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BOLESLAW WYSLOUCH:
Let's get started.
00:00:25.030 --> 00:00:27.690
So today hopefully
will be a busy day,
00:00:27.690 --> 00:00:34.860
with lots of interesting
insights into how things work.
00:00:34.860 --> 00:00:38.020
We talked about coupled
oscillators last time.
00:00:38.020 --> 00:00:41.850
We developed a
formalism in which
00:00:41.850 --> 00:00:46.860
we can find the most general
motion of oscillators.
00:00:46.860 --> 00:00:52.650
So let's remind ourselves what
are the coupled oscillators.
00:00:52.650 --> 00:00:55.920
Coupled oscillators, there
are many examples of them,
00:00:55.920 --> 00:00:59.310
and they have more or less
the following features.
00:00:59.310 --> 00:01:00.840
You have something
that oscillates--
00:01:00.840 --> 00:01:03.320
for example, a pendulum.
00:01:03.320 --> 00:01:08.250
You have to have more than one,
because for coupled oscillators
00:01:08.250 --> 00:01:10.560
you have to have at least two.
00:01:10.560 --> 00:01:14.100
So let's say you
have two oscillators.
00:01:14.100 --> 00:01:18.610
So each of them
is an oscillator,
00:01:18.610 --> 00:01:23.670
which in, for example, in
the limit of small angles,
00:01:23.670 --> 00:01:26.250
small displacement
angles, undergoes
00:01:26.250 --> 00:01:29.640
a pure harmonic motion
with some frequencies.
00:01:29.640 --> 00:01:32.610
And then you couple them
through various means.
00:01:32.610 --> 00:01:36.150
So for example, two masses
connected by a spring
00:01:36.150 --> 00:01:38.760
is an example of a
coupled oscillator.
00:01:38.760 --> 00:01:43.770
We could have two masses on
a track and another track,
00:01:43.770 --> 00:01:47.070
also connected by
several springs.
00:01:47.070 --> 00:01:49.740
This is also an example
of a coupled oscillator.
00:01:49.740 --> 00:01:54.122
Each of those masses
undergoes harmonic motion,
00:01:54.122 --> 00:01:56.580
and they are connected together
such that the motion of one
00:01:56.580 --> 00:01:58.560
affects motion of the other.
00:01:58.560 --> 00:02:00.750
You can have slightly
more complicated pendula.
00:02:00.750 --> 00:02:07.120
For example, you can hang
one pendula from the other.
00:02:07.120 --> 00:02:09.639
Each of them-- again, in the
limit of small oscillations--
00:02:09.639 --> 00:02:11.610
will undergo harmonic motion.
00:02:11.610 --> 00:02:14.890
And they are coupled together
because they are supported one
00:02:14.890 --> 00:02:17.020
on top of each other.
00:02:17.020 --> 00:02:18.840
And you can have--
00:02:18.840 --> 00:02:25.900
we have another example
of two tuning forks
00:02:25.900 --> 00:02:28.240
sitting on some sort of boxes.
00:02:28.240 --> 00:02:31.840
Each of them was an oscillator,
with audible oscillating
00:02:31.840 --> 00:02:38.350
frequency, and by putting them
next to each other they coupled
00:02:38.350 --> 00:02:41.590
through the sound waves
transmitted through the air.
00:02:41.590 --> 00:02:45.760
So one of them felt the
oscillations in the other one.
00:02:45.760 --> 00:02:47.700
This was an example of
coupled oscillation.
00:02:47.700 --> 00:02:51.100
Two masses and the thing.
00:02:51.100 --> 00:02:54.535
You can build oscillators
out of electronics.
00:02:54.535 --> 00:02:57.160
Some capacitor and
inductor together,
00:02:57.160 --> 00:02:58.270
with a little bit of--
00:02:58.270 --> 00:02:59.410
maybe without resistors.
00:02:59.410 --> 00:03:00.760
You have two of those.
00:03:00.760 --> 00:03:02.680
They constitute a
coupled oscillator
00:03:02.680 --> 00:03:04.690
if you put a wire between them.
00:03:04.690 --> 00:03:06.329
So there are many,
many examples.
00:03:06.329 --> 00:03:07.870
And of course, these
are all examples
00:03:07.870 --> 00:03:10.480
in which you have two
oscillating bodies,
00:03:10.480 --> 00:03:14.290
but it's very easy to have three
or more oscillating bodies.
00:03:14.290 --> 00:03:17.530
Then basically the
features of the system
00:03:17.530 --> 00:03:20.400
are the same, except the math
becomes more complicated,
00:03:20.400 --> 00:03:23.480
and we have more types of
oscillations you can have.
00:03:23.480 --> 00:03:27.100
And there's a couple
of characteristics
00:03:27.100 --> 00:03:30.220
which are the same for
all oscillating systems.
00:03:30.220 --> 00:03:31.690
And it's very
important to remember
00:03:31.690 --> 00:03:33.790
that we are learning
on one example,
00:03:33.790 --> 00:03:35.740
but it applies to very many.
00:03:35.740 --> 00:03:38.170
Number one, any motion--
00:03:38.170 --> 00:03:41.520
I can maybe summarize it here.
00:03:41.520 --> 00:03:49.070
So if you look at the motion of
an oscillator, you can have--
00:03:49.070 --> 00:03:52.290
let's say arbitrary oscillation.
00:03:52.290 --> 00:03:55.130
Arbitrary excitation.
00:03:58.300 --> 00:04:01.150
Excitation means I--
00:04:01.150 --> 00:04:05.630
I kick it in some sort
of arbitrary mode.
00:04:05.630 --> 00:04:08.480
I just come in and set up
some initial condition such
00:04:08.480 --> 00:04:10.270
that things are moving.
00:04:10.270 --> 00:04:14.140
And motion in this
arbitrary assertion
00:04:14.140 --> 00:04:16.930
is actually-- looks
pretty chaotic.
00:04:16.930 --> 00:04:21.040
It looks pretty
variable, changing.
00:04:21.040 --> 00:04:23.860
It's difficult to
understand what's going on.
00:04:23.860 --> 00:04:26.380
So And it clearly
doesn't look harmonic.
00:04:26.380 --> 00:04:27.400
Non-harmonic.
00:04:30.740 --> 00:04:33.110
There is no obvious
single frequency
00:04:33.110 --> 00:04:36.780
that is driving the system.
00:04:36.780 --> 00:04:40.640
If you look at
amplitude of the objects
00:04:40.640 --> 00:04:43.070
here-- for example, two
pendula, pendulum one and two.
00:04:43.070 --> 00:04:46.040
At any given moment of
time they are oscillating,
00:04:46.040 --> 00:04:48.020
there's a characteristic
amplitude.
00:04:48.020 --> 00:04:50.570
But what we saw is
that motion changes,
00:04:50.570 --> 00:04:53.300
looks like things are flowing
from one to the other.
00:04:53.300 --> 00:04:54.740
One of them has
a high amplitude.
00:04:54.740 --> 00:04:58.170
After some time, it cools
down, the other one grows.
00:04:58.170 --> 00:05:01.965
So the amplitudes
are changing in time.
00:05:01.965 --> 00:05:02.840
So they are variable.
00:05:10.570 --> 00:05:11.185
Are variable.
00:05:14.060 --> 00:05:17.540
And also, we didn't
calculate things exactly,
00:05:17.540 --> 00:05:21.302
but you know from study of
a single oscillator that
00:05:21.302 --> 00:05:23.510
if the things are moving,
it has a certain amplitude,
00:05:23.510 --> 00:05:26.180
there's certain energy
involved-- with some potential,
00:05:26.180 --> 00:05:27.350
some kinetic--
00:05:27.350 --> 00:05:29.820
and it's proportional to
the square of amplitude.
00:05:29.820 --> 00:05:32.690
So it's clear that energy
is moving from one pendulum
00:05:32.690 --> 00:05:34.100
to the other.
00:05:34.100 --> 00:05:36.470
This one was
oscillating like crazy.
00:05:36.470 --> 00:05:38.630
So all energy was sitting here.
00:05:38.630 --> 00:05:40.660
After some time,
this one stopped.
00:05:40.660 --> 00:05:42.080
So its energy is zero.
00:05:42.080 --> 00:05:44.870
And the other one was
oscillating like crazy.
00:05:44.870 --> 00:05:47.330
So the energy's flowing
from one to another.
00:05:47.330 --> 00:05:50.330
It's not sitting in one
place, but it's flowing.
00:05:50.330 --> 00:05:53.420
This one has lots of
energy right now, but now
00:05:53.420 --> 00:05:55.020
that one is picking up.
00:05:55.020 --> 00:05:59.430
So the energy-- you see
the energy flowing here.
00:05:59.430 --> 00:06:01.320
And this one will
eventually stop--
00:06:01.320 --> 00:06:03.030
well, this is a pretty
crappy oscillator,
00:06:03.030 --> 00:06:05.040
but it will eventually
stop, and this one
00:06:05.040 --> 00:06:07.950
will have all the energy.
00:06:07.950 --> 00:06:10.080
And this is, again,
characteristic in every system.
00:06:10.080 --> 00:06:13.650
We can see energy flowing
around from one to the other,
00:06:13.650 --> 00:06:15.270
growing, stopping.
00:06:15.270 --> 00:06:19.986
So it's-- in general, in
the most general case,
00:06:19.986 --> 00:06:23.840
it's a complicated system.
00:06:23.840 --> 00:06:31.100
Energy is migrating
between different masses.
00:06:31.100 --> 00:06:34.400
However, every single one
of those coupled oscillating
00:06:34.400 --> 00:06:35.990
systems has a magic.
00:06:35.990 --> 00:06:39.980
There's a magic involved, namely
the existence of normal modes.
00:06:39.980 --> 00:06:43.400
Every single coupled oscillator
system has normal modes,
00:06:43.400 --> 00:06:45.260
and those modes are beautiful.
00:06:45.260 --> 00:06:50.630
Those modes are-- everything
is moving in sync.
00:06:50.630 --> 00:06:59.540
So this is normal
mode excitation.
00:06:59.540 --> 00:07:02.440
There's a very special
way, a special setting
00:07:02.440 --> 00:07:07.090
of initial conditions, that
leads to the-- that results
00:07:07.090 --> 00:07:10.090
in a pure harmonic motion.
00:07:10.090 --> 00:07:19.640
So this is a harmonic motion,
with a certain frequency omega,
00:07:19.640 --> 00:07:22.640
characteristic frequency
for this particular motion.
00:07:22.640 --> 00:07:26.130
The amplitudes remain fixed.
00:07:26.130 --> 00:07:29.070
Once you set initial
conditions, you get it moving,
00:07:29.070 --> 00:07:31.500
everything is moving,
simple harmonic motion
00:07:31.500 --> 00:07:33.970
means its amplitude is constant.
00:07:33.970 --> 00:07:36.750
So if I-- and remember,
for example, this system.
00:07:36.750 --> 00:07:39.750
It was something like this.
00:07:39.750 --> 00:07:44.610
Symmetric or
antisymmetric motion.
00:07:44.610 --> 00:07:46.710
And if not for the
friction, the amplitudes
00:07:46.710 --> 00:07:49.080
would remain constant
forever, if it
00:07:49.080 --> 00:07:50.310
will be a perfect oscillator.
00:07:50.310 --> 00:07:53.580
So amplitudes-- in fact, it's
not amplitudes themselves,
00:07:53.580 --> 00:07:55.260
but amplitude ratio.
00:07:55.260 --> 00:07:59.670
The ratio of amplitude
between the different elements
00:07:59.670 --> 00:08:02.110
in the system is constant.
00:08:07.910 --> 00:08:10.220
So in a sense, every
harmonic motion
00:08:10.220 --> 00:08:13.010
has a characteristic shape.
00:08:13.010 --> 00:08:17.060
And then by-- since everything
is constant, nothing changes,
00:08:17.060 --> 00:08:20.800
this energy stays
in the place it is.
00:08:25.830 --> 00:08:28.439
So energy is--
once you put energy
00:08:28.439 --> 00:08:30.730
to mass number one, mass
number two, mass number three,
00:08:30.730 --> 00:08:32.159
the energy sits there.
00:08:32.159 --> 00:08:34.799
The energies are
constant, as the system
00:08:34.799 --> 00:08:36.510
undergoes harmonic motion.
00:08:36.510 --> 00:08:39.330
Energy does not migrate.
00:08:39.330 --> 00:08:42.539
So this is a very nice-- and
there is another beautiful
00:08:42.539 --> 00:08:48.960
feature, that any arbitrary
excitation can be made out
00:08:48.960 --> 00:08:53.227
of some linear sum--
00:08:56.149 --> 00:09:00.250
sum of normal modes.
00:09:02.980 --> 00:09:05.780
Linear sum, of
superposition of normal.
00:09:05.780 --> 00:09:09.410
Any arbitrary excitation with
all its complicated motion
00:09:09.410 --> 00:09:12.220
can be made into
some of normal modes.
00:09:12.220 --> 00:09:15.450
So since normal modes are
easy and simple and beautiful,
00:09:15.450 --> 00:09:20.520
the description of motion of
any coupled oscillator, the best
00:09:20.520 --> 00:09:23.070
way to approach it
is to decompose it,
00:09:23.070 --> 00:09:25.740
to find all possible
normal modes,
00:09:25.740 --> 00:09:29.010
and then decompose the initial
condition to correspond
00:09:29.010 --> 00:09:31.140
to this linear sum
of normal modes.
00:09:31.140 --> 00:09:33.660
Once you know the normal
modes, you add them up,
00:09:33.660 --> 00:09:37.590
and then you can predict
exactly the motion.
00:09:37.590 --> 00:09:39.900
And this is what we've done.
00:09:39.900 --> 00:09:42.530
So we have a--
00:09:42.530 --> 00:09:46.590
we have introduced a
mathematic mechanism
00:09:46.590 --> 00:09:50.910
in which we put all the
information about forces
00:09:50.910 --> 00:09:54.120
and masses in the system in
some sort of matrix form.
00:09:54.120 --> 00:09:56.430
In our example, it was
a two by two matrix,
00:09:56.430 --> 00:09:58.740
but if we have three
masses or four masses,
00:09:58.740 --> 00:10:01.320
the dimensionality of the
matrix will have to grow.
00:10:01.320 --> 00:10:03.730
But the equation
will remain the same.
00:10:03.730 --> 00:10:08.070
So this equation of motion,
we rework it a little bit.
00:10:08.070 --> 00:10:10.530
Since we are looking
for normal modes,
00:10:10.530 --> 00:10:14.230
we know that normal modes occur
with this one single frequency.
00:10:14.230 --> 00:10:19.260
So we postulate an
oscillation with a frequency.
00:10:19.260 --> 00:10:20.220
We plug it in.
00:10:20.220 --> 00:10:23.870
We obtain a simple
algebraic equation.
00:10:23.870 --> 00:10:25.290
Doesn't have any
time dependence,
00:10:25.290 --> 00:10:27.030
doesn't have any exponents.
00:10:27.030 --> 00:10:30.600
It's a simple algebraic
equation, basically a set
00:10:30.600 --> 00:10:35.930
of linear equations, which
we can solve and find
00:10:35.930 --> 00:10:39.170
the eigenvalue, or the
characteristic frequency
00:10:39.170 --> 00:10:40.510
for normal modes.
00:10:40.510 --> 00:10:43.190
And you can show that the number
of those frequencies in general
00:10:43.190 --> 00:10:46.400
is equal to the number of
masses involved in the system.
00:10:46.400 --> 00:10:49.130
And you solve it,
and then once you
00:10:49.130 --> 00:10:51.590
know the characteristic
frequencies,
00:10:51.590 --> 00:10:55.250
then you can find shape, you
can find the eigenvectors.
00:10:55.250 --> 00:10:59.510
What is the ratio of amplitudes
which corresponds to the mode.
00:10:59.510 --> 00:11:02.970
And in case of our two pendula,
there are two of such things.
00:11:02.970 --> 00:11:06.500
One is where both
amplitudes are equal,
00:11:06.500 --> 00:11:08.870
and this corresponds
to oscillation
00:11:08.870 --> 00:11:13.670
in which two pendola are
moving parallel to each other,
00:11:13.670 --> 00:11:15.860
with a spring being--
00:11:15.860 --> 00:11:18.350
not paying any roll.
00:11:18.350 --> 00:11:19.490
So this is one mode.
00:11:19.490 --> 00:11:21.570
And then amplitude is--
00:11:21.570 --> 00:11:25.070
as I said, any given moment is
the same, so the ratio is 1.
00:11:25.070 --> 00:11:28.800
And then you have a motion
in which the two pendula
00:11:28.800 --> 00:11:30.350
are going against each other.
00:11:30.350 --> 00:11:32.420
So any given moment
of time, they're
00:11:32.420 --> 00:11:35.103
in their negative position,
so the ratio is minus 1.
00:11:38.430 --> 00:11:41.650
The motion of one of
them can be obtained
00:11:41.650 --> 00:11:43.525
by looking at where
the first one is
00:11:43.525 --> 00:11:45.880
and multiplying by minus 1.
00:11:45.880 --> 00:11:48.190
So these are the two modes,
and any arbitrary-- any
00:11:48.190 --> 00:11:52.420
complicated, nasty excitation
with things moving around
00:11:52.420 --> 00:11:56.780
is a linear sum of
the oscillation.
00:11:56.780 --> 00:11:57.660
So we know that.
00:11:57.660 --> 00:11:59.380
We've worked it out.
00:11:59.380 --> 00:12:02.820
We used this example.
00:12:02.820 --> 00:12:06.430
And by the way, today, we'll
be using two examples--
00:12:06.430 --> 00:12:09.940
one which is the same thing
with two pendula and the spring,
00:12:09.940 --> 00:12:11.510
and the other one
with two masses,
00:12:11.510 --> 00:12:14.020
or maybe later three masses.
00:12:14.020 --> 00:12:20.730
And the exact values of
coefficients in matrix k
00:12:20.730 --> 00:12:23.870
are different in
two different cases.
00:12:23.870 --> 00:12:29.510
But in all types of other
motion, the shape of motion,
00:12:29.510 --> 00:12:32.340
the behavior of the
system is identical.
00:12:32.340 --> 00:12:36.390
So the solutions to the
two cases are identical.
00:12:36.390 --> 00:12:40.130
The difference is basically
numerical in how the spring
00:12:40.130 --> 00:12:41.960
constants and masses come in.
00:12:41.960 --> 00:12:45.080
So we can in fact
treat those two systems
00:12:45.080 --> 00:12:46.190
completely the same.
00:12:46.190 --> 00:12:48.050
So I'll be jumping
from one to another,
00:12:48.050 --> 00:12:49.190
but we don't have to worry.
00:12:49.190 --> 00:12:52.700
But let's now look
on the system.
00:12:52.700 --> 00:12:54.470
So what we are trying
to do today is,
00:12:54.470 --> 00:13:00.040
we are trying to
apply external force
00:13:00.040 --> 00:13:02.609
so we'll have a driven
coupled oscillator.
00:13:02.609 --> 00:13:04.150
And I assume that
you know everything
00:13:04.150 --> 00:13:06.040
about driven oscillators.
00:13:06.040 --> 00:13:09.300
So the idea was that you
come with an external .
00:13:09.300 --> 00:13:11.980
In 8.03, we assumed
that this external force
00:13:11.980 --> 00:13:13.160
is harmonic force.
00:13:13.160 --> 00:13:15.490
So there's a
characteristic frequency
00:13:15.490 --> 00:13:18.364
which is given by external--
00:13:18.364 --> 00:13:19.030
let's say by me.
00:13:19.030 --> 00:13:22.450
It has nothing to do with normal
frequencies of the system.
00:13:22.450 --> 00:13:25.000
It's an external frequency,
omega d, which I apply.
00:13:25.000 --> 00:13:26.740
Driven frequency.
00:13:26.740 --> 00:13:29.990
And then I look at how
the system responds.
00:13:29.990 --> 00:13:32.090
And I look for steady
state oscillations--
00:13:32.090 --> 00:13:35.540
the ones where everything
oscillates with the same driven
00:13:35.540 --> 00:13:36.680
frequency--
00:13:36.680 --> 00:13:38.270
trying to look for solutions.
00:13:38.270 --> 00:13:42.110
And as you know from a
single oscillator, what
00:13:42.110 --> 00:13:45.550
we were calculating is what is
the the response of the system?
00:13:45.550 --> 00:13:46.790
What is the amplitude?
00:13:46.790 --> 00:13:49.410
And the certain
frequencies that--
00:13:49.410 --> 00:13:52.310
you wiggle it and the
system doesn't do anything,
00:13:52.310 --> 00:13:55.507
but if you apply a certain
resonant frequency,
00:13:55.507 --> 00:13:56.840
then the response is very large.
00:13:56.840 --> 00:13:59.240
The system starts moving
like crazy, et cetera.
00:13:59.240 --> 00:14:01.550
And the same type of
thing will happen here,
00:14:01.550 --> 00:14:04.480
except that we have
multiple frequencies.
00:14:04.480 --> 00:14:06.950
So there will be a
possibility of a resonance
00:14:06.950 --> 00:14:08.640
for several frequencies.
00:14:08.640 --> 00:14:10.820
All right?
00:14:10.820 --> 00:14:16.090
So let me quickly set this up.
00:14:16.090 --> 00:14:17.320
Just-- yeah.
00:14:17.320 --> 00:14:18.165
Doesn't matter.
00:14:18.165 --> 00:14:20.790
So there were some--
00:14:20.790 --> 00:14:23.940
let's just start
working on the example.
00:14:23.940 --> 00:14:26.820
So just a reminder,
this is our system.
00:14:26.820 --> 00:14:29.500
A pendula of some
length L. There
00:14:29.500 --> 00:14:33.930
are two identical
masses, M. There
00:14:33.930 --> 00:14:37.320
is a spring of constant k.
00:14:37.320 --> 00:14:41.000
They are all-- and
for simplicity, we
00:14:41.000 --> 00:14:44.630
assume that we are all in
Earth's gravitational field.
00:14:44.630 --> 00:14:46.460
So we don't have to
worry about traveling
00:14:46.460 --> 00:14:49.730
to Jupiter or the moon.
00:14:49.730 --> 00:14:53.720
And-- except that
the difference will
00:14:53.720 --> 00:14:58.490
be that we apply an external
force to one of those masses.
00:14:58.490 --> 00:15:02.962
How, it doesn't matter, but
there is an external force F--
00:15:02.962 --> 00:15:08.960
F with subscript d,
which is equal to some--
00:15:08.960 --> 00:15:13.540
it has some amplitude
F0 cosine omega
00:15:13.540 --> 00:15:19.610
d times t, along
the x direction.
00:15:19.610 --> 00:15:23.570
And this is applied to mass one.
00:15:23.570 --> 00:15:25.000
OK.
00:15:25.000 --> 00:15:28.225
And there is a little
bit of just a warning.
00:15:30.880 --> 00:15:37.069
We will be assuming that there
is no damping in the system.
00:15:37.069 --> 00:15:38.860
For the single oscillator,
there was always
00:15:38.860 --> 00:15:40.370
a little bit of damping.
00:15:40.370 --> 00:15:43.360
So between you and me, remember
there's always a little
00:15:43.360 --> 00:15:43.870
damping.
00:15:43.870 --> 00:15:46.270
So in case we need damping--
00:15:46.270 --> 00:15:48.580
it will come in
and will help us,
00:15:48.580 --> 00:15:50.860
but if we try to use
damping in calculations,
00:15:50.860 --> 00:15:52.600
calculations become horrendous.
00:15:52.600 --> 00:15:54.730
So for the purpose
of calculations,
00:15:54.730 --> 00:15:56.710
we will ignore damping.
00:15:56.710 --> 00:15:57.430
It'll get some.
00:15:57.430 --> 00:16:01.780
But if things go bad with the
results, like dividing by 0,
00:16:01.780 --> 00:16:04.510
then we will bring in damping
and say no no, it's not so bad.
00:16:04.510 --> 00:16:05.630
Damping helps you.
00:16:05.630 --> 00:16:06.990
We are not dividing by 0.
00:16:06.990 --> 00:16:09.340
OK?
00:16:09.340 --> 00:16:12.250
So let's write those
equations of motions.
00:16:12.250 --> 00:16:13.375
Equations of motion.
00:16:16.370 --> 00:16:21.810
So we have-- so the forces
and accelerations on mass one
00:16:21.810 --> 00:16:23.650
is the same as before.
00:16:23.650 --> 00:16:25.750
There was a spring.
00:16:25.750 --> 00:16:29.170
There is mg over l.
00:16:29.170 --> 00:16:33.580
That's the pendulum by itself.
00:16:33.580 --> 00:16:36.220
Depending on position x1.
00:16:36.220 --> 00:16:40.090
There is the influence of
a spring, which depends
00:16:40.090 --> 00:16:43.090
on where spring number two is.
00:16:43.090 --> 00:16:49.540
And, plus, there is this new
driven term, F0 cosine omega
00:16:49.540 --> 00:16:53.870
d times t, where omega
d is fixed, arbitrary,
00:16:53.870 --> 00:16:54.730
externally given.
00:16:54.730 --> 00:17:00.130
So both F0 and omega d are
decided by somebody outside
00:17:00.130 --> 00:17:01.630
of the system.
00:17:01.630 --> 00:17:08.138
Now, the second mass
M X2 dot dot, is--
00:17:08.138 --> 00:17:14.079
actually has feels position
of x1, through the spring.
00:17:14.079 --> 00:17:21.730
And there is this-- its own
pendulum effect plus a string,
00:17:21.730 --> 00:17:23.560
depending on position x2.
00:17:23.560 --> 00:17:25.720
Interestingly, there
is no force here,
00:17:25.720 --> 00:17:29.360
because the force is
applied to mass one.
00:17:29.360 --> 00:17:32.290
So mass two a priori doesn't
know anything about the force.
00:17:32.290 --> 00:17:35.110
But of course it will
know through the coupling.
00:17:35.110 --> 00:17:35.610
Yes?
00:17:35.610 --> 00:17:37.882
Questions?
00:17:37.882 --> 00:17:40.350
Anybody have questions so far?
00:17:40.350 --> 00:17:42.990
So it's the same as
before, with the addition
00:17:42.990 --> 00:17:45.430
of this external force.
00:17:45.430 --> 00:17:51.820
Again, this is writing all
coordinates one by one.
00:17:51.820 --> 00:17:54.940
We immediately switch
to matrix form.
00:17:54.940 --> 00:17:59.170
We write it MX double
dot, where X is the same
00:17:59.170 --> 00:18:02.185
as we defined before, minus KX.
00:18:05.210 --> 00:18:09.590
I think I will stop writing
these kind of thick lines.
00:18:09.590 --> 00:18:12.160
But for now, let me--
00:18:12.160 --> 00:18:17.350
F cosine omega d times t.
00:18:17.350 --> 00:18:21.370
So this is now a matrix
equation for the vector XD.
00:18:21.370 --> 00:18:24.000
And let's remind ourselves
what those matrices are.
00:18:24.000 --> 00:18:30.120
Matrix M is M 0
0 M. This is just
00:18:30.120 --> 00:18:32.740
mass of the individual systems.
00:18:32.740 --> 00:18:38.650
We use M minus 1, which
is 1 over M, 1 over M,
00:18:38.650 --> 00:18:41.830
and diagonal 0 and 0.
00:18:41.830 --> 00:18:43.720
So this carries
information about masses,
00:18:43.720 --> 00:18:45.670
inertia of the system.
00:18:45.670 --> 00:18:50.680
Matrix K contains information
about all the springs
00:18:50.680 --> 00:18:53.860
in the system, and
some pendula effects.
00:18:53.860 --> 00:19:01.480
So we have a k plus
mg over l, minus k,
00:19:01.480 --> 00:19:09.220
minus k, k plus mg over l.
00:19:09.220 --> 00:19:11.500
And now there is
this new thing, which
00:19:11.500 --> 00:19:24.540
is this vector F. Vector F
is equal to F0 0 cosine omega
00:19:24.540 --> 00:19:27.110
d times t.
00:19:27.110 --> 00:19:30.290
So this is in a vector
form, this external force,
00:19:30.290 --> 00:19:33.900
which is applied only
to mass number one.
00:19:33.900 --> 00:19:34.400
OK?
00:19:37.990 --> 00:19:40.570
So these are the elements
which are plugged in.
00:19:40.570 --> 00:19:43.060
So now the question is, what
do you want to do with this?
00:19:43.060 --> 00:19:45.720
So we have the
equation of motion.
00:19:45.720 --> 00:19:48.690
And so what do we do with this?
00:19:48.690 --> 00:19:56.320
So there are two steps
that we have to do.
00:19:56.320 --> 00:20:00.090
Number one, we have
to remind ourselves
00:20:00.090 --> 00:20:04.200
what are the normal modes
of the system, in case--
00:20:04.200 --> 00:20:05.280
because we will need--
00:20:05.280 --> 00:20:10.792
the information about normal
modes will come in as--
00:20:10.792 --> 00:20:15.120
into solutions for
a driven motion.
00:20:15.120 --> 00:20:18.030
So let's remind
ourselves what this was.
00:20:18.030 --> 00:20:19.282
Well, this was a solution.
00:20:19.282 --> 00:20:20.865
I'll just rewrite
it very quickly such
00:20:20.865 --> 00:20:22.156
that we have it for the record.
00:20:24.535 --> 00:20:26.200
It should fit here.
00:20:26.200 --> 00:20:27.390
Now let's try.
00:20:27.390 --> 00:20:29.280
So there were two solutions.
00:20:29.280 --> 00:20:33.510
There was omega 1 squared,
which was equal to g over l.
00:20:33.510 --> 00:20:39.320
And the corresponding normal
mode was a symmetric one.
00:20:39.320 --> 00:20:41.570
It was 1, 1.
00:20:41.570 --> 00:20:42.070
OK.
00:20:42.070 --> 00:20:46.360
So this was one
type of solution,
00:20:46.360 --> 00:20:48.950
where the two masses
were moving together.
00:20:48.950 --> 00:20:54.650
There was a second frequency
which was equal to g over l.
00:20:54.650 --> 00:20:59.410
The square of it was
equal plus 2k over m.
00:20:59.410 --> 00:21:01.960
And this was the
characteristic normal frequency
00:21:01.960 --> 00:21:05.230
for the second type
of oscillation, which
00:21:05.230 --> 00:21:08.290
you can write it 1, minus 1.
00:21:08.290 --> 00:21:12.580
And the criterion for when we
were looking for solutions,
00:21:12.580 --> 00:21:17.110
we would find them by
calculating the determinant
00:21:17.110 --> 00:21:19.230
of this two by two matrix.
00:21:19.230 --> 00:21:29.360
It was the determinant of m
minus 1 k minus omega squared
00:21:29.360 --> 00:21:33.650
times unit matrix
was equal to 0.
00:21:33.650 --> 00:21:37.590
So this was the
equation that had
00:21:37.590 --> 00:21:41.730
to be satisfied for frequencies
corresponding to normal modes
00:21:41.730 --> 00:21:44.610
with zero external force.
00:21:44.610 --> 00:21:46.800
Interestingly, if you
do the calculations,
00:21:46.800 --> 00:21:48.110
it turns out you can--
00:21:48.110 --> 00:21:51.110
algebraically, you can write--
00:21:51.110 --> 00:21:53.020
after you know the
solution itself,
00:21:53.020 --> 00:21:56.420
you can write it in
a very compact way.
00:21:56.420 --> 00:21:59.910
So this determinant can be
written in the following way--
00:21:59.910 --> 00:22:03.790
omega squared minus
omega 1 squared,
00:22:03.790 --> 00:22:09.830
times omega squared
minus omega 2 squared.
00:22:09.830 --> 00:22:13.620
And this is-- the
condition was zero.
00:22:13.620 --> 00:22:20.840
And you see explicitly that this
is a fourth order in frequency
00:22:20.840 --> 00:22:23.270
equation, fourth order
frequency, which is
00:22:23.270 --> 00:22:27.890
0 for omega 1 and for omega 2.
00:22:27.890 --> 00:22:29.260
In a very explicit way.
00:22:29.260 --> 00:22:35.360
So this is a nice,
compact form of writing
00:22:35.360 --> 00:22:38.005
this particular
eigenvalue equation.
00:22:41.120 --> 00:22:46.420
And again, as a reminder,
the motion of the system--
00:22:46.420 --> 00:22:49.930
the most general motion of the
system with no external force
00:22:49.930 --> 00:22:54.790
was a superposition of
those two oscillations,
00:22:54.790 --> 00:22:57.320
which we can write as
some sort of amplitude--
00:22:57.320 --> 00:23:04.990
1, 1 cosine omega
1 t plus phi 1,
00:23:04.990 --> 00:23:14.690
plus beta 1, minus 1 cosine
omega 2 t plus phi 2.
00:23:14.690 --> 00:23:20.070
So this is oscillations of
two different frequencies.
00:23:20.070 --> 00:23:21.990
This is the shape
of oscillations,
00:23:21.990 --> 00:23:25.660
the relative amplitude
of one versus the other.
00:23:25.660 --> 00:23:28.680
And then there's the overall
amplitude alpha and beta,
00:23:28.680 --> 00:23:30.260
which has to be determined.
00:23:30.260 --> 00:23:32.880
And then there are
arbitrary phases.
00:23:32.880 --> 00:23:38.900
So there are in
fact four numbers,
00:23:38.900 --> 00:23:43.270
which can be determined from
four initial conditions.
00:23:43.270 --> 00:23:45.980
So typically two positions
for the two masses,
00:23:45.980 --> 00:23:48.230
and two initial
velocities for two masses.
00:23:48.230 --> 00:23:51.360
So everything matches.
00:23:51.360 --> 00:23:53.420
So this a so-called
homogeneous equation.
00:23:59.246 --> 00:24:00.320
Homogeneous solution.
00:24:06.350 --> 00:24:08.280
What about driven solution?
00:24:08.280 --> 00:24:14.310
Driven solution, as we remember
from a single oscillator,
00:24:14.310 --> 00:24:20.030
results in a motion in which
all the elements in the system
00:24:20.030 --> 00:24:22.590
are oscillating at
the same frequency,
00:24:22.590 --> 00:24:24.930
and that's the driven frequency.
00:24:24.930 --> 00:24:25.500
It's a fact.
00:24:25.500 --> 00:24:28.950
I come in, I apply
100 Hertz frequency,
00:24:28.950 --> 00:24:30.960
and everybody oscillates
on the 100 frequency.
00:24:30.960 --> 00:24:35.110
That's the solution for a
driven oscillating system.
00:24:35.110 --> 00:24:37.230
And we saw it for a
one-dimensional oscillator,
00:24:37.230 --> 00:24:38.563
and we will see it here as well.
00:24:38.563 --> 00:24:40.640
There's one frequency, omega d.
00:24:40.640 --> 00:24:44.310
So we will be now looking for
a solution which corresponds
00:24:44.310 --> 00:24:48.210
to the oscillation of the system
with this external frequency,
00:24:48.210 --> 00:24:50.700
which a priori is
not the same as one
00:24:50.700 --> 00:24:53.000
of the normal frequencies.
00:24:53.000 --> 00:24:56.600
So the complete motion of the
system consists of two parts.
00:24:56.600 --> 00:25:01.150
One is this homogeneous
self-oscillating motion
00:25:01.150 --> 00:25:03.330
with two characteristic
frequencies.
00:25:03.330 --> 00:25:06.050
And there will be a
second type of motion,
00:25:06.050 --> 00:25:08.130
which is a driven one.
00:25:08.130 --> 00:25:12.650
So how do we go
about solving that?
00:25:12.650 --> 00:25:16.200
So equations of motions of
course will be the same.
00:25:16.200 --> 00:25:21.810
The solution, the way that we
solve it will be very similar.
00:25:21.810 --> 00:25:23.840
So lets try-- start working.
00:25:23.840 --> 00:25:29.790
Maybe we can work on
those blackboards here.
00:25:29.790 --> 00:25:31.350
So what is going on?
00:25:34.830 --> 00:25:43.100
So we know that if we apply
external frequency omega d,
00:25:43.100 --> 00:25:49.250
everybody in the system, all
the elements will be oscillating
00:25:49.250 --> 00:25:51.700
with the same frequency.
00:25:55.130 --> 00:26:01.420
So we can then
introduce a variable Z,
00:26:01.420 --> 00:26:08.130
which will be defined
B e to the i omega d t.
00:26:08.130 --> 00:26:09.980
This will be the
oscillating term.
00:26:09.980 --> 00:26:14.120
And this will be the amplitude
of oscillation, which we'll try
00:26:14.120 --> 00:26:16.550
to make real for simplicity.
00:26:16.550 --> 00:26:19.640
And then we plug this
into the equation
00:26:19.640 --> 00:26:23.090
of motion, which is listed
up there on the screen.
00:26:23.090 --> 00:26:29.990
So the equation of motion
is Z dot dot plus M minus 1
00:26:29.990 --> 00:26:43.400
K times Z is equal to now M
minus 1 force e to i omega d t.
00:26:43.400 --> 00:26:48.990
You see our external
force is F cosine
00:26:48.990 --> 00:26:52.920
omega d t, with a vector 1, 0.
00:26:52.920 --> 00:26:57.310
But of course, in the complex
notation, this is exponent.
00:26:57.310 --> 00:27:00.280
So this is the challenge,
what we would like to have.
00:27:00.280 --> 00:27:04.460
And we assume that all the
elements in the system--
00:27:04.460 --> 00:27:06.700
position,
acceleration-- oscillate
00:27:06.700 --> 00:27:09.610
at the same frequency omega d.
00:27:09.610 --> 00:27:15.130
If you do that,
then the equations
00:27:15.130 --> 00:27:19.780
become somewhat simpler, because
the oscillating term drops out.
00:27:19.780 --> 00:27:23.620
So when you plug this type
of solution into here,
00:27:23.620 --> 00:27:27.310
what you get is minus
omega d squared--
00:27:27.310 --> 00:27:30.660
that's from second
differentiation with respect
00:27:30.660 --> 00:27:32.170
to time--
00:27:32.170 --> 00:27:45.010
plus M minus 1 K, multiplying
vector B e to i omega d t.
00:27:45.010 --> 00:27:54.570
This must be equal to M
minus one F e to i omega d t.
00:27:54.570 --> 00:27:57.540
This is vector B,
this is vector F.
00:27:57.540 --> 00:28:00.000
And there is this
oscillating term.
00:28:00.000 --> 00:28:02.670
But both sides oscillate
at the same frequency.
00:28:02.670 --> 00:28:03.750
That's what we assume.
00:28:03.750 --> 00:28:09.340
So we can simply
divide by this, and we
00:28:09.340 --> 00:28:14.590
are left with an
equation that equates
00:28:14.590 --> 00:28:16.360
what's going on
in the oscillating
00:28:16.360 --> 00:28:20.440
system with the external force.
00:28:20.440 --> 00:28:24.820
So now, let's see
here what is known
00:28:24.820 --> 00:28:30.490
and what is unknown
in this equation.
00:28:30.490 --> 00:28:35.140
M minus 1 K carries information
about the construction built
00:28:35.140 --> 00:28:37.360
of the system of accelerators.
00:28:37.360 --> 00:28:42.280
Strength of springs, masses,
gravitational field, et cetera.
00:28:42.280 --> 00:28:43.150
So this is fixed.
00:28:43.150 --> 00:28:45.370
This is given.
00:28:45.370 --> 00:28:49.490
Omega d is the external driving
frequency, and it's also given.
00:28:49.490 --> 00:28:51.090
It's a number.
00:28:51.090 --> 00:28:52.800
I said this is externally given.
00:28:52.800 --> 00:28:55.020
I just set it at some computer.
00:28:55.020 --> 00:28:56.940
Say 100 Hertz, and it's
driven at 100 Hertz.
00:28:56.940 --> 00:28:57.690
So we know that.
00:28:57.690 --> 00:29:00.660
We know exactly
what this number is.
00:29:00.660 --> 00:29:03.270
External force, we
know what it is.
00:29:03.270 --> 00:29:03.970
We defined it.
00:29:03.970 --> 00:29:05.040
It's F0.
00:29:05.040 --> 00:29:07.090
We know what its magnitude--
00:29:07.090 --> 00:29:15.570
so everything is known except
for vector B. And vector B are
00:29:15.570 --> 00:29:18.300
the amplitudes of oscillation--
00:29:18.300 --> 00:29:21.790
remember, everything
oscillates at omega d--
00:29:21.790 --> 00:29:25.650
of mass one and mass two.
00:29:25.650 --> 00:29:29.480
So in general, if I
apply external force,
00:29:29.480 --> 00:29:33.230
this guy will oscillate
with some amplitude.
00:29:33.230 --> 00:29:36.230
That guy with some amplitude,
a priori different.
00:29:36.230 --> 00:29:39.630
And this will be
B1, this will be B2.
00:29:39.630 --> 00:29:41.460
And we don't know
that at this stage.
00:29:41.460 --> 00:29:44.820
So this equation will
allow us to find it.
00:29:47.420 --> 00:29:50.120
And it is possible because--
00:29:50.120 --> 00:29:53.960
this is actually a very
straightforward equation.
00:29:53.960 --> 00:30:04.950
It contains-- actually, to
be very precise, I have to--
00:30:04.950 --> 00:30:06.720
this is a number,
this is a matrix.
00:30:06.720 --> 00:30:11.200
So I have to put a
unit matrix right here.
00:30:11.200 --> 00:30:13.700
So it's omega d
times unit matrix
00:30:13.700 --> 00:30:20.400
plus this matrix that carries
information about the system.
00:30:20.400 --> 00:30:23.480
And so we can write
this down again
00:30:23.480 --> 00:30:26.820
in some sort of more open
way, for our specific case.
00:30:26.820 --> 00:30:32.390
So this will be k
over m plus g over l,
00:30:32.390 --> 00:30:42.360
minus omega d squared, minus
k over m, minus k over m,
00:30:42.360 --> 00:30:50.770
k over m, plus g over l,
minus omega d squared.
00:30:50.770 --> 00:30:53.770
So this is this matrix here.
00:30:53.770 --> 00:30:57.820
This matrix is applied to
vector B, which is our unknown.
00:30:57.820 --> 00:31:01.175
Let's call it B1 and B2.
00:31:01.175 --> 00:31:04.600
These are the amplitudes
of oscillations
00:31:04.600 --> 00:31:07.900
of individual elements
in our system.
00:31:07.900 --> 00:31:11.465
And this is equal to m--
00:31:11.465 --> 00:31:16.720
the inverted mass matrix
times vector F, which--
00:31:16.720 --> 00:31:18.760
without its
oscillating part, which
00:31:18.760 --> 00:31:22.830
is simply F0 over m and 0.
00:31:25.080 --> 00:31:25.580
All right.
00:31:25.580 --> 00:31:28.820
So this is the task
in question, and we
00:31:28.820 --> 00:31:33.740
have to find out those
two values depending
00:31:33.740 --> 00:31:39.490
on these parameters and the
strength of force, et cetera.
00:31:39.490 --> 00:31:41.240
So this is actually
not a big deal.
00:31:41.240 --> 00:31:45.422
It's a two by two equation, two
equations with two unknowns.
00:31:45.422 --> 00:31:46.630
We solve it, and we are done.
00:31:49.720 --> 00:31:55.390
However, we want to
learn a little bit
00:31:55.390 --> 00:32:00.980
about slightly more general
ways of calculating things.
00:32:00.980 --> 00:32:05.851
So let's call this one matrix
E, with some funny double vector
00:32:05.851 --> 00:32:06.350
sign.
00:32:06.350 --> 00:32:09.890
Let's call this one vector B,
and let's call this one vector
00:32:09.890 --> 00:32:13.620
D, because we will use this--
00:32:13.620 --> 00:32:15.000
use it later.
00:32:15.000 --> 00:32:17.970
And what we are
trying to do is, we
00:32:17.970 --> 00:32:27.510
are trying to use the
so-called Cramer's rule to find
00:32:27.510 --> 00:32:29.580
those coefficients B1 and B2.
00:32:29.580 --> 00:32:32.170
And for some historical
reasons, 8.03 really
00:32:32.170 --> 00:32:33.840
likes Cramer's rule.
00:32:33.840 --> 00:32:36.390
I like MATLAB or Mathematica.
00:32:36.390 --> 00:32:41.340
I just plug things in, and it
crunches out and calculates.
00:32:41.340 --> 00:32:43.800
But it turns out
that for two by two,
00:32:43.800 --> 00:32:45.570
you can always do it quickly.
00:32:45.570 --> 00:32:48.500
Even for three by three, if
you just sit down and do it,
00:32:48.500 --> 00:32:49.740
you can actually work it out.
00:32:49.740 --> 00:32:50.920
It's not scary.
00:32:50.920 --> 00:32:53.640
By five by five--
00:32:53.640 --> 00:32:57.180
but even four by four, I'm sure
you are mighty students who
00:32:57.180 --> 00:33:01.040
can just do it in the exam.
00:33:01.040 --> 00:33:04.830
I have never seen an 8.03
exam with four masses,
00:33:04.830 --> 00:33:06.880
unless they're
general questions.
00:33:06.880 --> 00:33:08.790
But three-- well...
00:33:08.790 --> 00:33:10.140
All right.
00:33:10.140 --> 00:33:13.620
So do we go about
finding this B1 and B2?
00:33:13.620 --> 00:33:17.485
Because, again, this is a
simple two by two question.
00:33:23.600 --> 00:33:27.630
So maybe just to again
bring it even closer
00:33:27.630 --> 00:33:29.730
to what we are used
to, let me just quickly
00:33:29.730 --> 00:33:32.740
write this down as a set
of two by two equations.
00:33:32.740 --> 00:33:39.300
So there is a coefficient here,
k over m plus g over l minus
00:33:39.300 --> 00:33:42.330
omega d squared, which
is-- this is a number,
00:33:42.330 --> 00:33:53.190
times B1 minus k over m times
B2 is equal to F0 over m minus k
00:33:53.190 --> 00:34:03.720
over m B1 plus k over m plus g
over l minus omega d squared is
00:34:03.720 --> 00:34:05.190
equal to 0--
00:34:05.190 --> 00:34:08.280
times B2 is equal to 0.
00:34:08.280 --> 00:34:11.310
So you see two equations
with two unknowns.
00:34:11.310 --> 00:34:13.699
Couple of coefficients,
all fixed.
00:34:13.699 --> 00:34:16.489
You can eliminate variables.
00:34:16.489 --> 00:34:19.010
You can calculate B2
from here, plug it into--
00:34:19.010 --> 00:34:21.600
you can work it
out if you want to.
00:34:21.600 --> 00:34:24.740
However, there is,
again, a better way.
00:34:24.740 --> 00:34:30.170
It's Cramer's rule or method.
00:34:34.650 --> 00:34:36.880
Should have known if
it's method or rule.
00:34:36.880 --> 00:34:37.409
Rule.
00:34:37.409 --> 00:34:38.560
Right.
00:34:38.560 --> 00:34:41.090
And so the way you do
it is the following.
00:34:41.090 --> 00:34:47.320
So you look at those questions--
you calculate all kinds
00:34:47.320 --> 00:34:52.840
of determinants, and by taking
the set of two equations
00:34:52.840 --> 00:34:54.190
and plugging into--
00:34:54.190 --> 00:34:56.890
replacing columns in the matrix.
00:34:56.890 --> 00:35:03.700
So B1, what you do is you take
the original matrix, which
00:35:03.700 --> 00:35:08.810
is here, and you replace the
first column of the matrix
00:35:08.810 --> 00:35:12.400
with vector B. So you--
00:35:12.400 --> 00:35:16.730
no wait, with-- sorry, with
vector D. Take this matrix,
00:35:16.730 --> 00:35:18.160
and you plug in this.
00:35:18.160 --> 00:35:19.180
So what you do is--
00:35:19.180 --> 00:35:19.990
so it turns out--
00:35:22.620 --> 00:35:25.050
so B1 can be
explicitly calculated,
00:35:25.050 --> 00:35:29.430
but taking the determinant
of the first column replaced,
00:35:29.430 --> 00:35:35.310
F0 over M0, and keeping the
second column, which is minus
00:35:35.310 --> 00:35:38.485
k over m.
00:35:38.485 --> 00:35:47.910
m and then k over m plus g
over l minus omega d squared.
00:35:47.910 --> 00:35:51.680
So this is-- you
calculate the determinant
00:35:51.680 --> 00:35:54.601
of this thing, where-- original
matrix with the first column
00:35:54.601 --> 00:35:55.100
replaced.
00:35:55.100 --> 00:35:58.940
And you divide it
by the determinant
00:35:58.940 --> 00:36:00.090
of the original matrix.
00:36:00.090 --> 00:36:05.310
Let's call it E. So you
calculate this determinant
00:36:05.310 --> 00:36:08.850
again for the frequency omega d.
00:36:08.850 --> 00:36:12.630
So this can be written very
nicely, in a very compact way.
00:36:12.630 --> 00:36:13.830
This determinant is easy.
00:36:13.830 --> 00:36:15.720
It's just this times that.
00:36:15.720 --> 00:36:25.500
So have 0 over m multiplying
k over n plus g over l
00:36:25.500 --> 00:36:28.950
minus on I got the squared
remember this is a given
00:36:28.950 --> 00:36:33.720
number divided by n Here
comes this nice compact form
00:36:33.720 --> 00:36:38.360
for the determinant, which is
omega d squared minus omega 1
00:36:38.360 --> 00:36:45.492
squared, times omega d
squared minus omega 2 squared,
00:36:45.492 --> 00:36:54.350
where omega 1 and omega 2 were
the normal mode frequencies.
00:36:54.350 --> 00:36:54.850
Yes?
00:36:54.850 --> 00:36:57.516
AUDIENCE: Where are you getting
the minus k in the [INAUDIBLE]??
00:37:01.022 --> 00:37:03.225
BOLESLAW WYSLOUCH: This one?
00:37:03.225 --> 00:37:04.215
AUDIENCE: Yeah.
00:37:04.215 --> 00:37:04.734
[INAUDIBLE]
00:37:04.734 --> 00:37:05.900
BOLESLAW WYSLOUCH: This one?
00:37:05.900 --> 00:37:07.530
This is the second column.
00:37:07.530 --> 00:37:09.141
See?
00:37:09.141 --> 00:37:16.020
I'm taking-- so this is the
first column, second column.
00:37:16.020 --> 00:37:18.870
I take the first
column, I replace it
00:37:18.870 --> 00:37:22.080
with driven equation--
with a solution.
00:37:22.080 --> 00:37:23.430
I plug it here.
00:37:23.430 --> 00:37:24.820
So I have F0 for M0.
00:37:27.350 --> 00:37:29.570
And I keep the second column.
00:37:29.570 --> 00:37:30.070
All right?
00:37:30.070 --> 00:37:31.500
That's for the
first coefficient.
00:37:31.500 --> 00:37:33.250
For the second coefficient
what you do is,
00:37:33.250 --> 00:37:38.276
you put a driving term here
and you keep the first column.
00:37:38.276 --> 00:37:40.230
All right?
00:37:40.230 --> 00:37:43.290
So this is actually an
explicit solution for B1.
00:37:43.290 --> 00:37:48.350
This is magnitude
of oscillations
00:37:48.350 --> 00:37:52.020
of the first element.
00:37:52.020 --> 00:37:53.610
And you can do the
same thing for B2.
00:38:02.020 --> 00:38:04.080
And I'm not trying
to prove anything,
00:38:04.080 --> 00:38:06.381
I'm not trying to
derive anything.
00:38:06.381 --> 00:38:07.130
I'm just using it.
00:38:07.130 --> 00:38:09.730
And I'll show you a nice
slide with this to summarize.
00:38:09.730 --> 00:38:16.810
So B2 is the determinant of--
00:38:16.810 --> 00:38:19.240
I keep the first column.
00:38:19.240 --> 00:38:24.790
It's k over m plus g
over l, minus omega d
00:38:24.790 --> 00:38:28.590
squared, minus k over m.
00:38:28.590 --> 00:38:29.690
That's the first column.
00:38:29.690 --> 00:38:36.340
And I'm plugging in F0
over M here, and 0 here.
00:38:36.340 --> 00:38:42.520
So this is-- and divided by
omega d squared minus omega 1
00:38:42.520 --> 00:38:47.940
squared times omega d squared
minus omega 2 squared.
00:38:47.940 --> 00:38:51.100
That's the determinant
of the original matrix.
00:38:51.100 --> 00:38:56.650
And this one is also very
simple It's this time this is 0.
00:38:56.650 --> 00:38:57.550
I have minus that.
00:38:57.550 --> 00:39:04.360
So I simply have F0 k
over m squared divided
00:39:04.360 --> 00:39:11.320
by omega d squared minus
omega 1 squared, omega d
00:39:11.320 --> 00:39:15.475
squared minus omega 2 squared.
00:39:15.475 --> 00:39:17.070
All right.
00:39:17.070 --> 00:39:19.170
So we have those
things, and also what?
00:39:19.170 --> 00:39:21.950
Do you see anything
happening here?
00:39:21.950 --> 00:39:25.780
Yeah, there are some numbers,
but what do they mean?
00:39:25.780 --> 00:39:26.530
What does it mean?
00:39:26.530 --> 00:39:28.545
Yes, we can calculate it.
00:39:28.545 --> 00:39:29.270
You can trust me.
00:39:29.270 --> 00:39:30.392
These are the--
00:39:30.392 --> 00:39:31.850
I'm not sure that
you can trust it,
00:39:31.850 --> 00:39:33.960
but most likely these
are good results.
00:39:33.960 --> 00:39:38.270
And so we know the oscillation
of the first mass, oscillation
00:39:38.270 --> 00:39:43.580
of the second mass as they are
driven by the external force.
00:39:43.580 --> 00:39:47.640
Now, one of the
interesting things to do
00:39:47.640 --> 00:39:49.530
is to try to see
what's going on.
00:39:49.530 --> 00:39:53.460
One of the-- when we
talked about normal modes,
00:39:53.460 --> 00:39:56.820
the ratio of amplitudes
carried information.
00:39:56.820 --> 00:39:58.865
Remember, we had those
two different modes.
00:39:58.865 --> 00:40:02.180
Either amplitudes were the same,
or they were opposite sign.
00:40:02.180 --> 00:40:05.860
So let's ask ourselves, what
is the ratio of B1 and B2?
00:40:05.860 --> 00:40:07.490
So let's just divide
one by the other.
00:40:11.800 --> 00:40:14.790
So let's do B1 over B2.
00:40:14.790 --> 00:40:17.550
Let's see if we learn
anything from this.
00:40:17.550 --> 00:40:23.370
If you divide B1 over B2,
this bottom cancels out,
00:40:23.370 --> 00:40:31.650
and I have k over m plus
g over l minus omega
00:40:31.650 --> 00:40:36.910
d squared over k over m.
00:40:40.190 --> 00:40:42.470
And-- yeah.
00:40:42.470 --> 00:40:47.270
So now comes the
interesting question.
00:40:47.270 --> 00:40:51.495
This omega d can be anything.
00:40:54.340 --> 00:41:01.490
So let's say omega d is-- so we
can analyze it different ways.
00:41:01.490 --> 00:41:03.911
So for example,
when omega d is--
00:41:03.911 --> 00:41:06.160
you can look at small, large,
and so I can compare it.
00:41:06.160 --> 00:41:08.150
But one of the
interesting places to look
00:41:08.150 --> 00:41:13.060
is, what happens when omega
is very close to one of the--
00:41:13.060 --> 00:41:16.360
to the characteristic
frequencies?
00:41:16.360 --> 00:41:19.600
Because, remember, when we
analyzed a single driven
00:41:19.600 --> 00:41:21.880
oscillator, the
real cool stuff was
00:41:21.880 --> 00:41:25.430
happening when you are near
the resonant frequency.
00:41:25.430 --> 00:41:28.450
Things, you know, the bridges
broke down, et cetera.
00:41:28.450 --> 00:41:31.120
So let's see if we can do
something similar here.
00:41:31.120 --> 00:41:32.260
Now we have two choices.
00:41:32.260 --> 00:41:34.700
We have omega 1, omega 2.
00:41:34.700 --> 00:41:40.230
So let's see what happens
if I plug in omega 1.
00:41:40.230 --> 00:41:44.550
Omega d being very,
very close to omega 1.
00:41:44.550 --> 00:41:46.350
Let's say equal to omega 1.
00:41:46.350 --> 00:41:54.510
Omega 1 is-- omega 1
squared was g over l.
00:41:54.510 --> 00:41:59.840
So if I plug omega 1 here, I
have k over m plus g over l.
00:41:59.840 --> 00:42:07.410
So I have k over m plus
g over l, minus g over l,
00:42:07.410 --> 00:42:13.120
divide by k over m,
which is equal to what?
00:42:13.120 --> 00:42:14.460
Those two terms cancels.
00:42:14.460 --> 00:42:17.110
k over m, it's plus 1.
00:42:17.110 --> 00:42:17.990
That's interesting.
00:42:17.990 --> 00:42:24.000
So if I drive at a frequency
which corresponds to omega 1--
00:42:24.000 --> 00:42:28.440
and omega 1 was the
oscillation where both masses
00:42:28.440 --> 00:42:31.150
were going together.
00:42:31.150 --> 00:42:33.270
So the characteristic
normal mode
00:42:33.270 --> 00:42:36.520
had the ratio of two
masses equal to one.
00:42:36.520 --> 00:42:41.580
And here I'm getting the system
to drive at this type of mode.
00:42:41.580 --> 00:42:44.880
Again, I have-- the
driven amplitudes
00:42:44.880 --> 00:42:47.390
are the ratio is equal to one.
00:42:52.150 --> 00:42:58.630
So what happens if I drive
at omega d close to omega 2?
00:42:58.630 --> 00:43:09.690
Omega 2 squared was equal
to g over l plus 2k over m.
00:43:09.690 --> 00:43:16.274
If I plug it in here, I get
that the ratio is minus 1.
00:43:16.274 --> 00:43:20.470
Again, the ratio is strikingly
similar to the ratio
00:43:20.470 --> 00:43:25.130
of the normal mode corresponding
to frequency omega 2.
00:43:25.130 --> 00:43:28.240
So it's like I'm inducing
those oscillations.
00:43:32.070 --> 00:43:35.380
So what does this all mean?
00:43:35.380 --> 00:43:38.040
There's, by the way,
a little catch here
00:43:38.040 --> 00:43:40.770
for all of your mathematicians.
00:43:40.770 --> 00:43:44.730
What happens to equations if I
set omega d equal to minus 1--
00:43:44.730 --> 00:43:48.337
to omega 1, for example?
00:43:48.337 --> 00:43:50.170
I just plugged it here,
and nobody screamed.
00:43:50.170 --> 00:43:52.250
But there was something
fishy about what I did.
00:43:52.250 --> 00:43:52.750
Yes?
00:43:52.750 --> 00:43:56.439
AUDIENCE: --coefficient
[INAUDIBLE]
00:43:56.439 --> 00:43:57.772
BOLESLAW WYSLOUCH: If you took--
00:43:57.772 --> 00:43:58.734
AUDIENCE: Oh, sorry.
00:43:58.734 --> 00:44:01.035
[INAUDIBLE]
00:44:01.035 --> 00:44:02.160
BOLESLAW WYSLOUCH: Exactly.
00:44:02.160 --> 00:44:07.200
So the ratio of the two was one,
but both of them were infinite.
00:44:07.200 --> 00:44:09.410
So infinite divided by
infinite equals what?
00:44:09.410 --> 00:44:11.280
I mean, this happens.
00:44:11.280 --> 00:44:12.080
So what's going on?
00:44:12.080 --> 00:44:12.900
Why can I do it?
00:44:12.900 --> 00:44:17.204
One-- we should
not really scream.
00:44:17.204 --> 00:44:17.971
Damping.
00:44:17.971 --> 00:44:18.470
Exactly.
00:44:18.470 --> 00:44:20.233
This is where the
damping comes in.
00:44:20.233 --> 00:44:22.804
So the amplitude is enormous,
but it's not infinite,
00:44:22.804 --> 00:44:24.470
because there's always
a little damping.
00:44:24.470 --> 00:44:26.810
The system will
not go to infinity.
00:44:26.810 --> 00:44:30.200
So in real life, there's
a little term here
00:44:30.200 --> 00:44:32.290
that makes sure things
don't blow up completely.
00:44:32.290 --> 00:44:33.539
There's a little damping here.
00:44:33.539 --> 00:44:34.052
Yes?
00:44:34.052 --> 00:44:35.820
AUDIENCE: Does it
at all matter--
00:44:35.820 --> 00:44:37.327
also the fact that
those equations
00:44:37.327 --> 00:44:39.701
are inexact in the
first place, because we
00:44:39.701 --> 00:44:40.784
had made theta smaller--
00:44:40.784 --> 00:44:41.700
BOLESLAW WYSLOUCH: No.
00:44:41.700 --> 00:44:43.860
That's not-- no.
00:44:43.860 --> 00:44:45.630
This doesn't actually matter.
00:44:45.630 --> 00:44:53.000
It's the absence of damping that
makes things look nonphysical.
00:44:53.000 --> 00:44:57.320
AUDIENCE: But as the frequency--
as the amplitude increases,
00:44:57.320 --> 00:45:00.200
when we're in resonance,
eventually those equations
00:45:00.200 --> 00:45:02.120
wouldn't hold any
longer, and perhaps--
00:45:02.120 --> 00:45:03.703
BOLESLAW WYSLOUCH:
Yeah, that's right.
00:45:03.703 --> 00:45:06.850
But you could-- that's true.
00:45:06.850 --> 00:45:08.570
That's true.
00:45:08.570 --> 00:45:13.650
But you can come up with, for
example, an electronic system
00:45:13.650 --> 00:45:15.870
which has a huge range of--
00:45:15.870 --> 00:45:18.150
enormous range of possibilities.
00:45:18.150 --> 00:45:21.090
And then-- or of amplitudes.
00:45:21.090 --> 00:45:26.950
Many, many-- so the damping is
much more important in that.
00:45:26.950 --> 00:45:29.560
So in reality, there is some
damping here and so forth.
00:45:29.560 --> 00:45:30.060
All right.
00:45:30.060 --> 00:45:32.310
So why don't we do,
now, the following.
00:45:32.310 --> 00:45:40.200
So let's try to see
how this all works out.
00:45:40.200 --> 00:45:42.360
First of all, such
that we can get
00:45:42.360 --> 00:45:45.490
started, I will make
a sketch for you.
00:45:45.490 --> 00:45:51.860
I'll calculate these formulas--
00:45:51.860 --> 00:45:59.740
just a second-- and display
you as a function of frequency,
00:45:59.740 --> 00:46:02.604
such that we can
analyze what's going on.
00:46:02.604 --> 00:46:03.270
So where is it--
00:46:06.140 --> 00:46:06.705
OK.
00:46:06.705 --> 00:46:08.580
It's still slow.
00:46:08.580 --> 00:46:11.490
All right.
00:46:11.490 --> 00:46:15.610
So this is what those--
00:46:15.610 --> 00:46:16.930
OK, so let's say--
00:46:16.930 --> 00:46:22.090
I don't know which is which,
but let's say B1 is the red one,
00:46:22.090 --> 00:46:23.860
B2 is the blue
one, or vice versa.
00:46:23.860 --> 00:46:25.770
It doesn't matter.
00:46:25.770 --> 00:46:28.690
These are the numbers which
I plug in for some values
00:46:28.690 --> 00:46:30.410
for some system.
00:46:30.410 --> 00:46:31.320
So we see that--
00:46:31.320 --> 00:46:34.790
and this is as a
function of frequency.
00:46:34.790 --> 00:46:38.890
So first of all, you see
a characteristic frequency
00:46:38.890 --> 00:46:41.350
around one, characteristic
frequency around three
00:46:41.350 --> 00:46:42.940
on my plot.
00:46:42.940 --> 00:46:46.520
And in the region in the
vicinity of frequency number
00:46:46.520 --> 00:46:50.860
one, you see that
both the blue and red,
00:46:50.860 --> 00:46:53.950
the individual amplitudes
are basically close together.
00:46:53.950 --> 00:46:56.940
So the ratio is close to one.
00:46:56.940 --> 00:46:59.250
If you look at this plot,
you should believe me
00:46:59.250 --> 00:47:01.650
that it's plausible
that if you are
00:47:01.650 --> 00:47:03.870
very close to the
frequency, basically
00:47:03.870 --> 00:47:08.460
the red and blue
will move together.
00:47:08.460 --> 00:47:12.960
If you go around the
second frequency,
00:47:12.960 --> 00:47:17.360
you see that red goes up,
blue goes down, or vice versa
00:47:17.360 --> 00:47:19.260
on the other side.
00:47:19.260 --> 00:47:22.170
So the ratio is minus 1.
00:47:22.170 --> 00:47:24.920
So this plot actually
carries in formation.
00:47:24.920 --> 00:47:27.710
And in fact, what
you see also is
00:47:27.710 --> 00:47:32.580
that there is some sort
of resonant behavior.
00:47:32.580 --> 00:47:35.000
So the amplitudes
are enormous if you
00:47:35.000 --> 00:47:39.660
are close to any of those
characteristic frequencies,
00:47:39.660 --> 00:47:42.790
but they're much smaller
if you're further out.
00:47:42.790 --> 00:47:47.300
There is some motion, but
not as pronounced as when
00:47:47.300 --> 00:47:51.120
you're at the right
driving frequencies.
00:47:51.120 --> 00:47:51.620
All right.
00:47:51.620 --> 00:47:54.650
So let's try to see it.
00:47:54.650 --> 00:47:55.940
Why not?
00:47:55.940 --> 00:47:57.530
So let me go to another system--
00:47:57.530 --> 00:47:59.870
a system which
consists of two masses,
00:47:59.870 --> 00:48:06.500
has the same type of behavior,
slightly different parameters.
00:48:06.500 --> 00:48:09.960
There is no g here, but
everything looks the same.
00:48:09.960 --> 00:48:12.980
It's just much easier to show.
00:48:12.980 --> 00:48:16.320
And I can remove
most of damping.
00:48:19.250 --> 00:48:22.430
And you'll see there
are again two modes, one
00:48:22.430 --> 00:48:24.890
which is like this--
00:48:24.890 --> 00:48:28.160
that's number one,
that slow motion.
00:48:28.160 --> 00:48:30.050
They move together.
00:48:30.050 --> 00:48:34.160
And the other one, which
is like this, where
00:48:34.160 --> 00:48:35.960
the amplitudes are minus 1.
00:48:35.960 --> 00:48:40.240
This is the
frequency number two.
00:48:40.240 --> 00:48:42.080
So now let's try to drive it.
00:48:46.040 --> 00:48:47.030
How do I drive it?
00:48:47.030 --> 00:48:52.440
I have some sort of engine here
which is applying frequency.
00:48:52.440 --> 00:49:01.530
So let's start with some
sort of slow motion.
00:49:08.930 --> 00:49:12.110
So you see they are
moving a little bit.
00:49:12.110 --> 00:49:14.800
Very small, minimally.
00:49:14.800 --> 00:49:16.910
Just a tiny motion.
00:49:16.910 --> 00:49:21.080
But they're kind of together,
more or less, right?
00:49:21.080 --> 00:49:22.980
Slowly, but together.
00:49:22.980 --> 00:49:24.050
And this is what--
00:49:24.050 --> 00:49:27.040
this is this area here.
00:49:27.040 --> 00:49:28.970
I don't know if you see that.
00:49:28.970 --> 00:49:31.100
This is this area.
00:49:31.100 --> 00:49:34.310
I'm driving at a
very slow frequency.
00:49:34.310 --> 00:49:36.020
I'm somewhere here.
00:49:36.020 --> 00:49:41.120
The two masses kind of go
together, but very slowly.
00:49:41.120 --> 00:49:43.130
So let me now crank
up the frequency
00:49:43.130 --> 00:49:47.900
and try to be in the
region of oscillation.
00:49:53.732 --> 00:49:56.570
So you see?
00:49:56.570 --> 00:50:00.050
All I did is I
changed frequency.
00:50:00.050 --> 00:50:01.460
The effect is enormous.
00:50:01.460 --> 00:50:02.456
I'm somewhere here now.
00:50:06.936 --> 00:50:07.436
You see?
00:50:10.440 --> 00:50:12.000
Enormous resonance.
00:50:12.000 --> 00:50:13.740
And very soon, I
will hit the limit.
00:50:13.740 --> 00:50:15.790
The system will break.
00:50:15.790 --> 00:50:17.740
OK, so we are somewhere here.
00:50:17.740 --> 00:50:20.370
I'm driving it.
00:50:20.370 --> 00:50:23.550
Interestingly, this really
looks like a harmonic motion
00:50:23.550 --> 00:50:24.600
of first type.
00:50:24.600 --> 00:50:27.030
There is no other things.
00:50:27.030 --> 00:50:32.290
OK, so now let's swing
by and get to this area.
00:50:32.290 --> 00:50:41.559
So all I'm doing is, I quickly
change frequency to- this one.
00:50:48.780 --> 00:50:51.090
So now what you
see is that there
00:50:51.090 --> 00:50:53.910
were some random
initial conditions, so
00:50:53.910 --> 00:50:56.220
we have a homogeneous
equation going,
00:50:56.220 --> 00:50:57.900
but the driven is coming in.
00:50:57.900 --> 00:51:00.600
All I did is I
changed frequency.
00:51:00.600 --> 00:51:06.796
And suddenly the system knows
that it has to go like that.
00:51:06.796 --> 00:51:07.792
Isn't that cool?
00:51:12.590 --> 00:51:16.320
So this is the region here.
00:51:16.320 --> 00:51:21.150
And all I'm doing is I'm
bringing the amplitude up,
00:51:21.150 --> 00:51:25.790
because this is close to zero.
00:51:25.790 --> 00:51:28.650
And then I'm keeping
the ratios close
00:51:28.650 --> 00:51:31.130
to the characteristic modes.
00:51:31.130 --> 00:51:40.436
So I think-- to be honest,
this is one of the coolest--
00:51:40.436 --> 00:51:42.060
all I'm doing, just
changing frequency.
00:51:42.060 --> 00:51:45.060
And the system just
responds and starts
00:51:45.060 --> 00:51:48.840
going with a resonance
of one particular mode.
00:51:48.840 --> 00:51:52.440
So imagine a system
that has 1,000 masses,
00:51:52.440 --> 00:51:54.541
and you come in with
1,000 frequencies.
00:51:54.541 --> 00:51:56.790
You tune one frequency, and
suddenly everything starts
00:51:56.790 --> 00:51:59.970
oscillating in one go.
00:51:59.970 --> 00:52:02.290
And imagine you have
multiple buildings,
00:52:02.290 --> 00:52:05.100
each with different frequency,
and there's an earthquake.
00:52:05.100 --> 00:52:06.837
And the frequency is
of a certain type,
00:52:06.837 --> 00:52:09.420
and one building collapses, and
all the other ones are happily
00:52:09.420 --> 00:52:10.000
standing.
00:52:10.000 --> 00:52:10.500
Why?
00:52:10.500 --> 00:52:14.010
Because the earthquake just
happened to hit the frequency
00:52:14.010 --> 00:52:17.400
that corresponded to one
of the normal frequencies
00:52:17.400 --> 00:52:19.660
of that particular building.
00:52:19.660 --> 00:52:24.045
And it's an extremely
powerful trick.
00:52:24.045 --> 00:52:27.480
It fishes out normal modes
through this driving thing.
00:52:27.480 --> 00:52:30.450
And we are able to
calculate it explicitly.
00:52:30.450 --> 00:52:33.930
So now what I will do is,
I will modify the system
00:52:33.930 --> 00:52:38.120
and I will make it into
a three mass thing, which
00:52:38.120 --> 00:52:41.340
will have a somewhat
more complicated set
00:52:41.340 --> 00:52:44.250
of normal modes.
00:52:44.250 --> 00:52:46.800
And then I will show
you that I can in fact
00:52:46.800 --> 00:52:48.810
go with three
different frequencies,
00:52:48.810 --> 00:52:53.840
and pull out those
even complicated modes.
00:52:53.840 --> 00:52:55.030
So this will be it.
00:52:55.030 --> 00:52:57.610
So this is a three mass system.
00:52:57.610 --> 00:53:02.890
Now before, since we didn't
calculate it, what I will do
00:53:02.890 --> 00:53:10.060
is, I'll go to the web and I
will pull out a nice example.
00:53:10.060 --> 00:53:13.330
Let me go to my bookmarks.
00:53:13.330 --> 00:53:15.860
Normal modes.
00:53:15.860 --> 00:53:21.240
So this is a nice
applet from Colorado.
00:53:21.240 --> 00:53:23.980
And you can--
00:53:23.980 --> 00:53:27.100
I suppose preso ENG will
send you links, et cetera.
00:53:27.100 --> 00:53:31.810
You can simulate-- you
can do everything with it.
00:53:31.810 --> 00:53:35.900
So it has two masses.
00:53:35.900 --> 00:53:40.790
It has different amplitudes,
different normal modes.
00:53:40.790 --> 00:53:42.930
And you can see nothing happens.
00:53:42.930 --> 00:53:45.230
So I have to give it
some initial condition.
00:53:45.230 --> 00:53:49.220
Sorry, I have to
change polarization.
00:53:49.220 --> 00:53:50.210
Where is polarization?
00:53:50.210 --> 00:53:51.920
Here.
00:53:51.920 --> 00:53:54.670
I give it some
initial condition.
00:53:54.670 --> 00:53:56.790
So this is basically
what you just saw.
00:53:56.790 --> 00:54:02.010
I'm just demonstrating to you
that this applet looks the same
00:54:02.010 --> 00:54:03.370
as our track.
00:54:03.370 --> 00:54:06.112
So this is you can
see normal modes.
00:54:06.112 --> 00:54:07.570
It's a combination
of normal modes.
00:54:07.570 --> 00:54:12.792
There's one which is first
frequency, second frequency.
00:54:12.792 --> 00:54:16.400
This is first normal mode.
00:54:16.400 --> 00:54:18.580
This is second normal mode.
00:54:18.580 --> 00:54:21.740
You can very quickly
see what happens.
00:54:21.740 --> 00:54:23.270
So this is what
we just looked at.
00:54:23.270 --> 00:54:26.310
This is what we calculated,
more or less, and so on.
00:54:26.310 --> 00:54:28.370
Now I want to show you
three masses where things
00:54:28.370 --> 00:54:29.840
are somewhat more complicated.
00:54:29.840 --> 00:54:32.150
In general, three normal modes.
00:54:32.150 --> 00:54:35.750
For the three mass elements, the
first normal mode is like that.
00:54:35.750 --> 00:54:38.950
All the three masses
move together.
00:54:38.950 --> 00:54:43.000
And slightly different--
the ratio of amplitudes
00:54:43.000 --> 00:54:45.090
is slightly different.
00:54:45.090 --> 00:54:48.300
The second mode of operation
is actually quite interesting.
00:54:48.300 --> 00:54:51.330
The central mass is
stationary, and those two
00:54:51.330 --> 00:54:56.130
are going forth and
back, like this.
00:54:56.130 --> 00:54:59.760
And then I have
a third frequency
00:54:59.760 --> 00:55:04.230
where the middle one is
going double the distance,
00:55:04.230 --> 00:55:07.080
and the two other
ones are going up.
00:55:07.080 --> 00:55:09.270
So this is the
third normal mode.
00:55:13.210 --> 00:55:13.710
All right.
00:55:13.710 --> 00:55:17.520
So this is the system which
we now have standing here.
00:55:17.520 --> 00:55:19.260
Let's quickly see if
it works in reality.
00:55:29.410 --> 00:55:32.280
So this is the first--
00:55:35.010 --> 00:55:36.930
so this is the first mode.
00:55:45.740 --> 00:55:49.230
This is the second one.
00:55:49.230 --> 00:55:50.110
All right.
00:55:50.110 --> 00:55:51.260
And the third one will be--
00:55:58.410 --> 00:56:02.900
Sometimes I do five of them,
and then it's really difficult.
00:56:02.900 --> 00:56:03.400
OK.
00:56:03.400 --> 00:56:10.690
But-- so we have a computer
model, we have a real model.
00:56:10.690 --> 00:56:14.320
Let's now do the calculation
of the frequencies, the ratios,
00:56:14.320 --> 00:56:16.765
such that we can
see what happens.
00:56:16.765 --> 00:56:20.350
So I'm coming here, I'm
changing mass to three.
00:56:20.350 --> 00:56:26.100
I'm running my-- the
terminal calculating thingy.
00:56:26.100 --> 00:56:26.855
OK.
00:56:26.855 --> 00:56:27.910
It's very slow.
00:56:27.910 --> 00:56:29.360
It's busy, busy, busy.
00:56:29.360 --> 00:56:32.400
Imagine-- OK.
00:56:32.400 --> 00:56:33.490
Spectacularly slow.
00:56:33.490 --> 00:56:34.530
Where is it?
00:56:34.530 --> 00:56:37.030
I hope it's not--
oh, here it is.
00:56:37.030 --> 00:56:40.250
OK, so this is
what's coming out.
00:56:40.250 --> 00:56:43.480
So this is the same
calculation as we
00:56:43.480 --> 00:56:46.810
did, except for three masses.
00:56:46.810 --> 00:56:48.610
So what do we have here?
00:56:48.610 --> 00:56:50.110
Where's my pointer?
00:56:50.110 --> 00:56:54.100
So we have, again, three
characteristic frequencies,
00:56:54.100 --> 00:56:58.130
we have three masses, and
the same type of behavior.
00:56:58.130 --> 00:57:00.560
See, if you are far
away from resonance,
00:57:00.560 --> 00:57:05.230
if you have very low frequency,
everybody goes together.
00:57:05.230 --> 00:57:06.770
I haven't shown
you this one here,
00:57:06.770 --> 00:57:08.492
which is also interesting.
00:57:08.492 --> 00:57:10.150
I'll show you in a second.
00:57:10.150 --> 00:57:12.280
And then-- so
presumably if you are
00:57:12.280 --> 00:57:15.550
close to the first frequency,
you see all three of them
00:57:15.550 --> 00:57:17.170
go together.
00:57:17.170 --> 00:57:19.840
And this is the first mode.
00:57:19.840 --> 00:57:22.510
So I should see, if I
set the proper frequency,
00:57:22.510 --> 00:57:25.000
the thing should respond
in mode number one.
00:57:25.000 --> 00:57:29.050
This is the one where two of
them go opposite to each other,
00:57:29.050 --> 00:57:31.440
and the red one is stationary.
00:57:31.440 --> 00:57:33.020
It doesn't move.
00:57:33.020 --> 00:57:35.620
And then you have those
things where they're
00:57:35.620 --> 00:57:36.700
kind of more complicated.
00:57:36.700 --> 00:57:39.940
It's difficult to
read them from here.
00:57:39.940 --> 00:57:42.310
And I can do it for
more masses, et cetera.
00:57:42.310 --> 00:57:44.245
So generally it's calculable.
00:57:44.245 --> 00:57:47.000
It can be calculated and can
be actually demonstrated.
00:57:47.000 --> 00:57:49.720
So let's try it.
00:57:49.720 --> 00:57:52.705
So-- 32.
00:57:59.700 --> 00:58:01.730
So there's this magic
frequency number one.
00:58:06.944 --> 00:58:10.060
I'm setting frequency
by turning a knob.
00:58:10.060 --> 00:58:11.100
That's omega d.
00:58:11.100 --> 00:58:14.230
I'm a supervisor
of this operation.
00:58:14.230 --> 00:58:20.530
It stops because of other
reasons, but it will continue.
00:58:20.530 --> 00:58:24.375
Then I go to 56.
00:58:34.360 --> 00:58:37.370
By the way, remember
that every--
00:58:37.370 --> 00:58:39.420
this is the particular solution.
00:58:39.420 --> 00:58:42.280
This is a steady state
distillation with omega d.
00:58:42.280 --> 00:58:44.830
But we also have all those
homogeneous solutions, which
00:58:44.830 --> 00:58:46.840
have to die down with damping.
00:58:46.840 --> 00:58:50.200
Remember, it's a combination
of homogeneous plus particular.
00:58:50.200 --> 00:58:52.420
So the motion is actually
a little bit distorted
00:58:52.420 --> 00:58:56.200
because we have this homogeneous
stuff hanging around.
00:58:56.200 --> 00:58:59.320
But hopefully, if I can start it
with little homogeneous stuff,
00:58:59.320 --> 00:59:01.820
it will be better.
00:59:01.820 --> 00:59:02.320
So you see?
00:59:05.050 --> 00:59:06.160
Pretty cool.
00:59:06.160 --> 00:59:07.466
Almost there.
00:59:07.466 --> 00:59:10.846
It's almost in assembly.
00:59:10.846 --> 00:59:11.870
Then it kind of stops.
00:59:14.636 --> 00:59:15.310
You see?
00:59:15.310 --> 00:59:17.380
I get two of those
going forth and back,
00:59:17.380 --> 00:59:19.790
more or less, and
this one going.
00:59:19.790 --> 00:59:21.700
I could probably
tune the frequency
00:59:21.700 --> 00:59:23.890
a little bit higher or lower.
00:59:23.890 --> 00:59:27.520
I'm not exactly at the
right place, but I'm close.
00:59:27.520 --> 00:59:34.030
And now let's go to the
last one, which is 68
00:59:34.030 --> 00:59:36.896
according to my helpers here.
00:59:44.640 --> 00:59:45.610
You see?
00:59:45.610 --> 00:59:48.310
This one goes opposite
phase, and those two
00:59:48.310 --> 00:59:51.480
more or less together.
00:59:51.480 --> 00:59:54.820
Then they keep going.
00:59:54.820 --> 00:59:57.750
See now, those two move a
little bit forth and back,
00:59:57.750 --> 00:59:58.790
but they are in phase.
00:59:58.790 --> 00:59:59.760
They move together.
00:59:59.760 --> 01:00:01.490
The ratio is 1.
01:00:01.490 --> 01:00:03.796
And this one-- the
ratio is minus 2.
01:00:08.500 --> 01:00:09.000
Right?
01:00:09.000 --> 01:00:10.371
Make sense?
01:00:10.371 --> 01:00:11.120
That's the beauty.
01:00:11.120 --> 01:00:15.200
You drive it at some frequency,
and those normal modes pop out.
01:00:15.200 --> 01:00:17.200
It's actually very, very cool.
01:00:17.200 --> 01:00:23.900
And as I said, you encounter
those type of behaviors
01:00:23.900 --> 01:00:24.490
very often.
01:00:24.490 --> 01:00:27.160
Sometimes you drive a car and
something starts vibrating,
01:00:27.160 --> 01:00:30.940
it's just because the
car driving on the road
01:00:30.940 --> 01:00:33.280
creates a frequency, provides
a driving frequency which
01:00:33.280 --> 01:00:37.290
corresponds to oscillation
frequency or some piece of--
01:00:37.290 --> 01:00:37.920
old car.
01:00:37.920 --> 01:00:39.430
Usually it happens in old cars.
01:00:42.120 --> 01:00:46.470
So I think that's
the message we can--
01:00:46.470 --> 01:00:49.330
and we have all the machinery
to be able to do it.
01:00:49.330 --> 01:00:52.240
We can set up any
matrix at K, which
01:00:52.240 --> 01:00:55.240
has information about all the
forces acting on anything,
01:00:55.240 --> 01:00:57.640
and we can set matrix
M with the masses.
01:00:57.640 --> 01:01:01.870
We can put it all together,
we can find normal modes,
01:01:01.870 --> 01:01:03.880
and then we can use
Cramer's equation
01:01:03.880 --> 01:01:07.870
to take care of the
arbitrary external forces.
01:01:07.870 --> 01:01:09.740
And what comes out, just as a--
01:01:09.740 --> 01:01:14.380
for summary, for
future reference,
01:01:14.380 --> 01:01:18.250
the oscillation
of the system is--
01:01:18.250 --> 01:01:20.110
this is conveniently written.
01:01:20.110 --> 01:01:24.250
This is vector X.
In general, this
01:01:24.250 --> 01:01:34.700
homogeneous solution this plus
the particular solution, which
01:01:34.700 --> 01:01:41.480
is plus vector B, which
is very important.
01:01:41.480 --> 01:01:45.780
Vector B depends on
the driving frequency.
01:01:45.780 --> 01:01:48.950
Those amplitudes of a particular
solution during emotions
01:01:48.950 --> 01:01:51.040
are dependent on
driving frequency.
01:01:51.040 --> 01:01:57.220
Cosine omega d times t.
01:01:57.220 --> 01:02:01.500
So in the most
general situation,
01:02:01.500 --> 01:02:04.100
we have some homogeneous
solution here,
01:02:04.100 --> 01:02:07.310
and there is this
driven solution
01:02:07.310 --> 01:02:11.120
which we observed in action,
with proper amplitudes.
01:02:11.120 --> 01:02:13.160
So in fact, what
you've seen is the sum
01:02:13.160 --> 01:02:16.230
of both, because this depends
on the initial conditions.
01:02:16.230 --> 01:02:20.720
Now, in reality, as with
a single oscillator,
01:02:20.720 --> 01:02:23.120
this homogeneous equation,
there's always a little
01:02:23.120 --> 01:02:25.730
damping, which we ignore it.
01:02:25.730 --> 01:02:28.190
And the damping
comes in, and it only
01:02:28.190 --> 01:02:29.840
affects the
homogeneous solution.
01:02:29.840 --> 01:02:33.530
So this part will
eventually die down,
01:02:33.530 --> 01:02:36.840
whereas a driven
solution is always there.
01:02:36.840 --> 01:02:40.400
There's external force that
is driving the system forever
01:02:40.400 --> 01:02:41.850
and ever.
01:02:41.850 --> 01:02:45.860
So this part, this steady
state or particular solution
01:02:45.860 --> 01:02:49.110
will remain forever,
because there's
01:02:49.110 --> 01:02:52.170
an external source of energy
which will always provide it.
01:02:52.170 --> 01:02:53.940
So these guys will die down.
01:02:53.940 --> 01:02:57.090
And of course, because of
damping the exact value
01:02:57.090 --> 01:02:59.790
of coefficients B will
be slightly modified,
01:02:59.790 --> 01:03:03.692
because as you know from the
from a one oscillator example,
01:03:03.692 --> 01:03:05.400
the presence of damping
actually slightly
01:03:05.400 --> 01:03:07.280
modifies the frequency.
01:03:07.280 --> 01:03:08.880
Whereas here, we--
for simplicity--
01:03:08.880 --> 01:03:11.640
if we introduce damping
here, those calculations
01:03:11.640 --> 01:03:13.440
are really amazing.
01:03:13.440 --> 01:03:16.410
So we don't want to do it.
01:03:16.410 --> 01:03:17.100
All right.
01:03:17.100 --> 01:03:18.120
Any questions about it?
01:03:22.950 --> 01:03:23.782
Yes.
01:03:23.782 --> 01:03:26.480
AUDIENCE: If we were doing
Cramer's rule with a three
01:03:26.480 --> 01:03:30.510
by three matrix, would we
only replace the column
01:03:30.510 --> 01:03:34.069
that corresponds to the B that
we're trying to find, and then
01:03:34.069 --> 01:03:34.860
keep the other two?
01:03:34.860 --> 01:03:36.160
BOLESLAW WYSLOUCH: Yes.
01:03:36.160 --> 01:03:38.130
So it's always-- you'll
be doing always that.
01:03:38.130 --> 01:03:40.980
In fact, I should have
some slides from Yen-Jie
01:03:40.980 --> 01:03:43.830
on Cramer's rule.
01:03:43.830 --> 01:03:45.859
Let's see.
01:03:45.859 --> 01:03:46.358
OK.
01:03:46.358 --> 01:03:51.597
So this is some
reminder of last time.
01:03:51.597 --> 01:03:53.305
So this is Cramer's--
there's Mr. Cramer.
01:03:55.840 --> 01:03:59.170
So this is an example of what--
01:03:59.170 --> 01:04:02.730
this is the two by
two, three by three.
01:04:02.730 --> 01:04:03.230
OK?
01:04:03.230 --> 01:04:05.210
That's what you do.
01:04:05.210 --> 01:04:06.200
Question?
01:04:06.200 --> 01:04:10.655
AUDIENCE: So it makes sense that
the Cramer's rule [INAUDIBLE],,
01:04:10.655 --> 01:04:12.635
but what does that mean
for physical system?
01:04:16.610 --> 01:04:19.840
BOLESLAW WYSLOUCH:
Well, basically--
01:04:19.840 --> 01:04:22.800
so the Cramer's rule
is Cramer's rule.
01:04:22.800 --> 01:04:24.650
The question is
what do you plug in?
01:04:24.650 --> 01:04:27.770
And what you plug in
depends on the omega d.
01:04:27.770 --> 01:04:30.790
So it is true that if you
insist on plugging in omega
01:04:30.790 --> 01:04:33.950
d exactly equal to one of
the normal frequencies,
01:04:33.950 --> 01:04:38.290
then things blow
up mathematically.
01:04:38.290 --> 01:04:41.620
In reality, there is--
01:04:41.620 --> 01:04:43.960
this is the situation
of resonance.
01:04:43.960 --> 01:04:46.150
So as I discussed this
before, in reality
01:04:46.150 --> 01:04:49.070
there is a little
bit of damping.
01:04:49.070 --> 01:04:51.500
So those equations
have to be modified.
01:04:51.500 --> 01:04:54.710
There will be some small
additional terms here
01:04:54.710 --> 01:04:57.680
that will prevent this from
being exactly equal to 0.
01:04:57.680 --> 01:04:59.510
So this will be a
very large number.
01:04:59.510 --> 01:05:02.026
The amplitude will be enormous.
01:05:02.026 --> 01:05:03.650
If I would have a
little bit more time,
01:05:03.650 --> 01:05:05.120
I'll fiddle with
frequency, I could actually
01:05:05.120 --> 01:05:07.850
break the system, because those
masses would be just swinging
01:05:07.850 --> 01:05:10.870
forth and back like crazy.
01:05:10.870 --> 01:05:14.310
So you basically go out
of limit of the system.
01:05:14.310 --> 01:05:16.560
So physically, there's always
a little bit of damping.
01:05:16.560 --> 01:05:18.460
You do not divide by zero.
01:05:18.460 --> 01:05:22.660
On the other hand, it's so
close that, for simplicity
01:05:22.660 --> 01:05:26.110
and for most of the-- to get
a feeling of what's going on,
01:05:26.110 --> 01:05:27.870
it's OK to ignore it.
01:05:27.870 --> 01:05:31.720
Just have to make sure
you don't divide by 0.
01:05:31.720 --> 01:05:34.715
So you can do this Cramer's
rule with arbitrary omega d.
01:05:34.715 --> 01:05:37.900
Make sure you don't
divide by 0, you solve it,
01:05:37.900 --> 01:05:41.040
and then you can
interpret what's going on.
01:05:41.040 --> 01:05:43.700
Again, Cramer's rule has
nothing to do with physics.
01:05:43.700 --> 01:05:46.464
It's just a way to solve
those matrix equations.
01:05:46.464 --> 01:05:48.130
As I say, you can do
it anyway you want.
01:05:48.130 --> 01:05:52.000
Two by two, you can do it
by elimination of variables.
01:05:52.000 --> 01:05:55.240
Five by five I do by
running a MATLAB program.
01:05:55.240 --> 01:05:56.450
Anything you want.
01:05:56.450 --> 01:06:01.575
But for some historical reasons,
8.03 always does Cramer's rule.
01:06:01.575 --> 01:06:02.220
All right?
01:06:02.220 --> 01:06:06.110
And, yeah, it's useful,
especially for three by three.
01:06:06.110 --> 01:06:06.610
All right?
01:06:09.271 --> 01:06:09.770
OK.
01:06:09.770 --> 01:06:12.832
So I have to start
a new chapter.
01:06:12.832 --> 01:06:14.665
I'm much slower than
the engine, by the way.
01:06:14.665 --> 01:06:15.831
I don't know if you noticed.
01:06:18.270 --> 01:06:20.440
And that is the--
01:06:20.440 --> 01:06:22.710
there's a very
interesting trick that you
01:06:22.710 --> 01:06:29.670
can do which is of an absolutely
fundamental nature in physics,
01:06:29.670 --> 01:06:32.310
which has to do with symmetry.
01:06:32.310 --> 01:06:36.900
You see, many things
are symmetric.
01:06:36.900 --> 01:06:39.040
There's a circular symmetry.
01:06:39.040 --> 01:06:41.110
There's a left and
right symmetry.
01:06:41.110 --> 01:06:44.800
Example, two little
smiley faces are
01:06:44.800 --> 01:06:47.040
mirror images of each other.
01:06:47.040 --> 01:06:50.250
There is some-- this
thing is symmetric
01:06:50.250 --> 01:06:52.620
along this vertical axis.
01:06:52.620 --> 01:06:55.665
This one is symmetric around
rotations by 30 degrees.
01:06:58.230 --> 01:07:01.470
That house seems to be
symmetric along this way.
01:07:01.470 --> 01:07:04.660
This is part of our
experiment in Switzerland,
01:07:04.660 --> 01:07:07.155
also kind of symmetric
in the picture.
01:07:09.624 --> 01:07:12.290
The rotational symmetry, there's
reflection symmetry, et cetera.
01:07:12.290 --> 01:07:16.210
It turns out, if you have
a system that is symmetric,
01:07:16.210 --> 01:07:19.570
then the normal modes
are also symmetric.
01:07:19.570 --> 01:07:23.500
And there's a way to dig out
normal modes just by looking
01:07:23.500 --> 01:07:24.610
at symmetry of the system.
01:07:27.460 --> 01:07:29.870
So let me explain
exactly what this means.
01:07:29.870 --> 01:07:34.520
So let's take our system here--
01:07:34.520 --> 01:07:39.890
OK, so we have one
mass, the other mass.
01:07:39.890 --> 01:07:43.000
There is a spring here.
01:07:43.000 --> 01:07:46.520
This one is x1, this one is x2.
01:07:46.520 --> 01:07:48.590
If I take a reflection
of a system--
01:07:48.590 --> 01:07:53.170
let's say this mass is
displaced by some distance.
01:07:53.170 --> 01:07:56.330
Some x2.
01:07:56.330 --> 01:07:57.620
This one's some x1.
01:07:57.620 --> 01:07:59.280
If I do the
following transform--
01:07:59.280 --> 01:08:08.780
I replace x1 with minus
x2, and x2 with minus x1,
01:08:08.780 --> 01:08:09.890
this is mirror symmetry.
01:08:16.700 --> 01:08:20.050
I basically flip
this thing around.
01:08:20.050 --> 01:08:25.149
In other words, what I do here
is I look at the system here--
01:08:25.149 --> 01:08:28.850
hello-- and I go to
the other system.
01:08:28.850 --> 01:08:29.540
Hello.
01:08:29.540 --> 01:08:30.040
Right?
01:08:30.040 --> 01:08:31.081
I did a mirror transform.
01:08:31.081 --> 01:08:33.750
I looked at it from
this side, that side.
01:08:33.750 --> 01:08:35.890
Now, when I look
at it I see the one
01:08:35.890 --> 01:08:37.359
on the left, one on the right.
01:08:37.359 --> 01:08:40.210
I call this one x1, this one x2.
01:08:40.210 --> 01:08:41.950
It's oscillating.
01:08:41.950 --> 01:08:46.029
You are looking at it, this
is your x1, this is your x2.
01:08:46.029 --> 01:08:47.960
When I move this one, is it--
01:08:47.960 --> 01:08:50.229
it's your negative x1.
01:08:50.229 --> 01:08:53.399
For me this is positive x2.
01:08:53.399 --> 01:08:56.920
This one is positive x2 for you.
01:08:56.920 --> 01:09:00.160
It's negative x1 for me.
01:09:00.160 --> 01:09:01.630
Do we see a different system?
01:09:01.630 --> 01:09:03.130
Does it have different
oscillations?
01:09:03.130 --> 01:09:04.250
Does it have a
different frequency?
01:09:04.250 --> 01:09:04.750
No.
01:09:04.750 --> 01:09:05.689
It's identical.
01:09:05.689 --> 01:09:07.640
They're completely identical.
01:09:07.640 --> 01:09:09.520
So the physics of
those two pendula
01:09:09.520 --> 01:09:11.590
doesn't depend on if
he's working on it
01:09:11.590 --> 01:09:13.000
or if I'm working on.
01:09:13.000 --> 01:09:14.439
That's the whole thing.
01:09:14.439 --> 01:09:17.529
And this is how you
write it mathematically.
01:09:17.529 --> 01:09:20.920
And if you have a
solution which--
01:09:20.920 --> 01:09:27.640
x1 of t, which consists of
some sort of x1 of t, x2 of t.
01:09:27.640 --> 01:09:30.180
Let's say we find it.
01:09:30.180 --> 01:09:31.240
Now it's over there.
01:09:31.240 --> 01:09:34.490
We know alphas,
betas and everything.
01:09:34.490 --> 01:09:38.029
Because of the symmetry, I
know that for sure the equation
01:09:38.029 --> 01:09:41.180
which looks like this-- x1--
01:09:41.180 --> 01:09:42.420
no, it's not x1.
01:09:42.420 --> 01:09:44.270
It's x of t.
01:09:44.270 --> 01:09:46.208
That's the vector x of t.
01:09:46.208 --> 01:09:51.170
I have another one with a tilde,
which is identical functions,
01:09:51.170 --> 01:09:56.990
everything is dependent, except
that this one is minus x2 of t
01:09:56.990 --> 01:10:00.290
minus x1 of t.
01:10:00.290 --> 01:10:04.520
And I know for sure that if
this is the correct solution,
01:10:04.520 --> 01:10:07.230
this is also a correct solution.
01:10:07.230 --> 01:10:07.980
Why?
01:10:07.980 --> 01:10:14.370
Because he did x,
and I did x tilde.
01:10:14.370 --> 01:10:16.200
But the system is the same.
01:10:16.200 --> 01:10:17.220
Completely identical.
01:10:17.220 --> 01:10:22.020
And you don't have to know
anything about masses, lengths,
01:10:22.020 --> 01:10:23.790
springs, anything like that.
01:10:23.790 --> 01:10:25.890
Just the symmetry.
01:10:25.890 --> 01:10:26.730
All right.
01:10:26.730 --> 01:10:29.940
How do you write
it in matrix form?
01:10:29.940 --> 01:10:37.690
You introduce a symmetry
matrix, S, which is 0, minus 1,
01:10:37.690 --> 01:10:40.400
minus 1, 0.
01:10:40.400 --> 01:10:48.584
And then x tilde of t is
simply equal S, x of t.
01:10:48.584 --> 01:10:49.500
And we can check that.
01:10:49.500 --> 01:10:53.590
That's simple you just
multiply the vector by 0,
01:10:53.590 --> 01:10:57.370
minus 1, minus 1, 0, and
you get the same thing.
01:10:57.370 --> 01:11:02.200
Turns out-- and if
this is symmetry,
01:11:02.200 --> 01:11:04.660
if this is a solution,
this is also a solution.
01:11:04.660 --> 01:11:08.670
So we can make solutions
by multiplying by matrix S.
01:11:08.670 --> 01:11:10.030
So what does it mean?
01:11:10.030 --> 01:11:14.115
So let's look at
our motion equation.
01:11:17.880 --> 01:11:19.970
The original motion
equation was--
01:11:19.970 --> 01:11:23.810
equation of motion was minus
1 k matrix times x of t.
01:11:23.810 --> 01:11:26.740
This is what we use
to find solutions.
01:11:26.740 --> 01:11:28.550
Usual thing, normal
modes, et cetera.
01:11:28.550 --> 01:11:32.970
Let's multiply both sides by
matrix S. I can take any matrix
01:11:32.970 --> 01:11:35.230
and multiply by both sides.
01:11:35.230 --> 01:11:39.391
So I get here S X
double dot of t.
01:11:39.391 --> 01:11:43.300
And of course, S
is a fixed matrix,
01:11:43.300 --> 01:11:45.710
so it survives differentiation.
01:11:45.710 --> 01:11:51.910
And this is equal to minus
S M minus 1 k x of t.
01:11:51.910 --> 01:11:54.970
Just multiply both sides by S.
01:11:54.970 --> 01:12:07.286
However, if MS is equal to
SM, and KS is equal to SK--
01:12:11.110 --> 01:12:14.160
in general matrices, the
multiplication of matrices
01:12:14.160 --> 01:12:15.990
matters.
01:12:15.990 --> 01:12:18.970
But it turns out that if
the system is symmetric,
01:12:18.970 --> 01:12:24.130
if you multiply mass M
by S, you just replace--
01:12:24.130 --> 01:12:26.840
it will just change
position of two masses.
01:12:26.840 --> 01:12:28.520
So nothing changes.
01:12:28.520 --> 01:12:34.000
Also, if the forces are the
same, then multiplying mass S,
01:12:34.000 --> 01:12:34.850
you flip things.
01:12:34.850 --> 01:12:36.110
Nothing changes.
01:12:36.110 --> 01:12:38.480
And mathematically, it
means that the order
01:12:38.480 --> 01:12:41.420
of multiplication
does not matter.
01:12:41.420 --> 01:12:42.870
It means that they
are commuting.
01:12:42.870 --> 01:12:50.740
And of course, M minus 1
S is equal to S M minus 1.
01:12:50.740 --> 01:12:55.820
If this is the case, then I
can plug it into equations
01:12:55.820 --> 01:12:58.408
and see what happens.
01:13:07.870 --> 01:13:16.420
So I can take this equation,
and I can take this S here
01:13:16.420 --> 01:13:18.260
and I can just move it around.
01:13:18.260 --> 01:13:21.790
I can flip it with M1 position,
because the order doesn't
01:13:21.790 --> 01:13:22.340
matter.
01:13:22.340 --> 01:13:23.650
So I can bring it here.
01:13:23.650 --> 01:13:25.780
And I can flip it with K,
because the order doesn't
01:13:25.780 --> 01:13:26.280
matter.
01:13:26.280 --> 01:13:28.180
I can bring it here.
01:13:28.180 --> 01:13:31.080
So after using those
features, I get
01:13:31.080 --> 01:13:40.580
that S X dot dot is equal
to minus M minus 1 K S X,
01:13:40.580 --> 01:13:44.810
which means that X dot dot--
01:13:44.810 --> 01:13:47.430
remember, this was--
01:13:47.430 --> 01:13:48.920
I'm using this expression.
01:13:48.920 --> 01:13:53.440
I'm just-- S times a
variable x gives me X tilde.
01:13:53.440 --> 01:14:00.090
X tilde dot dot is equal to
minus M minus 1 k X tilde.
01:14:00.090 --> 01:14:01.770
X tilde.
01:14:01.770 --> 01:14:04.415
Which basically proves--
this is a proof--
01:14:04.415 --> 01:14:06.930
that x tilde is a solution.
01:14:06.930 --> 01:14:12.100
So if a system is symmetric,
it means that it commutes--
01:14:12.100 --> 01:14:20.010
that mass and K matrices
commute, and you can--
01:14:20.010 --> 01:14:22.530
and this means that
this holds true.
01:14:22.530 --> 01:14:24.870
If I have one solution,
the symmetric solution
01:14:24.870 --> 01:14:27.290
is also there.
01:14:27.290 --> 01:14:28.150
All right?
01:14:32.000 --> 01:14:34.354
Let's say x-- yes?
01:14:34.354 --> 01:14:36.764
AUDIENCE: So in the
center equation,
01:14:36.764 --> 01:14:42.080
you introduced negative S.
I didn't really get that.
01:14:42.080 --> 01:14:45.040
BOLESLAW WYSLOUCH: So this
negative is simply the--
01:14:45.040 --> 01:14:47.200
Hooke's law.
01:14:47.200 --> 01:14:49.371
This is this minus sign here.
01:14:49.371 --> 01:14:52.294
AUDIENCE: Yeah, but where
did the S come from in the--
01:14:52.294 --> 01:14:54.460
BOLESLAW WYSLOUCH: Oh, I
multiplied both sides by S.
01:14:54.460 --> 01:14:55.600
AUDIENCE: Oh, OK.
01:14:55.600 --> 01:14:57.225
BOLESLAW WYSLOUCH:
I just brought the S
01:14:57.225 --> 01:14:58.500
and I put it here.
01:14:58.500 --> 01:15:02.950
S, X dot dot, and S after--
01:15:02.950 --> 01:15:07.360
minus commutes with S, so
I kind of shifted my minus.
01:15:07.360 --> 01:15:10.540
But then I waited before
I hit the matrices,
01:15:10.540 --> 01:15:12.180
because I wanted to discuss.
01:15:12.180 --> 01:15:12.680
OK?
01:15:15.300 --> 01:15:18.850
So now comes the
interesting question.
01:15:18.850 --> 01:15:22.070
Let's say X is a normal mode.
01:15:22.070 --> 01:15:22.846
Right?
01:15:22.846 --> 01:15:23.720
We have normal modes.
01:15:23.720 --> 01:15:25.850
Let's say X is a normal mode.
01:15:25.850 --> 01:15:29.630
It oscillates with
a certain frequency.
01:15:29.630 --> 01:15:31.990
So I have X of t.
01:15:31.990 --> 01:15:33.560
Let's say it's equal to--
01:15:33.560 --> 01:15:36.510
let's say it's a
normal mode number one.
01:15:36.510 --> 01:15:39.880
Cosine omega 1 t.
01:15:42.500 --> 01:15:46.390
And we know that X tilde
is also a solution.
01:15:46.390 --> 01:15:54.180
So what happens to mode number
one when I apply matrix S?
01:15:54.180 --> 01:15:59.140
So X tilde-- so matrix
is a constant number.
01:15:59.140 --> 01:16:00.640
It's just a couple
of numbers I just
01:16:00.640 --> 01:16:01.806
reshuffle things, et cetera.
01:16:01.806 --> 01:16:04.630
Try So if I have X, which is
oscillating with frequency
01:16:04.630 --> 01:16:08.290
omega 1, if I multiply
by some numbers
01:16:08.290 --> 01:16:10.780
and reshuffle things
around, it will also
01:16:10.780 --> 01:16:12.910
be oscillating in number one.
01:16:12.910 --> 01:16:16.310
So it will be also
the same normal mode.
01:16:16.310 --> 01:16:20.380
So if I take matrix S, I
apply it to the normal mode,
01:16:20.380 --> 01:16:23.080
I will get the same
normal mode, with maybe
01:16:23.080 --> 01:16:25.310
a different coefficient.
01:16:25.310 --> 01:16:26.770
Linear coefficient.
01:16:26.770 --> 01:16:29.020
Plus, minus, maybe some
factor, something like that.
01:16:31.670 --> 01:16:34.170
So if this is the solution,
it means automatically
01:16:34.170 --> 01:16:43.540
that X tilde is proportional
to A1 cosine omega 1 t.
01:16:43.540 --> 01:16:46.540
And the same is
true for omega 2.
01:16:46.540 --> 01:16:48.190
So the only way
this is possible,
01:16:48.190 --> 01:16:52.290
since cosine is the
same in both cases--
01:16:52.290 --> 01:16:54.160
matrix S to normal
solution gives me
01:16:54.160 --> 01:16:56.700
normal solution with some sign.
01:16:56.700 --> 01:16:59.010
So the only way this can
work, matrix S actually
01:16:59.010 --> 01:17:02.010
works on vectors, on A1.
01:17:02.010 --> 01:17:04.830
This is just an
oscillating factor.
01:17:04.830 --> 01:17:13.110
So we know for sure that
S A1 must be proportional.
01:17:13.110 --> 01:17:13.680
to A1.
01:17:17.840 --> 01:17:24.238
Similarly, S times A2
is proportional to A2.
01:17:28.110 --> 01:17:32.530
So let's try to see with
our own eyes if this works.
01:17:32.530 --> 01:17:36.420
So let's say S is
0, minus 1, minus 1,
01:17:36.420 --> 01:17:42.820
0, times 1, 1 is equal to what?
01:17:42.820 --> 01:17:47.160
0 minus 1, I get minus 1 here.
01:17:47.160 --> 01:17:50.340
This one, I get
minus 1 here, which
01:17:50.340 --> 01:17:57.780
is equal to minus 1 times
1, 1, which is vector A. So
01:17:57.780 --> 01:18:04.850
vector 1, 1, which is our
first mode of oscillation,
01:18:04.850 --> 01:18:09.410
is when you apply the
matrix S, you get a minus 1
01:18:09.410 --> 01:18:12.020
the same thing.
01:18:12.020 --> 01:18:19.144
And similarly, if you do the
same thing with matrix S--
01:18:19.144 --> 01:18:21.890
so you see, the simple
symmetric matrix
01:18:21.890 --> 01:18:24.530
consisting of 0s
and minus 1s has
01:18:24.530 --> 01:18:26.600
something to do with
our solutions, which
01:18:26.600 --> 01:18:27.500
is kind of amazing.
01:18:30.020 --> 01:18:36.980
So if I have 0, minus 1, minus
1, 0, I multiply by 1, minus 1,
01:18:36.980 --> 01:18:43.740
I get 1 here, I
get minus 1 here.
01:18:48.800 --> 01:18:49.740
Just a moment.
01:18:49.740 --> 01:18:52.494
Something is not right.
01:18:52.494 --> 01:18:53.410
Something's not right.
01:18:53.410 --> 01:18:54.040
No, it should be--
01:18:54.040 --> 01:18:54.978
AUDIENCE: [INAUDIBLE]
01:18:54.978 --> 01:18:55.936
BOLESLAW WYSLOUCH: Hmm?
01:18:55.936 --> 01:18:57.358
AUDIENCE: [INAUDIBLE]
01:18:57.358 --> 01:18:59.760
AUDIENCE: It's 1, minus 1.
01:18:59.760 --> 01:19:01.010
BOLESLAW WYSLOUCH: 1, minus 1.
01:19:01.010 --> 01:19:01.664
Yes.
01:19:01.664 --> 01:19:03.080
I don't know how
to multiply here.
01:19:03.080 --> 01:19:05.280
I should be fine.
01:19:05.280 --> 01:19:06.799
OK.
01:19:06.799 --> 01:19:07.340
That's right.
01:19:07.340 --> 01:19:09.006
This is-- sorry, this
is 1, because it's
01:19:09.006 --> 01:19:11.870
minus 1 times minus 1, and
this is-- yeah, that's right.
01:19:11.870 --> 01:19:17.060
Which is 1 times 1, minus 1.
01:19:17.060 --> 01:19:18.590
So this is something that--
01:19:18.590 --> 01:19:21.930
I get the same vector
multiplied by plus 1.
01:19:21.930 --> 01:19:23.270
So this is, of course--
01:19:23.270 --> 01:19:27.150
these are eigenvectors
and eigenvalues.
01:19:27.150 --> 01:19:29.870
So the matrix S has
two eigenvectors,
01:19:29.870 --> 01:19:32.690
one with eigenvalue of plus
one, the other one plus 2.
01:19:32.690 --> 01:19:38.196
So we have an equation SA
is equal to beta times A,
01:19:38.196 --> 01:19:41.250
and beta is--
01:19:44.910 --> 01:19:46.310
OK.
01:19:46.310 --> 01:19:48.150
So this is something--
01:19:48.150 --> 01:19:51.680
so it turns out-- and I don't
think I have time to prove it,
01:19:51.680 --> 01:19:53.770
but it turns out
you can prove it--
01:19:53.770 --> 01:19:56.760
if I would have
another three minutes--
01:19:56.760 --> 01:20:01.350
you can prove it that the
eigenvalues of matrix S--
01:20:01.350 --> 01:20:06.300
eigenvectors, sorry,
eigenvectors of matrix S
01:20:06.300 --> 01:20:11.310
are the same as eigenvectors
of the full motion matrix.
01:20:14.120 --> 01:20:22.260
So in other words, our
motion matrix M minus 1 K--
01:20:22.260 --> 01:20:23.160
this is the matrix.
01:20:23.160 --> 01:20:30.930
Then we have a matrix
S. And normal modes are,
01:20:30.930 --> 01:20:34.830
you have a normal frequency
and they have a shape.
01:20:34.830 --> 01:20:38.400
You have a normal vector,
the ratio of amplitudes.
01:20:38.400 --> 01:20:46.560
And turns out that eigenvectors
here, so the A's are the same.
01:20:46.560 --> 01:20:50.780
And again, I don't
have time to show it,
01:20:50.780 --> 01:20:53.620
but you can show that
this is the case.
01:20:53.620 --> 01:20:57.990
So if you have a
symmetry in the system,
01:20:57.990 --> 01:21:04.770
then you can simply find
eigenvectors of the thing
01:21:04.770 --> 01:21:07.170
to obtain the normal modes.
01:21:07.170 --> 01:21:14.870
So if I look at my two pendula
here, the symmetry is this way,
01:21:14.870 --> 01:21:19.880
so I have to have one which
is fully symmetric, like this,
01:21:19.880 --> 01:21:21.830
and I have another one
which is antisymmetric.
01:21:21.830 --> 01:21:26.042
Plus 1, minus 1,
plus 1, minus 1.
01:21:26.042 --> 01:21:28.280
Similarly, here I have--
01:21:28.280 --> 01:21:30.330
let's say if I have
two masses, there
01:21:30.330 --> 01:21:32.650
is one motion
which is like this,
01:21:32.650 --> 01:21:34.880
and one motion which
is like that, because
01:21:34.880 --> 01:21:37.860
of the mirror symmetry.
01:21:37.860 --> 01:21:40.410
And you can show that if you
have some other symmetries,
01:21:40.410 --> 01:21:42.360
like on a circle
et cetera, that you
01:21:42.360 --> 01:21:43.800
have a similar type of fact.
01:21:43.800 --> 01:21:47.040
So you can build up on
this symmetry argument.
01:21:47.040 --> 01:21:53.880
And finding eigenvectors of a
matrix 0, minus 1, minus 1, 0
01:21:53.880 --> 01:21:56.940
is infinitely simpler
than finding matrix
01:21:56.940 --> 01:22:00.290
with G's and K's and
everything, right?
01:22:00.290 --> 01:22:00.790
All right.
01:22:00.790 --> 01:22:08.120
So thank you very much, and
I hope this was educational.