eigenvalue problem,
01:14:15.240 --> 01:14:18.720
or the determinant
equal to zero problem,
01:14:18.720 --> 01:14:20.940
you arrive at that
there are only
01:14:20.940 --> 01:14:25.740
three possible
values of omega which
01:14:25.740 --> 01:14:28.620
can make the
determinant of M minus 1
01:14:28.620 --> 01:14:32.600
K minus omega
squared I equal to 0.
01:14:32.600 --> 01:14:35.550
What are the three?
01:14:35.550 --> 01:14:38.710
k/m, 2k/m, and 0.
01:14:42.260 --> 01:14:45.150
If you look at this
value, then we'll say,
01:14:45.150 --> 01:14:49.800
this is essentially what we
actually argued before, right?
01:14:49.800 --> 01:14:55.000
Omega A squared is equal
to 4k over 2m is 2k over m.
01:14:55.000 --> 01:14:55.930
Wow.
01:14:55.930 --> 01:14:56.560
We got it.
01:15:00.320 --> 01:15:03.210
The second one is,
we think about really
01:15:03.210 --> 01:15:04.930
keep a straight
question in my head
01:15:04.930 --> 01:15:07.460
and understand this system.
01:15:07.460 --> 01:15:12.260
The second identified
normal more is having omega
01:15:12.260 --> 01:15:14.090
squared be equal to k/m.
01:15:14.090 --> 01:15:18.650
I got this also here magically,
after all those magics.
01:15:18.650 --> 01:15:24.960
And finally, the third one,
the math also knows physics.
01:15:24.960 --> 01:15:29.700
It also predicted that this is
one mode which have oscillation
01:15:29.700 --> 01:15:32.430
frequency of 0.
01:15:32.430 --> 01:15:34.924
Isn't that amazing to you?
01:15:39.690 --> 01:15:43.900
But that also gives
us a sense of safety.
01:15:43.900 --> 01:15:48.320
Because I can now add
10 pendulums, or 10
01:15:48.320 --> 01:15:51.080
coupled system to
your homework, and you
01:15:51.080 --> 01:15:52.210
will be able to solve it.
01:15:55.350 --> 01:15:56.790
So very good.
01:15:56.790 --> 01:16:00.530
This example seems
to be complicated.
01:16:00.530 --> 01:16:03.570
But the what I want to say,
I have one minute left,
01:16:03.570 --> 01:16:05.790
is that what we
have been doing is
01:16:05.790 --> 01:16:11.370
to write the equation of
motion based on force diagram.
01:16:11.370 --> 01:16:15.660
Then I convert that
to matrix format,
01:16:15.660 --> 01:16:19.090
and X double dot
equal to minus KX.
01:16:19.090 --> 01:16:21.270
Then I follow the
whole procedure,
01:16:21.270 --> 01:16:24.110
solve the eigenvalue problem.
01:16:24.110 --> 01:16:30.060
Then I will be able to figure
out what are the possible omega
01:16:30.060 --> 01:16:37.555
values which can satisfy
this eigenvalue problem
01:16:37.555 --> 01:16:38.920
or this determinant.
01:16:38.920 --> 01:16:42.630
M minus 1 K minus omega
squared I equal to 0 problem.
01:16:42.630 --> 01:16:46.130
And after solving
all those, you will
01:16:46.130 --> 01:16:51.000
be able to solve the
corresponding so-called normal
01:16:51.000 --> 01:16:52.530
mode frequencies.
01:16:52.530 --> 01:16:53.760
You can solve it.
01:16:53.760 --> 01:16:57.700
And of course, you can plug
those normal mode frequencies
01:16:57.700 --> 01:17:01.170
back in, then you
will be able to drive
01:17:01.170 --> 01:17:05.430
the relative amplitude,
A1, A2, and A3.
01:17:05.430 --> 01:17:07.890
So what we have
we learned today?
01:17:07.890 --> 01:17:11.670
We have learned how
to predict the motion
01:17:11.670 --> 01:17:14.460
of coupled oscillators.
01:17:14.460 --> 01:17:15.900
That's really cool.
01:17:15.900 --> 01:17:18.540
And then next time,
we are going to learn
01:17:18.540 --> 01:17:22.050
a special kind of motion in
coupled oscillators, which
01:17:22.050 --> 01:17:23.850
is the big phenomena.
01:17:23.850 --> 01:17:26.710
And also, what will
happen if I start to drive
01:17:26.710 --> 01:17:28.110
the coupled oscillators?
01:17:28.110 --> 01:17:31.830
So I will be here if
you have any questions
01:17:31.830 --> 01:17:34.130
about the lecture.
01:17:34.130 --> 01:17:36.080
Thank you very much.