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PROFESSOR: So what's important
the last couple of lectures?
00:00:26.835 --> 00:00:28.232
Christina.
00:00:28.232 --> 00:00:29.440
AUDIENCE: Transfer functions.
00:00:29.440 --> 00:00:31.364
PROFESSOR: Transfer
functions, all right.
00:00:40.503 --> 00:00:42.154
How about something else?
00:00:42.154 --> 00:00:43.320
AUDIENCE: Modal coordinates.
00:00:43.320 --> 00:00:44.528
PROFESSOR: Modal coordinates.
00:00:47.020 --> 00:00:54.910
And I'll expand that to call
this modal analysis in general.
00:00:54.910 --> 00:00:55.520
Anything else?
00:01:15.220 --> 00:01:17.320
Well, think more about that.
00:01:17.320 --> 00:01:21.660
And are there any questions
for the week, anything muddy,
00:01:21.660 --> 00:01:23.710
fuzzy, not quite
clear to you that if I
00:01:23.710 --> 00:01:26.294
have time to say a
few words about today
00:01:26.294 --> 00:01:27.460
you'd like me to talk about?
00:01:30.412 --> 00:01:32.380
AUDIENCE: So I guess I'm
a little bit confused
00:01:32.380 --> 00:01:37.220
with what we did in
class, the modal analysis.
00:01:37.220 --> 00:01:44.505
Is that just like-- so when we
did originally Newton's law,
00:01:44.505 --> 00:01:47.900
and then we did the [INAUDIBLE].
00:01:50.810 --> 00:01:52.426
Is that sort of
similar here where
00:01:52.426 --> 00:01:55.478
we can do either the
things in motion,
00:01:55.478 --> 00:02:00.160
or we can do
[INAUDIBLE] analysis?
00:02:00.160 --> 00:02:02.925
PROFESSOR: Is modal analysis
an alternative way of--
00:02:02.925 --> 00:02:03.800
AUDIENCE: [INAUDIBLE]
00:02:07.150 --> 00:02:10.389
PROFESSOR: The analogy
isn't too helpful.
00:02:10.389 --> 00:02:15.010
But modal analysis is
one way of attacking
00:02:15.010 --> 00:02:18.150
the equations of
motion that describe
00:02:18.150 --> 00:02:20.450
vibration of linear systems.
00:02:20.450 --> 00:02:23.570
You can work through
the entire analysis
00:02:23.570 --> 00:02:31.680
and figure out the solution
to the equations of motion.
00:02:31.680 --> 00:02:35.820
And usually with these multiple
degree of freedom vibration
00:02:35.820 --> 00:02:42.680
problems, you can cast
them as a mass matrix
00:02:42.680 --> 00:02:46.430
times an acceleration vector
plus a stiffness matrix
00:02:46.430 --> 00:02:51.670
times a displacement vector
equals some forcing function.
00:02:51.670 --> 00:02:53.570
So these are-- and
we linearize them.
00:02:53.570 --> 00:02:55.920
And we know they're
for vibration problems.
00:02:55.920 --> 00:02:59.410
We can solve these equations
using modal analysis.
00:02:59.410 --> 00:03:03.440
Or we can solve them
brute force directly
00:03:03.440 --> 00:03:09.350
without breaking the results
into their modal contributions.
00:03:09.350 --> 00:03:12.490
And so there really are
two kinds of approaches
00:03:12.490 --> 00:03:14.884
you can use to doing it.
00:03:14.884 --> 00:03:16.050
Any other kind of questions?
00:03:16.050 --> 00:03:16.550
Yeah.
00:03:16.550 --> 00:03:19.620
AUDIENCE: So for a
definition of mode shape,
00:03:19.620 --> 00:03:22.840
is mode shape just the
ratio of the two amplitudes?
00:03:22.840 --> 00:03:25.970
PROFESSOR: In the case of a
two degree of freedom system,
00:03:25.970 --> 00:03:42.010
so mode shapes for mode n of
an n degree of freedom system--
00:03:42.010 --> 00:03:44.860
like let's say n equals 3 here.
00:03:44.860 --> 00:03:53.730
So for mode n, the mode
shape you always have
00:03:53.730 --> 00:03:55.960
to present some normalization.
00:03:55.960 --> 00:03:58.420
I do the normalization
oftentimes by just saying, OK,
00:03:58.420 --> 00:04:00.420
I'm going to make the top
coordinate-- I'm going
00:04:00.420 --> 00:04:01.810
to give it unit amplitude.
00:04:01.810 --> 00:04:04.760
So I'm going to take a1
and divide it by itself.
00:04:04.760 --> 00:04:06.020
And that'll give me a 1.
00:04:06.020 --> 00:04:12.180
Then every other one, this
becomes a2 over a1, a3 over a1.
00:04:12.180 --> 00:04:23.570
The mode shape says if this
is then 1 minus 1/2 and a 1/4,
00:04:23.570 --> 00:04:28.270
it says that if
generalized coordinate
00:04:28.270 --> 00:04:33.120
1 moves a unit amount,
generalized coordinate 2
00:04:33.120 --> 00:04:40.050
will move half of that in
its negative direction.
00:04:40.050 --> 00:04:41.630
This could be an
x, a displacement,
00:04:41.630 --> 00:04:43.820
and that could be a rotation.
00:04:43.820 --> 00:04:47.130
But they all have positively
defined directions.
00:04:47.130 --> 00:04:51.150
So if there's a positive 1
at x1, there's a minus 1/2
00:04:51.150 --> 00:04:54.350
at generalized coordinate
2, and there's a plus 1/4 at
00:04:54.350 --> 00:04:55.870
generalized coordinate 3.
00:04:55.870 --> 00:05:00.570
And that ratio stays
constant for the mode.
00:05:00.570 --> 00:05:04.910
And so if it's an initial
condition problem,
00:05:04.910 --> 00:05:08.120
and you set it up so
that you actually give it
00:05:08.120 --> 00:05:12.110
an exhibit, just an
initial displacement,
00:05:12.110 --> 00:05:14.770
of exactly that shape, it'll
sit there and vibrate only
00:05:14.770 --> 00:05:15.900
in that mode.
00:05:15.900 --> 00:05:19.150
And you'll notice
that proportion,
00:05:19.150 --> 00:05:22.590
even as it dies out with
damping, of the first
00:05:22.590 --> 00:05:25.210
to the other two stays
exactly constant.
00:05:25.210 --> 00:05:26.560
That's what a mode is.
00:05:26.560 --> 00:05:28.210
And it's a character.
00:05:28.210 --> 00:05:32.440
It's a property of the system.
00:05:32.440 --> 00:05:34.050
The natural frequency
is a property.
00:05:34.050 --> 00:05:38.790
And the shape of every mode
is a property of that system.
00:05:38.790 --> 00:05:42.100
OK, anything else, questions?
00:05:42.100 --> 00:05:44.380
Next lecture, we'll do
more modal analysis.
00:05:44.380 --> 00:05:47.460
We did the response to
initial conditions yesterday.
00:05:47.460 --> 00:05:51.480
Tuesday, we'll talk about the
response to external forces.
00:05:51.480 --> 00:05:54.360
And that will give us a chance
to review it and post it
00:05:54.360 --> 00:05:55.700
on Stellar.
00:05:55.700 --> 00:05:57.970
Last night, I put up a
little two page sheet
00:05:57.970 --> 00:06:00.375
that is a cookbook,
how to do the procedure
00:06:00.375 --> 00:06:02.070
of a modal analysis.
00:06:02.070 --> 00:06:04.770
It's very cookbook, just step
by step, bang, bang, bang.
00:06:04.770 --> 00:06:07.837
And everything falls out.
00:06:07.837 --> 00:06:08.920
Any other questions, yeah?
00:06:08.920 --> 00:06:12.340
AUDIENCE: So is that where
we get the mode shape from?
00:06:12.340 --> 00:06:14.220
PROFESSOR: You
get the mode shape
00:06:14.220 --> 00:06:19.430
by solving the
characteristic-- if you're
00:06:19.430 --> 00:06:22.671
doing it by hand, by solving the
characteristic equation, which
00:06:22.671 --> 00:06:24.420
we're going to get
some practice at today.
00:06:24.420 --> 00:06:26.200
So I think this is
the exercise of today,
00:06:26.200 --> 00:06:28.725
is how to get natural
frequencies and mode shapes.
00:06:28.725 --> 00:06:32.256
AUDIENCE: Then you'd
be actually controlling
00:06:32.256 --> 00:06:36.594
the amount that-- isn't the
mode shape actually drawing
00:06:36.594 --> 00:06:38.827
some sort of graph?
00:06:38.827 --> 00:06:41.160
PROFESSOR: As a graph that
shows the displacement of it,
00:06:41.160 --> 00:06:42.970
or it shows the time history?
00:06:42.970 --> 00:06:45.761
AUDIENCE: Either, I guess.
00:06:45.761 --> 00:06:49.058
Because in the book,
they have mode shape.
00:06:49.058 --> 00:06:51.510
But actually using this,
they draw a certain graph.
00:06:51.510 --> 00:06:52.718
PROFESSOR: Sometimes they do.
00:06:52.718 --> 00:06:55.480
If it's easy, like for
a vibrating string,
00:06:55.480 --> 00:06:57.414
it's easy to draw
the mode shape.
00:06:57.414 --> 00:06:59.580
AUDIENCE: And the piece
that they ask us to do that.
00:06:59.580 --> 00:07:01.746
PROFESSOR: Yeah, and so
vibrating string, first mode
00:07:01.746 --> 00:07:03.562
vibration looks like
half a sign wave.
00:07:03.562 --> 00:07:05.270
The mode shape is just
half the sine wave
00:07:05.270 --> 00:07:08.280
going up and down in space.
00:07:08.280 --> 00:07:13.140
For this system, the mode shape
has two different amplitudes
00:07:13.140 --> 00:07:15.160
at these two bobs.
00:07:15.160 --> 00:07:18.190
And their relative motion we're
going to figure that out today.
00:07:18.190 --> 00:07:20.350
OK, we better get going
or we won't finish.
00:07:20.350 --> 00:07:23.430
So good questions.
00:07:23.430 --> 00:07:34.590
And here's this system
simplified, just into two
00:07:34.590 --> 00:07:37.410
bobs and massless strings.
00:07:37.410 --> 00:07:40.100
Here's the equations
of motion for that.
00:07:40.100 --> 00:07:41.680
They've been linearized already.
00:07:41.680 --> 00:07:43.970
There's no sine
thetas or anything.
00:07:43.970 --> 00:07:46.890
And part of the linearization--
so this assumes,
00:07:46.890 --> 00:07:50.050
then, small amplitudes, right?
00:07:50.050 --> 00:07:53.530
Not only did the sine--
the torque for a pendulum,
00:07:53.530 --> 00:07:56.460
you get this mgL
sine theta term.
00:07:56.460 --> 00:07:59.080
And so the linearization,
you say small angles,
00:07:59.080 --> 00:08:00.920
sine theta becomes theta.
00:08:00.920 --> 00:08:06.846
So that's where this g/L sine
theta has become theta term.
00:08:06.846 --> 00:08:08.220
There's another
assumption that's
00:08:08.220 --> 00:08:09.610
been made to model this system.
00:08:09.610 --> 00:08:11.892
We're essentially
modeling the system.
00:08:11.892 --> 00:08:13.350
And that is for
small angles, we're
00:08:13.350 --> 00:08:16.700
assuming the spring
remains horizontal.
00:08:16.700 --> 00:08:20.230
Because the spring puts
a force on the rod.
00:08:20.230 --> 00:08:22.440
And what you want is
torque about that point.
00:08:22.440 --> 00:08:24.150
These two equations
are the equations
00:08:24.150 --> 00:08:28.190
for torque about that
point and about that point.
00:08:28.190 --> 00:08:32.300
And the torque is the spring
force times that distance.
00:08:32.300 --> 00:08:35.020
And if they're perpendicular,
that's just one times
00:08:35.020 --> 00:08:35.520
the other.
00:08:35.520 --> 00:08:37.144
But if it doesn't
remain perpendicular,
00:08:37.144 --> 00:08:39.570
then you'd have
to take a cosine.
00:08:39.570 --> 00:08:41.260
And it gets really messy.
00:08:41.260 --> 00:08:43.419
So another part of this
small angle approximation
00:08:43.419 --> 00:08:47.010
is that spring stays horizontal.
00:08:47.010 --> 00:08:48.520
There's your equation of motion.
00:08:48.520 --> 00:08:52.270
And you can see what's
happened here is that this
00:08:52.270 --> 00:08:58.070
used to have an m1 L1 squared.
00:08:58.070 --> 00:09:01.050
This is the mass moment
of inertia of that mass.
00:09:01.050 --> 00:09:05.080
We've divided through the
equation by m1 L1 squared.
00:09:05.080 --> 00:09:08.100
And that's put things in
the bottom here like that.
00:09:08.100 --> 00:09:12.260
And it just makes this actually
a little easier to work with.
00:09:12.260 --> 00:09:17.590
The mass matrix turns out
to be 1, 1 when you do this.
00:09:17.590 --> 00:09:19.600
But it's still the same
two equations of motion.
00:09:19.600 --> 00:09:25.150
I've just divided through by m1
L squared, and this one by m2 L
00:09:25.150 --> 00:09:26.960
squared.
00:09:26.960 --> 00:09:29.130
So these are your
equations of motion.
00:09:29.130 --> 00:09:34.180
And your job is to
come up-- first of all,
00:09:34.180 --> 00:09:36.527
just put them in matrix form.
00:09:36.527 --> 00:09:38.110
And you're going to
do this in groups.
00:09:38.110 --> 00:09:46.200
Today we've got five,
10, and 18 people.
00:09:46.200 --> 00:09:48.740
So that's probably
four groups of four.
00:09:48.740 --> 00:09:51.730
Work together in groups quickly.
00:09:51.730 --> 00:09:54.040
Let's put up the
equations of motion
00:09:54.040 --> 00:09:56.536
from this in matrix form.
00:09:56.536 --> 00:10:00.022
AUDIENCE: [INAUDIBLE]
00:11:06.330 --> 00:11:09.200
PROFESSOR: In your
lower left matrix?
00:11:09.200 --> 00:11:14.020
Is the-- oh, the k.
00:11:14.020 --> 00:11:16.290
I'm worried about the m.
00:11:16.290 --> 00:11:19.290
AUDIENCE: The k's don't have
any symbols. it's just m2.
00:11:19.290 --> 00:11:21.453
PROFESSOR: There you go.
00:11:21.453 --> 00:11:22.974
Yeah, it's just one.
00:11:22.974 --> 00:11:23.765
There's a single k.
00:11:23.765 --> 00:11:25.920
There's only one spring,
so we just call it k.
00:11:53.860 --> 00:11:56.610
All right, you're all
looking pretty good to me.
00:11:56.610 --> 00:11:59.534
And the last hour, we
did this, and it went up,
00:11:59.534 --> 00:12:00.950
and I looked at
it, and I just had
00:12:00.950 --> 00:12:03.930
one of these moments of just,
uhh, cognitive dissonance.
00:12:03.930 --> 00:12:08.570
It just was, how can
that possibly be?
00:12:08.570 --> 00:12:16.000
And the reason was
this stiffness matrix.
00:12:16.000 --> 00:12:20.820
This matrix is not diagonal--
excuse me, wrong word.
00:12:20.820 --> 00:12:24.580
This matrix is not symmetric.
00:12:24.580 --> 00:12:29.740
And linear stiffness matrices
are always symmetric.
00:12:29.740 --> 00:12:32.690
So I saw this, and I
said, what has gone wrong?
00:12:35.670 --> 00:12:37.730
Every one of you
had the same answer.
00:12:37.730 --> 00:12:41.330
Every one of you, your stiffness
matrices are not symmetric.
00:12:41.330 --> 00:12:46.470
It's minus k over m1
and minus k over m2.
00:12:46.470 --> 00:12:49.260
And I about had heart
failure, and I've
00:12:49.260 --> 00:12:51.019
been doing this for 40 years.
00:12:51.019 --> 00:12:52.310
And it's always been symmetric.
00:12:52.310 --> 00:12:57.320
And all of a sudden, you guys
unanimously get asymmetric.
00:12:57.320 --> 00:13:02.020
So I figured it out
at great relief.
00:13:02.020 --> 00:13:08.380
We divided through by m1 L1
squared in the first equation,
00:13:08.380 --> 00:13:12.380
and m2 L2 squared in
the second equation.
00:13:12.380 --> 00:13:16.500
The stiffness matrix, the
true stiffness matrix,
00:13:16.500 --> 00:13:19.770
hasn't been divided
through by the mL squareds.
00:13:19.770 --> 00:13:22.762
And it is indeed symmetric.
00:13:22.762 --> 00:13:24.970
It's one of the things that's
really helpful to know.
00:13:24.970 --> 00:13:27.020
Because when you're
working these things out,
00:13:27.020 --> 00:13:27.940
you know they're symmetric.
00:13:27.940 --> 00:13:29.648
You don't have to
evaluate all the terms.
00:13:29.648 --> 00:13:32.710
You only have to do the
diagonals and half of the ones.
00:13:32.710 --> 00:13:34.700
The mass matrices are
usually symmetric, too.
00:13:34.700 --> 00:13:36.780
There's maybe some
exceptions if you
00:13:36.780 --> 00:13:38.780
start putting in gyroscopes
and stuff like that.
00:13:38.780 --> 00:13:42.260
But then it's not very linear.
00:13:42.260 --> 00:13:45.220
So your next task
is to-- the way
00:13:45.220 --> 00:13:47.500
we find natural
frequencies and mode
00:13:47.500 --> 00:13:51.020
shapes is kind of the
hard way, using algebra,
00:13:51.020 --> 00:13:54.730
is to find what we call the
characteristic equation.
00:13:54.730 --> 00:13:57.120
It turns out to
be a fourth order
00:13:57.120 --> 00:13:59.450
equation, omega to the fourth.
00:13:59.450 --> 00:14:00.850
Find it.
00:14:00.850 --> 00:14:03.470
Don't solve it, just write down
the characteristic equation
00:14:03.470 --> 00:14:05.917
for this thing.
00:14:05.917 --> 00:14:08.250
And as soon as you have that,
come up and write it down.
00:14:12.710 --> 00:14:14.530
So remember, you
basically have matrices.
00:14:14.530 --> 00:14:15.904
This is what you're
working with.
00:14:15.904 --> 00:14:17.720
You've got something
of this form.
00:14:17.720 --> 00:14:20.630
We've divided through by
some stuff to collect terms.
00:14:20.630 --> 00:14:24.880
But how do you go about going
from the equations of motion
00:14:24.880 --> 00:14:29.415
to your algebraic equation
in omega to the fourth?
00:14:41.070 --> 00:14:44.019
I just want you to write up
the characteristic equation.
00:14:44.019 --> 00:14:46.310
You're going to get a quadratic
in omega to the fourth.
00:14:46.310 --> 00:14:47.934
Just go up and write
down the equation.
00:14:52.060 --> 00:14:55.350
AUDIENCE: So do you
want us to expand?
00:14:55.350 --> 00:14:57.440
PROFESSOR: By expand,
what do you mean?
00:14:57.440 --> 00:15:00.230
AUDIENCE: [INAUDIBLE]
multiply this by this?
00:15:00.230 --> 00:15:02.560
But we could also
expand that out.
00:15:02.560 --> 00:15:04.640
PROFESSOR: Well, I want
an equation in omega
00:15:04.640 --> 00:15:05.900
to the fourth.
00:15:05.900 --> 00:15:07.100
Just write one up there.
00:15:07.100 --> 00:15:13.960
And if you want to simplify the
writing, you can let h equal--
00:15:13.960 --> 00:15:15.950
and let's see, I forgot
the tell you something,
00:15:15.950 --> 00:15:17.550
a key assumption
here, a key thing.
00:15:17.550 --> 00:15:19.880
I was going to make
it easier for you.
00:15:19.880 --> 00:15:20.680
Let--
00:15:20.680 --> 00:15:23.246
AUDIENCE: Are you going
to say the m's are equal?
00:15:23.246 --> 00:15:26.850
PROFESSOR: Yeah,
sorry, my mistake.
00:15:26.850 --> 00:15:27.790
Let the m's be equal.
00:15:27.790 --> 00:15:29.280
Then it makes it a lot easier.
00:15:29.280 --> 00:15:35.310
And then you can say,
call h g/L plus k/m.
00:15:35.310 --> 00:15:39.000
And it'll kind of make the
equation a whole lot easier
00:15:39.000 --> 00:15:39.590
to write.
00:15:43.076 --> 00:15:45.566
AUDIENCE: [INAUDIBLE]
00:16:36.880 --> 00:16:40.305
PROFESSOR: Great, OK, people
are getting all the same answers
00:16:40.305 --> 00:16:41.720
here, good.
00:16:41.720 --> 00:16:47.390
So the next step is I will
give you-- if you solve this,
00:16:47.390 --> 00:16:55.250
you get omega 1 is g/L.
Omega 1 squared is g/L.
00:16:55.250 --> 00:17:03.900
And omega 2 squared
is g/L plus 2k over m.
00:17:03.900 --> 00:17:05.810
Those are your two
natural frequencies.
00:17:05.810 --> 00:17:08.989
So now, find the mode shapes.
00:17:11.650 --> 00:17:13.000
Find the mode shapes.
00:17:13.000 --> 00:17:15.335
If you know the natural
frequencies, now you go back in
00:17:15.335 --> 00:17:19.740
and you get a-- pardon?
00:17:19.740 --> 00:17:21.450
Yeah, keep h.
00:17:21.450 --> 00:17:26.670
I think h will-- things will
fall out rather quickly.
00:17:26.670 --> 00:17:31.410
Well, I like to
normalize the mode shapes
00:17:31.410 --> 00:17:34.290
so the top one is 1.
00:17:34.290 --> 00:17:36.730
And so the top element
of the mode shape
00:17:36.730 --> 00:17:41.000
vector I call a1, or
u1, down through un.
00:17:41.000 --> 00:17:44.150
And so if you divide out
each one by the top one,
00:17:44.150 --> 00:17:45.620
the top one becomes 1.
00:17:45.620 --> 00:17:47.570
The second one
becomes a2 over a1.
00:17:47.570 --> 00:17:49.570
The third one, if it's
three degrees of freedom,
00:17:49.570 --> 00:17:50.620
would be a3 over a1.
00:17:53.480 --> 00:17:58.350
The fact that this-- you
just solve this equation.
00:17:58.350 --> 00:18:00.860
To get that characteristic
determinant,
00:18:00.860 --> 00:18:04.250
you said this was true.
00:18:04.250 --> 00:18:07.850
And that's a particular
kind of situation
00:18:07.850 --> 00:18:11.530
where you have a set
of linear homogeneous
00:18:11.530 --> 00:18:13.336
equations equal to 0.
00:18:13.336 --> 00:18:14.335
They're not independent.
00:18:14.335 --> 00:18:17.550
Remember from algebra when
you have little equations
00:18:17.550 --> 00:18:22.115
with constant coefficients, and
they're equal to a constant,
00:18:22.115 --> 00:18:26.090
and the constants are
all 0, means they are not
00:18:26.090 --> 00:18:27.230
independent equations.
00:18:27.230 --> 00:18:29.670
So this is two equations
and two unknowns.
00:18:29.670 --> 00:18:31.350
And they're not independent.
00:18:31.350 --> 00:18:34.040
You won't be able to solve
for unique values of a1, u1,
00:18:34.040 --> 00:18:34.710
and u2.
00:18:34.710 --> 00:18:37.840
You'd only be able
to get the ratio.
00:18:37.840 --> 00:18:41.700
So basically you
found this value.
00:18:41.700 --> 00:18:44.190
You're going to plug in
a value for omega squared
00:18:44.190 --> 00:18:48.310
and find out the values
of a1 and a2 that work.
00:18:48.310 --> 00:18:49.650
That's what the mode shapes are.
00:18:52.800 --> 00:18:55.970
Let me pull you back
together here, and let's
00:18:55.970 --> 00:18:58.410
run through this
kind of quickly.
00:18:58.410 --> 00:19:01.860
You've basically
solved this equation.
00:19:01.860 --> 00:19:04.550
You've said, I'm going
to assume this thing has
00:19:04.550 --> 00:19:10.320
a mode shape and a
frequency, a harmonic-- it's
00:19:10.320 --> 00:19:11.670
going to vibrate.
00:19:11.670 --> 00:19:12.270
It's undamped.
00:19:12.270 --> 00:19:15.550
So either you could write this
as cosine omega t, sine omega
00:19:15.550 --> 00:19:17.220
t, e to the i omega t.
00:19:17.220 --> 00:19:20.310
But it has some constants
out here called the shape.
00:19:20.310 --> 00:19:26.670
You plug that into the
equations of motion,
00:19:26.670 --> 00:19:28.805
you're going to get
back this expression.
00:19:28.805 --> 00:19:31.080
The two derivatives,
the theta double dots,
00:19:31.080 --> 00:19:33.720
give you the minus
omega squareds.
00:19:33.720 --> 00:19:35.440
And you can write it like that.
00:19:35.440 --> 00:19:38.950
You can throw away the
e to the i omega t.
00:19:38.950 --> 00:19:43.120
In this case, I've expand this.
m is just that 1, 1 matrix.
00:19:43.120 --> 00:19:46.890
So this part looks like this.
00:19:46.890 --> 00:19:49.380
The k matrix looks like this.
00:19:49.380 --> 00:19:51.730
You can add them
element by element.
00:19:51.730 --> 00:19:57.440
So this is minus omega
squared plus h, minus k/m.
00:19:57.440 --> 00:20:00.770
This one is minus k/m, and
this term is minus omega
00:20:00.770 --> 00:20:05.100
squared plus h again times
u1 or u2, the two elements
00:20:05.100 --> 00:20:07.520
that we're looking for.
00:20:07.520 --> 00:20:08.920
That's two equations.
00:20:08.920 --> 00:20:10.840
This is just now an
algebraic equation,
00:20:10.840 --> 00:20:16.840
two algebraic equations-- this
time u1 minus k/m times u2.
00:20:16.840 --> 00:20:20.000
And the second equation is that.
00:20:20.000 --> 00:20:22.180
And we know there's only
two equations, and not
00:20:22.180 --> 00:20:23.280
linearly independent.
00:20:23.280 --> 00:20:25.450
So you actually only
have one useful equation.
00:20:25.450 --> 00:20:29.360
If this is a three by three,
they're not independent.
00:20:29.360 --> 00:20:33.441
But you need to use
two out of the three.
00:20:33.441 --> 00:20:33.940
Yeah.
00:20:33.940 --> 00:20:35.370
AUDIENCE: How do you
get the second equation?
00:20:35.370 --> 00:20:37.747
PROFESSOR: The second
one, I take the first one,
00:20:37.747 --> 00:20:38.830
and I multiply it up here.
00:20:38.830 --> 00:20:42.060
And that gives me an equation,
this times u1 plus this times
00:20:42.060 --> 00:20:42.630
u2.
00:20:42.630 --> 00:20:43.960
That's equation one.
00:20:43.960 --> 00:20:47.320
I take this, and I
multiply it by that.
00:20:47.320 --> 00:20:52.630
And I get minus k/m u2.
00:20:52.630 --> 00:21:00.300
And this is u1.
00:21:00.300 --> 00:21:01.410
This is the omega squared.
00:21:01.410 --> 00:21:02.990
I plugged in the
natural frequency.
00:21:02.990 --> 00:21:04.230
So we solved for it.
00:21:04.230 --> 00:21:06.840
AUDIENCE: Oh, OK, OK.
00:21:06.840 --> 00:21:09.690
But for the top one, you didn't
plug in the natural frequency.
00:21:09.690 --> 00:21:11.513
PROFESSOR: Oh,
minus omega squared.
00:21:14.110 --> 00:21:17.730
OK, I'm picking one of the
natural frequencies, g/L.
00:21:17.730 --> 00:21:24.530
So this up here is g/L
plus h u1, that one.
00:21:24.530 --> 00:21:26.850
And down here it's g/L h u2.
00:21:26.850 --> 00:21:29.785
And you see, these two
equations turn out to be--
00:21:29.785 --> 00:21:31.790
AUDIENCE: So this
is just for omega 1?
00:21:31.790 --> 00:21:33.830
PROFESSOR: Yeah, you
only do it one at a time.
00:21:33.830 --> 00:21:36.900
You just plug in one of
the natural frequencies.
00:21:36.900 --> 00:21:38.692
And you get two equations
and two unknowns.
00:21:38.692 --> 00:21:40.108
And they're going
to each give you
00:21:40.108 --> 00:21:41.730
exactly the same information.
00:21:41.730 --> 00:21:43.860
They're not
independent any longer.
00:21:43.860 --> 00:21:45.215
So you solve this one.
00:21:48.560 --> 00:21:50.560
h has what in it?
00:21:50.560 --> 00:21:56.860
A plus g/L and a plus k/m.
00:21:56.860 --> 00:21:58.820
The g/L is canceled.
00:21:58.820 --> 00:22:06.910
And you're left with k/m
u1 minus k/m u2 equals 0.
00:22:06.910 --> 00:22:11.390
And this implies
that u1 equals u2.
00:22:17.550 --> 00:22:19.610
This is the g/L in this.
00:22:19.610 --> 00:22:22.620
I'm going to plug
in the-- that is h.
00:22:22.620 --> 00:22:23.810
I just put it in here.
00:22:23.810 --> 00:22:26.140
I've plugged in one of the
natural frequencies squared.
00:22:26.140 --> 00:22:28.360
The g/L parts cancel.
00:22:28.360 --> 00:22:32.610
This is plus k/m
u1 minus k/m u2.
00:22:32.610 --> 00:22:36.710
That means k/m u1 equals k/m u2.
00:22:36.710 --> 00:22:38.190
Cancel out the k/m's.
00:22:38.190 --> 00:22:38.735
I get that.
00:22:38.735 --> 00:22:41.370
It just tells me that
that's the answer.
00:22:41.370 --> 00:22:46.400
And if that's the answer,
the vector looks like u1, u2.
00:22:46.400 --> 00:22:52.790
That's also equal to-- factor
out a u1-- u1 times 1 and u2
00:22:52.790 --> 00:22:54.780
over u1.
00:22:54.780 --> 00:22:59.710
That's the mode shape
there, 1 and u2 over u1.
00:22:59.710 --> 00:23:02.820
That's how you factor it out to
put it in the normalized form.
00:23:02.820 --> 00:23:04.970
You just take whatever's
in the top one and divide
00:23:04.970 --> 00:23:06.530
everything in the column by it.
00:23:10.700 --> 00:23:13.220
So to do the second
equation, you just
00:23:13.220 --> 00:23:17.600
now go back to this, plug
in for omega 2 squared.
00:23:17.600 --> 00:23:23.140
The second natural frequency
is g/L plus 2k over m.
00:23:23.140 --> 00:23:29.200
This is g/L minus k/m.
00:23:29.200 --> 00:23:32.850
The k over m's cancel.
00:23:32.850 --> 00:23:35.322
And you're just left with g/L's.
00:23:35.322 --> 00:23:36.780
You work this
through, you're going
00:23:36.780 --> 00:23:41.260
to find out that u2
equals minus u1 if you
00:23:41.260 --> 00:23:42.240
put in the second one.
00:23:46.180 --> 00:23:47.298
Yeah.
00:23:47.298 --> 00:23:54.110
AUDIENCE: So we have a1 over a2,
b1 over b2, being equal to 1,
00:23:54.110 --> 00:23:54.610
obviously.
00:23:54.610 --> 00:23:56.400
Because b1 is equal to b2.
00:23:56.400 --> 00:23:57.882
But I don't
understand the format
00:23:57.882 --> 00:23:59.215
that you're writing it up there.
00:23:59.215 --> 00:24:03.145
Because if you plug back in the
u, then you just get u1 over u2
00:24:03.145 --> 00:24:04.540
is equal to u1 over u2.
00:24:04.540 --> 00:24:06.205
And that's
self-explanatory, right?
00:24:06.205 --> 00:24:07.830
PROFESSOR: Well,
when you solve this,
00:24:07.830 --> 00:24:13.520
you find that k/m
u1 equals k/m u2.
00:24:13.520 --> 00:24:16.625
That's just saying in this
case they are equal, period.
00:24:16.625 --> 00:24:18.730
And that's all you learn.
00:24:18.730 --> 00:24:20.074
You only have two equations.
00:24:20.074 --> 00:24:21.240
And they're not independent.
00:24:21.240 --> 00:24:23.300
So it means you only
have one equation.
00:24:23.300 --> 00:24:25.280
And you don't learn--
you're not able to solve
00:24:25.280 --> 00:24:27.550
for numeric values of u1 and u2.
00:24:27.550 --> 00:24:29.860
At best, you can get u1 over u2.
00:24:29.860 --> 00:24:32.300
AUDIENCE: Yeah, because
they're equal to [INAUDIBLE].
00:24:32.300 --> 00:24:36.330
PROFESSOR: Right, and in fact,
you can just solve this for u2
00:24:36.330 --> 00:24:38.261
over u1, and it would be 1.
00:24:38.261 --> 00:24:40.094
AUDIENCE: I just don't
understand the format
00:24:40.094 --> 00:24:41.500
you're writing it in up there.
00:24:41.500 --> 00:24:44.100
PROFESSOR: Well, that's
back in my vector format.
00:24:44.100 --> 00:24:47.390
I want to write the
mode shape as a vector.
00:24:47.390 --> 00:24:50.210
You solve for u1.
00:24:50.210 --> 00:24:51.817
So you can actually do this.
00:24:51.817 --> 00:24:52.900
You know that that's true.
00:24:52.900 --> 00:24:54.485
You could say, well,
the top one is 1,
00:24:54.485 --> 00:24:56.110
and the bottom one
is the same as that.
00:24:56.110 --> 00:24:58.205
So the mode shape
for mode one is that.
00:25:03.295 --> 00:25:07.440
AUDIENCE: In the
answers, it says omega 1,
00:25:07.440 --> 00:25:09.340
and the ratio of a1 and a2 is 1.
00:25:09.340 --> 00:25:12.359
What are the a's?
00:25:12.359 --> 00:25:13.400
PROFESSOR: I'm using u's.
00:25:13.400 --> 00:25:14.870
And they used a's.
00:25:14.870 --> 00:25:17.000
They're the elements of
the mode shape vector.
00:25:20.600 --> 00:25:23.880
So if we had put in the other
natural frequency instead
00:25:23.880 --> 00:25:37.770
of-- I may have just done
too many steps here at once.
00:25:47.000 --> 00:25:50.650
So our first equation
looks like minus omega
00:25:50.650 --> 00:25:57.620
squared plus h minus k/m.
00:26:02.280 --> 00:26:07.010
And the second
equation is minus k/m.
00:26:07.010 --> 00:26:13.810
And over here is minus
omega squared plus h again.
00:26:13.810 --> 00:26:15.020
That's our matrix.
00:26:15.020 --> 00:26:17.850
When we add these two
matrices together,
00:26:17.850 --> 00:26:20.100
the minus omega
squared m plus k,
00:26:20.100 --> 00:26:23.300
that's what it looks like when
you add elements together.
00:26:23.300 --> 00:26:26.340
And these are two
equations and two unknowns.
00:26:26.340 --> 00:26:31.500
We're looking for the answer for
some u1, u2 multiplied by this.
00:26:31.500 --> 00:26:33.220
So you multiply this out.
00:26:33.220 --> 00:26:42.690
You get minus omega squared
plus h u1 minus k/m u2.
00:26:42.690 --> 00:26:46.110
And now let's plug in the
second natural frequency.
00:26:46.110 --> 00:26:50.790
So omega 2 squared is that.
00:26:50.790 --> 00:26:52.650
And solve for the
second mode now.
00:26:52.650 --> 00:26:54.530
Plug it in here.
00:26:54.530 --> 00:27:02.520
We get minus g/L minus
2k over m plus h,
00:27:02.520 --> 00:27:13.945
which is g/L, plus k/m plus, and
that's times u1, plus k/m u2.
00:27:17.230 --> 00:27:19.920
All it's equal to 0.
00:27:19.920 --> 00:27:24.425
This cancels this, this,
and this, plus 1 minus 2.
00:27:24.425 --> 00:27:28.558
I get a minus k/m out of this.
00:27:28.558 --> 00:27:40.156
So I have this whole thing gives
me minus k/m u1 plus k/m u2.
00:27:43.407 --> 00:27:45.073
AUDIENCE: It's supposed
to be minus k/m.
00:27:48.264 --> 00:27:49.430
PROFESSOR: I agree with you.
00:27:49.430 --> 00:27:50.680
Where did I make the mistake?
00:27:50.680 --> 00:27:53.795
AUDIENCE: The very first
line, it's minus k/m u2.
00:27:53.795 --> 00:27:56.680
PROFESSOR: Ah, good, all right.
00:28:01.440 --> 00:28:04.070
And this is equal to 0.
00:28:04.070 --> 00:28:06.170
The k over m's cancel out.
00:28:06.170 --> 00:28:13.570
And you find that this implies
that u1 equals minus u2.
00:28:13.570 --> 00:28:20.790
And so the mode shape for mode
two you could write as u1.
00:28:20.790 --> 00:28:24.490
And u2 is minus u1, if you will.
00:28:24.490 --> 00:28:26.370
And it's obvious
if you factor out
00:28:26.370 --> 00:28:32.690
u1 you get u1 and
a 1 minus 1, right?
00:28:32.690 --> 00:28:35.970
So now you've solved for
the second mode shape.
00:28:35.970 --> 00:28:37.820
If this had been more
than a two by two,
00:28:37.820 --> 00:28:43.990
like a three degree
of freedom system,
00:28:43.990 --> 00:28:46.740
you'd have to use two
of the three equations
00:28:46.740 --> 00:28:48.250
to get the ratio.
00:28:48.250 --> 00:28:51.090
You'd get two equations
and three unknowns.
00:28:51.090 --> 00:28:54.650
And at best, you can
find the two of them
00:28:54.650 --> 00:28:56.010
in terms of the third.
00:28:56.010 --> 00:28:57.400
And the third one would be u1.
00:28:57.400 --> 00:29:00.790
You'd just divide
all the others by u1.
00:29:00.790 --> 00:29:02.110
So you only get their ratios.
00:29:02.110 --> 00:29:06.890
OK, so those are the
natural frequencies.
00:29:06.890 --> 00:29:07.985
Those are the mode shapes.
00:29:10.640 --> 00:29:11.682
Let's demonstrate it.
00:29:11.682 --> 00:29:12.390
Let's look at it.
00:29:12.390 --> 00:29:14.450
This is basically the system.
00:29:14.450 --> 00:29:19.880
And you now know the
mode shapes, the two mode
00:29:19.880 --> 00:29:26.600
shapes of this thing, are 1,
1 for the first mode and 1,
00:29:26.600 --> 00:29:29.080
minus 1 for the second mode.
00:29:31.820 --> 00:29:36.550
And the modal expansion
theorem depends on this fact
00:29:36.550 --> 00:29:45.790
that the mode shapes form
a complete independent set
00:29:45.790 --> 00:29:49.780
of vectors that are
orthogonal to one another.
00:29:49.780 --> 00:29:53.630
A weighted sum of all the
mode shades of the system
00:29:53.630 --> 00:29:58.130
can represent any allowable
motion of the system.
00:29:58.130 --> 00:29:59.570
Any possible motion
of the system
00:29:59.570 --> 00:30:04.430
can be written as a weight
of-- so any way I can displace
00:30:04.430 --> 00:30:07.160
these things, a little bit
here, a little bit there,
00:30:07.160 --> 00:30:10.860
I can write that position as a
weighted sum of those two mode
00:30:10.860 --> 00:30:12.420
shapes.
00:30:12.420 --> 00:30:16.290
So let's say I want to
have initial conditions
00:30:16.290 --> 00:30:22.240
on displacement, on r
theta, 1 0 and theta
00:30:22.240 --> 00:30:25.620
2 0, no initial velocities,
just initial angles.
00:30:25.620 --> 00:30:33.960
And I want that initial
condition vector to be 1, 0.
00:30:33.960 --> 00:30:38.510
I'm claiming I can write
that as a sum of something 1,
00:30:38.510 --> 00:30:43.000
1 plus another
something 1, minus 1.
00:30:43.000 --> 00:30:45.940
So what do c1 and c2 have
to be to give me that?
00:30:49.209 --> 00:30:53.060
You ought to be able to kind of
do that by inspection, almost.
00:30:53.060 --> 00:30:54.670
For example, just
try them equal.
00:30:54.670 --> 00:30:56.975
And what happens?
00:30:56.975 --> 00:30:57.850
AUDIENCE: [INAUDIBLE]
00:31:03.140 --> 00:31:05.530
PROFESSOR: So 1/2
and 1/2, right?
00:31:05.530 --> 00:31:14.210
Great, so to satisfy this, it's
1/2 1, 1 plus 1/2 1, minus 1.
00:31:14.210 --> 00:31:15.760
And I've just
illustrated the fact
00:31:15.760 --> 00:31:18.020
that you can represent
an arbitrary deflection,
00:31:18.020 --> 00:31:20.100
allowable deflection
of the system,
00:31:20.100 --> 00:31:23.200
as a weighted sum of
the two mode shapes.
00:31:23.200 --> 00:31:30.310
So that says if I make
the initial-- now,
00:31:30.310 --> 00:31:35.435
this says that theta 1
is 1, and theta 2 is 0.
00:31:35.435 --> 00:31:38.440
And what it's telling
us is by modal analysis,
00:31:38.440 --> 00:31:40.720
that means the
resulting motion should
00:31:40.720 --> 00:31:44.230
look like the response
to initial conditions
00:31:44.230 --> 00:31:47.780
where I have equal amounts
of each of those two modes.
00:31:47.780 --> 00:31:50.110
So there ought to be some
vibration at omega 1,
00:31:50.110 --> 00:31:54.340
and there ought to be equal
amount of vibration at omega 2.
00:31:54.340 --> 00:31:56.610
Agreed?
00:31:56.610 --> 00:32:00.030
OK, let's try it.
00:32:00.030 --> 00:32:04.290
I'll hold one of these in place,
and I'll deflect the other one
00:32:04.290 --> 00:32:05.250
and let go.
00:32:12.695 --> 00:32:16.710
Now, that one's stationary,
and this one's moving.
00:32:16.710 --> 00:32:19.100
And a few cycles later,
this one will be stationary,
00:32:19.100 --> 00:32:21.860
and that one will be moving.
00:32:21.860 --> 00:32:24.140
So what you're observing,
could that motion,
00:32:24.140 --> 00:32:27.050
what you're seeing,
possibly be a mode,
00:32:27.050 --> 00:32:29.720
a natural mode by itself?
00:32:29.720 --> 00:32:31.380
No, because the
different proportions
00:32:31.380 --> 00:32:32.940
are changing, right?
00:32:32.940 --> 00:32:35.620
This is the sum of
two modes-- exactly
00:32:35.620 --> 00:32:37.880
as you said, equal
amounts of each.
00:32:37.880 --> 00:32:40.100
And when you do
that, and you have
00:32:40.100 --> 00:32:45.540
two things of equal amplitude,
two cosine omega t like terms
00:32:45.540 --> 00:32:48.860
of equal amplitude and
different frequencies,
00:32:48.860 --> 00:32:51.410
you get a phenomenon
known as beating.
00:32:51.410 --> 00:32:53.910
That's what this is.
00:32:53.910 --> 00:32:58.094
Beating means that one vibrates,
and then it will get small.
00:32:58.094 --> 00:33:00.510
And then you'll see it build
up again, and then get small.
00:33:00.510 --> 00:33:02.190
So if you watch
either one of these,
00:33:02.190 --> 00:33:06.360
it's doing large and stopping.
00:33:06.360 --> 00:33:08.700
And the other one is doing
the same thing, but actually
00:33:08.700 --> 00:33:11.410
90 degrees out of phase.
00:33:11.410 --> 00:33:21.870
So what we're really seeing is
that the motion of this system,
00:33:21.870 --> 00:33:36.240
this is 1, 1, 1/2 and 1, 1
cosine omega 1 t plus a 1/2 1,
00:33:36.240 --> 00:33:41.550
minus 1 cosine omega 2 t.
00:33:41.550 --> 00:33:44.710
That's the total
response of the system.
00:33:44.710 --> 00:33:46.670
Let's look at the
first line of this.
00:33:46.670 --> 00:33:56.590
This says 1/2 cosine omega
1 t plus 1/2 cosine omega
00:33:56.590 --> 00:34:03.270
2 t should be equal to the
motion we call theta 1 of t.
00:34:03.270 --> 00:34:05.220
That's the first row here.
00:34:05.220 --> 00:34:09.250
This is theta 1, theta 2.
00:34:09.250 --> 00:34:13.460
So the first row of this,
the first equation, is that.
00:34:13.460 --> 00:34:17.630
And two equal amplitude cosines
of different frequencies
00:34:17.630 --> 00:34:24.690
you can write as cosine
omega 2 minus omega 1
00:34:24.690 --> 00:34:36.790
over 2 times t times cosine
omega 1 plus omega 2 over 2 t.
00:34:40.989 --> 00:34:50.300
This beat phenomena that you're
seeing when you plot this--
00:34:50.300 --> 00:34:54.300
and this is now for theta 1.
00:34:54.300 --> 00:34:55.990
It starts off at some amplitude.
00:35:04.680 --> 00:35:07.910
And the actual motion you
see, if you watch it, it
00:35:07.910 --> 00:35:11.420
does what I'm drawing right now.
00:35:14.420 --> 00:35:19.510
This envelope is this term,
the difference frequency
00:35:19.510 --> 00:35:21.700
divided by 2.
00:35:21.700 --> 00:35:26.340
What's inside is that term.
00:35:26.340 --> 00:35:28.490
And this is called beating.
00:35:28.490 --> 00:35:29.920
This is the equation
for beating.
00:35:29.920 --> 00:35:33.755
When you add two equal
amplitude cosines together,
00:35:33.755 --> 00:35:35.630
they give you something
that looks like that.
00:35:35.630 --> 00:35:38.530
And one period of
the beat is how long
00:35:38.530 --> 00:35:41.430
it takes to go through one
full cycle from here to here.
00:35:41.430 --> 00:35:45.010
That's one period of the beat.
00:35:45.010 --> 00:35:45.654
Yeah.
00:35:45.654 --> 00:35:47.195
AUDIENCE: How did
you get those terms
00:35:47.195 --> 00:35:49.800
with the omega 2 [INAUDIBLE]?
00:35:49.800 --> 00:35:52.320
PROFESSOR: Well, that's
just trig identity.
00:35:52.320 --> 00:35:54.740
You add-- you just
go back to your trig,
00:35:54.740 --> 00:35:58.270
take cosine of a
plus cosine of b.
00:35:58.270 --> 00:36:00.750
You'll find out you can do that.
00:36:00.750 --> 00:36:03.550
I don't have time
to go through that.
00:36:03.550 --> 00:36:08.870
And the other one is
a 1/2 cosine omega 1
00:36:08.870 --> 00:36:14.510
t minus 1/2 cosine omega 2 t.
00:36:14.510 --> 00:36:20.920
And that turns out to be
sine omega 2 minus omega 1
00:36:20.920 --> 00:36:32.060
over 2 t times sine omega 1 plus
omega 2 over 2 quantity times
00:36:32.060 --> 00:36:32.870
time.
00:36:32.870 --> 00:36:35.960
And that means that
the theta 2 coordinate,
00:36:35.960 --> 00:36:41.140
what it looks like when you draw
it, is 90 degrees out of phase.
00:36:41.140 --> 00:36:47.530
It's the same thing, but it
starts at 0 here and beats.
00:36:47.530 --> 00:36:50.620
But it starts 0 where
that one started here.
00:36:50.620 --> 00:36:52.730
And that's what--
if you look at one.
00:36:55.475 --> 00:37:01.900
So if that's theta 1 and this
is theta 2, when one of them
00:37:01.900 --> 00:37:06.629
stops, that's one of these
lines, one of these equations.
00:37:06.629 --> 00:37:08.170
And the other equation
is 90 degrees.
00:37:08.170 --> 00:37:11.250
So that one's stopped right now.
00:37:11.250 --> 00:37:16.390
Pi over 2 later, this
one will be stopped.
00:37:16.390 --> 00:37:20.990
So the beat behaves
like sine in one case,
00:37:20.990 --> 00:37:23.780
and behaves like cosine
in the other case.
00:37:26.740 --> 00:37:29.798
One of these needs a minus sign.
00:37:29.798 --> 00:37:33.458
That's a plus, minus and
plus, minus and plus.
00:37:36.552 --> 00:37:39.590
Now what if you have
unequal amounts?
00:37:39.590 --> 00:37:45.810
We have exactly equal
amounts of the two modes.
00:37:45.810 --> 00:37:47.760
If you have unequal
amounts of the two modes,
00:37:47.760 --> 00:37:50.040
then it's not going
to be 1/2 and 1/2.
00:37:50.040 --> 00:37:53.310
What if it's like 1 and 0.1?
00:37:53.310 --> 00:37:55.020
Then do you see full beats?
00:37:55.020 --> 00:37:57.660
Does one come to a stop?
00:37:57.660 --> 00:37:59.710
If they're not
equal, you'll find
00:37:59.710 --> 00:38:03.830
that the sums of the two
motions, the envelope
00:38:03.830 --> 00:38:04.850
will look like this.
00:38:10.550 --> 00:38:12.820
It'll be modulated.
00:38:12.820 --> 00:38:14.450
Each one's motion
will look like that.
00:38:14.450 --> 00:38:15.990
It'll never go to 0 quite.
00:38:18.780 --> 00:38:20.596
But that's beating.
00:38:20.596 --> 00:38:21.885
How are we doing on time?
00:38:21.885 --> 00:38:23.480
We're good.
00:38:23.480 --> 00:38:29.830
And let's see, we said
the one natural mode is
00:38:29.830 --> 00:38:31.530
the two opposite one another.
00:38:31.530 --> 00:38:34.890
So if I start, go equal
amounts in opposite directions
00:38:34.890 --> 00:38:37.880
and let go, it ought to
just sit there and vibrate
00:38:37.880 --> 00:38:41.470
all day like that-- no beats.
00:38:41.470 --> 00:38:46.511
Because it's the 1,
0 case over there.
00:38:46.511 --> 00:38:48.010
And actually, this
is the 0, 1 case.
00:38:48.010 --> 00:38:49.690
This is 1, minus 1.
00:38:49.690 --> 00:38:53.340
I gave an initial displacement
in exactly the shape
00:38:53.340 --> 00:38:54.630
of the second mode.
00:38:54.630 --> 00:38:57.980
And that means the
two contributions
00:38:57.980 --> 00:39:00.630
are that one of
those constants is 0.
00:39:00.630 --> 00:39:02.220
There's no mode one in this.
00:39:02.220 --> 00:39:03.370
It's only mode two.
00:39:03.370 --> 00:39:06.080
And it'll sit here all day long
and vibrate just in mode two.
00:39:06.080 --> 00:39:10.570
And if I displace it in
the shape of mode one only,
00:39:10.570 --> 00:39:11.320
it'll vibrate.
00:39:11.320 --> 00:39:12.361
And that one's harder do.
00:39:12.361 --> 00:39:16.800
Because I have to move it
exactly the same amount.
00:39:16.800 --> 00:39:18.960
That's mode one.
00:39:18.960 --> 00:39:20.980
And it should vibrate
all day long on mode one.
00:39:20.980 --> 00:39:23.886
Because now there's no mode
two involved in that one.
00:39:23.886 --> 00:39:26.010
You see it's already getting
a little out of phase.
00:39:26.010 --> 00:39:29.640
It's hard for me to move my
hands exactly the same amount.
00:39:29.640 --> 00:39:34.530
So I have a little bit of
the other mode in there.
00:39:34.530 --> 00:39:37.810
But you'll never see either one
of these come to a full stop.
00:39:37.810 --> 00:39:42.454
It's actually doing that when
you get a little contamination
00:39:42.454 --> 00:39:43.620
of the second mode in there.
00:39:49.100 --> 00:39:53.730
OK, so I've posted on Stellar
a little two-page sheet
00:39:53.730 --> 00:39:57.375
that just gives you step by
step how to do a modal analysis.
00:39:57.375 --> 00:39:58.680
It's just cookbook.
00:39:58.680 --> 00:39:59.661
Modal analysis is easy.
00:39:59.661 --> 00:40:00.910
You do it all on the computer.
00:40:00.910 --> 00:40:04.020
You just put your m
matrix and your k matrix,
00:40:04.020 --> 00:40:05.625
and you just crank stuff out.
00:40:08.640 --> 00:40:10.760
We did multimodal
analysis yesterday
00:40:10.760 --> 00:40:12.230
just from initial conditions.
00:40:12.230 --> 00:40:13.890
Next Tuesday, we'll
do modal analysis
00:40:13.890 --> 00:40:16.890
assuming you've got harmonic
excitation and steady state
00:40:16.890 --> 00:40:20.600
vibration and do
that kind of thing.