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PROFESSOR: This week is my
second pair of lectures.
00:00:24.420 --> 00:00:28.540
Last week the two lectures were
about first order differential
00:00:28.540 --> 00:00:32.340
equations, and this
week second order.
00:00:32.340 --> 00:00:38.360
Those are the two big topics
in differential equations.
00:00:38.360 --> 00:00:46.020
Let me start with most
basic second order equation.
00:00:46.020 --> 00:00:51.330
We see the second derivative
and the function itself,
00:00:51.330 --> 00:00:57.740
and we don't see yet the
first derivative term.
00:00:57.740 --> 00:01:06.030
This is the nice case, when I
just have y double prime and y.
00:01:06.030 --> 00:01:10.870
In general, I-- I'm taking
constant coefficients today.
00:01:10.870 --> 00:01:16.150
Because if the coefficients
depend on time,
00:01:16.150 --> 00:01:18.930
the problem gets
much, much harder now.
00:01:18.930 --> 00:01:21.560
So let's stay with
constant coefficients,
00:01:21.560 --> 00:01:28.640
meaning we have a mass, for
example, we have a spring.
00:01:28.640 --> 00:01:32.700
The stiffness of the
spring is k, the mass is m,
00:01:32.700 --> 00:01:42.310
and the y, the
unknown displacement,
00:01:42.310 --> 00:01:44.910
tells us the
movement of the mass.
00:01:44.910 --> 00:01:46.170
The classical problem.
00:01:46.170 --> 00:01:47.630
You will have seen it before.
00:01:50.240 --> 00:01:52.550
Because you have an
exam this afternoon,
00:01:52.550 --> 00:01:55.580
I wanted to start with
things that-- they
00:01:55.580 --> 00:01:57.590
are about second
order equations,
00:01:57.590 --> 00:02:03.550
but they're still close to
the exam idea, particularly
00:02:03.550 --> 00:02:07.670
the idea of exponentials.
00:02:07.670 --> 00:02:11.870
With constant coefficients,
that's the fundamental message.
00:02:11.870 --> 00:02:15.400
Exponentials in,
exponentials out.
00:02:15.400 --> 00:02:20.360
But it's not quite
so clear when we
00:02:20.360 --> 00:02:24.110
had first order,
y prime equal ay,
00:02:24.110 --> 00:02:28.630
we knew that the exponent was a.
00:02:28.630 --> 00:02:31.746
The solution was e to the at.
00:02:31.746 --> 00:02:36.580
Now we've got second
derivatives coming in,
00:02:36.580 --> 00:02:41.400
and it won't be so much
e to the at type thing.
00:02:41.400 --> 00:02:47.850
Either the at was growth for a
positive, decay for a negative.
00:02:47.850 --> 00:02:51.580
Now we're going to
see oscillation.
00:02:51.580 --> 00:02:55.250
It's still exponentials,
but oscillation.
00:02:55.250 --> 00:02:59.450
Things going up and down,
things going around.
00:02:59.450 --> 00:03:01.430
Harmonic motion, you call it.
00:03:01.430 --> 00:03:04.230
Sines and cosines.
00:03:04.230 --> 00:03:10.130
And sines and cosines connect
to complex exponentials.
00:03:10.130 --> 00:03:14.500
So that instead of e to the
at-- so now oscillations--
00:03:14.500 --> 00:03:21.630
they're going to be coming
from e to the i omega t.
00:03:21.630 --> 00:03:24.535
In other words, instead of an a,
we're going to have an i omega.
00:03:27.100 --> 00:03:30.940
Or, if we like to
stay real, we can
00:03:30.940 --> 00:03:36.320
stay with cos--
cosine-- and sine.
00:03:39.800 --> 00:03:43.360
And actually, I've written
two real guys there,
00:03:43.360 --> 00:03:45.770
so I better have
two complex ones.
00:03:45.770 --> 00:03:49.420
And it will turn out
to be plus or minus.
00:03:49.420 --> 00:03:51.650
There are two frequencies there.
00:03:51.650 --> 00:03:57.540
Plus i omega, minus i omega, and
they turn into cosine and sine.
00:03:57.540 --> 00:04:04.400
So in this case,
with no damping term,
00:04:04.400 --> 00:04:08.360
we can stay entirely real
without creating any problems.
00:04:08.360 --> 00:04:12.340
We can work with
cosines and sines.
00:04:12.340 --> 00:04:16.040
The first question
is, what's omega.
00:04:16.040 --> 00:04:22.010
What is the frequency
of oscillation.
00:04:22.010 --> 00:04:24.580
And of course, another
similar picture
00:04:24.580 --> 00:04:27.400
would be a pendulum,
a linear pendulum,
00:04:27.400 --> 00:04:32.030
swinging side to
side, keeping time,
00:04:32.030 --> 00:04:34.095
because that frequency
will stay constant.
00:04:38.420 --> 00:04:41.560
Always I'll start with zero
on the right hand side.
00:04:44.430 --> 00:04:46.660
Just look at these equations.
00:04:46.660 --> 00:04:47.820
Constants there.
00:04:47.820 --> 00:04:50.690
I'm looking for solutions.
00:04:50.690 --> 00:04:54.920
And I'm looking
for null solutions,
00:04:54.920 --> 00:05:02.570
looking for the natural motion
of this spring, the natural up
00:05:02.570 --> 00:05:05.780
and down motion of this spring.
00:05:05.780 --> 00:05:07.630
Classical problem.
00:05:07.630 --> 00:05:11.830
Won't be brand new, but
it's the right starting
00:05:11.830 --> 00:05:18.500
point for the full
second order equation.
00:05:18.500 --> 00:05:22.400
It'll get a little
complicated on Wednesday,
00:05:22.400 --> 00:05:25.050
when damping gets in there.
00:05:25.050 --> 00:05:29.700
The formula got a little
messy, because you've
00:05:29.700 --> 00:05:34.470
got a mass-- you've got
an m-- and a k, still,
00:05:34.470 --> 00:05:37.340
but you also will have
a damping constant.
00:05:40.240 --> 00:05:43.510
Then complex numbers
really come in.
00:05:43.510 --> 00:05:47.570
Here they're optional.
00:05:47.570 --> 00:05:51.420
So this is my equation to solve.
00:05:55.280 --> 00:06:00.230
Because we don't have
a first derivative,
00:06:00.230 --> 00:06:02.620
a cosine will solve that.
00:06:02.620 --> 00:06:06.190
So let me look for-- I
could look for exponentials.
00:06:06.190 --> 00:06:08.940
Maybe I should do
that first, look
00:06:08.940 --> 00:06:10.780
for an exponential solution.
00:06:10.780 --> 00:06:12.560
Yeah, that's a good idea.
00:06:12.560 --> 00:06:18.810
And let me not
jump ahead to know
00:06:18.810 --> 00:06:22.620
that the exponent has
that i omega form.
00:06:22.620 --> 00:06:24.430
Let me discover it.
00:06:24.430 --> 00:06:30.850
So I look for no
solutions-- because I
00:06:30.850 --> 00:06:34.900
have that zero there--
no solutions of the form
00:06:34.900 --> 00:06:38.995
e to the st, some exponent.
00:06:41.510 --> 00:06:43.500
Plug it in.
00:06:43.500 --> 00:06:48.740
That's the message with
constant coefficients.
00:06:48.740 --> 00:06:51.690
Look for exponentials,
substitute them in,
00:06:51.690 --> 00:06:54.010
discover what s will be.
00:06:54.010 --> 00:06:55.210
So let's just do that.
00:06:58.285 --> 00:07:00.700
This is the most basic step.
00:07:00.700 --> 00:07:03.430
For null solutions
will be exponentials,
00:07:03.430 --> 00:07:05.660
I substitute into the equation.
00:07:05.660 --> 00:07:09.730
I get ms squared
from two derivatives.
00:07:09.730 --> 00:07:11.570
It will bring s down twice.
00:07:14.550 --> 00:07:20.860
This is just ke to the st, and
I'm looking for null solutions.
00:07:20.860 --> 00:07:21.640
Zero.
00:07:21.640 --> 00:07:22.630
No forcing.
00:07:22.630 --> 00:07:26.540
So this is undamped, unforced.
00:07:26.540 --> 00:07:29.960
Undamped, unforced.
00:07:29.960 --> 00:07:32.780
Natural motion.
00:07:32.780 --> 00:07:35.710
What do I do now?
00:07:35.710 --> 00:07:40.190
Plugged in an exponential,
got this equation.
00:07:40.190 --> 00:07:44.290
And the beauty is that
the exponentials cancel.
00:07:44.290 --> 00:07:48.290
An exponential is never zero,
so I can safely divide by it.
00:07:48.290 --> 00:07:55.120
So I cancel those, and I get
ms squared plus k equals zero.
00:07:55.120 --> 00:07:57.940
The key equation-- and
it's so simple-- it's
00:07:57.940 --> 00:08:02.050
just we're doing algebra now.
00:08:02.050 --> 00:08:06.730
The calculus, the derivative
we took when we plugged it in,
00:08:06.730 --> 00:08:09.200
but now it's an
algebra question.
00:08:09.200 --> 00:08:13.210
And of course, solving
that system is easy.
00:08:13.210 --> 00:08:17.110
There's no s term,
no damping term.
00:08:17.110 --> 00:08:26.160
So the frequency, s, is-- put k
on the other side, divide by n.
00:08:26.160 --> 00:08:33.320
s is-- s squared, let's say--
is k over m-- is minus k over m.
00:08:33.320 --> 00:08:34.730
Critical point.
00:08:39.850 --> 00:08:42.900
That tells me, with
that minus sign there,
00:08:42.900 --> 00:08:48.840
that s is an imaginary number.
00:08:48.840 --> 00:08:54.990
A complex number has a real
part and an imaginary part.
00:08:54.990 --> 00:08:58.130
In this case, all imaginary.
00:08:58.130 --> 00:08:59.260
No real part at all.
00:09:03.310 --> 00:09:07.380
It's natural to think-- s
is the square root of that,
00:09:07.380 --> 00:09:11.870
so I'm going to write--
everybody writes-- s,
00:09:11.870 --> 00:09:17.080
the frequency s, is i omega.
00:09:17.080 --> 00:09:22.220
So if I plug that in, I have
omega squared equal k over m.
00:09:22.220 --> 00:09:26.030
i squared and the minus
1 deal with each other.
00:09:26.030 --> 00:09:30.970
So the frequency omega-- here
is the great fact-- square
00:09:30.970 --> 00:09:31.980
root of k over m.
00:09:34.520 --> 00:09:36.190
That's-- yes?
00:09:36.190 --> 00:09:38.146
AUDIENCE: What
difference does having
00:09:38.146 --> 00:09:42.058
imaginary parts to answer
affect the oscillation?
00:09:42.058 --> 00:09:43.560
PROFESSOR: To-- OK.
00:09:46.270 --> 00:09:49.430
Oscillation, just
pure oscillation--
00:09:49.430 --> 00:09:53.140
which is what we would
see here with no damping--
00:09:53.140 --> 00:10:04.370
is the frequency, e to the--
the solution-- the displacement,
00:10:04.370 --> 00:10:08.350
I could write-- the
displacement up and down,
00:10:08.350 --> 00:10:19.310
y, will involve e to the i
omega t, and e to the minus i
00:10:19.310 --> 00:10:19.940
omega t.
00:10:22.530 --> 00:10:24.280
We've got second order equation.
00:10:24.280 --> 00:10:29.300
Let me just go back
to that key point.
00:10:29.300 --> 00:10:30.980
When we have second
order equations,
00:10:30.980 --> 00:10:33.070
we look for two-- we
expect and we want
00:10:33.070 --> 00:10:36.160
and we need two-- solutions.
00:10:36.160 --> 00:10:38.730
There will be two.
00:10:38.730 --> 00:10:42.250
I didn't put it
here, and I should.
00:10:42.250 --> 00:10:48.140
s, the frequency, is
plus or minus i omega,
00:10:48.140 --> 00:10:53.500
because in both cases when we
square, it comes out right.
00:10:53.500 --> 00:10:57.440
So we get two frequencies,
and here they are.
00:11:01.480 --> 00:11:05.370
So let's see how to
answer your question.
00:11:05.370 --> 00:11:11.110
The presence of this
i is only telling me
00:11:11.110 --> 00:11:18.110
that, essentially, I've
got sines and cosines.
00:11:18.110 --> 00:11:23.720
That's really what-- when it's a
pure imaginary number-- I would
00:11:23.720 --> 00:11:26.090
call that a pure
imaginary number,
00:11:26.090 --> 00:11:34.450
there's no real part at
all-- then equally cos omega
00:11:34.450 --> 00:11:37.250
t and sine omega t.
00:11:40.490 --> 00:11:43.910
I can now, if I want, go real.
00:11:43.910 --> 00:11:50.860
I can say, OK, these were
the general null solutions.
00:11:50.860 --> 00:11:52.750
Let me put this down, then.
00:11:52.750 --> 00:11:55.930
The null solution-- I'm
looking only right now
00:11:55.930 --> 00:12:00.690
at null solutions-- is
some combination of e
00:12:00.690 --> 00:12:07.950
to the i omega t and e
to the minus i omega t.
00:12:07.950 --> 00:12:12.260
That's what we got from
plugging in e to the st,
00:12:12.260 --> 00:12:15.550
discovering that s was
an imaginary number,
00:12:15.550 --> 00:12:17.150
and we got these guys.
00:12:17.150 --> 00:12:26.150
But equally-- equally--
yn is a combination
00:12:26.150 --> 00:12:33.010
of cos omega t and sine omega t.
00:12:36.300 --> 00:12:38.490
And maybe you'll
like those better.
00:12:38.490 --> 00:12:42.200
I think everybody practically
likes those better.
00:12:42.200 --> 00:12:47.050
Do you see that these guys
are the same as these guys?
00:12:47.050 --> 00:12:51.210
The c's are a little
different because, well, we
00:12:51.210 --> 00:12:55.650
know that we can switch
from one to the other.
00:12:55.650 --> 00:13:02.210
We remember that basic fact
that e to the i omega t
00:13:02.210 --> 00:13:09.630
is cos omega t plus
i times sine omega t.
00:13:09.630 --> 00:13:13.370
You're used to maybe seeing that
omega t as theta, e to the i
00:13:13.370 --> 00:13:17.470
theta is cos theta
plus i sine theta.
00:13:17.470 --> 00:13:21.900
And e to the minus i
omega t, of course,
00:13:21.900 --> 00:13:28.157
is cos omega t minus
i sine omega t.
00:13:31.079 --> 00:13:34.140
I hope you won't think
I'm filling the blackboard
00:13:34.140 --> 00:13:39.900
with formulas, because I'm
really just writing down--
00:13:39.900 --> 00:13:41.570
well anyway, they're
beautiful formulas.
00:13:44.120 --> 00:13:50.550
So if I have these guys, then
I have these and vice versa.
00:13:50.550 --> 00:13:53.200
If I have cos--
how would you write
00:13:53.200 --> 00:13:58.910
cos omega t using
the exponentials?
00:13:58.910 --> 00:14:01.750
I want to just see
totally clearly
00:14:01.750 --> 00:14:07.600
that I can go back and forth
between complex imaginary
00:14:07.600 --> 00:14:10.860
exponentials and
cosines and sines.
00:14:10.860 --> 00:14:17.210
So how would I, I want to
go in the opposite direction
00:14:17.210 --> 00:14:24.820
and write the cosine and the
sine as combinations of these,
00:14:24.820 --> 00:14:28.040
just to show if I've
got combinations of one,
00:14:28.040 --> 00:14:29.870
I've got combinations
of the other.
00:14:29.870 --> 00:14:33.980
Combinations of these are the
same as combinations of those.
00:14:33.980 --> 00:14:37.271
So what is cos omega t
in terms of these guys?
00:14:37.271 --> 00:14:39.270
AUDIENCE: [INAUDIBLE]
some of them divided by 2?
00:14:39.270 --> 00:14:40.700
PROFESSOR: Exactly.
00:14:40.700 --> 00:14:44.630
If I add those two,
this part cancels.
00:14:44.630 --> 00:14:48.130
I've got two of these, so I
have to divide by 2, as you say.
00:14:48.130 --> 00:14:52.395
It's a half of the first
plus a half of the second.
00:14:55.890 --> 00:14:57.770
And how about sine omega t?
00:14:57.770 --> 00:15:00.510
Sine omega t is always
slightly more annoying,
00:15:00.510 --> 00:15:04.600
because it's the one-- it's
the imaginary part that
00:15:04.600 --> 00:15:06.430
brings in an i.
00:15:06.430 --> 00:15:08.010
What would be the same formula?
00:15:08.010 --> 00:15:10.880
How could I produce sine
omega t out of that?
00:15:14.361 --> 00:15:14.860
Yes?
00:15:14.860 --> 00:15:16.772
AUDIENCE: The difference
divided by 2i.
00:15:16.772 --> 00:15:17.670
PROFESSOR: Yes.
00:15:17.670 --> 00:15:21.160
If I take the difference,
that'll cancel the cosines.
00:15:21.160 --> 00:15:23.340
So I'm going to take
e to the i omega
00:15:23.340 --> 00:15:29.290
t minus e-- minus e to
the minus i omega t.
00:15:29.290 --> 00:15:30.710
Take the difference.
00:15:30.710 --> 00:15:36.240
But then I've got 2i
multiplying this sine.
00:15:36.240 --> 00:15:39.880
Up here I had a 2, but now I've
got-- when I take the subtract,
00:15:39.880 --> 00:15:42.790
these i's are in there,
so I divide by 2i.
00:15:48.630 --> 00:15:52.880
So this just tells me
that I can go either way.
00:15:57.530 --> 00:16:00.930
Next time, we'll see what
happens when there is damping
00:16:00.930 --> 00:16:09.680
and there are complex numbers
instead of pure i omegas.
00:16:12.480 --> 00:16:14.180
We're golden here.
00:16:14.180 --> 00:16:21.745
We've found the great
quantity with the right units.
00:16:21.745 --> 00:16:26.210
The right units of
omega are 1 over time.
00:16:26.210 --> 00:16:31.610
Actually the units are
radians per second,
00:16:31.610 --> 00:16:35.390
would be the typical
appropriate unit.
00:16:35.390 --> 00:16:36.470
Radians per second.
00:16:45.760 --> 00:16:48.720
I'll use the word
frequency for that,
00:16:48.720 --> 00:16:52.950
but there's another
definition of frequency,
00:16:52.950 --> 00:16:54.130
cycles per second.
00:16:57.760 --> 00:17:02.260
I just want to think about
steady motion around a circle.
00:17:02.260 --> 00:17:04.859
So this tells me how
many radians per second.
00:17:04.859 --> 00:17:12.420
And if this is 2pi-- if
omega happened to be 2pi--
00:17:12.420 --> 00:17:15.099
then I would go once
around the circle.
00:17:15.099 --> 00:17:19.940
If omega was 2pi, then
when t reached one,
00:17:19.940 --> 00:17:22.950
I would be around the circle.
00:17:22.950 --> 00:17:25.920
Let me draw a
circle in a minute.
00:17:25.920 --> 00:17:32.160
So there's a 2pi here hiding
behind the word radians.
00:17:32.160 --> 00:17:38.450
And in many cases,
you'll want also
00:17:38.450 --> 00:17:41.680
a definition in
cycles per second.
00:17:41.680 --> 00:17:50.700
So f is omega
divided by the 2pi,
00:17:50.700 --> 00:17:53.000
and that's in cycles per second.
00:17:55.940 --> 00:17:58.270
Full revolutions per second.
00:17:58.270 --> 00:18:00.950
And that's hertz.
00:18:00.950 --> 00:18:09.310
I think I misspoke last
time in confusing these two,
00:18:09.310 --> 00:18:10.910
so let's get them straight here.
00:18:14.450 --> 00:18:20.040
There's no complicated math in
here, it's just a factor 2pi,
00:18:20.040 --> 00:18:23.200
but of course that
factor is important.
00:18:23.200 --> 00:18:31.520
So a typical frequency
in everyday life
00:18:31.520 --> 00:18:38.110
would be like f, 60 cycles
per second, 120pi radians
00:18:38.110 --> 00:18:39.400
per second.
00:18:39.400 --> 00:18:41.080
So I'm going around in a circle.
00:18:43.700 --> 00:18:50.750
Now I'm ready to have
initial conditions.
00:18:50.750 --> 00:18:55.860
This connects, again,
to the afternoon exam.
00:18:55.860 --> 00:19:01.310
We found the general solution
with some constants, like here.
00:19:03.910 --> 00:19:06.120
Let's keep that real form.
00:19:06.120 --> 00:19:09.240
And now those constants
get determined
00:19:09.240 --> 00:19:11.660
by the initial conditions.
00:19:11.660 --> 00:19:18.240
Conditions plural, because
we have an initial position,
00:19:18.240 --> 00:19:24.310
like I stretch it-- maybe
I stretch it and let go.
00:19:24.310 --> 00:19:26.990
Maybe I stretch the
spring and then I let go.
00:19:26.990 --> 00:19:29.040
What happens?
00:19:29.040 --> 00:19:33.010
By stretching it, I'm giving
it an initial displacement.
00:19:33.010 --> 00:19:35.300
And I'm giving it
zero initial velocity,
00:19:35.300 --> 00:19:38.050
because I stretched
it and just let go.
00:19:38.050 --> 00:19:41.250
Another possibility
would be to strike it.
00:19:41.250 --> 00:19:43.860
If I hit that
mass, that would be
00:19:43.860 --> 00:19:46.050
a different initial condition.
00:19:46.050 --> 00:19:47.570
What would be the
initial condition
00:19:47.570 --> 00:19:50.850
if it's sitting there in
equilibrium quietly minding
00:19:50.850 --> 00:19:54.210
its own business and I hit it?
00:19:54.210 --> 00:19:59.090
Then I've given it
an initial velocity,
00:19:59.090 --> 00:20:01.630
with initial displacement zero.
00:20:01.630 --> 00:20:04.360
So those would be the two
extreme possibilities.
00:20:04.360 --> 00:20:10.380
Pull it down, let
go, or strike it
00:20:10.380 --> 00:20:14.470
when it's sitting
in equilibrium.
00:20:14.470 --> 00:20:16.195
Anyway, we've got two
initial conditions.
00:20:19.520 --> 00:20:23.970
You see why-- y double
prime is showing up
00:20:23.970 --> 00:20:28.750
because essentially
we've got Newton's Law.
00:20:28.750 --> 00:20:31.610
This is Newton's Law.
00:20:31.610 --> 00:20:37.690
Mass times acceleration is
equal to minus ky-- that's
00:20:37.690 --> 00:20:42.580
the, with the minus sign,
and the all-important minus
00:20:42.580 --> 00:20:45.780
sign, that's the acceleration.
00:20:45.780 --> 00:20:48.240
That's a force, sorry.
00:20:48.240 --> 00:20:52.860
Mass, acceleration, this thing
with a minus sign is the force,
00:20:52.860 --> 00:20:54.510
and the force is pulling back.
00:20:54.510 --> 00:20:59.425
If y is stretching,
the force is restoring.
00:21:04.890 --> 00:21:08.780
Let me just go ahead
with what you know.
00:21:08.780 --> 00:21:10.100
The initial conditions.
00:21:10.100 --> 00:21:17.380
And I want to solve my double
prime plus ky equals 0.
00:21:17.380 --> 00:21:28.340
So I'm still talking about
the unforced with given
00:21:28.340 --> 00:21:32.254
y of 0 and y prime of 0.
00:21:36.840 --> 00:21:38.800
Just think for a moment.
00:21:38.800 --> 00:21:41.730
Could you do that?
00:21:41.730 --> 00:21:44.710
This is the most basic
second order equation.
00:21:44.710 --> 00:21:48.030
We know what the
solutions look like.
00:21:48.030 --> 00:21:52.120
Let's do this one in a
box, cosines and sines.
00:21:52.120 --> 00:21:54.400
We know what omega is.
00:21:54.400 --> 00:21:57.740
Omega had to be square
root of k over m.
00:21:57.740 --> 00:22:01.830
Then the equation was solved.
00:22:01.830 --> 00:22:04.800
All I've got left
is to get c1 and c2.
00:22:07.440 --> 00:22:10.900
All I have left is to
match-- choose c1 and c2
00:22:10.900 --> 00:22:13.650
to match the two
initial conditions.
00:22:13.650 --> 00:22:14.950
So let me just do that.
00:22:17.700 --> 00:22:19.260
What are c2 and c2?
00:22:22.880 --> 00:22:27.470
At time zero, I have to match
an initial displacement.
00:22:27.470 --> 00:22:33.440
So at time zero, this is a
1, cosine of zero is a one,
00:22:33.440 --> 00:22:34.760
and that's a zero.
00:22:34.760 --> 00:22:41.450
So at t equals 0, I have y of 0.
00:22:44.180 --> 00:22:48.160
The displacement
matches c1 times cosine
00:22:48.160 --> 00:22:53.540
of omega t, which is
a 1, plus c2 times 0.
00:22:53.540 --> 00:22:55.901
I'll put it in there
plus c2 times 0.
00:22:58.490 --> 00:23:01.770
C1 cos 0 plus c2 sine 0.
00:23:04.750 --> 00:23:08.320
I've learned c1.
00:23:08.320 --> 00:23:10.620
And also-- what do I do next?
00:23:14.980 --> 00:23:18.130
I want to get c2.
00:23:18.130 --> 00:23:22.340
And where is c2 coming from?
00:23:22.340 --> 00:23:25.130
Now I would like to know
what's the coefficient
00:23:25.130 --> 00:23:29.100
of the-- the initial
conditions are supposed
00:23:29.100 --> 00:23:31.450
to determine that coefficient.
00:23:31.450 --> 00:23:34.630
It'll be that initial
condition that determines it.
00:23:34.630 --> 00:23:37.570
y prime of 0.
00:23:37.570 --> 00:23:42.140
The initial velocity should
match the derivative.
00:23:42.140 --> 00:23:43.525
OK, so what's the derivative?
00:23:48.200 --> 00:23:50.440
y prime.
00:23:50.440 --> 00:23:54.460
So the derivative of the
cosine will be a sine.
00:23:54.460 --> 00:23:57.320
And that will disappear
at t equals 0.
00:23:57.320 --> 00:23:59.230
The derivative of
this sine will be
00:23:59.230 --> 00:24:03.170
a cosine with a factor omega.
00:24:03.170 --> 00:24:11.650
So I'll have y prime of 0
will be the c2 omega cos of 0.
00:24:17.220 --> 00:24:17.975
Which is what?
00:24:26.100 --> 00:24:27.010
That tells me c2.
00:24:31.550 --> 00:24:39.590
You could do all this
without my pointing the way.
00:24:39.590 --> 00:24:42.160
I'm solving this equation.
00:24:42.160 --> 00:24:47.750
I have the solution in general
form with two constants.
00:24:47.750 --> 00:24:49.870
Now I'm determining
those constants,
00:24:49.870 --> 00:24:54.100
and cosine and sine just
determine them perfectly
00:24:54.100 --> 00:25:02.870
because cosine is 1 and
sine is 0 at the start.
00:25:02.870 --> 00:25:07.960
So we've got the answer.
00:25:07.960 --> 00:25:17.920
The solution is y of t is
c1 is y of 0 cos omega t,
00:25:17.920 --> 00:25:21.320
and c2 is-- now
you'll notice, c2
00:25:21.320 --> 00:25:25.720
is y prime at 0
divided by omega.
00:25:25.720 --> 00:25:30.850
y prime of 0 divided
by omega sine omega t.
00:25:30.850 --> 00:25:32.570
There we go.
00:25:32.570 --> 00:25:34.030
Finished.
00:25:34.030 --> 00:25:35.420
Finished.
00:25:35.420 --> 00:25:37.850
Unforced problem solved.
00:25:42.300 --> 00:25:44.520
Everybody in this room
could get to that point.
00:25:49.370 --> 00:25:52.280
Let me make some
comments about that.
00:25:52.280 --> 00:25:56.440
It's a combination
of cosine and sine.
00:25:56.440 --> 00:26:00.780
They're both running at
the same frequency, omega.
00:26:00.780 --> 00:26:04.780
I'm going to give a special
name to that frequency, omega,
00:26:04.780 --> 00:26:09.140
this famous formula,
all-important.
00:26:09.140 --> 00:26:10.950
Lots of physics in that formula.
00:26:14.070 --> 00:26:18.080
I call that the
natural frequency,
00:26:18.080 --> 00:26:27.170
because the next step will be
to drive the system by a driving
00:26:27.170 --> 00:26:31.540
frequency, which would
be different from omega.
00:26:31.540 --> 00:26:34.150
So we need to--
we've got 2 omegas.
00:26:34.150 --> 00:26:37.630
Actually when I first
wrote the book I thought,
00:26:37.630 --> 00:26:40.580
we've got to keep
these two separate.
00:26:40.580 --> 00:26:42.270
Everybody has to
keep them separate.
00:26:42.270 --> 00:26:46.570
My first attempt was to use
little omega and big omega
00:26:46.570 --> 00:26:49.310
for the two.
00:26:49.310 --> 00:26:53.890
I concluded after
looking at it for a while
00:26:53.890 --> 00:26:57.500
that it was better to
be more conventional.
00:26:57.500 --> 00:27:00.320
People had figured out
a good way to do it.
00:27:00.320 --> 00:27:04.980
And the good way is to call
this the natural frequency
00:27:04.980 --> 00:27:07.450
and put a subscript, m.
00:27:07.450 --> 00:27:11.670
So all the omegas that
you see on this board
00:27:11.670 --> 00:27:14.640
should be omega n.
00:27:14.640 --> 00:27:20.520
I can change them all, but
let me just change it here.
00:27:20.520 --> 00:27:22.490
I'll change omega n.
00:27:25.990 --> 00:27:27.240
So that's omega n.
00:27:27.240 --> 00:27:31.240
We've only got that
one omega right now,
00:27:31.240 --> 00:27:34.700
because we don't have
a driving term yet.
00:27:34.700 --> 00:27:37.850
So natural frequency
has the advantage,
00:27:37.850 --> 00:27:45.130
which kind of made me smile,
that the n stands for natural,
00:27:45.130 --> 00:27:47.910
and everybody calls it
the natural frequency.
00:27:47.910 --> 00:27:51.780
And n also stands
for null, and we're
00:27:51.780 --> 00:27:55.130
talking here about
the null solution,
00:27:55.130 --> 00:27:56.950
because there's no forcing.
00:27:56.950 --> 00:28:00.510
So I could have
subscripts on all the y's.
00:28:00.510 --> 00:28:02.930
Eventually I'll need
subscripts on the y
00:28:02.930 --> 00:28:07.640
to separate what we've done.
00:28:07.640 --> 00:28:11.785
This is really yn of
t, the null solution.
00:28:15.220 --> 00:28:17.670
Good?
00:28:17.670 --> 00:28:20.180
Now we could take one more step.
00:28:23.110 --> 00:28:26.730
This is a combination
of cosine and sine.
00:28:26.730 --> 00:28:29.590
And we learned
last time that that
00:28:29.590 --> 00:28:35.160
could be put in a polar form,
but I don't plan to do this.
00:28:35.160 --> 00:28:37.590
Let me just say I could do it.
00:28:37.590 --> 00:28:43.180
This would be some
amplitude, some gain--
00:28:43.180 --> 00:28:46.925
maybe g for gain--
no, a for amplitude
00:28:46.925 --> 00:28:53.620
is good-- times-- what is this
second optional form, which I'm
00:28:53.620 --> 00:28:57.220
just going to write here,
say that we could do it,
00:28:57.220 --> 00:29:01.200
remember a little about it,
but not make a big deal-- what
00:29:01.200 --> 00:29:02.400
is it I'm after here?
00:29:05.170 --> 00:29:09.620
I'm looking to write this
combination of cosine and sine,
00:29:09.620 --> 00:29:14.450
which is two oscillations, a
cosine curve and a sine curve,
00:29:14.450 --> 00:29:17.080
but with the same frequency.
00:29:17.080 --> 00:29:22.180
Then I can combine them
into a single cosine,
00:29:22.180 --> 00:29:27.200
a single cosine of omega nt.
00:29:27.200 --> 00:29:31.590
But now what else have
I got in this form?
00:29:31.590 --> 00:29:35.900
There's a phase
shift, minus phi.
00:29:35.900 --> 00:29:38.670
Thanks.
00:29:38.670 --> 00:29:43.400
So there's an a phi, two
constants, or there's y of 0,
00:29:43.400 --> 00:29:47.060
y prime of 0, two constants.
00:29:47.060 --> 00:29:50.940
Let me not write again the
formula for a or for phi,
00:29:50.940 --> 00:29:52.640
I don't plan to do
anything with it.
00:29:52.640 --> 00:29:55.210
It just could be done.
00:29:55.210 --> 00:29:59.280
In other words, what
we've done so far
00:29:59.280 --> 00:30:08.690
is just to see that the
single spring oscillates
00:30:08.690 --> 00:30:10.500
with the frequency omega n.
00:30:10.500 --> 00:30:12.370
That's really what we've done.
00:30:12.370 --> 00:30:16.023
A single spring oscillates
with a frequency omega n.
00:30:21.220 --> 00:30:26.500
Saying that makes me
think, let me look ahead
00:30:26.500 --> 00:30:31.790
to the linear algebra
part of the course.
00:30:31.790 --> 00:30:35.730
So where is linear
algebra going to come in?
00:30:35.730 --> 00:30:38.880
It's going to come in
for a system of springs.
00:30:38.880 --> 00:30:40.990
When I have another spring.
00:30:40.990 --> 00:30:44.580
Can I draw another
spring and another mass
00:30:44.580 --> 00:30:45.750
and another spring?
00:30:45.750 --> 00:30:50.550
Say six springs, six masses.
00:30:50.550 --> 00:30:55.630
Then-- and they could be
different k's, different m's,
00:30:55.630 --> 00:30:56.260
or not.
00:30:59.730 --> 00:31:03.040
Then we've got six
displacements--
00:31:03.040 --> 00:31:07.490
six differential equations--
coupled together,
00:31:07.490 --> 00:31:10.610
because the whole system
is coupled together.
00:31:10.610 --> 00:31:14.160
So what happens at that point?
00:31:14.160 --> 00:31:18.030
That's the point where linear
algebra, where matrices
00:31:18.030 --> 00:31:20.210
are coming in.
00:31:20.210 --> 00:31:23.030
You want to see what's
the point of matrices.
00:31:23.030 --> 00:31:27.160
It's not a separate
course by any means.
00:31:27.160 --> 00:31:33.350
It's a most necessary part,
because a single spring happens
00:31:33.350 --> 00:31:38.380
in reality but also
systems today are coupled.
00:31:38.380 --> 00:31:40.960
Big, actually, there
are many, many things.
00:31:40.960 --> 00:31:44.580
You have an electric
circuit with thousands
00:31:44.580 --> 00:31:47.610
or tens of thousands
of elements.
00:31:47.610 --> 00:31:51.640
You have a coupled
system with many gears,
00:31:51.640 --> 00:31:54.190
many oscillations going on.
00:31:56.780 --> 00:32:01.190
So we need matrices
at that point.
00:32:01.190 --> 00:32:07.270
Can I even just add one more
word about the language?
00:32:07.270 --> 00:32:11.730
When we had-- here
we have a frequency
00:32:11.730 --> 00:32:14.150
of motion for one spring.
00:32:14.150 --> 00:32:18.910
What are we going to have for
two springs or six springs?
00:32:18.910 --> 00:32:24.005
The motion will be a combination
of six different frequencies.
00:32:26.950 --> 00:32:30.500
And so you'll see that it's
a much more interesting,
00:32:30.500 --> 00:32:34.800
much more not so simple motion.
00:32:34.800 --> 00:32:38.960
A combination of six
pure frequencies.
00:32:38.960 --> 00:32:41.680
And those frequencies
are determined
00:32:41.680 --> 00:32:45.810
from the six eigenvalues
of the matrix.
00:32:45.810 --> 00:32:49.490
I'm just using that
word looking ahead.
00:32:49.490 --> 00:32:55.260
We will have a 6x6 matrix to
describe the coupled system.
00:32:55.260 --> 00:32:57.820
That matrix will
have six eigenvalues.
00:32:57.820 --> 00:33:00.910
It will tell us six
natural frequencies,
00:33:00.910 --> 00:33:05.460
and our solution will be
a combination of all six
00:33:05.460 --> 00:33:07.370
oscillations.
00:33:07.370 --> 00:33:10.210
Here, it's 1.
00:33:10.210 --> 00:33:11.080
Here it's 1.
00:33:11.080 --> 00:33:12.290
That spring is not there.
00:33:20.130 --> 00:33:23.070
So the problem
we've solved now is
00:33:23.070 --> 00:33:29.070
the fundamental, basic problem,
and I have to-- next step
00:33:29.070 --> 00:33:33.000
is forcing.
00:33:33.000 --> 00:33:38.640
I now want to add a force
that drives the motion.
00:33:43.710 --> 00:33:46.660
In general, it could be
any function of time.
00:33:50.840 --> 00:33:51.980
Calling it f of t.
00:33:51.980 --> 00:33:54.500
So that's what I'm
going to put in now.
00:33:54.500 --> 00:33:59.350
But in reality, very,
very, very often f of t
00:33:59.350 --> 00:34:04.630
is also a simple
harmonic motion.
00:34:04.630 --> 00:34:08.409
It's also a cosine.
00:34:08.409 --> 00:34:11.810
But at a different frequency,
at a driving frequency.
00:34:11.810 --> 00:34:13.840
So I'm going to--
the next equation
00:34:13.840 --> 00:34:24.300
to solve is to put in cosine--
let's stay real for now--
00:34:24.300 --> 00:34:30.210
at another, driving frequency.
00:34:30.210 --> 00:34:31.494
At a driving frequency.
00:34:34.300 --> 00:34:36.360
And of course, it could
have an amplitude.
00:34:36.360 --> 00:34:41.270
But let me take that amplitude
as 1 to keep things simple.
00:34:41.270 --> 00:34:46.920
So now I'm talking
about forced motion.
00:34:46.920 --> 00:34:49.830
Can we solve it?
00:34:49.830 --> 00:34:52.040
How can we solve this equation?
00:34:52.040 --> 00:35:00.020
Let me take out the 0 or-- take
out the 0-- equals cosine omega
00:35:00.020 --> 00:35:00.520
t.
00:35:04.352 --> 00:35:05.310
With a different omega.
00:35:08.170 --> 00:35:11.970
If the two omegas were the same,
if the driving frequency is
00:35:11.970 --> 00:35:18.270
the same as the
natural frequency,
00:35:18.270 --> 00:35:22.500
the formulas have to
be slightly adjusted.
00:35:22.500 --> 00:35:28.250
There's still an answer,
but it's a case of resonance
00:35:28.250 --> 00:35:30.100
and you have to look separately.
00:35:30.100 --> 00:35:32.290
But let's say, no.
00:35:32.290 --> 00:35:35.230
Let's say omega d is
different from omega n.
00:35:35.230 --> 00:35:36.680
How are you going to solve this?
00:35:40.480 --> 00:35:42.520
I have to think myself.
00:35:42.520 --> 00:35:47.030
How do I solve that.
00:35:47.030 --> 00:35:48.604
Let's start a fresh board.
00:35:52.400 --> 00:36:00.070
my double prime plus
ky equals cosine
00:36:00.070 --> 00:36:05.210
of omega dt-- or often,
I won't put the d.
00:36:05.210 --> 00:36:07.910
I don't have to
put the d anymore.
00:36:07.910 --> 00:36:11.320
Omega will now represent
the driving frequency,
00:36:11.320 --> 00:36:19.290
because I've got omega
n, the natural frequency,
00:36:19.290 --> 00:36:22.926
as the square root of k over m.
00:36:29.080 --> 00:36:31.550
What am I looking for now?
00:36:31.550 --> 00:36:33.490
I found the null solution.
00:36:33.490 --> 00:36:36.830
I'm looking for a
particular solution.
00:36:36.830 --> 00:36:39.920
I'm trying to keep the
whole thing systematic.
00:36:39.920 --> 00:36:42.630
Null solutions are
now dealt with.
00:36:42.630 --> 00:36:44.945
Took a little more
time than just ce
00:36:44.945 --> 00:36:48.540
to the at for first
order equations,
00:36:48.540 --> 00:36:54.100
because we've now got a
two-dimensional collection
00:36:54.100 --> 00:36:56.580
of null solutions,
but we've got them.
00:36:56.580 --> 00:37:01.090
Now I'm taking a forcing term.
00:37:01.090 --> 00:37:06.480
So I'm looking for a
particular solution.
00:37:06.480 --> 00:37:09.230
I'm looking for any
solution to this equation.
00:37:09.230 --> 00:37:12.930
I'm looking for
a particular guy.
00:37:12.930 --> 00:37:14.855
What do you suggest?
00:37:18.230 --> 00:37:22.830
Again, it's a neat
problem because
00:37:22.830 --> 00:37:29.130
of that particular forcing
term, a cosine, an oscillation.
00:37:29.130 --> 00:37:50.730
So I'm going to look for yp is
some gain times [INAUDIBLE].
00:37:50.730 --> 00:37:53.470
This is the next and,
fortunately, a highly, highly
00:37:53.470 --> 00:37:59.620
important case, in which
the particular solution has
00:37:59.620 --> 00:38:03.280
the same form as
the forcing term.
00:38:03.280 --> 00:38:05.430
It's just a multiple
of the forcing term.
00:38:05.430 --> 00:38:07.400
That's best possible.
00:38:07.400 --> 00:38:15.150
That's best possible, is to have
the forcing term reveal to me--
00:38:15.150 --> 00:38:21.200
the forcing term immediately
reveals a particular solution.
00:38:23.890 --> 00:38:26.460
Once I know what I'm
looking for, what do I do?
00:38:26.460 --> 00:38:28.220
Substitute it in.
00:38:28.220 --> 00:38:33.090
So I substitute that
particular solution in here.
00:38:33.090 --> 00:38:36.580
And notice everything
is going to be
00:38:36.580 --> 00:38:45.030
a cosine, mgy double prime.
00:38:45.030 --> 00:38:52.200
So what do I get when I
plug this in for that guy?
00:38:52.200 --> 00:38:57.340
I want to-- you can do
it quickly, but let's
00:38:57.340 --> 00:39:00.290
stay together and do it
together, because we can
00:39:00.290 --> 00:39:01.540
with this case.
00:39:01.540 --> 00:39:04.740
What happens when I plug that in
and take its second derivative?
00:39:07.410 --> 00:39:09.030
I get the g.
00:39:09.030 --> 00:39:10.810
And then what's the
second derivative?
00:39:10.810 --> 00:39:12.960
AUDIENCE: [INAUDIBLE].
00:39:12.960 --> 00:39:16.960
PROFESSOR: We have a negative,
because two derivatives
00:39:16.960 --> 00:39:22.880
of the cosine bring out a minus
omega d, will come out twice.
00:39:22.880 --> 00:39:25.480
And I'll keep writing
omega d for a moment,
00:39:25.480 --> 00:39:27.710
but then I'll give up on the d.
00:39:27.710 --> 00:39:32.100
Cosine of omega dt.
00:39:32.100 --> 00:39:39.930
And then k times this,
g cosine of omega dt,
00:39:39.930 --> 00:39:43.640
equals the forcing term,
cosine of omega dt.
00:39:46.420 --> 00:39:49.270
It worked.
00:39:49.270 --> 00:39:55.290
This is one of that small
family of nice functions
00:39:55.290 --> 00:40:04.230
where the solution has the
same form as the function.
00:40:04.230 --> 00:40:11.720
Actually that list of what
you could call best possible
00:40:11.720 --> 00:40:17.490
forcing functions, where the
form of the forcing function
00:40:17.490 --> 00:40:23.820
tells you the form
of the solution.
00:40:23.820 --> 00:40:25.085
That's a small family.
00:40:27.930 --> 00:40:30.640
But it's fortunately
a very important one.
00:40:30.640 --> 00:40:34.070
Cosines, sines are
included, and we'll
00:40:34.070 --> 00:40:37.730
see all the other guys
that are included.
00:40:37.730 --> 00:40:41.510
Most forcing functions
we couldn't just
00:40:41.510 --> 00:40:47.380
assume that the solution
had the same form.
00:40:47.380 --> 00:40:48.910
It's only these nice ones.
00:40:48.910 --> 00:40:51.250
But cosines are nice.
00:40:51.250 --> 00:40:52.300
So what do I do now?
00:40:55.160 --> 00:40:57.150
Everything is
multiplying cosines,
00:40:57.150 --> 00:41:01.970
so I just look at--
I have minus m omega
00:41:01.970 --> 00:41:08.410
squared g-- g is going to factor
out-- minus m omega squared
00:41:08.410 --> 00:41:13.300
and a k times g.
00:41:13.300 --> 00:41:18.020
Let me remove that
off for the moment.
00:41:18.020 --> 00:41:24.990
I'm canceling cosine omega,
so my right hand side is 1.
00:41:28.910 --> 00:41:29.960
That's it.
00:41:38.110 --> 00:41:42.190
We looked for a solution
with that simple format,
00:41:42.190 --> 00:41:43.820
and we found it.
00:41:43.820 --> 00:41:45.940
Now we know g, the gain.
00:41:45.940 --> 00:41:53.800
So the solution is-- this
is g is 1 over k minus m
00:41:53.800 --> 00:41:59.270
omega squared times
cosine of omega t.
00:41:59.270 --> 00:42:01.690
And omega is omega d.
00:42:01.690 --> 00:42:03.680
Omega is omega d now.
00:42:10.026 --> 00:42:11.150
Does that look good to you?
00:42:13.840 --> 00:42:26.520
This is the periodic solution
going at the driving.
00:42:26.520 --> 00:42:31.130
This is what the-- this g is
the gain, the driving force.
00:42:31.130 --> 00:42:35.710
The driving force is 1
times cosine omega d,
00:42:35.710 --> 00:42:38.430
then that 1 gets
multiplied by this number.
00:42:38.430 --> 00:42:40.990
This is, you could say,
the amplifying factor.
00:42:44.390 --> 00:42:49.080
I guess frequency response
would be the right word.
00:42:49.080 --> 00:42:52.280
Can I bring in that
word, response, again?
00:42:52.280 --> 00:42:54.450
Response is a word
for a solution.
00:42:54.450 --> 00:42:58.920
It's what comes out.
00:42:58.920 --> 00:43:04.700
When the input is
this, a pure frequency,
00:43:04.700 --> 00:43:08.850
the output, the response,
is a pure frequency--
00:43:08.850 --> 00:43:13.780
same frequency, of course--
multiplied by that.
00:43:13.780 --> 00:43:21.350
That is the frequency
response factor.
00:43:24.800 --> 00:43:32.670
Notice we could write that
a cool way, by remembering
00:43:32.670 --> 00:43:36.580
that omega squared--
that's wrong as it stands.
00:43:36.580 --> 00:43:39.150
What have I forgotten
in writing k
00:43:39.150 --> 00:43:44.580
minus m omega squared
in that denominator?
00:43:44.580 --> 00:43:49.630
I forgot a subscript,
which is n.
00:43:49.630 --> 00:43:51.111
Which is n.
00:43:51.111 --> 00:43:53.190
This is n.
00:43:53.190 --> 00:43:56.750
This is-- is that right?
00:43:56.750 --> 00:43:57.250
No.
00:43:57.250 --> 00:43:58.690
Is it?
00:43:58.690 --> 00:43:59.760
Or is it d?
00:43:59.760 --> 00:44:01.350
Maybe I didn't make a mistake.
00:44:01.350 --> 00:44:02.700
Is it d?
00:44:02.700 --> 00:44:06.370
You're seeing a kind
of critical moment.
00:44:06.370 --> 00:44:07.278
Which is it?
00:44:07.278 --> 00:44:09.718
AUDIENCE: [INAUDIBLE].
00:44:09.718 --> 00:44:11.011
PROFESSOR: It's d, isn't it?
00:44:11.011 --> 00:44:11.510
Yeah.
00:44:11.510 --> 00:44:12.220
It's d.
00:44:12.220 --> 00:44:12.720
Sorry.
00:44:12.720 --> 00:44:13.400
It's d.
00:44:15.910 --> 00:44:21.220
But when I see this and remember
what omega n squared is--
00:44:21.220 --> 00:44:25.370
omega n squared is
k over m-- I can
00:44:25.370 --> 00:44:35.390
see that I can get an omega--
I can use this in here
00:44:35.390 --> 00:44:38.420
to make it even
more interesting.
00:44:38.420 --> 00:44:45.590
So it'll be equals-- let me
get this box ready-- cosine
00:44:45.590 --> 00:44:53.170
of omega dt divided by-- now
I just want to rewrite that.
00:44:57.440 --> 00:44:58.924
I want to take out an m.
00:45:01.590 --> 00:45:06.060
I'm going to write this
as m times k over m.
00:45:06.060 --> 00:45:08.780
m times k over m.
00:45:08.780 --> 00:45:10.580
Safe to do that.
00:45:10.580 --> 00:45:13.060
Now I have a factor, m,
that I can bring out.
00:45:13.060 --> 00:45:15.100
And what is m multiplying?
00:45:15.100 --> 00:45:17.710
That's the neat thing.
00:45:17.710 --> 00:45:20.680
What is m multiplying?
00:45:20.680 --> 00:45:25.745
k over m is-- omega n squared.
00:45:28.600 --> 00:45:31.330
And this is minus
m omega d squared.
00:45:31.330 --> 00:45:34.570
Minus omega d squared.
00:45:34.570 --> 00:45:36.460
That's pretty terrific.
00:45:40.120 --> 00:45:47.250
The gain is this multiplier,
1 over m, times that.
00:45:47.250 --> 00:45:50.970
And we see that the gain
is bigger and bigger when
00:45:50.970 --> 00:45:53.760
the frequency is near
the natural frequency.
00:45:53.760 --> 00:45:58.000
And of course everybody has seen
the pictures of that bridge--
00:45:58.000 --> 00:46:00.100
wherever the heck
was that bridge?
00:46:00.100 --> 00:46:02.672
Somewhere in the
Northwest, I think.
00:46:02.672 --> 00:46:04.740
You know the bridge
I'm talking about?
00:46:04.740 --> 00:46:06.484
AUDIENCE: [INAUDIBLE]
Tacoma, Washington.
00:46:06.484 --> 00:46:08.400
PROFESSOR: Yeah, I think
Tacoma, that's right.
00:46:08.400 --> 00:46:09.670
The Tacoma Narrows Bridge.
00:46:09.670 --> 00:46:10.170
Right.
00:46:10.170 --> 00:46:11.690
Tacoma, Washington.
00:46:11.690 --> 00:46:16.950
Where the natural--
when you build a bridge,
00:46:16.950 --> 00:46:19.220
you've built in a
natural frequency.
00:46:19.220 --> 00:46:23.080
And then when traffic comes,
it's doing a driving frequency.
00:46:23.080 --> 00:46:25.930
And if you haven't got
those two well-separated,
00:46:25.930 --> 00:46:29.600
you're in trouble,
as this shows.
00:46:29.600 --> 00:46:38.960
Or similarly, when an
architect designs a skyscraper,
00:46:38.960 --> 00:46:43.250
there's going to be a
frequency of oscillation,
00:46:43.250 --> 00:46:48.170
a natural frequency, at
which that skyscraper swings.
00:46:48.170 --> 00:46:51.230
And then there's wind.
00:46:51.230 --> 00:46:57.870
Actually I talked yesterday
to the-- by chance,
00:46:57.870 --> 00:47:04.440
the math department is not a
very party-going department,
00:47:04.440 --> 00:47:07.510
but once a year we let it out.
00:47:07.510 --> 00:47:11.830
And so we had our party
at Endicott house out
00:47:11.830 --> 00:47:16.490
in the suburbs, and
all the usual people--
00:47:16.490 --> 00:47:20.660
that's all the professors
I know-- came, of course.
00:47:20.660 --> 00:47:23.030
But also, there was
a really cool person.
00:47:23.030 --> 00:47:27.590
He's the key architect
for Building 2.
00:47:27.590 --> 00:47:32.560
You've noticed that
Building 2 is under wraps
00:47:32.560 --> 00:47:34.940
and we're moved out.
00:47:34.940 --> 00:47:39.170
And we move back
in January 2016.
00:47:39.170 --> 00:47:41.730
So we've been out a
year and a quarter
00:47:41.730 --> 00:47:45.150
and we have another year
and a quarter to go.
00:47:45.150 --> 00:47:47.060
It's going to be cool.
00:47:47.060 --> 00:47:49.720
And you may say,
well, who cares.
00:47:49.720 --> 00:47:53.870
But the key point is
Building 1 is next,
00:47:53.870 --> 00:47:57.360
and Building 1 is going to
have the same cool addition
00:47:57.360 --> 00:47:59.150
of a fourth floor.
00:47:59.150 --> 00:48:05.730
We're putting in a fourth floor,
which all the-- Buildings 3, 4,
00:48:05.730 --> 00:48:11.865
5, 6 go up to four, but
Buildings 2 and 1 stopped
00:48:11.865 --> 00:48:12.910
at the third floor.
00:48:12.910 --> 00:48:15.780
But there's a lot of space
up there under the roof.
00:48:15.780 --> 00:48:20.170
And they've
discovered they could
00:48:20.170 --> 00:48:23.090
put a fourth floor up there.
00:48:23.090 --> 00:48:26.330
Here was one interesting
thing, though.
00:48:26.330 --> 00:48:30.670
These buildings that we're
sitting in are sinking.
00:48:30.670 --> 00:48:38.410
You know that MIT was built
on marshy land, just the way
00:48:38.410 --> 00:48:41.030
the Back Bay-- which
is like the greatest
00:48:41.030 --> 00:48:45.510
idea in the history of Boston,
the Back Bay and the dam
00:48:45.510 --> 00:48:50.320
that makes the Charles
River beautiful--
00:48:50.320 --> 00:48:53.820
was built by bringing
in trainloads of earth
00:48:53.820 --> 00:48:54.840
from Needham.
00:48:54.840 --> 00:49:00.020
So whole mountains and hills in
Needham have come into Boston
00:49:00.020 --> 00:49:01.190
and come here.
00:49:01.190 --> 00:49:02.420
So anyway, we're sinking.
00:49:06.160 --> 00:49:10.605
You may say something like
3/16 of an inch a year
00:49:10.605 --> 00:49:14.490
is not something to
worry about, but now it's
00:49:14.490 --> 00:49:20.620
been more than 100 years that
these buildings have been here.
00:49:20.620 --> 00:49:24.650
Anyway, not good to sink faster.
00:49:24.650 --> 00:49:28.170
So the weight had
to be controlled.
00:49:28.170 --> 00:49:31.890
So by putting in a
fourth floor, that
00:49:31.890 --> 00:49:35.950
put in a lot a new weight,
and faster sinking,
00:49:35.950 --> 00:49:39.290
probably by some formula here.
00:49:39.290 --> 00:49:42.340
Probably there.
00:49:42.340 --> 00:49:44.720
So the weight had to
get subtracted out.
00:49:44.720 --> 00:49:48.410
It turns out that
the ceiling, the roof
00:49:48.410 --> 00:49:52.260
to Building 1-- Building 2
and no doubt to Building 1--
00:49:52.260 --> 00:49:56.420
was more than a foot
thick of concrete.
00:49:56.420 --> 00:49:58.150
Really heavy.
00:49:58.150 --> 00:50:02.896
And some more asbestos
probably, which we don't want
00:50:02.896 --> 00:50:03.520
to think about.
00:50:06.040 --> 00:50:07.410
That's much reduced.
00:50:07.410 --> 00:50:10.700
A whole lot of weight
came out of the roof.
00:50:10.700 --> 00:50:14.870
I think they probably did
the calculation right,
00:50:14.870 --> 00:50:17.210
so we won't get
rain coming through,
00:50:17.210 --> 00:50:23.960
but it won't weigh as much and
the fourth floor is acceptable.
00:50:23.960 --> 00:50:28.540
All this was a big decision
by MIT to pay for that,
00:50:28.540 --> 00:50:32.270
or to raise money and pay
for the new fourth floor.
00:50:32.270 --> 00:50:34.030
But it's going to be fantastic.
00:50:34.030 --> 00:50:36.580
And it'll be fantastic
in Building 1 also.
00:50:39.820 --> 00:50:44.240
So all that is discussion
of that formula.
00:50:47.210 --> 00:50:53.650
That's the frequency response,
this factor to frequency,
00:50:53.650 --> 00:50:57.070
omega d, or omega,
is this factor.
00:51:01.510 --> 00:51:08.330
I guess I should say
something about resonance.
00:51:08.330 --> 00:51:13.230
What happens when that
formula breaks down?
00:51:13.230 --> 00:51:20.320
When the driving force
equals the natural frequency,
00:51:20.320 --> 00:51:24.035
then we're dividing by 0,
and something is different.
00:51:24.035 --> 00:51:27.990
The formula isn't right anymore.
00:51:27.990 --> 00:51:32.760
What enters in the formula-- let
me just tell you what enters,
00:51:32.760 --> 00:51:35.600
and then we'll see it
in a simple example.
00:51:38.840 --> 00:51:43.340
When I have this repeated
thing, two things are equal,
00:51:43.340 --> 00:51:45.230
what tends to
happen is a factor,
00:51:45.230 --> 00:51:47.840
an extra factor, t, appears.
00:51:47.840 --> 00:51:54.100
So an extra factor, t, will
appear in the case omega n
00:51:54.100 --> 00:51:55.520
equal omega d.
00:51:58.730 --> 00:52:03.580
The solution, y,
will be some factor,
00:52:03.580 --> 00:52:06.540
I'll still call it g-- no,
I don't want to call it g,
00:52:06.540 --> 00:52:08.560
let me call it a.
00:52:08.560 --> 00:52:14.900
There'll be a factor, t,
times cosine of omega t.
00:52:14.900 --> 00:52:17.470
So in this case, there's
really only one frequency.
00:52:17.470 --> 00:52:18.590
We're driving it.
00:52:18.590 --> 00:52:22.990
So the oscillation grows.
00:52:22.990 --> 00:52:28.400
As you know, when you
push a child on a swing,
00:52:28.400 --> 00:52:31.090
the whole point of
pushing that child
00:52:31.090 --> 00:52:35.240
is to push at the
natural frequency.
00:52:35.240 --> 00:52:38.860
You wait for this swing
to swing back naturally
00:52:38.860 --> 00:52:43.110
and you drive it again
with that-- at that--
00:52:43.110 --> 00:52:44.600
maintain that frequency.
00:52:44.600 --> 00:52:47.390
And of course you
see the amplitude--
00:52:47.390 --> 00:52:50.110
the child swing
higher and higher.
00:52:50.110 --> 00:52:59.380
Presumably you stop pushing
before disaster for that child.
00:52:59.380 --> 00:53:03.190
But that's a case of resonance.
00:53:03.190 --> 00:53:06.600
And it's what happened in
the Tacoma Narrows Bridge,
00:53:06.600 --> 00:53:11.210
and there was nothing
to-- nobody stopped,
00:53:11.210 --> 00:53:12.960
traffic just kept coming.
00:53:12.960 --> 00:53:14.650
The movie is amazing,
because there's
00:53:14.650 --> 00:53:19.730
one car that shows up after
it's already swinging wildly,
00:53:19.730 --> 00:53:23.620
some crazy person
still driving across.
00:53:23.620 --> 00:53:28.310
And you might think, OK,
that's ancient history.
00:53:28.310 --> 00:53:30.030
But you know the
bridge in London,
00:53:30.030 --> 00:53:34.610
the pedestrian bridge, the
Millennium Bridge-- it's just
00:53:34.610 --> 00:53:39.440
a walking bridge
across the Thames--
00:53:39.440 --> 00:53:44.280
a big feature of modern London,
and it had the same problem.
00:53:44.280 --> 00:53:45.030
It was swaying.
00:53:45.030 --> 00:53:47.520
People could not walk across.
00:53:47.520 --> 00:53:48.940
They couldn't keep
their balance.
00:53:48.940 --> 00:53:51.010
So they had change it.
00:53:51.010 --> 00:53:56.845
So it's not trivial
to anticipate.
00:54:02.030 --> 00:54:07.060
So now we've solved it-- we've
solved the the null equation
00:54:07.060 --> 00:54:14.620
with no force, and
we've solved the driving
00:54:14.620 --> 00:54:16.580
force equal to a cosine.
00:54:16.580 --> 00:54:19.780
And of course, we
could do a sine.
00:54:19.780 --> 00:54:23.880
What other driving
force should we do?
00:54:23.880 --> 00:54:27.690
I think we should
do a delta function.
00:54:27.690 --> 00:54:35.340
I think we have to understand
the fundamental solution is
00:54:35.340 --> 00:54:37.890
the case when, if
we can solve it--
00:54:37.890 --> 00:54:40.910
there's always this general
rule, if we can solve
00:54:40.910 --> 00:54:43.840
with a delta function,
that will give us
00:54:43.840 --> 00:54:49.300
a formula for every
driving force,
00:54:49.300 --> 00:54:53.880
because every function is some
combination of delta functions.
00:54:53.880 --> 00:54:57.130
So if we could do
it with a delta--
00:54:57.130 --> 00:54:59.830
really the great
right hand sides
00:54:59.830 --> 00:55:04.660
are-- well, cosines
and sines I'll
00:55:04.660 --> 00:55:06.910
include as great
right hand sides.
00:55:06.910 --> 00:55:09.820
Those are the
exponentials in disguise.
00:55:09.820 --> 00:55:11.590
So the great right
hand sides are
00:55:11.590 --> 00:55:17.660
really exponentials at
different frequencies and delta
00:55:17.660 --> 00:55:19.350
functions.
00:55:19.350 --> 00:55:20.830
Delta of impulses.
00:55:20.830 --> 00:55:23.890
So now I want to find
the impulse response.
00:55:23.890 --> 00:55:29.006
That's the next--
that's really a job.
00:55:31.900 --> 00:55:35.040
At this point, in these
last 20 minutes when
00:55:35.040 --> 00:55:46.310
I solve my double prime plus ky
equal a delta function-- well,
00:55:46.310 --> 00:55:51.760
what I was going to say was
I'm now taking you to something
00:55:51.760 --> 00:55:56.240
that you won't see on
the exam this afternoon.
00:55:56.240 --> 00:55:59.680
But maybe you will.
00:55:59.680 --> 00:56:02.180
Delta function, right hand side.
00:56:02.180 --> 00:56:03.810
I haven't seen it yet.
00:56:03.810 --> 00:56:05.070
Or I haven't looked recently.
00:56:05.070 --> 00:56:09.710
You won't see second
derivatives, I guess.
00:56:09.710 --> 00:56:10.585
So what is it?
00:56:15.760 --> 00:56:20.880
So now this is of the
form with an f of t,
00:56:20.880 --> 00:56:23.060
a very special f of t.
00:56:23.060 --> 00:56:26.960
And that very special f of
t makes that extremely easy
00:56:26.960 --> 00:56:27.460
to solve.
00:56:31.670 --> 00:56:34.550
That's really my
point here, is it
00:56:34.550 --> 00:56:36.770
it's going to be a
cinch to solve that,
00:56:36.770 --> 00:56:38.770
and we practically
have done it already
00:56:38.770 --> 00:56:43.900
to solve that with
a delta function.
00:56:43.900 --> 00:56:51.570
And the reason is
sort of physical.
00:56:51.570 --> 00:56:55.630
We have here our spring.
00:56:55.630 --> 00:56:59.540
And what am I doing
with that force?
00:56:59.540 --> 00:57:02.755
I'm hitting the mass.
00:57:02.755 --> 00:57:03.970
I'm striking the mass.
00:57:09.170 --> 00:57:12.840
Let me say, and I'll write
it on the board, the point I
00:57:12.840 --> 00:57:15.920
want to make about this.
00:57:15.920 --> 00:57:20.710
That point is that this
equation with a delta function
00:57:20.710 --> 00:57:27.930
force starting from 0-- say,
y of 0 equal y prime of 0
00:57:27.930 --> 00:57:33.560
equals 0, let's give
it starting from rest--
00:57:33.560 --> 00:57:37.090
it starts from
rest by hitting it.
00:57:37.090 --> 00:57:46.050
And that hit, that impulse,
is in no time at all.
00:57:46.050 --> 00:57:47.580
It's not stretched out.
00:57:47.580 --> 00:57:49.650
It's hit over one second.
00:57:49.650 --> 00:57:51.890
So this has the same solution.
00:57:51.890 --> 00:57:53.860
This is the beauty.
00:57:53.860 --> 00:57:57.930
This is why we can
solve it so easily.
00:57:57.930 --> 00:58:08.110
Same solution as-- let me write
it and see what you think--
00:58:08.110 --> 00:58:13.420
as my double prime
plus ky equal 0.
00:58:13.420 --> 00:58:15.370
We know how to solve those.
00:58:15.370 --> 00:58:19.860
With-- it's still, when
I hit it-- when I hit it,
00:58:19.860 --> 00:58:23.250
what happens in
that split second?
00:58:23.250 --> 00:58:26.690
In that split second, it
doesn't have time to move.
00:58:26.690 --> 00:58:27.600
It doesn't move.
00:58:27.600 --> 00:58:31.420
It still has y of 0 equals 0.
00:58:31.420 --> 00:58:36.640
But in that split second,
we've given it a velocity.
00:58:36.640 --> 00:58:38.710
We've given it a velocity.
00:58:38.710 --> 00:58:43.120
And that velocity
will be y prime.
00:58:43.120 --> 00:58:48.610
The initial velocity is 1--
because here I had a 1--
00:58:48.610 --> 00:58:50.300
over an m.
00:58:50.300 --> 00:58:52.380
We have to have the units right.
00:58:59.740 --> 00:59:03.210
So here's a point, and
we will stay with it.
00:59:03.210 --> 00:59:05.860
We'll come back to
this point next time.
00:59:05.860 --> 00:59:09.000
Maybe the first thing
for you to take in
00:59:09.000 --> 00:59:14.280
is the fact that it's
such a nice thing.
00:59:14.280 --> 00:59:18.330
We have this equation with
this mysterious delta function,
00:59:18.330 --> 00:59:20.210
and I'm saying that
the solution is
00:59:20.210 --> 00:59:25.820
the same as this
equation with no force,
00:59:25.820 --> 00:59:27.880
but starting from a mass.
00:59:31.510 --> 00:59:36.670
I'm tempted to take an
example to make this point.
00:59:36.670 --> 00:59:42.010
Let me take an example where the
whole thing is a lot simpler.
00:59:42.010 --> 00:59:44.325
y double prime equal delta of t.
00:59:47.670 --> 00:59:52.230
I've taken the spring away, so
the k is gone, the mass is 1.
00:59:52.230 --> 00:59:55.410
What's the solution to y
double prime equal delta of t?
00:59:55.410 --> 01:00:02.660
If we concentrate on this
example, we're good for today.
01:00:02.660 --> 01:00:09.790
So my point is the
same solution as-- now,
01:00:09.790 --> 01:00:12.090
what's the other problem?
01:00:12.090 --> 01:00:16.490
I'm just repeating here, but
making it simple by taking k
01:00:16.490 --> 01:00:18.410
equals 0 and m equal 1.
01:00:18.410 --> 01:00:25.070
So the same solution as y double
prime equals 0, with y of 0
01:00:25.070 --> 01:00:29.005
equal what, and y
prime of 0 equal what.
01:00:31.830 --> 01:00:36.570
I just wanted to repeat
here what I've said there,
01:00:36.570 --> 01:00:40.930
and then we'll solve it and
we'll see that it's all true.
01:00:40.930 --> 01:00:48.190
If I look for a solution to
y double prime equal delta
01:00:48.190 --> 01:00:58.780
starting from 0-- this
was starting from 0--
01:00:58.780 --> 01:01:03.740
if I say that's the same
as this, what should y of 0
01:01:03.740 --> 01:01:04.240
be here?
01:01:04.240 --> 01:01:04.912
AUDIENCE: Zero.
01:01:04.912 --> 01:01:05.870
PROFESSOR: Zero, right.
01:01:05.870 --> 01:01:09.040
It hasn't had time to move.
01:01:09.040 --> 01:01:10.500
It hasn't had time to move.
01:01:10.500 --> 01:01:15.250
But in that instant,
what happened to y prime?
01:01:15.250 --> 01:01:17.360
It jumped to 1.
01:01:17.360 --> 01:01:18.499
That's right.
01:01:18.499 --> 01:01:19.040
That's right.
01:01:19.040 --> 01:01:20.570
Exactly.
01:01:20.570 --> 01:01:24.700
Now just solve that
equation for me.
01:01:24.700 --> 01:01:25.850
Solve this example for me.
01:01:29.340 --> 01:01:31.450
Suppose y double prime-- yeah.
01:01:31.450 --> 01:01:32.370
Here we go.
01:01:32.370 --> 01:01:35.940
What's the solution if
y double prime is 0?
01:01:35.940 --> 01:01:42.230
What are the solutions to
y double prime equals 0?
01:01:42.230 --> 01:01:43.640
AUDIENCE: [INAUDIBLE].
01:01:43.640 --> 01:01:47.100
PROFESSOR: Constant and linear.
01:01:47.100 --> 01:01:50.620
a plus bt, right, have
second derivative 0.
01:01:50.620 --> 01:01:57.250
Now what's the solution that
starts from 0 that kills the a
01:01:57.250 --> 01:01:58.600
and has slope 1?
01:01:58.600 --> 01:02:00.274
What's the answer
to that question?
01:02:00.274 --> 01:02:01.190
AUDIENCE: [INAUDIBLE].
01:02:01.190 --> 01:02:02.460
PROFESSOR: t.
01:02:02.460 --> 01:02:03.150
t.
01:02:03.150 --> 01:02:08.370
The solution to this
equation is a ramp.
01:02:08.370 --> 01:02:12.310
It's zero everything in this
course, is zero up until time
01:02:12.310 --> 01:02:13.780
0.
01:02:13.780 --> 01:02:19.290
At time 0, in this example,
all the action happens.
01:02:19.290 --> 01:02:20.730
Everything happens.
01:02:20.730 --> 01:02:25.500
And what happens is it
gets a velocity of 1,
01:02:25.500 --> 01:02:30.640
and the solution
is y equal to t.
01:02:30.640 --> 01:02:33.160
y is 0 here, of course.
01:02:33.160 --> 01:02:36.590
At that point, that's the
key point, t equals 0,
01:02:36.590 --> 01:02:42.040
right there-- it gets a slope.
01:02:45.430 --> 01:02:47.500
We don't have a step function.
01:02:47.500 --> 01:02:49.520
There's no jump in y.
01:02:49.520 --> 01:02:53.460
The jump is in y prime, the
y prime the velocity jumped
01:02:53.460 --> 01:02:55.980
from 0 to 1.
01:02:55.980 --> 01:03:00.030
That's exactly-- I think when
I introduced delta functions
01:03:00.030 --> 01:03:02.180
and drew a picture.
01:03:02.180 --> 01:03:05.440
What is the derivative, the
first derivative, y prime,
01:03:05.440 --> 01:03:06.360
for that guy?
01:03:06.360 --> 01:03:11.050
Let's just review, because this
is what we've seen already.
01:03:11.050 --> 01:03:12.245
The first derivative is--
01:03:12.245 --> 01:03:13.370
AUDIENCE: [INAUDIBLE] step.
01:03:13.370 --> 01:03:14.790
PROFESSOR: A step.
01:03:14.790 --> 01:03:19.930
And the second
derivative is delta.
01:03:19.930 --> 01:03:24.080
The second derivative of this is
the first derivative of a step.
01:03:24.080 --> 01:03:30.050
The derivative of a step is 0
everywhere except at the step,
01:03:30.050 --> 01:03:32.200
at the jump when it jumps to 1.
01:03:32.200 --> 01:03:36.120
So that's the solution
in this example.
01:03:36.120 --> 01:03:41.485
And now to end the lecture,
let's solve it in this example.
01:03:44.200 --> 01:03:52.940
Again, let me just say-- why
do I like this forcing term?
01:03:52.940 --> 01:03:55.230
Mathematically, I
like it because, if I
01:03:55.230 --> 01:03:58.050
can solve that guy--
as we're doing,
01:03:58.050 --> 01:04:00.940
we are solving it-- if
I can solve that one,
01:04:00.940 --> 01:04:04.140
I can solve all forces.
01:04:04.140 --> 01:04:10.250
Over here, I could solve when
I had a very happy f of t,
01:04:10.250 --> 01:04:13.965
a perfect f of t, where
I could guess the answer
01:04:13.965 --> 01:04:16.070
and push through.
01:04:16.070 --> 01:04:20.380
Now with a delta, I can
build everything out
01:04:20.380 --> 01:04:21.570
of delta functions.
01:04:21.570 --> 01:04:24.270
That's why I like
it mathematically.
01:04:24.270 --> 01:04:25.760
Why do I like it physically?
01:04:25.760 --> 01:04:32.330
Because it's a very physical
thing to have an impulse.
01:04:32.330 --> 01:04:35.610
That happens in real
time, in real things.
01:04:35.610 --> 01:04:40.010
And by the way, let's just,
before I write down any more
01:04:40.010 --> 01:04:46.420
formula, what
would-- I would like
01:04:46.420 --> 01:04:51.300
to be able to solve it
for a step function.
01:04:51.300 --> 01:04:55.599
[? Heavy thud. ?] I would like
to be able to do that one.
01:04:55.599 --> 01:04:57.140
I'm going to have
to erase something,
01:04:57.140 --> 01:04:59.650
or I'll write it right
above just for the moment.
01:04:59.650 --> 01:05:07.680
I would also like to
solve my double prime plus
01:05:07.680 --> 01:05:10.309
ky equal a step function.
01:05:18.820 --> 01:05:24.360
So I would call the solution,
y, the step response.
01:05:24.360 --> 01:05:26.600
And what would be a
step function start?
01:05:26.600 --> 01:05:30.040
A step function start would
be like turning a switch.
01:05:30.040 --> 01:05:32.390
Suddenly things happen.
01:05:32.390 --> 01:05:40.030
That's forcing by a step, so I'm
looking for the step response.
01:05:40.030 --> 01:05:42.940
And how do you think
these two are related?
01:05:47.140 --> 01:05:50.160
I look at the relation
at the right hand sides.
01:05:50.160 --> 01:05:56.966
What's the relation of
this step to the delta?
01:05:56.966 --> 01:05:57.465
Yeah?
01:05:57.465 --> 01:05:58.673
AUDIENCE: One's a derivative.
01:05:58.673 --> 01:06:00.870
PROFESSOR: One's a
derivative of the other.
01:06:00.870 --> 01:06:03.320
And we've got linear equations.
01:06:03.320 --> 01:06:06.240
So the right hand sides.
01:06:06.240 --> 01:06:12.310
The step response, y step,
and the delta response,
01:06:12.310 --> 01:06:16.320
y delta-- I'll use a
different letter for this
01:06:16.320 --> 01:06:19.570
because it's so important.
01:06:19.570 --> 01:06:21.270
One is the derivative
of the other.
01:06:24.280 --> 01:06:26.040
The great thing about
linear equations
01:06:26.040 --> 01:06:29.950
is we have linear equations,
differentiation, integration.
01:06:29.950 --> 01:06:31.180
Those are linear operations.
01:06:34.460 --> 01:06:38.990
The step function is just
like a steady-- anyway.
01:06:38.990 --> 01:06:42.400
I was going to-- I won't--
is the integral of the delta.
01:06:42.400 --> 01:06:44.810
Step function is the
integral of the delta,
01:06:44.810 --> 01:06:48.205
so the step response is the
integral of the delta response.
01:06:51.220 --> 01:06:54.660
I guess to finish the
lecture, why don't we
01:06:54.660 --> 01:06:57.020
solve this problem,
which looks tricky
01:06:57.020 --> 01:06:59.030
because it's got a delta.
01:06:59.030 --> 01:07:01.240
Instead, we'll
solve this problem,
01:07:01.240 --> 01:07:03.130
which doesn't look
tricky at all.
01:07:03.130 --> 01:07:06.510
It's exactly what we
started the lecture with.
01:07:06.510 --> 01:07:10.320
Zero forcing and some
initial conditions.
01:07:10.320 --> 01:07:13.840
So let me just
finally make space
01:07:13.840 --> 01:07:16.690
for the big deal
from today's lecture,
01:07:16.690 --> 01:07:25.810
which would be the fundamental
solution with a force
01:07:25.810 --> 01:07:27.900
by a delta.
01:07:27.900 --> 01:07:30.310
I'm just going to write
down the answer when
01:07:30.310 --> 01:07:31.270
you tell me what it is.
01:07:39.214 --> 01:07:41.000
What's the answer to that?
01:07:41.000 --> 01:07:44.120
What's the solution
to this second order
01:07:44.120 --> 01:07:48.130
constant coefficient
unforced equation
01:07:48.130 --> 01:07:50.235
with those initial conditions?
01:07:53.630 --> 01:07:55.160
We probably had it here.
01:07:55.160 --> 01:07:56.950
I may just have erased it.
01:07:56.950 --> 01:07:58.950
But now let's get it.
01:07:58.950 --> 01:08:02.350
So y is y delta.
01:08:02.350 --> 01:08:05.220
This is the impulse response.
01:08:12.350 --> 01:08:15.895
y of t-- and I'll give
it later another name.
01:08:18.790 --> 01:08:24.319
So here's a perfect
review question.
01:08:24.319 --> 01:08:28.020
What's the solution
to this problem?
01:08:28.020 --> 01:08:30.740
Everybody remembers-- what
are the solutions, what's
01:08:30.740 --> 01:08:34.370
the general form for the
solution to the equation?
01:08:34.370 --> 01:08:37.080
I'm reviewing today's lecture.
01:08:37.080 --> 01:08:40.619
The solution to that
equation looks like what?
01:08:40.619 --> 01:08:42.585
AUDIENCE: [INAUDIBLE].
01:08:42.585 --> 01:08:46.930
PROFESSOR: It's a cosine
and a sine, right.
01:08:46.930 --> 01:08:52.050
And then how much of a cosine do
we have and how much of a sine
01:08:52.050 --> 01:08:53.120
do we have?
01:08:53.120 --> 01:08:58.189
The initial condition will
tell me how much of a cosine
01:08:58.189 --> 01:08:59.590
we have.
01:08:59.590 --> 01:09:01.680
And what's the answer?
01:09:01.680 --> 01:09:03.270
None?
01:09:03.270 --> 01:09:05.090
No cosine.
01:09:05.090 --> 01:09:08.399
This condition, this
initial velocity,
01:09:08.399 --> 01:09:10.729
will tell me how much
of a sine we have,
01:09:10.729 --> 01:09:15.399
because the sines are the things
that have initial velocities.
01:09:15.399 --> 01:09:21.859
So it would be a sine
of-- the sine of what?
01:09:21.859 --> 01:09:25.670
Square root of k over m, right?
01:09:25.670 --> 01:09:28.359
omega nt, right?
01:09:28.359 --> 01:09:30.320
And what's the number?
01:09:33.575 --> 01:09:40.170
What's the number so
this has the right-- let
01:09:40.170 --> 01:09:41.580
me write again what I want.
01:09:41.580 --> 01:09:45.500
I want y prime at
0 to be 1 over m.
01:09:52.450 --> 01:09:54.320
What's the number
that I put in there?
01:09:54.320 --> 01:09:56.990
I've got something, its
derivative, at zero.
01:09:59.630 --> 01:10:04.540
This is some number-- I'll call
it little a for the moment,
01:10:04.540 --> 01:10:06.110
but I want to find
out what it is.
01:10:10.630 --> 01:10:11.250
Are we right?
01:10:11.250 --> 01:10:12.436
Yeah?
01:10:12.436 --> 01:10:13.328
I think we're right.
01:10:15.996 --> 01:10:16.496
Yeah.
01:10:19.600 --> 01:10:25.360
The derivative is at zero, so
I just plug that into here,
01:10:25.360 --> 01:10:27.760
take the derivative at zero--
of course that makes it
01:10:27.760 --> 01:10:32.400
a cosine, which will be 1-- but
it also brings out that factor.
01:10:32.400 --> 01:10:38.140
So a times-- well, that
factor will be 1 over m,
01:10:38.140 --> 01:10:41.700
and that tells me
what a has to be.
01:10:41.700 --> 01:10:43.830
Well, this is omega.
01:10:43.830 --> 01:10:47.020
So a is-- this is
omega a equal m.
01:10:47.020 --> 01:10:51.460
This is 1 over m omega.
01:10:51.460 --> 01:10:52.445
And that's omega.
01:10:57.420 --> 01:11:03.070
Sorry I'm erasing
stuff which I-- this is
01:11:03.070 --> 01:11:04.990
the formula I'm after.
01:11:04.990 --> 01:11:06.790
Sine omega t over omega.
01:11:12.034 --> 01:11:12.825
I think we're good.
01:11:12.825 --> 01:11:13.880
Are we?
01:11:13.880 --> 01:11:14.440
Yeah?
01:11:14.440 --> 01:11:15.280
Yeah.
01:11:15.280 --> 01:11:20.880
I'll come back to
this in-- Wednesday
01:11:20.880 --> 01:11:25.310
is my day to move
to damping terms.
01:11:25.310 --> 01:11:32.490
I've intentionally stayed
with undamped equations
01:11:32.490 --> 01:11:37.310
here, because you're thinking
about that level of equation.
01:11:37.310 --> 01:11:40.320
Damping is going to
bring in new stuff,
01:11:40.320 --> 01:11:43.850
and that should
wait till Wednesday.
01:11:43.850 --> 01:11:46.550
Shall I recap today?
01:11:46.550 --> 01:11:50.540
I'll just recap today,
and then we're done.
01:11:50.540 --> 01:11:56.770
Today started with
the unforced equation.
01:12:01.370 --> 01:12:06.660
We solved it by assuming--
by not thinking ahead,
01:12:06.660 --> 01:12:09.920
just assume I have
an exponential,
01:12:09.920 --> 01:12:11.620
because the beauty
of exponentials
01:12:11.620 --> 01:12:17.020
is, when I plug it in,
the exponential cancels.
01:12:17.020 --> 01:12:20.080
And that told me that
s was pure imaginary.
01:12:20.080 --> 01:12:23.920
It told me that it had this
form, e to the i omega t.
01:12:23.920 --> 01:12:25.420
And there were two s's.
01:12:25.420 --> 01:12:27.495
Two possible s's,
plus and minus.
01:12:30.730 --> 01:12:37.072
I get to make a little comment
about this example here.
01:12:42.970 --> 01:12:43.910
What was omega?
01:12:43.910 --> 01:12:49.010
What's the natural
frequency in this problem?
01:12:49.010 --> 01:12:50.970
What's the natural
frequency here?
01:12:53.970 --> 01:13:00.510
I guess this is a case where--
what's the natural frequency?
01:13:00.510 --> 01:13:05.900
I guess this is a case where m
is 1 and k is 0, is that right?
01:13:05.900 --> 01:13:10.540
This does fit into that pattern,
but it's a little special.
01:13:10.540 --> 01:13:13.590
This is a case where
m is 1 and k is 0.
01:13:13.590 --> 01:13:15.320
So what's the natural
frequency in this?
01:13:15.320 --> 01:13:16.200
AUDIENCE: Zero.
01:13:16.200 --> 01:13:18.320
PROFESSOR: Zero.
01:13:18.320 --> 01:13:20.100
Zero.
01:13:20.100 --> 01:13:25.880
This is a crazy
case of resonance.
01:13:25.880 --> 01:13:29.130
It's a case in which the natural
frequency and the driving
01:13:29.130 --> 01:13:35.440
frequency, say in this--
I'll have to do it here--
01:13:35.440 --> 01:13:40.720
this simplest of all equations
is, in a way, special.
01:13:40.720 --> 01:13:46.910
It's a case when the
natural frequency is zero
01:13:46.910 --> 01:13:51.900
and the driving frequency
is zero and they're equal.
01:13:51.900 --> 01:13:56.170
And what happens with resonance?
01:13:56.170 --> 01:14:00.010
What's the new
formula, the new term
01:14:00.010 --> 01:14:02.550
that comes in with resonance?
01:14:02.550 --> 01:14:03.250
It's t.
01:14:07.410 --> 01:14:09.510
You saw it happen
for this example,
01:14:09.510 --> 01:14:11.970
and we didn't have to
use the word resonance.
01:14:11.970 --> 01:14:15.460
We knew that we had a ramp.
01:14:15.460 --> 01:14:18.370
We just used the word
ramp, not resonance.
01:14:18.370 --> 01:14:21.280
But this is a case of resonance.
01:14:21.280 --> 01:14:24.880
When omega n is zero
and omega d is zero.
01:14:27.570 --> 01:14:30.790
And the factor t up here.
01:14:30.790 --> 01:14:31.350
Anyway.
01:14:31.350 --> 01:14:34.040
Just that small comment there.
01:14:34.040 --> 01:14:37.220
And now, just going
back to the recap.
01:14:37.220 --> 01:14:41.070
The recap was, we
tried exponentials.
01:14:41.070 --> 01:14:44.050
We learned that they
were pure oscillations.
01:14:44.050 --> 01:14:49.760
We realized that we could do
cosines and sines instead,
01:14:49.760 --> 01:14:50.780
and we did.
01:14:50.780 --> 01:14:52.840
And we took off.
01:14:52.840 --> 01:14:56.170
We got the formula.
01:14:56.170 --> 01:15:02.760
Then of course the-- so this
is section 2.1 of the book.
01:15:02.760 --> 01:15:08.290
And it goes through all
those steps carefully.
01:15:08.290 --> 01:15:12.290
Section 2.2 of the book tells
us about complex numbers,
01:15:12.290 --> 01:15:15.510
and section 2.3
brings damping in.
01:15:15.510 --> 01:15:18.730
So that's what's
coming next time.
01:15:18.730 --> 01:15:20.480
So the recap again.
01:15:20.480 --> 01:15:26.810
We found the null solution, we
found a particular solution--
01:15:26.810 --> 01:15:28.950
oh there's just one
comment I want to make,
01:15:28.950 --> 01:15:29.896
and then I'm done.
01:15:33.210 --> 01:15:36.160
Where was our
particular solution?
01:15:40.170 --> 01:15:41.180
Yeah.
01:15:41.180 --> 01:15:42.860
This was our
particular solution.
01:15:47.940 --> 01:15:49.940
Here's my comment.
01:15:49.940 --> 01:15:52.190
Here's my comment.
01:15:52.190 --> 01:15:56.720
Suppose I want to solve
this basic equation starting
01:15:56.720 --> 01:16:01.720
from a given y of 0
and a y prime of 0.
01:16:01.720 --> 01:16:03.770
I'm going to do it in
two parts, I think.
01:16:03.770 --> 01:16:06.430
I've got the null
solution, and I've
01:16:06.430 --> 01:16:10.080
got this particular solution.
01:16:10.080 --> 01:16:15.450
Now here's my point.
01:16:15.450 --> 01:16:22.760
If I want to get y of
0-- how shall I say this.
01:16:22.760 --> 01:16:25.390
You can't just put together--
it's an easy mistake
01:16:25.390 --> 01:16:29.500
to make-- solve
the null equation
01:16:29.500 --> 01:16:33.410
with the initial
conditions and then
01:16:33.410 --> 01:16:35.510
add in the particular solution.
01:16:35.510 --> 01:16:39.800
You'd think, I just
followed all the rules.
01:16:39.800 --> 01:16:46.360
But this particular solution
that you added in has a-- at t
01:16:46.360 --> 01:16:51.120
equals 0, it's not zero.
01:16:51.120 --> 01:16:53.000
So you have to change.
01:16:53.000 --> 01:17:00.400
So the correct thing,
the correct yp plus yn--
01:17:00.400 --> 01:17:04.720
let me make that point.
01:17:04.720 --> 01:17:05.390
Just a warning.
01:17:10.670 --> 01:17:12.740
So in words the
warning is, remember
01:17:12.740 --> 01:17:16.360
that the particular solution
has some initial condition--
01:17:16.360 --> 01:17:19.380
in that case, g-- and
then that is going
01:17:19.380 --> 01:17:23.690
to affect the right
null solution.
01:17:23.690 --> 01:17:31.190
So again, y is y null
plus y particular-- plus y
01:17:31.190 --> 01:17:40.670
of particular-- so it's some c1
cos omega nt plus some c2 sine
01:17:40.670 --> 01:17:47.748
omega nt plus this particular
guy, g, cosine of omega dt.
01:17:52.900 --> 01:17:53.990
All correct.
01:17:53.990 --> 01:17:55.610
All correct.
01:17:55.610 --> 01:17:58.190
But now, put in the
initial conditions.
01:17:58.190 --> 01:17:59.612
y of 0 is given.
01:18:02.930 --> 01:18:05.760
And what do I get on
the right hand side
01:18:05.760 --> 01:18:08.860
when I put in t equals 0?
01:18:08.860 --> 01:18:13.810
I get c1 here.
01:18:13.810 --> 01:18:17.020
What do I get when I
put t equals 0 in there?
01:18:17.020 --> 01:18:17.800
Nothing.
01:18:17.800 --> 01:18:21.043
What do I get when I
put t equals 0 in here?
01:18:21.043 --> 01:18:21.543
g.
01:18:26.360 --> 01:18:28.860
So it's not c1 equal
y of 0 anymore.
01:18:28.860 --> 01:18:33.440
That's the easy mistake
that I'm correcting.
01:18:33.440 --> 01:18:41.640
When you put in this
particular solution,
01:18:41.640 --> 01:18:44.830
it has an initial value.
01:18:44.830 --> 01:18:47.980
That initial value is
going to come in here.
01:18:47.980 --> 01:18:55.240
So c1, then, the correct
c1 is y of 0 minus g.
01:18:55.240 --> 01:18:56.230
End of story.
01:18:56.230 --> 01:19:02.140
Just don't be too quick to just
add the two pieces and think
01:19:02.140 --> 01:19:04.700
you can do them
completely separately,
01:19:04.700 --> 01:19:06.657
because you're
putting them together.
01:19:06.657 --> 01:19:08.240
And then you have
to put them together
01:19:08.240 --> 01:19:10.390
in the initial condition.