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PROFESSOR: So morning.
00:00:23.150 --> 00:00:29.810
This is my fourth lecture
on differential equations,
00:00:29.810 --> 00:00:31.230
that part of the course.
00:00:31.230 --> 00:00:35.690
And I haven't said anything
about the textbook.
00:00:35.690 --> 00:00:38.810
That's Differential
Equations and Linear Algebra.
00:00:38.810 --> 00:00:44.310
I wrote it because so
many courses like this one
00:00:44.310 --> 00:00:47.930
want to combine
those two topics.
00:00:47.930 --> 00:00:51.760
They're the two major topics
of undergraduate math,
00:00:51.760 --> 00:00:56.220
after calculus, the two major
directions, and the book
00:00:56.220 --> 00:00:57.735
connects those two directions.
00:00:57.735 --> 00:01:05.300
And I just wanted to give
you the website for lots
00:01:05.300 --> 00:01:07.110
of things connected
with the book.
00:01:07.110 --> 00:01:12.340
So it's the math website, dela
for differential equations
00:01:12.340 --> 00:01:14.270
and linear algebra.
00:01:14.270 --> 00:01:23.370
And so today is differential
equations, second order,
00:01:23.370 --> 00:01:29.610
with a damping term, with
a first derivative term.
00:01:29.610 --> 00:01:37.020
So that in many engineering
problems, those coefficients
00:01:37.020 --> 00:01:41.420
A, B, C would have
the meaning of mass,
00:01:41.420 --> 00:01:45.880
damping, and stiffness.
00:01:45.880 --> 00:01:47.410
Mass, damping, and stiffness.
00:01:47.410 --> 00:01:52.910
And physically, we know
what the mass comes from.
00:01:52.910 --> 00:01:57.640
The stiffness comes from a
spring, as I drew before.
00:01:57.640 --> 00:02:02.660
And traditionally we
describe the source
00:02:02.660 --> 00:02:05.920
of damping as a dashpot.
00:02:05.920 --> 00:02:09.320
I guess in my whole life
I've never seen a dashpot,
00:02:09.320 --> 00:02:15.770
but maybe it's-- think of a
piston going up and down within
00:02:15.770 --> 00:02:20.770
a cylinder of oil or
something, with resistance.
00:02:20.770 --> 00:02:25.230
OK, so that's the
left-hand side.
00:02:25.230 --> 00:02:28.610
Linear constant
coefficients still.
00:02:28.610 --> 00:02:35.910
We don't have formulas,
it's not easy to see what's
00:02:35.910 --> 00:02:38.620
happening when
things are varying,
00:02:38.620 --> 00:02:43.100
when the equations non-linear,
so this is the starting place.
00:02:43.100 --> 00:02:45.380
OK.
00:02:45.380 --> 00:02:46.650
With some forcing.
00:02:46.650 --> 00:02:50.650
So this is damped forced motion.
00:02:50.650 --> 00:02:54.230
This is the ultimate
within linear equations.
00:02:54.230 --> 00:02:54.980
OK.
00:02:54.980 --> 00:02:56.900
And now, what's the forcing?
00:02:56.900 --> 00:03:04.770
So now, always we solve
first for the null solution.
00:03:04.770 --> 00:03:09.200
With no force, what are
the natural motions?
00:03:09.200 --> 00:03:12.080
And we'll find a
formula for those.
00:03:12.080 --> 00:03:16.760
Then the big one, the
special right-hand side
00:03:16.760 --> 00:03:18.165
is always an exponential.
00:03:20.670 --> 00:03:24.590
So this is going to be y is
going to be the null solution
00:03:24.590 --> 00:03:28.190
yn, and we'll get
a formula for that.
00:03:28.190 --> 00:03:31.000
This, the exponentials.
00:03:31.000 --> 00:03:36.410
Always, the response to an
exponential is an exponential.
00:03:36.410 --> 00:03:40.890
That's like the most important
fact in getting solutions.
00:03:40.890 --> 00:03:46.050
So the response will
be some ye to the st
00:03:46.050 --> 00:03:50.960
at the same frequency--
well, I say frequency.
00:03:50.960 --> 00:03:55.470
If s is a real number that
would be a growth or a decay,
00:03:55.470 --> 00:03:58.540
but very frequently s
is an imaginary number,
00:03:58.540 --> 00:04:00.120
like last time.
00:04:00.120 --> 00:04:06.080
And this is comes from a
rotation or an oscillation.
00:04:06.080 --> 00:04:11.210
And then, the
complete picture comes
00:04:11.210 --> 00:04:14.590
from being able to solve
it with an impulse.
00:04:14.590 --> 00:04:17.560
So then y is the
impulse response,
00:04:17.560 --> 00:04:23.320
which I write as g of t.
00:04:23.320 --> 00:04:25.940
Let me introduce
just that letter g.
00:04:25.940 --> 00:04:30.120
That stands for growth factor
in first order equations
00:04:30.120 --> 00:04:35.150
stands for Green's
function, so that word Green
00:04:35.150 --> 00:04:37.310
is getting in here, his name.
00:04:37.310 --> 00:04:43.170
And it represents
the impulse response.
00:04:43.170 --> 00:04:46.610
OK, one, two, three.
00:04:46.610 --> 00:04:52.250
And then the point of doing this
one is that then we can do it,
00:04:52.250 --> 00:04:55.270
then we can get a
formula for any f of t.
00:04:55.270 --> 00:04:57.710
So this is the pattern
that we followed
00:04:57.710 --> 00:05:00.620
for first order equations.
00:05:00.620 --> 00:05:02.980
We followed it for
second order equations
00:05:02.980 --> 00:05:04.790
that didn't have damping.
00:05:04.790 --> 00:05:10.630
And now we're doing the
big one with damping.
00:05:10.630 --> 00:05:12.910
So what's the
damping going to do?
00:05:12.910 --> 00:05:16.110
What's going to be
effect of damping?
00:05:16.110 --> 00:05:23.820
Say if I had a right side
of one, just a unit force?
00:05:23.820 --> 00:05:25.300
The damping.
00:05:25.300 --> 00:05:29.240
So I'll still have
oscillation-- you'll
00:05:29.240 --> 00:05:31.260
see this-- I'll still
have oscillation,
00:05:31.260 --> 00:05:34.310
but its amplitude damps out.
00:05:34.310 --> 00:05:37.240
Like that like everything
we know, like something
00:05:37.240 --> 00:05:42.090
swinging back and
forth, but friction
00:05:42.090 --> 00:05:45.390
is it is eventually
damping out that motion.
00:05:45.390 --> 00:05:50.570
So it'll be oscillating
with exponentially
00:05:50.570 --> 00:05:52.160
decaying amplitude.
00:05:52.160 --> 00:05:53.730
Let's find this solution.
00:05:53.730 --> 00:05:56.430
OK.
00:05:56.430 --> 00:06:01.210
How do we find
the null solution?
00:06:01.210 --> 00:06:04.480
I've got constant
coefficients here.
00:06:04.480 --> 00:06:11.900
So the nice, the right thing
to look at is exponentials.
00:06:11.900 --> 00:06:15.110
In first order equations
it was e to the at.
00:06:15.110 --> 00:06:17.110
Here, we'll have
to see what it is.
00:06:17.110 --> 00:06:22.330
So I'm going to try
a particular-- I'm
00:06:22.330 --> 00:06:25.310
going to look for the
right exponentials.
00:06:25.310 --> 00:06:29.770
Certain exponentials
will be null solution.
00:06:29.770 --> 00:06:30.460
Can I do it?
00:06:30.460 --> 00:06:37.040
So I take that
equation and put in y
00:06:37.040 --> 00:06:40.550
equal e to the st. Remember
now, I'm doing 0 here
00:06:40.550 --> 00:06:43.740
so it's not the same s
as the right-hand side.
00:06:43.740 --> 00:06:44.500
OK.
00:06:44.500 --> 00:06:47.550
What happens if
I put in-- so I'm
00:06:47.550 --> 00:06:49.560
looking for the null solution.
00:06:49.560 --> 00:06:54.950
I'll try y equal e to
the st. Plug it in.
00:06:54.950 --> 00:06:57.510
I get m.
00:06:57.510 --> 00:07:05.660
Two derivatives gives me
s squared e to the st, b.
00:07:05.660 --> 00:07:09.565
One derivative gives
me an s, e to the st,
00:07:09.565 --> 00:07:14.430
and k gives me k e
to the st. And that's
00:07:14.430 --> 00:07:18.440
supposed to equal to 0
for the null solution.
00:07:18.440 --> 00:07:20.840
OK.
00:07:20.840 --> 00:07:27.640
Have we made a good
guess at what will work?
00:07:27.640 --> 00:07:31.224
Yes, because I can
cancel e to the st,
00:07:31.224 --> 00:07:34.600
and I come to the most
important equation.
00:07:34.600 --> 00:07:36.840
The equation that
governs everything
00:07:36.840 --> 00:07:43.430
in this whole
lecture is ms squared
00:07:43.430 --> 00:07:49.500
plus bs plus k equals 0.
00:07:49.500 --> 00:07:53.950
That's called the characteristic
equation, or it has many names.
00:07:53.950 --> 00:07:56.890
It's obviously the big deal.
00:07:56.890 --> 00:07:58.970
It's the thing.
00:07:58.970 --> 00:08:09.590
It's an equation for s, for
the special frequencies that
00:08:09.590 --> 00:08:13.920
solve the null
equation with no force.
00:08:13.920 --> 00:08:14.930
OK.
00:08:14.930 --> 00:08:18.900
And how many values
of s do we expect?
00:08:18.900 --> 00:08:20.230
Two.
00:08:20.230 --> 00:08:33.080
So I expect solutions s1 and
s2, and then my null solution
00:08:33.080 --> 00:08:41.400
is e to the s1t is
a null solution.
00:08:41.400 --> 00:08:46.700
E to the s2t is another one.
00:08:46.700 --> 00:08:48.920
My equation is linear.
00:08:48.920 --> 00:08:51.840
I can multiply that
by any constant.
00:08:51.840 --> 00:08:54.770
I can multiply this
by any constant.
00:08:54.770 --> 00:09:00.410
And I can superimpose, i
can add, I can combine,
00:09:00.410 --> 00:09:04.930
because I can do
linear algebra here.
00:09:04.930 --> 00:09:08.260
This is the most important
operation in linear algebra,
00:09:08.260 --> 00:09:12.050
multiply things by
constants and add.
00:09:14.900 --> 00:09:16.920
That's called a
linear combination.
00:09:16.920 --> 00:09:20.525
It's the basic operation
in linear algebra
00:09:20.525 --> 00:09:24.140
and it's a basic operation
here, because we're
00:09:24.140 --> 00:09:27.170
doing linear algebra
with functions.
00:09:27.170 --> 00:09:29.530
Do you see that that's it?
00:09:29.530 --> 00:09:32.520
We've got it, except we
should really write a formula
00:09:32.520 --> 00:09:36.220
and draw some pictures
to show s1 and s2.
00:09:39.820 --> 00:09:45.170
What would be the formula for
s1 and s2, to the two solutions?
00:09:45.170 --> 00:09:53.420
We remember that from school,
it's the quadratic formula.
00:09:53.420 --> 00:09:55.500
The two solutions to this.
00:09:55.500 --> 00:09:58.254
Everybody remember this one?
00:09:58.254 --> 00:10:01.330
Let's see if I do.
00:10:01.330 --> 00:10:05.670
Does it start with-- a minus b.
00:10:05.670 --> 00:10:08.900
And then there's going to be a
denominator that I'll remember,
00:10:08.900 --> 00:10:09.615
which is 2a.
00:10:12.150 --> 00:10:12.650
No.
00:10:12.650 --> 00:10:16.530
I said a but I mean m.
00:10:16.530 --> 00:10:18.730
2m.
00:10:18.730 --> 00:10:25.590
And now this is plus or
minus-- what goes into here?
00:10:25.590 --> 00:10:28.800
B squared minus 4ac.
00:10:28.800 --> 00:10:34.065
The key quantity in this whole
business, b squared minus 4ac.
00:10:37.340 --> 00:10:40.820
Ah, what is it?
00:10:40.820 --> 00:10:41.890
Mk, thank you.
00:10:41.890 --> 00:10:43.661
4mk.
00:10:43.661 --> 00:10:44.160
Great.
00:10:46.990 --> 00:10:53.440
And can I just remark, a
little remark about units.
00:10:53.440 --> 00:10:55.950
B squared has the
same units as 4mk.
00:10:55.950 --> 00:11:01.010
It has to or such a
formula would be crazy.
00:11:01.010 --> 00:11:04.390
So we will see that
actually there's
00:11:04.390 --> 00:11:07.220
something called the
damping ratio that
00:11:07.220 --> 00:11:10.900
involves the ratio
of these guys.
00:11:10.900 --> 00:11:13.100
OK, that's the formula.
00:11:13.100 --> 00:11:19.720
But it's not like-- OK
it's got a square root,
00:11:19.720 --> 00:11:21.790
and what's the point
of-- the thing we
00:11:21.790 --> 00:11:26.500
have to remember with this
square root is that if it's
00:11:26.500 --> 00:11:30.080
the square root of
a positive number
00:11:30.080 --> 00:11:34.790
then we have a plus or
minus, ordinary real numbers.
00:11:34.790 --> 00:11:38.860
If this thing is
negative, then what?
00:11:38.860 --> 00:11:44.970
What's up if b squared
is smaller than 4mk?
00:11:44.970 --> 00:11:49.730
So if b squared is-- if there's
not much damping, if it's
00:11:49.730 --> 00:11:53.380
underdamped, b squared
would be smaller than 4mk.
00:11:53.380 --> 00:11:59.120
And what then?
00:11:59.120 --> 00:12:01.670
We've got a negative
number here.
00:12:01.670 --> 00:12:05.890
When we take its square root
we have an imaginary number.
00:12:05.890 --> 00:12:08.470
That's oscillation.
00:12:08.470 --> 00:12:11.720
Under-damping is going
to show oscillation.
00:12:11.720 --> 00:12:14.920
Let me draw this.
00:12:14.920 --> 00:12:19.520
Let me draw that curve for
different choices of m, b,
00:12:19.520 --> 00:12:20.490
and k.
00:12:20.490 --> 00:12:24.530
This is a good picture to see.
00:12:24.530 --> 00:12:35.004
So let me draw first of all
one that has b equals 0.
00:12:35.004 --> 00:12:37.360
OK.
00:12:37.360 --> 00:12:40.670
So there's a curve.
00:12:40.670 --> 00:12:44.950
What I'm drawing
is, up on this curve
00:12:44.950 --> 00:12:49.930
is this ms squared
plus bs plus k.
00:12:49.930 --> 00:12:54.790
Ms squared and bs and k.
00:12:54.790 --> 00:13:00.760
And here, this one is one
with no damping at all.
00:13:00.760 --> 00:13:08.110
This is just s squared
plus 0 s plus 1.
00:13:08.110 --> 00:13:13.530
That's what that curve is That's
the example we saw before.
00:13:13.530 --> 00:13:15.686
Y double prime plus y.
00:13:15.686 --> 00:13:18.010
Y double prime plus y equals 0.
00:13:18.010 --> 00:13:22.000
This is coming from Y double
prime plus y equals 0.
00:13:22.000 --> 00:13:25.170
This is pure oscillation.
00:13:29.970 --> 00:13:30.740
OK.
00:13:30.740 --> 00:13:33.540
Now let me bring
in some damping.
00:13:33.540 --> 00:13:38.170
Now, as I bring in damping
the curve will move.
00:13:38.170 --> 00:13:42.880
And it takes a little patience
to see where it moves.
00:13:42.880 --> 00:13:50.830
Let me have a little
damping a little damping,
00:13:50.830 --> 00:13:57.210
so s squared plus 1 s plus 1.
00:13:57.210 --> 00:14:03.230
I think the curve--
so this is s here.
00:14:03.230 --> 00:14:07.250
And it's a parabola.
00:14:07.250 --> 00:14:09.600
Everything I draw
here is a parabola,
00:14:09.600 --> 00:14:11.970
is just that different
parabolas come
00:14:11.970 --> 00:14:14.470
from different choices of b.
00:14:14.470 --> 00:14:17.810
So I think that
this choice, I think
00:14:17.810 --> 00:14:22.220
it goes a little bit like that.
00:14:22.220 --> 00:14:25.090
That's the next one.
00:14:25.090 --> 00:14:28.160
It goes down a bit.
00:14:28.160 --> 00:14:35.156
Now, let me do s
squared plus 2s plus 1.
00:14:35.156 --> 00:14:41.040
Now, let's see, I really
should stop for a moment
00:14:41.040 --> 00:14:48.390
and solve the equation, find the
roots for each of these guys.
00:14:48.390 --> 00:14:52.340
And then I'm going to have
an s squared plus 3s plus 1.
00:14:54.980 --> 00:15:04.310
So we're doing something very
straightforward, parabolas.
00:15:04.310 --> 00:15:07.640
But it shows us the
different possibilities.
00:15:07.640 --> 00:15:10.360
And we could give them names.
00:15:10.360 --> 00:15:11.870
OK.
00:15:11.870 --> 00:15:13.740
So I would call
this one undamped.
00:15:18.500 --> 00:15:21.480
And what are the roots
of that equation?
00:15:21.480 --> 00:15:25.170
S squared plus 0
s plus 1 equals 0.
00:15:25.170 --> 00:15:30.410
What are the s1 and
s2 for that guy.
00:15:30.410 --> 00:15:35.420
So the roots of s squared
plus 1 equals 0 are?
00:15:35.420 --> 00:15:38.230
Stay with me here.
00:15:38.230 --> 00:15:41.200
S squared plus 1
equals 0, that's
00:15:41.200 --> 00:15:42.485
the equation that has roots.
00:15:42.485 --> 00:15:44.150
STUDENT: I and minus i
00:15:44.150 --> 00:15:46.500
PROFESSOR: I and minus i.
00:15:46.500 --> 00:15:47.780
I and minus i.
00:15:47.780 --> 00:15:50.760
So this is undamped.
00:15:50.760 --> 00:15:55.120
S1 is i and s2 is minus i.
00:15:55.120 --> 00:15:57.870
That's the pure oscillation.
00:15:57.870 --> 00:16:05.280
Pure oscillation, that's
the case where b is 0 here,
00:16:05.280 --> 00:16:09.860
I have a square root
of a negative number,
00:16:09.860 --> 00:16:12.370
and it gives me
plus or minus 2i,
00:16:12.370 --> 00:16:15.710
and then the 2s cancel
and I get plus or minus i.
00:16:15.710 --> 00:16:19.530
So I can always go
back to this but I'll
00:16:19.530 --> 00:16:23.010
try to choose numbers
that come out nicely.
00:16:23.010 --> 00:16:26.250
Now, what happens
with some damping?
00:16:26.250 --> 00:16:27.530
This guy.
00:16:27.530 --> 00:16:30.480
What are the roots for this one?
00:16:30.480 --> 00:16:33.190
Well, I better use this formula.
00:16:33.190 --> 00:16:37.880
Now, I'm always keeping
m equal 1 and k equal 1,
00:16:37.880 --> 00:16:40.630
in all those samples.
00:16:40.630 --> 00:16:43.690
But now b has increased to 1.
00:16:47.070 --> 00:16:52.720
So if b is 1, what do I have?
00:16:52.720 --> 00:16:56.660
I have s is-- the
roots are minus 1,
00:16:56.660 --> 00:17:02.350
plus or minus the square root--
And it's going to be just 2
00:17:02.350 --> 00:17:03.860
down below.
00:17:03.860 --> 00:17:07.650
What's in the square root,
the all important square root.
00:17:07.650 --> 00:17:13.420
When m and k and b are all 1.
00:17:13.420 --> 00:17:16.230
Just do the calculation
with me, so you see it.
00:17:16.230 --> 00:17:17.329
It's negative?
00:17:17.329 --> 00:17:18.183
STUDENT: Three.
00:17:18.183 --> 00:17:19.099
PROFESSOR: Negative 3.
00:17:19.099 --> 00:17:20.359
So what's the point there?
00:17:23.810 --> 00:17:28.470
The point is, this is going to
be square root of 3i or minus
00:17:28.470 --> 00:17:29.250
i.
00:17:29.250 --> 00:17:34.200
We have oscillation at a
frequency square root of 3,
00:17:34.200 --> 00:17:38.880
and we have decay
from s minus one-half.
00:17:38.880 --> 00:17:44.900
The real part of this is
giving us the drop off.
00:17:44.900 --> 00:17:49.000
We didn't have any drop
off at all in this case.
00:17:49.000 --> 00:17:50.810
They were pure imaginary.
00:17:50.810 --> 00:17:57.960
Now the s1 and s2 are
whatever I have a minus 1,
00:17:57.960 --> 00:18:02.330
plus or minus square
root of 3i over 2.
00:18:05.000 --> 00:18:08.220
Those are the roots.
00:18:08.220 --> 00:18:09.790
All right, I'm
ready for this guy,
00:18:09.790 --> 00:18:12.090
and it's particularly nice.
00:18:12.090 --> 00:18:13.270
It's particularly nice.
00:18:13.270 --> 00:18:18.030
What do you see for s
squared plus 2s plus 1?
00:18:18.030 --> 00:18:24.220
When you see this parabola,
now b has moved up to 2.
00:18:24.220 --> 00:18:27.410
What's up with that one?
00:18:27.410 --> 00:18:31.980
It's going to be--
let's see-- that looks
00:18:31.980 --> 00:18:34.420
like a perfect square to me.
00:18:34.420 --> 00:18:38.520
Right, inside the
square root is 0.
00:18:38.520 --> 00:18:39.820
Right, exactly.
00:18:39.820 --> 00:18:44.430
Inside the square root, b
squared is 4, and 4mk is 4,
00:18:44.430 --> 00:18:46.910
so I have 0 inside
this square root.
00:18:46.910 --> 00:18:56.270
So now I have s equals minus
1 plus or minus 0 over 2.
00:18:56.270 --> 00:19:00.644
That's when this was the
case, when b moved up to 2.
00:19:00.644 --> 00:19:03.370
STUDENT: [INAUDIBLE] minus 2.
00:19:03.370 --> 00:19:05.046
PROFESSOR: I'm messing it up?
00:19:05.046 --> 00:19:08.240
STUDENT: [INAUDIBLE] equals
minus 2, plus or minus 0.
00:19:08.240 --> 00:19:10.180
PROFESSOR: Minus
2 plus or minus 0.
00:19:10.180 --> 00:19:12.020
Thank you.
00:19:12.020 --> 00:19:13.130
So what do I have?
00:19:16.620 --> 00:19:19.450
What are the roots of this guy?
00:19:22.600 --> 00:19:24.080
Negative 1.
00:19:24.080 --> 00:19:26.690
What's the other one?
00:19:26.690 --> 00:19:28.000
Negative 1.
00:19:28.000 --> 00:19:29.420
A double root.
00:19:29.420 --> 00:19:31.220
It's critical damping.
00:19:31.220 --> 00:19:33.440
Critical damping--
it's not underdamped.
00:19:33.440 --> 00:19:34.540
It's not overdamped.
00:19:34.540 --> 00:19:37.210
It's right on the borderline.
00:19:37.210 --> 00:19:43.060
And I see that, when you
first saw quadratics,
00:19:43.060 --> 00:19:47.140
before anybody brought
up that awful formula,
00:19:47.140 --> 00:19:52.990
you would have factored this
into s plus 1 squared equals 0,
00:19:52.990 --> 00:19:57.610
and you would have discovered
that s was minus 1 twice.
00:20:01.540 --> 00:20:02.170
A double root.
00:20:02.170 --> 00:20:13.764
So the picture there would
be-- there's minus 1.
00:20:13.764 --> 00:20:15.960
Yeah.
00:20:15.960 --> 00:20:22.600
Everybody recognize that this,
we're hitting 0, height 0.
00:20:22.600 --> 00:20:26.490
We're hitting 0 twice
at s equal minus 1.
00:20:29.090 --> 00:20:31.620
So this is now the case.
00:20:31.620 --> 00:20:34.530
This was b equal 0, no damping.
00:20:34.530 --> 00:20:37.590
This was b equal
1, under-damping.
00:20:37.590 --> 00:20:42.060
This is b equal 2, critical
damping, just on the border.
00:20:42.060 --> 00:20:45.040
And what do you
think the s squared
00:20:45.040 --> 00:20:48.210
plus 3s plus 1 is
going to look like.
00:20:48.210 --> 00:20:52.370
Again, we can find it.
00:20:52.370 --> 00:20:58.050
No, let me do the s squared
plus-- let me take b equal 3
00:20:58.050 --> 00:21:01.620
and find the roots
and draw the picture.
00:21:01.620 --> 00:21:06.220
If you're with, me that
picture and these formulas
00:21:06.220 --> 00:21:10.950
tell you the difference
between these four cases.
00:21:10.950 --> 00:21:12.520
So what do I get?
00:21:12.520 --> 00:21:19.550
Minus 3 plus or minus the
square root of, 9 minus 4,
00:21:19.550 --> 00:21:22.510
is 5, over 2.
00:21:26.160 --> 00:21:30.370
So I have two negative roots.
00:21:30.370 --> 00:21:32.520
I have decay.
00:21:32.520 --> 00:21:35.670
I have decay at a fast
rate and a slow rate
00:21:35.670 --> 00:21:38.280
but both are giving decay.
00:21:38.280 --> 00:21:44.660
So the curve now is coming
down here and back up there,
00:21:44.660 --> 00:21:51.350
and it hits there, and
it's got the two roots.
00:21:51.350 --> 00:21:54.290
These two roots are x and x.
00:22:01.810 --> 00:22:04.380
So these are s1 and s2.
00:22:04.380 --> 00:22:06.510
Let me copy them over here.
00:22:06.510 --> 00:22:12.190
S1 and s2 are minus
3 plus or minus
00:22:12.190 --> 00:22:16.610
the square root of 5 over 2.
00:22:16.610 --> 00:22:17.110
Yeah.
00:22:17.110 --> 00:22:18.800
Real.
00:22:18.800 --> 00:22:19.800
Real roots.
00:22:19.800 --> 00:22:22.030
So this is two real roots.
00:22:22.030 --> 00:22:24.140
This is a double real root.
00:22:24.140 --> 00:22:26.490
This is two complex roots.
00:22:26.490 --> 00:22:29.650
And this is two pure
imaginary roots.
00:22:29.650 --> 00:22:31.950
The four possibilities.
00:22:31.950 --> 00:22:33.850
The famous four.
00:22:33.850 --> 00:22:34.460
OK.
00:22:34.460 --> 00:22:38.460
And for me, that
picture is a-- so
00:22:38.460 --> 00:22:44.910
this was the b equal 3
curve, more overdamped.
00:22:44.910 --> 00:22:47.730
It's a little interesting
that overdamping
00:22:47.730 --> 00:22:51.110
has this root that's
pretty near 0.
00:22:51.110 --> 00:22:58.400
So overdamping doesn't mean
that you go to 0 real fast.
00:22:58.400 --> 00:23:01.860
Actually, the one
that goes fastest to 0
00:23:01.860 --> 00:23:03.470
is the critical damping.
00:23:03.470 --> 00:23:09.450
Then, as b grows, one
root gets closer to 0,
00:23:09.450 --> 00:23:13.570
so it's like slower
decay, and another root
00:23:13.570 --> 00:23:15.156
is going off to fast decay.
00:23:21.870 --> 00:23:26.050
I think you have
to know those guys,
00:23:26.050 --> 00:23:29.140
because the physically
that's very important,
00:23:29.140 --> 00:23:31.320
where's the damping.
00:23:31.320 --> 00:23:37.500
But we've now found the
null solution completely.
00:23:37.500 --> 00:23:41.550
The null solution
completely is--
00:23:41.550 --> 00:23:46.360
let me write it again
here-- is anything times e
00:23:46.360 --> 00:23:53.370
to the s1t and anything
times e to the s2t.
00:23:53.370 --> 00:23:56.620
And where do those constants,
c1 and c2 get decided?
00:23:59.450 --> 00:24:00.365
By the?
00:24:00.365 --> 00:24:01.240
STUDENT: [INAUDIBLE].
00:24:01.240 --> 00:24:03.011
PROFESSOR: Initial conditions.
00:24:03.011 --> 00:24:03.510
Right.
00:24:03.510 --> 00:24:08.150
These two conditions
determine c1 and c2.
00:24:08.150 --> 00:24:09.820
OK.
00:24:09.820 --> 00:24:12.520
Are we good?
00:24:12.520 --> 00:24:18.790
So the null solution has already
separated the different cases
00:24:18.790 --> 00:24:22.390
that depend on how much damping.
00:24:22.390 --> 00:24:26.530
I'm ready for number two.
00:24:26.530 --> 00:24:28.750
Number two now.
00:24:28.750 --> 00:24:30.140
OK.
00:24:30.140 --> 00:24:36.885
So the idea is I now have a
forcing term, some frequency
00:24:36.885 --> 00:24:41.190
e to the st. And I will
assume that it's not--
00:24:41.190 --> 00:24:45.080
s not equal s1 or s2.
00:24:49.320 --> 00:24:51.610
That's to make my life easy.
00:24:51.610 --> 00:24:55.570
Just as last time, the
formulas will break down.
00:24:55.570 --> 00:25:01.550
And I'll have to put
in something different
00:25:01.550 --> 00:25:05.030
if there's resonance,
if the driving force is
00:25:05.030 --> 00:25:09.960
at the same frequency
as a natural frequency.
00:25:09.960 --> 00:25:11.720
In that case,
there's a resonance.
00:25:11.720 --> 00:25:15.350
And the way you spot
a resonance formula
00:25:15.350 --> 00:25:18.220
is there's an extra factor of t.
00:25:18.220 --> 00:25:19.930
There's a growth of t.
00:25:19.930 --> 00:25:22.940
Up here, we're seeing
no factors of t.
00:25:22.940 --> 00:25:25.760
Over there in the
exponent, of course.
00:25:25.760 --> 00:25:31.440
I mean, down below, there
would be a t times e to the st.
00:25:31.440 --> 00:25:34.980
And the book does
that carefully.
00:25:38.240 --> 00:25:43.230
I don't see that it has a
place in the very first lecture
00:25:43.230 --> 00:25:45.930
on this topic.
00:25:45.930 --> 00:25:46.430
OK.
00:25:46.430 --> 00:25:50.180
So this is saying no resonance.
00:25:50.180 --> 00:25:51.310
All right.
00:25:51.310 --> 00:25:53.070
What's the solution then?
00:25:53.070 --> 00:25:57.070
The solution is a
multiple of e to the st.
00:25:57.070 --> 00:26:00.620
The input was e to the
st. The response is
00:26:00.620 --> 00:26:06.740
a multiple of e to st, so
that's the frequency response.
00:26:06.740 --> 00:26:10.220
Capital Y is telling
us the response to s.
00:26:10.220 --> 00:26:12.980
And it's a very, very
important function.
00:26:12.980 --> 00:26:15.040
It's called the
transfer function.
00:26:15.040 --> 00:26:18.490
It's just the key to everything.
00:26:18.490 --> 00:26:22.150
I probably say that
so often, that this
00:26:22.150 --> 00:26:23.460
is the key to everything.
00:26:23.460 --> 00:26:26.130
Well, it's partly
because I just have
00:26:26.130 --> 00:26:31.980
one lecture to do a big part
of differential equations,
00:26:31.980 --> 00:26:34.590
and it's got some key ideas.
00:26:34.590 --> 00:26:38.590
And the first key
idea is s1 and s2.
00:26:38.590 --> 00:26:44.170
And the second key idea
is Y. And let's go for it.
00:26:44.170 --> 00:26:48.040
All right, so this is
the s1 and s2 picture.
00:26:48.040 --> 00:26:56.200
I'll move that up, and
now, in the equation,
00:26:56.200 --> 00:27:03.640
try Y e to the st.
We hope it works.
00:27:03.640 --> 00:27:04.330
What is this?
00:27:04.330 --> 00:27:11.240
I'm trying, this is my solution,
and it's a particular solution
00:27:11.240 --> 00:27:12.720
now.
00:27:12.720 --> 00:27:14.000
I got the null solution.
00:27:14.000 --> 00:27:16.870
I've moved to a particular
00:27:16.870 --> 00:27:24.390
And this is when the force is
e to the st. In other words,
00:27:24.390 --> 00:27:28.690
I'll just say again, if we
have an exponential force,
00:27:28.690 --> 00:27:31.650
try an exponential response.
00:27:31.650 --> 00:27:37.490
1803 would call this the
exponential response formula.
00:27:37.490 --> 00:27:39.330
So you could use
the word exponential
00:27:39.330 --> 00:27:42.060
response, very appropriate.
00:27:42.060 --> 00:27:44.810
You could use the word
frequency response.
00:27:44.810 --> 00:27:47.650
That frequency response is
kind of the right word when
00:27:47.650 --> 00:27:53.860
s is an imaginary number giving
an oscillation frequency.
00:27:53.860 --> 00:27:56.500
OK, I'm going to plug that in.
00:27:56.500 --> 00:28:01.030
So into the differential
equation, m.
00:28:01.030 --> 00:28:09.970
Second derivative of that is s
squared Y e to the st, right?
00:28:09.970 --> 00:28:11.860
That's m Y double prime.
00:28:14.540 --> 00:28:17.360
The beauty of exponentials.
00:28:17.360 --> 00:28:20.520
Take every derivative,
just brings down an s.
00:28:20.520 --> 00:28:21.750
The next term.
00:28:21.750 --> 00:28:22.660
What's the next term?
00:28:22.660 --> 00:28:25.850
Maybe do it with
me, do it for me.
00:28:25.850 --> 00:28:33.906
What happens when I plug in this
as Y into b, so there'll be a b
00:28:33.906 --> 00:28:34.840
Y prime.
00:28:34.840 --> 00:28:35.950
So what's Y prime?
00:28:35.950 --> 00:28:37.734
STUDENT: [INAUDIBLE].
00:28:37.734 --> 00:28:45.196
PROFESSOR: s, Y, this
constant, e to the st.
00:28:45.196 --> 00:28:50.460
And then the final term
is k times this Y itself,
00:28:50.460 --> 00:28:53.453
no derivative, so it's
just Y e to the st,
00:28:53.453 --> 00:28:59.480
and that's matching e to
the st. That's the force.
00:28:59.480 --> 00:29:01.765
This is f.
00:29:01.765 --> 00:29:05.290
F, this is an exponential force.
00:29:05.290 --> 00:29:10.850
Of course, I could and
should have a constant here
00:29:10.850 --> 00:29:16.580
to give me the units of force.
00:29:16.580 --> 00:29:18.520
Let me just keep
the formula as clean
00:29:18.520 --> 00:29:22.990
as possible by taking
units, so that's one.
00:29:22.990 --> 00:29:25.295
OK what do I do?
00:29:25.295 --> 00:29:26.392
STUDENT: [INAUDIBLE].
00:29:26.392 --> 00:29:27.225
PROFESSOR: I cancel.
00:29:29.990 --> 00:29:36.604
The nice part of 287 is
canceling e to the st.
00:29:36.604 --> 00:29:39.760
That's the most fun you get.
00:29:43.010 --> 00:29:47.160
And now, I have Y's on the left.
00:29:47.160 --> 00:29:53.170
So Y is-- can I see what Y is?
00:29:53.170 --> 00:30:00.760
Y is this 1, divided by
the coefficient of Y.
00:30:00.760 --> 00:30:02.590
And what's the coefficient of Y?
00:30:02.590 --> 00:30:04.780
We know it.
00:30:04.780 --> 00:30:07.640
We've seen that coefficient.
00:30:07.640 --> 00:30:13.910
Y on the left is multiplied by
ms squared plus bs plus k ms
00:30:13.910 --> 00:30:19.870
squared, again, ms
squared, bs, and k.
00:30:19.870 --> 00:30:24.110
Multiplying Y, I divide by
that, so I put it down here.
00:30:24.110 --> 00:30:30.478
Ms squared plus bs plus k.
00:30:36.960 --> 00:30:44.790
And because s is not one
of the roots, that's not 0,
00:30:44.790 --> 00:30:47.640
so we're golden.
00:30:47.640 --> 00:30:52.170
That problem took five minutes.
00:30:52.170 --> 00:30:54.690
The null solution
took half an hour.
00:30:58.030 --> 00:31:02.020
The exponential
response is clear.
00:31:02.020 --> 00:31:05.780
And you can see
what it would be.
00:31:05.780 --> 00:31:07.820
And let's give it a name.
00:31:07.820 --> 00:31:09.900
This is the transfer function.
00:31:19.080 --> 00:31:20.670
Widely used name.
00:31:20.670 --> 00:31:25.740
Other names could be
given but that's the best.
00:31:25.740 --> 00:31:28.915
So it's a function of s.
00:31:28.915 --> 00:31:29.790
It's a function of s.
00:31:33.090 --> 00:31:37.870
And so, again, when
f is e to the st,
00:31:37.870 --> 00:31:44.360
it sort of transfers the
input into the output.
00:31:44.360 --> 00:31:47.840
That's the way I think
of the transfer function.
00:31:47.840 --> 00:31:49.950
Here is the input.
00:31:49.950 --> 00:31:53.740
The output is just multiplied
by the transfer function.
00:31:53.740 --> 00:31:57.630
And the transfer function is
just that nice expression.
00:31:57.630 --> 00:31:59.780
Just that nice expression.
00:31:59.780 --> 00:32:06.280
So we are golden
for a frequency,
00:32:06.280 --> 00:32:15.390
for a linear equation with an
exponential forcing function.
00:32:15.390 --> 00:32:17.120
What would be another example?
00:32:17.120 --> 00:32:22.040
I'm using mass,
dashpot, spring here.
00:32:22.040 --> 00:32:25.750
If this was in
electrical engineering,
00:32:25.750 --> 00:32:32.750
what three things would I be
using instead of mass, dashpot,
00:32:32.750 --> 00:32:33.750
spring.
00:32:33.750 --> 00:32:37.590
The three guys would be?
00:32:37.590 --> 00:32:39.200
What would correspond
to the dashpot?
00:32:43.220 --> 00:32:46.610
So can I just draw
here a little--
00:32:46.610 --> 00:32:50.430
let me put on a
low voltage and put
00:32:50.430 --> 00:32:56.990
on something that does this,
and then something like this.
00:32:56.990 --> 00:33:01.130
And then there's
something like this.
00:33:01.130 --> 00:33:07.020
And give me a break, tell
me what these things are.
00:33:07.020 --> 00:33:08.060
This guy is?
00:33:08.060 --> 00:33:08.960
STUDENT: Inductor.
00:33:08.960 --> 00:33:10.370
PROFESSOR: An inductor.
00:33:10.370 --> 00:33:11.681
This guy is a?
00:33:11.681 --> 00:33:12.430
STUDENT: Resistor.
00:33:12.430 --> 00:33:13.340
PROFESSOR: Resistor.
00:33:13.340 --> 00:33:16.880
Now that's the one
that's like damping.
00:33:16.880 --> 00:33:21.640
This resistor here
is like damping.
00:33:21.640 --> 00:33:24.300
Like the damping
term, or maybe 1/b.
00:33:24.300 --> 00:33:27.880
I'm not getting its
units right because I
00:33:27.880 --> 00:33:30.410
haven't got any
equation here at all.
00:33:30.410 --> 00:33:36.040
Resisting is-- there's
friction in that resistor.
00:33:36.040 --> 00:33:38.390
It burns up heat.
00:33:38.390 --> 00:33:42.020
And similarly, the
dashpot slows things down.
00:33:42.020 --> 00:33:43.380
And then this guy is a--
00:33:43.380 --> 00:33:44.300
STUDENT: [INAUDIBLE].
00:33:44.300 --> 00:33:46.970
PROFESSOR: Capacitor, right.
00:33:46.970 --> 00:33:53.170
In other words, you can do
the mechanical application
00:33:53.170 --> 00:33:59.080
and the electrical application
with exactly the same ideas,
00:33:59.080 --> 00:34:07.380
just a change of letter, and
of course, different units,
00:34:07.380 --> 00:34:09.730
but same problem.
00:34:09.730 --> 00:34:15.389
OK, so that's a comment
that you've seen before.
00:34:15.389 --> 00:34:16.960
What else do I
want to comment on?
00:34:16.960 --> 00:34:21.594
Because this example was
really so straightforward.
00:34:27.060 --> 00:34:32.560
I think what I want
to mention, and this
00:34:32.560 --> 00:34:39.080
is important, is that this
is the central starting
00:34:39.080 --> 00:34:43.159
point for the Laplace transform.
00:34:43.159 --> 00:34:51.409
So I can't do Laplace transforms
all today by any means,
00:34:51.409 --> 00:34:58.650
and so Professor Fry will talk
about the Laplace transform
00:34:58.650 --> 00:35:00.100
next week.
00:35:00.100 --> 00:35:04.570
But what is the point of
the Laplace transform?
00:35:04.570 --> 00:35:06.490
The point of the
Laplace transform
00:35:06.490 --> 00:35:18.890
is to get your money's worth
out of the simple formula
00:35:18.890 --> 00:35:21.165
for exponentials.
00:35:26.500 --> 00:35:30.280
Having an exponential there
turns the whole differential
00:35:30.280 --> 00:35:33.430
equation problem into
an algebra problem.
00:35:33.430 --> 00:35:37.220
We just have
quadratic equations.
00:35:37.220 --> 00:35:42.130
We just have a division
by a quadratic.
00:35:42.130 --> 00:35:45.590
That's the great thing
about the Laplace transform.
00:35:45.590 --> 00:35:48.900
It turns the t domain,
the time domain,
00:35:48.900 --> 00:35:53.290
where we have exponentials,
into the s domain,
00:35:53.290 --> 00:35:57.050
the exponent domain,
the frequency domain,
00:35:57.050 --> 00:35:59.930
where we just have quadratics.
00:35:59.930 --> 00:36:03.060
And then first order
equation's just linear.
00:36:03.060 --> 00:36:10.770
And we can even get from
second order of the quadratic
00:36:10.770 --> 00:36:14.190
to linear, because I
can factor that guy.
00:36:14.190 --> 00:36:18.360
If I factor into s minus
s1 times s minus s2,
00:36:18.360 --> 00:36:20.520
I've two linear pieces.
00:36:20.520 --> 00:36:23.030
And that's the first step
in the Laplace transform,
00:36:23.030 --> 00:36:24.310
in the algebra.
00:36:24.310 --> 00:36:27.320
So all of the algebra
in the Laplace transform
00:36:27.320 --> 00:36:34.760
is this algebra for e to the st.
00:36:34.760 --> 00:36:38.420
And then the job of the
Laplace transform-- and this
00:36:38.420 --> 00:36:41.120
is the tricky part.
00:36:41.120 --> 00:36:43.990
So let me even take a
little board space on that.
00:36:47.070 --> 00:36:51.590
So this is like a
heads up for next week.
00:36:51.590 --> 00:36:52.890
So the Laplace transform.
00:37:02.305 --> 00:37:02.805
OK.
00:37:07.730 --> 00:37:16.300
So for any forcing
function, f of t.
00:37:19.556 --> 00:37:20.930
That's the thing
that we're going
00:37:20.930 --> 00:37:25.920
to take the Laplace
transform of and the response
00:37:25.920 --> 00:37:33.920
and its response, y of t.
00:37:33.920 --> 00:37:36.860
OK.
00:37:36.860 --> 00:37:39.110
So here's the idea of
transforms in general.
00:37:43.220 --> 00:37:49.600
I choose some terrific
functions like exponentials.
00:37:49.600 --> 00:37:57.550
So I want to convert my
problem to exponentials.
00:38:00.280 --> 00:38:07.270
e to the st for all s, s between
let's say 0 and infinity.
00:38:12.170 --> 00:38:13.450
So what do I do?
00:38:13.450 --> 00:38:17.000
I take my function,
and I figure out
00:38:17.000 --> 00:38:20.580
how much of every
exponential is in it.
00:38:20.580 --> 00:38:22.950
That's the Laplace transform.
00:38:22.950 --> 00:38:27.240
I take my function
f of t, and I go
00:38:27.240 --> 00:38:33.130
to what you'll see next
week, its Laplace transform.
00:38:33.130 --> 00:38:36.510
Let me call it capital F
of s, or it's sometimes
00:38:36.510 --> 00:38:42.710
written the Laplace
transform of F of s.
00:38:42.710 --> 00:38:44.700
Something like that.
00:38:44.700 --> 00:38:46.760
I'm not going to do that.
00:38:46.760 --> 00:38:48.314
I'm not going there.
00:38:51.100 --> 00:38:57.430
So F of s is like the amount
of a particular exponential
00:38:57.430 --> 00:38:59.180
in my function.
00:38:59.180 --> 00:39:01.790
If my function is just the
sum of two exponentials,
00:39:01.790 --> 00:39:05.990
then the Laplace transform
just is a big bump on one
00:39:05.990 --> 00:39:07.670
and a big bump on the other.
00:39:07.670 --> 00:39:15.090
But most functions, like
some forcing function,
00:39:15.090 --> 00:39:20.000
has some e to the st is for
all for a whole range of s.
00:39:20.000 --> 00:39:22.990
So I figure out how
much of this is.
00:39:22.990 --> 00:39:26.040
OK now second step.
00:39:26.040 --> 00:39:29.100
Using linearity, I
can solve the problem
00:39:29.100 --> 00:39:33.980
for when the right
hand side is just that.
00:39:33.980 --> 00:39:41.810
Then solve for
right hand side, F
00:39:41.810 --> 00:39:47.450
of s, e to the st. Solve
for each frequency,
00:39:47.450 --> 00:39:49.340
each s separately.
00:39:49.340 --> 00:39:51.210
And what does that mean?
00:39:51.210 --> 00:39:54.940
That means just
dividing by this.
00:39:54.940 --> 00:39:57.110
So this was the easy step.
00:39:57.110 --> 00:40:00.440
So this is step one, is
take the Laplace transform.
00:40:00.440 --> 00:40:05.440
The Laplace transform tells you
how much of each exponential
00:40:05.440 --> 00:40:06.440
is in it.
00:40:06.440 --> 00:40:10.140
Now step two is a cinch.
00:40:10.140 --> 00:40:11.290
Step two is a cinch.
00:40:11.290 --> 00:40:15.270
I just multiply by
the transfer function.
00:40:15.270 --> 00:40:21.160
I divide by this, bs plus k.
00:40:21.160 --> 00:40:23.940
And now, what's step three.
00:40:23.940 --> 00:40:28.710
Step three, I have the solution
for each separate exponential,
00:40:28.710 --> 00:40:31.030
but I've got a whole
lots of exponentials,
00:40:31.030 --> 00:40:36.020
so I have to do an
inverse Laplace transform,
00:40:36.020 --> 00:40:44.390
add up, figure out what function
has this Laplace transform.
00:40:44.390 --> 00:40:48.840
And that's often the place
where the algebra gets harder.
00:40:51.430 --> 00:40:54.710
In principle, we can
do it for any F of t.
00:40:54.710 --> 00:40:56.940
We can take its
Laplace transform,
00:40:56.940 --> 00:41:00.810
we can solve for each
frequency in the transform,
00:41:00.810 --> 00:41:02.950
we can assemble
them all together.
00:41:02.950 --> 00:41:05.410
That's the inverse
Laplace transform,
00:41:05.410 --> 00:41:19.310
so this is the inverse Laplace
to get the solution y of t.
00:41:19.310 --> 00:41:26.610
So this is really the Laplace
transform of the solution,
00:41:26.610 --> 00:41:31.130
and we have to get back
to the solution itself.
00:41:31.130 --> 00:41:34.445
Can I just let you
think about those ideas?
00:41:37.690 --> 00:41:47.750
I'm not up to describing
the algebra here.
00:41:47.750 --> 00:41:52.560
The point is the
Laplace transform
00:41:52.560 --> 00:41:55.500
takes this into
separate exponentials.
00:41:55.500 --> 00:41:58.430
Each of those right hand
sides is simple algebra,
00:41:58.430 --> 00:42:00.040
divide by that.
00:42:00.040 --> 00:42:03.310
And then you have
the job of okay,
00:42:03.310 --> 00:42:08.000
what function has this Laplace
transform, to go backwards.
00:42:08.000 --> 00:42:09.650
And that's usually
the hard part.
00:42:09.650 --> 00:42:12.960
So people make tables
of Laplace transforms.
00:42:12.960 --> 00:42:16.705
Everybody remembers the Laplace
transforms of a few functions.
00:42:19.880 --> 00:42:24.720
You could say that
those few functions are
00:42:24.720 --> 00:42:29.740
the golden functions of
differential equations.
00:42:29.740 --> 00:42:32.120
The golden functions of
differential equations
00:42:32.120 --> 00:42:35.870
are the ones where you know
their Laplace transform
00:42:35.870 --> 00:42:38.240
and you can go back
and forth easily.
00:42:38.240 --> 00:42:41.360
And what are those
golden functions?
00:42:41.360 --> 00:42:45.470
Well, you might
guess exponentials,
00:42:45.470 --> 00:42:48.640
simple polynomials,
1 t, t squared.
00:42:48.640 --> 00:42:51.210
You can do Laplace
transform for those.
00:42:51.210 --> 00:42:53.650
What else do you
think would be nice?
00:42:53.650 --> 00:42:56.840
Cosine and sine.
00:42:56.840 --> 00:42:59.120
And you could multiply
those together.
00:42:59.120 --> 00:43:03.740
We could deal with
t times cosine t.
00:43:03.740 --> 00:43:05.510
So a little bit different.
00:43:05.510 --> 00:43:09.430
But having said that,
the reality is I've
00:43:09.430 --> 00:43:13.440
given you the whole
list of nice functions.
00:43:13.440 --> 00:43:16.080
Those functions show up
in every simple method
00:43:16.080 --> 00:43:17.830
for solving equations.
00:43:17.830 --> 00:43:21.980
There's a method called
undetermined coefficients
00:43:21.980 --> 00:43:23.900
and what does it amount to?
00:43:23.900 --> 00:43:26.490
I'm sorry, I don't have
time to say it this morning,
00:43:26.490 --> 00:43:28.680
but it can come later.
00:43:28.680 --> 00:43:29.500
Undetermined.
00:43:29.500 --> 00:43:31.820
It just means if
the right hand side
00:43:31.820 --> 00:43:33.650
is one of these
nice guys-- shall
00:43:33.650 --> 00:43:38.040
I write down again
the golden functions?
00:43:38.040 --> 00:43:42.440
e to the st is like
the platinum function.
00:43:42.440 --> 00:43:46.820
And then some golden functions
are like t and t squared
00:43:46.820 --> 00:43:48.700
and so on.
00:43:48.700 --> 00:43:54.600
Of course, when we get that,
we're close to cosine omega t,
00:43:54.600 --> 00:43:57.220
and sine omega t.
00:43:57.220 --> 00:44:01.410
All these guys, their
Laplace transforms are nice.
00:44:01.410 --> 00:44:03.950
We can deal with
them completely.
00:44:03.950 --> 00:44:07.680
Or multiply any
of those together.
00:44:07.680 --> 00:44:12.420
And when the right hand side is
one of these golden functions,
00:44:12.420 --> 00:44:14.480
you can write down the answer.
00:44:14.480 --> 00:44:18.350
We've focused on this one
because it's the platinum one.
00:44:18.350 --> 00:44:21.090
And we did these two
too, because they
00:44:21.090 --> 00:44:24.860
come from s equal i omega.
00:44:24.860 --> 00:44:29.680
And then these guys are
a little bit of juice.
00:44:29.680 --> 00:44:31.970
But that's it.
00:44:31.970 --> 00:44:34.260
I'm sorry the list isn't longer.
00:44:34.260 --> 00:44:36.680
It'd be nice to
have--and of course,
00:44:36.680 --> 00:44:44.860
people for centuries have
worked with the next hardest
00:44:44.860 --> 00:44:46.292
functions.
00:44:46.292 --> 00:44:48.335
You know, the silver functions.
00:44:50.880 --> 00:44:54.890
Famous functions have names
like Bessel's function
00:44:54.890 --> 00:44:56.670
or Legendre's function.
00:44:56.670 --> 00:45:01.200
Others where you
can get pretty far.
00:45:01.200 --> 00:45:03.540
Those are the best.
00:45:03.540 --> 00:45:07.110
Then you have the famous ones
where you get pretty far,
00:45:07.110 --> 00:45:10.710
in the web has the
Laplace transforms,
00:45:10.710 --> 00:45:15.380
and then you get the
general function, F of t.
00:45:15.380 --> 00:45:17.820
Oh, could you get
anywhere with delta of t?
00:45:17.820 --> 00:45:20.310
Oh, yes.
00:45:20.310 --> 00:45:24.440
Does it belong on the
list of golden functions?
00:45:24.440 --> 00:45:25.790
Yes, it does.
00:45:25.790 --> 00:45:28.980
I almost forgot it,
and it's like-- I'll
00:45:28.980 --> 00:45:33.982
call it-- Yeah, delta of t.
00:45:33.982 --> 00:45:36.982
Yeah, that's a beauty.
00:45:36.982 --> 00:45:38.475
That's a beauty.
00:45:38.475 --> 00:45:41.030
The Laplace transform
of delta of t
00:45:41.030 --> 00:45:43.550
happens to come out 1 over s.
00:45:43.550 --> 00:45:45.960
You can't ask for
more than that.
00:45:45.960 --> 00:45:46.770
Or maybe it's one.
00:45:49.810 --> 00:45:51.630
Yeah, it's probably one.
00:45:51.630 --> 00:45:53.810
Yeah, the Laplace
transform of that is one.
00:45:53.810 --> 00:45:54.660
Yeah.
00:45:54.660 --> 00:45:58.790
It's got all exponentials
in sort of, you could say,
00:45:58.790 --> 00:45:59.560
equal amounts.
00:45:59.560 --> 00:46:00.060
OK.
00:46:03.660 --> 00:46:07.720
So that's some thoughts about
that about Laplace transforms,
00:46:07.720 --> 00:46:12.870
just sort of the big picture
that takes the differential
00:46:12.870 --> 00:46:16.330
equation, turns it into
an algebra problem,
00:46:16.330 --> 00:46:18.470
and then at the end,
you have to get back,
00:46:18.470 --> 00:46:23.280
and that's the part
that's not always doable.
00:46:23.280 --> 00:46:24.080
OK.
00:46:24.080 --> 00:46:31.670
So what's left for
today is this guy.
00:46:31.670 --> 00:46:33.990
Now this one now.
00:46:33.990 --> 00:46:36.380
Have I got to that point?
00:46:36.380 --> 00:46:42.390
So this will be the final
ideas in this course,
00:46:42.390 --> 00:46:44.360
this four unit core elective.
00:46:51.310 --> 00:46:56.630
What is the impulse response
when there's damping?
00:46:56.630 --> 00:46:59.695
What's the impulse response
when there's damping?
00:46:59.695 --> 00:47:00.194
OK.
00:47:02.966 --> 00:47:03.890
OK.
00:47:03.890 --> 00:47:05.480
So that means that
I would really
00:47:05.480 --> 00:47:08.750
like to solve m
y double prime, b
00:47:08.750 --> 00:47:15.264
y prime, k Y equals an
impulse, delta of t.
00:47:23.380 --> 00:47:27.610
Because if I can solve this,
then I can solve everything.
00:47:27.610 --> 00:47:28.950
And I can solve this.
00:47:28.950 --> 00:47:31.470
It turns out to be easy.
00:47:31.470 --> 00:47:32.780
Now why is it easy?
00:47:35.900 --> 00:47:39.990
You might think, my gosh, we've
got second order equations
00:47:39.990 --> 00:47:46.650
here, we've got a delta
function there, where do we go?
00:47:46.650 --> 00:47:51.540
And so my advice is go this way.
00:47:51.540 --> 00:47:54.270
The solution to
that is the same.
00:47:54.270 --> 00:48:00.386
This with y of 0 equals 0
and y prime of 0 equal 0.
00:48:03.320 --> 00:48:15.140
So you have a spring, so again
we have a spring with a mass.
00:48:15.140 --> 00:48:18.740
And that spring is in a damper.
00:48:18.740 --> 00:48:21.870
So can I just, without
knowing what I'm doing,
00:48:21.870 --> 00:48:25.240
draw a damper around it.
00:48:25.240 --> 00:48:32.130
So the idea is I'm striking
that mass at time 0.
00:48:32.130 --> 00:48:35.390
Striking that mass at time 0.
00:48:35.390 --> 00:48:35.940
What happens?
00:48:38.570 --> 00:48:40.705
What happens immediately?
00:48:43.440 --> 00:48:45.250
And then I don't touch it again.
00:48:45.250 --> 00:48:47.060
I strike it and that's it.
00:48:47.060 --> 00:48:51.900
I've set off, and what have
I-- so my point is this
00:48:51.900 --> 00:49:11.140
has the same solution as m y
double prime plus b y prime.
00:49:11.140 --> 00:49:13.505
And Ky equals 0.
00:49:18.280 --> 00:49:21.810
Nothing happens beyond time 0.
00:49:21.810 --> 00:49:34.590
But what are y of 0-- Well,
let me give this a letter g,
00:49:34.590 --> 00:49:39.370
just to emphasize it's special
and deserves its own name.
00:49:39.370 --> 00:49:45.020
So now, what is the
starting position
00:49:45.020 --> 00:49:51.980
and the starting velocity
for this picture?
00:49:51.980 --> 00:49:58.090
So I'm saying that the y here
is the same as the g there.
00:49:58.090 --> 00:50:00.770
And really if you see
it physically then
00:50:00.770 --> 00:50:04.550
you see it best.
00:50:04.550 --> 00:50:09.610
So again, at the
instant t equals 0,
00:50:09.610 --> 00:50:12.630
I'm hitting that mass
with a unit impulse.
00:50:15.270 --> 00:50:20.230
So what is the position
of that mass, instantly
00:50:20.230 --> 00:50:21.340
after I've hit it?
00:50:21.340 --> 00:50:21.899
STUDENT: 0.
00:50:21.899 --> 00:50:22.690
PROFESSOR: Still 0.
00:50:22.690 --> 00:50:24.130
Good for you.
00:50:24.130 --> 00:50:25.520
Good for you.
00:50:25.520 --> 00:50:30.740
And what is the velocity
of that mass instantly
00:50:30.740 --> 00:50:31.777
after I've hit it?
00:50:31.777 --> 00:50:32.771
STUDENT: 1.
00:50:32.771 --> 00:50:36.660
PROFESSOR: 1 is essentially
the right answer.
00:50:36.660 --> 00:50:38.250
1 is the right answer.
00:50:38.250 --> 00:50:43.780
But the way I've set it up
is, there is an m there.
00:50:43.780 --> 00:50:48.310
If I put an m there,
which would be nicer,
00:50:48.310 --> 00:50:50.040
then the answer would be 1.
00:50:50.040 --> 00:50:54.500
But the way I've written
it here, I have an m there,
00:50:54.500 --> 00:50:59.870
and I haven't fixed the units.
00:50:59.870 --> 00:51:03.248
So that turns out to
give me a 1/m there.
00:51:07.160 --> 00:51:09.830
I could explain why
it's a 1/m, but let me
00:51:09.830 --> 00:51:15.870
just for the moment
say it's my fault.
00:51:15.870 --> 00:51:20.680
It's because I didn't get any
units right that we have 1/m.
00:51:20.680 --> 00:51:23.650
No big deal.
00:51:23.650 --> 00:51:28.170
But now, what's good here?
00:51:28.170 --> 00:51:29.240
What's good?
00:51:29.240 --> 00:51:32.990
This was a problem with a
mysterious delta function.
00:51:32.990 --> 00:51:35.560
This is a problem with 0.
00:51:35.560 --> 00:51:39.300
And the only price we're
paying is the impulse
00:51:39.300 --> 00:51:44.240
gave the mass a little velocity.
00:51:44.240 --> 00:51:46.490
And you can imagine
that the velocity gives
00:51:46.490 --> 00:51:52.930
it is 1/m because the strike
didn't tell us about that.
00:51:52.930 --> 00:51:59.760
So what I'm saying is we can
solve that equation for g.
00:51:59.760 --> 00:52:03.460
We can find g, and
in fact, we have.
00:52:03.460 --> 00:52:10.460
So this really brings
the lecture full circle.
00:52:10.460 --> 00:52:12.600
What do I have here?
00:52:12.600 --> 00:52:19.170
I have a null solution.
00:52:19.170 --> 00:52:21.850
So this g here is
a null solution.
00:52:21.850 --> 00:52:24.450
So what form does it have?
00:52:24.450 --> 00:52:28.380
What can you tell
me about g of t?
00:52:28.380 --> 00:52:30.490
And it's the same as y of t.
00:52:30.490 --> 00:52:34.565
So y is the same as g, and
what's the form for it?
00:52:39.100 --> 00:52:39.870
Yes?
00:52:39.870 --> 00:52:40.690
Tell me?
00:52:40.690 --> 00:52:42.650
I'm taking it back
to the very beginning
00:52:42.650 --> 00:52:46.164
of the lecture where
I solved it with a 0.
00:52:46.164 --> 00:52:47.526
STUDENT: [INAUDIBLE].
00:52:47.526 --> 00:52:49.490
PROFESSOR: C1, thanks.
00:52:49.490 --> 00:52:51.040
STUDENT: [INAUDIBLE].
00:52:51.040 --> 00:52:53.970
PROFESSOR: e to the?
00:52:53.970 --> 00:53:03.260
C1, e to the s1, t, the two
s's, s1 and s2, the special two
00:53:03.260 --> 00:53:11.642
s's, the special roots
of the key equation.
00:53:11.642 --> 00:53:19.260
That equation gives us s1
and s2, by this formula,
00:53:19.260 --> 00:53:24.590
or in a homework
or an exam problem,
00:53:24.590 --> 00:53:28.420
we hope that it would
come out easily.
00:53:28.420 --> 00:53:29.950
So this is that part of it.
00:53:29.950 --> 00:53:32.440
What's the rest of it?
00:53:32.440 --> 00:53:36.410
C2 e to the s2 t.
00:53:40.350 --> 00:53:46.820
The impulse response is the
particular null solution
00:53:46.820 --> 00:53:53.200
that starts with a shot,
starts with an impulse,
00:53:53.200 --> 00:53:55.010
starts with a strike.
00:53:55.010 --> 00:53:55.510
OK.
00:53:55.510 --> 00:53:59.250
So I just have to
find c1 and c2 here.
00:53:59.250 --> 00:54:00.170
All right?
00:54:00.170 --> 00:54:03.250
We've come back to
the basic problem
00:54:03.250 --> 00:54:05.100
in differential equations.
00:54:05.100 --> 00:54:06.360
We've got the solution.
00:54:06.360 --> 00:54:07.630
We've got two constants.
00:54:07.630 --> 00:54:09.590
We've got two equations.
00:54:09.590 --> 00:54:14.940
We just plug that function
into that equation.
00:54:14.940 --> 00:54:18.640
It gives us one
fact about c1, c2.
00:54:18.640 --> 00:54:20.420
This gives us the second fact.
00:54:20.420 --> 00:54:21.280
We solve them.
00:54:21.280 --> 00:54:24.740
Why don't I just
write down the answer?
00:54:24.740 --> 00:54:28.890
It turns out to
be e to the s1 t,
00:54:28.890 --> 00:54:32.410
and the other guy will come
in with an opposite sign e
00:54:32.410 --> 00:54:37.625
to the s2t over s1 minus s2.
00:54:40.330 --> 00:54:43.110
I think that gives us the c's.
00:54:46.910 --> 00:54:48.430
Oh, there'd be an m.
00:54:48.430 --> 00:54:52.060
There'd be an m
times this, because
00:54:52.060 --> 00:54:55.010
of my messing with units.
00:54:58.600 --> 00:55:01.340
So can we just check?
00:55:01.340 --> 00:55:04.540
At t equals 0, what
do I get out of this?
00:55:04.540 --> 00:55:06.380
STUDENT: 0.
00:55:06.380 --> 00:55:07.050
PROFESSOR: 0.
00:55:07.050 --> 00:55:08.760
What I'm supposed to.
00:55:08.760 --> 00:55:12.450
What's the derivative
at t equals 0?
00:55:12.450 --> 00:55:15.160
The derivative at t
equals 0, this derivative
00:55:15.160 --> 00:55:17.530
is going to bring down an s1.
00:55:17.530 --> 00:55:20.710
The derivative here
will bring down an s2.
00:55:20.710 --> 00:55:23.760
At t equals 0 the
exponentials will all
00:55:23.760 --> 00:55:27.520
be one so I'll just have
the s1 minus the s2.
00:55:27.520 --> 00:55:30.570
It'll cancel that and
it'll be 1 over m.
00:55:30.570 --> 00:55:36.850
So this is the neat formula
for the impulse response.
00:55:36.850 --> 00:55:40.540
That's the neat formula
for the impulse response.
00:55:40.540 --> 00:55:46.070
And then why-- can I use this
little corner of the board?
00:55:46.070 --> 00:55:49.380
Why do I want that
impulse response?
00:55:49.380 --> 00:55:51.830
What can I use it for?
00:55:51.830 --> 00:55:54.569
It gives me the answer,
not just for the impulse
00:55:54.569 --> 00:55:55.360
but for everything.
00:55:58.790 --> 00:56:08.730
The particular solution
is for any forces, force,
00:56:08.730 --> 00:56:14.790
I multiply by whatever the--
let me write the formula
00:56:14.790 --> 00:56:18.210
and I'll show you what it says.
00:56:18.210 --> 00:56:21.436
Yes, there is the formula.
00:56:21.436 --> 00:56:22.435
I'm sorry it's squeezed.
00:56:32.480 --> 00:56:35.900
But really, the
goal here was simply
00:56:35.900 --> 00:56:40.690
to get a handle on what
is the response to any f.
00:56:40.690 --> 00:56:43.090
And again, I look
at that this way.
00:56:43.090 --> 00:56:46.636
F of s is the input at time s.
00:56:46.636 --> 00:56:54.720
G is the growth factor over the
remaining time up until time t.
00:56:54.720 --> 00:57:01.690
So Y at time t, I take all
the inputs up to time t,
00:57:01.690 --> 00:57:06.080
And each input gets multiplied
by its growth factor.
00:57:06.080 --> 00:57:10.840
It was e to the a, t minus s
in the first order equation.
00:57:10.840 --> 00:57:14.460
Now we've got two exponentials.
00:57:14.460 --> 00:57:19.910
But that's the solution
of the general problem.
00:57:19.910 --> 00:57:26.950
So we have now in one
lecture completed a solution
00:57:26.950 --> 00:57:33.090
to the second order constant
coefficient differential
00:57:33.090 --> 00:57:33.898
equation.
00:57:33.898 --> 00:57:34.682
Right.
00:57:34.682 --> 00:57:35.182
Yeah.
00:57:37.900 --> 00:57:42.422
By finding the impulse response.
00:57:42.422 --> 00:57:44.827
Yes?
00:57:44.827 --> 00:57:47.232
STUDENT: [INAUDIBLE]
would we still
00:57:47.232 --> 00:57:50.610
be able to [INAUDIBLE]
if s1 is equal to s2?
00:57:50.610 --> 00:57:52.730
PROFESSOR: Ah, if s1 equals s2.
00:57:52.730 --> 00:57:58.130
That's the case where
formulas need a patch.
00:57:58.130 --> 00:57:59.090
They need a patch.
00:57:59.090 --> 00:58:05.235
If s1 equals s2, what
do you think happens?
00:58:08.130 --> 00:58:11.083
If s1 equal s2, everybody
sees, I have 0/0.
00:58:14.950 --> 00:58:19.170
And so this is like
a technical question
00:58:19.170 --> 00:58:21.990
that I wasn't going
to ask myself.
00:58:21.990 --> 00:58:23.820
You asked it.
00:58:23.820 --> 00:58:25.320
You're responsible.
00:58:25.320 --> 00:58:27.500
What do we do for 0/0?
00:58:32.120 --> 00:58:34.740
What did you learn in calculus?
00:58:34.740 --> 00:58:40.180
Who's the crazy guy who figured
out how to deal with 0/0?
00:58:40.180 --> 00:58:44.030
In a way, calculus is
all about 0/0, right?
00:58:44.030 --> 00:58:48.390
Delta y over delta x,
they're both headed to 0.
00:58:48.390 --> 00:58:51.860
And suppose you
have-- let me take
00:58:51.860 --> 00:58:55.980
the most famous example of 0/0.
00:58:55.980 --> 00:59:03.060
It's like sine x over
x, as x goes to 0.
00:59:03.060 --> 00:59:05.560
So x going to 0.
00:59:05.560 --> 00:59:08.265
Sine x goes to 0, x goes to 0.
00:59:11.580 --> 00:59:13.210
What's the answer, by the way?
00:59:13.210 --> 00:59:17.300
What happens to that
ratio as x goes to 0?
00:59:17.300 --> 00:59:20.360
This is maybe the
most famous example.
00:59:20.360 --> 00:59:24.350
Sine x over x, when x gets
very small, is close to?
00:59:24.350 --> 00:59:24.970
STUDENT: 1.
00:59:24.970 --> 00:59:26.530
PROFESSOR: 1, thanks.
00:59:26.530 --> 00:59:30.820
And now just help me out
with the name of the guy.
00:59:30.820 --> 00:59:35.490
It's a crazy spelling
name, and do you remember?
00:59:35.490 --> 00:59:37.630
L'Hopital.
00:59:37.630 --> 00:59:38.880
L'Hopital.
00:59:38.880 --> 00:59:40.690
Everybody despises him.
00:59:40.690 --> 00:59:42.980
Probably hi friends
despised him.
00:59:42.980 --> 00:59:48.920
But anyway, L'Hopital says
in a situation like that,
00:59:48.920 --> 00:59:52.950
when you're going to
0/0, you're allowed
00:59:52.950 --> 00:59:55.580
to do something
a little strange.
00:59:55.580 --> 00:59:58.470
You're allowed to take
the derivative of the top,
00:59:58.470 --> 01:00:01.260
so it has the same limit.
01:00:01.260 --> 01:00:03.810
Instead of looking
at this 0/0, you
01:00:03.810 --> 01:00:07.210
can take the derivative
of the top, cos x,
01:00:07.210 --> 01:00:10.600
divided by the derivative
of the bottom, 1.
01:00:10.600 --> 01:00:16.810
And now you can let x go to 0,
and you get the right answer.
01:00:16.810 --> 01:00:19.000
So this gave a 0/0.
01:00:19.000 --> 01:00:20.600
Unclear.
01:00:20.600 --> 01:00:21.850
Fuzzy.
01:00:21.850 --> 01:00:24.480
This gives-- what's
the right answer then?
01:00:24.480 --> 01:00:25.780
Just tell me again.
01:00:25.780 --> 01:00:27.850
When x goes to 0 this becomes?
01:00:27.850 --> 01:00:28.630
STUDENT: 1.
01:00:28.630 --> 01:00:29.171
PROFESSOR: 1.
01:00:29.171 --> 01:00:30.370
Right.
01:00:30.370 --> 01:00:31.910
So that's L'Hopital.
01:00:31.910 --> 01:00:34.890
So that's what I
would have to do here.
01:00:34.890 --> 01:00:37.560
I would take the
derivative of this,
01:00:37.560 --> 01:00:40.820
and the derivative of this--
the only sort of tricky part
01:00:40.820 --> 01:00:42.900
is it's the s derivative.
01:00:42.900 --> 01:00:46.470
It's s1 going to s2.
01:00:46.470 --> 01:00:49.220
Let me just tell you the result.
01:00:49.220 --> 01:00:50.760
Since you asked.
01:00:50.760 --> 01:00:52.760
A factor t comes out.
01:00:52.760 --> 01:00:59.220
It's t e to the s1, or s1 is the
same as s2, divided by the m.
01:01:02.680 --> 01:01:04.530
It actually looks simpler.
01:01:04.530 --> 01:01:05.450
There's only one.
01:01:05.450 --> 01:01:09.910
This is in the case s1 equal s2.
01:01:09.910 --> 01:01:11.840
So s1 is the same as s2.
01:01:11.840 --> 01:01:13.080
I just chose s1.
01:01:13.080 --> 01:01:13.580
Yeah.
01:01:16.520 --> 01:01:20.520
L'Hopital gives
a simpler answer.
01:01:20.520 --> 01:01:27.480
And it's got this suspicious
and recognizable factor t.
01:01:27.480 --> 01:01:29.090
That came from L'Hopital.
01:01:29.090 --> 01:01:31.280
OK, I won't do that stiff.
01:01:35.850 --> 01:01:43.800
So let me say again, we've
now done the second order
01:01:43.800 --> 01:01:48.160
constant coefficient
equation I do just
01:01:48.160 --> 01:01:56.090
have 10 minutes of
something to make it better.
01:01:56.090 --> 01:02:03.290
And that is that the famous
quadratic formula for s,
01:02:03.290 --> 01:02:08.800
for s1 and s2 is not beautiful.
01:02:08.800 --> 01:02:11.480
It's correct.
01:02:11.480 --> 01:02:13.050
It's correct.
01:02:13.050 --> 01:02:15.890
But it's a little bit of a mess.
01:02:15.890 --> 01:02:20.250
You've got three things, b
and m and k playing around.
01:02:20.250 --> 01:02:25.630
And we saw in this picture,
we saw all the differences.
01:02:25.630 --> 01:02:34.590
I guess in this example
I kept m1 and I kept k1,
01:02:34.590 --> 01:02:36.860
and I increased b.
01:02:36.860 --> 01:02:43.990
I could do other examples
where I increase the k,
01:02:43.990 --> 01:02:46.600
I make it stiffer and stiffer.
01:02:46.600 --> 01:02:48.260
All these examples.
01:02:48.260 --> 01:02:54.930
And engineers have
worked for 100 years
01:02:54.930 --> 01:03:00.740
to see, out of this formula what
are the important parameters,
01:03:00.740 --> 01:03:04.930
what are the important
numbers, and hopefully,
01:03:04.930 --> 01:03:06.950
where possible dimensionless.
01:03:06.950 --> 01:03:10.970
So I just want to-
the final minutes
01:03:10.970 --> 01:03:16.840
would be-- back to high school--
playing with this formula,
01:03:16.840 --> 01:03:20.330
to get better numbers in there.
01:03:20.330 --> 01:03:21.320
May I do that?
01:03:21.320 --> 01:03:23.120
I just think,
because then you'll
01:03:23.120 --> 01:03:29.530
see-- I learned this, actually,
so this is like something math
01:03:29.530 --> 01:03:31.910
professors have no reason to do.
01:03:31.910 --> 01:03:32.610
Look at that.
01:03:32.610 --> 01:03:33.890
That's the formula.
01:03:33.890 --> 01:03:35.590
End of story.
01:03:35.590 --> 01:03:41.330
But the Web, 1803
website, has a class
01:03:41.330 --> 01:03:44.570
in which Professor Miller
from the math department
01:03:44.570 --> 01:03:52.010
was teaching this subject,
doing these, exactly these,
01:03:52.010 --> 01:03:56.850
but also Professor
Vandiver from Engineering
01:03:56.850 --> 01:04:03.570
was putting in his suggestion
of what are the good parameters?
01:04:03.570 --> 01:04:08.060
What are the parameters
that engineers look at?
01:04:08.060 --> 01:04:10.070
So that would be
my final comment,
01:04:10.070 --> 01:04:13.340
and I won't do it as well
as Professor Vandiver did.
01:04:13.340 --> 01:04:16.620
But can I just
take that-- let me
01:04:16.620 --> 01:04:20.990
erase these two
special examples,
01:04:20.990 --> 01:04:27.000
and look at this question.
01:04:29.900 --> 01:04:31.320
Again, the book will do it.
01:04:34.380 --> 01:04:49.060
So one nice-- b/2m is a pretty
natural parameter to use.
01:04:49.060 --> 01:04:52.520
Let me introduce
that as one of them.
01:04:52.520 --> 01:04:59.790
I'm going to, by taking ratios
like b/2m, let me call that p.
01:04:59.790 --> 01:05:03.970
Let me call b/2m.
01:05:03.970 --> 01:05:10.450
So that's a ratio
of damping to mass.
01:05:10.450 --> 01:05:21.050
And then this has got
to come out simpler.
01:05:21.050 --> 01:05:23.120
What does that come out?
01:05:23.120 --> 01:05:25.200
If you'll allow me, I'm
going to open the book
01:05:25.200 --> 01:05:27.834
so I don't write the
wrong thing here.
01:05:30.550 --> 01:05:35.750
This is in the book, on page 99.
01:05:35.750 --> 01:05:39.960
The title is Better
Formulas for s1 and s2.
01:05:39.960 --> 01:05:42.310
Better Formulas for s1 and s2.
01:05:42.310 --> 01:05:44.500
And here's my first
better formula.
01:05:47.380 --> 01:05:52.180
You can see that I get a minus
b plus or minus the square root
01:05:52.180 --> 01:05:54.800
of something.
01:05:54.800 --> 01:05:57.965
And that something will
turn out to be p squared.
01:06:00.550 --> 01:06:03.570
And then it'll be
a minus something.
01:06:03.570 --> 01:06:05.930
And that something
will turn out to be
01:06:05.930 --> 01:06:08.140
the natural frequency squared.
01:06:08.140 --> 01:06:10.630
Isn't that nice?
01:06:10.630 --> 01:06:15.440
So what's the natural frequency?
01:06:15.440 --> 01:06:19.700
Somehow, the natural frequency's
coming in from this and this?
01:06:19.700 --> 01:06:23.870
And just remind me what
that second parameter is,
01:06:23.870 --> 01:06:27.480
omega n squared, the
natural frequency
01:06:27.480 --> 01:06:31.070
of oscillation with no damping.
01:06:31.070 --> 01:06:33.870
Tell me again what
that is, because that
01:06:33.870 --> 01:06:36.750
was the fundamental
ratio from last time.
01:06:36.750 --> 01:06:39.275
It's central to
all of engineering.
01:06:39.275 --> 01:06:39.775
It's?
01:06:39.775 --> 01:06:41.185
STUDENT: k/m.
01:06:41.185 --> 01:06:42.010
PROFESSOR: k/m.
01:06:42.010 --> 01:06:42.510
Thanks, k/m.
01:06:45.580 --> 01:06:48.520
So I believe-- and
maybe Professor Fry
01:06:48.520 --> 01:06:53.310
could make this assignment
a homework question,
01:06:53.310 --> 01:06:56.020
which is just algebra question.
01:06:56.020 --> 01:07:01.320
Everybody sees that I
have a minus p here.
01:07:01.320 --> 01:07:09.230
And with a little care you get
p squared, which is quite nice.
01:07:09.230 --> 01:07:10.150
Which is quite nice.
01:07:10.150 --> 01:07:15.000
And so we see that the
decision between overdamping--
01:07:15.000 --> 01:07:16.630
remember now?
01:07:16.630 --> 01:07:21.730
Overdamping is when you damped
so much that this became
01:07:21.730 --> 01:07:26.690
negative and you got an
imaginary number in there.
01:07:26.690 --> 01:07:29.370
Underdamping, it's
still positive.
01:07:29.370 --> 01:07:31.430
Overdamping, it's negative.
01:07:31.430 --> 01:07:33.930
And so really that separation
between overdamping
01:07:33.930 --> 01:07:39.124
and underdamping is the
ratio of p to omega n.
01:07:39.124 --> 01:07:43.590
P to omega n is
the damping ratio.
01:07:43.590 --> 01:07:44.360
I think.
01:07:44.360 --> 01:07:45.500
There may be a factor too.
01:07:45.500 --> 01:07:53.200
Let me try to-- everybody
sees that's the battle
01:07:53.200 --> 01:07:56.490
between these, if you
accept that formula,
01:07:56.490 --> 01:08:00.420
and if it's in the book
it's got to be right.
01:08:00.420 --> 01:08:02.920
OK.
01:08:02.920 --> 01:08:11.060
And so the damping
ratio is just that.
01:08:11.060 --> 01:08:12.620
That's the damping ratio.
01:08:12.620 --> 01:08:17.890
Now that's called zeta.
01:08:17.890 --> 01:08:19.260
The Greek letter zeta.
01:08:19.260 --> 01:08:24.300
I'm not Greek and not
good writing zeta.
01:08:24.300 --> 01:08:31.350
So I have unilaterally decided
to use a capital zeta, which
01:08:31.350 --> 01:08:39.870
is a Z. Zeta is the Greek letter
for Z. I could try to write it
01:08:39.870 --> 01:08:43.550
but you wouldn't be impressed.
01:08:43.550 --> 01:08:45.910
So it's that damping ratio.
01:08:45.910 --> 01:08:47.689
So now what does this mean?
01:08:47.689 --> 01:08:53.779
Z smaller than 1, z equal
to 1, c greater than 1?
01:08:53.779 --> 01:09:01.750
Tell me what those-- obviously,
when it's smaller than 1,
01:09:01.750 --> 01:09:05.705
p is smaller than omega n.
01:09:11.410 --> 01:09:14.180
Yeah, so what's going on here?
01:09:14.180 --> 01:09:16.800
Which one is underdamping,
which one is critical
01:09:16.800 --> 01:09:18.858
damping, which one
is overdamping?
01:09:23.080 --> 01:09:31.439
Because there's no
difficult stuff here.
01:09:31.439 --> 01:09:33.330
We're coasting in
the last minutes
01:09:33.330 --> 01:09:39.859
here by just choosing words and
notation that have turned out
01:09:39.859 --> 01:09:47.319
over a century to be more
revealing than b squared
01:09:47.319 --> 01:09:51.770
minus 4mk, and this Z.
01:09:51.770 --> 01:09:54.900
So z less than 1 will be what?
01:09:54.900 --> 01:09:58.370
Z less than 1 will be
p smaller than omega n.
01:09:58.370 --> 01:10:00.160
p smaller than this.
01:10:00.160 --> 01:10:02.480
What's the story
on that case then?
01:10:02.480 --> 01:10:04.930
STUDENT: [INAUDIBLE].
01:10:04.930 --> 01:10:10.820
PROFESSOR: That is underdamping,
I guess. p small has to go.
01:10:10.820 --> 01:10:14.880
p is like b, the damping.
01:10:14.880 --> 01:10:18.100
And small damping
is underdamping.
01:10:18.100 --> 01:10:19.010
So this is underdamp.
01:10:22.430 --> 01:10:25.110
Underdamp.
01:10:25.110 --> 01:10:27.410
And that's the
case, in which we're
01:10:27.410 --> 01:10:29.400
going to have some
imaginary stuff.
01:10:29.400 --> 01:10:31.680
We're going to have
some oscillation
01:10:31.680 --> 01:10:34.350
with the decay
coming from there.
01:10:34.350 --> 01:10:36.700
Now, what about z equal to 1?
01:10:36.700 --> 01:10:40.000
z equal to 1 means
p equals omega m,
01:10:40.000 --> 01:10:43.160
so that equals that so
it's a big 0 in there.
01:10:43.160 --> 01:10:44.286
What case is that?
01:10:44.286 --> 01:10:45.160
STUDENT: [INAUDIBLE].
01:10:45.160 --> 01:10:46.470
PROFESSOR: Critical damping.
01:10:46.470 --> 01:10:49.300
It's this case in that picture.
01:10:49.300 --> 01:10:54.890
It's that case with a
double 0, equal s's.
01:10:54.890 --> 01:10:58.390
Formulas that have to
take account of that, this
01:10:58.390 --> 01:10:59.040
is critical.
01:11:02.480 --> 01:11:08.530
And then finally, z
greater than 1 is what?
01:11:08.530 --> 01:11:09.590
Overdamped.
01:11:09.590 --> 01:11:10.890
Overdamping.
01:11:10.890 --> 01:11:13.800
Z bigger than 1 means p is big.
01:11:13.800 --> 01:11:19.895
P big means b is big, damping
is big, it's overdamped.
01:11:19.895 --> 01:11:20.395
Overdamped.
01:11:24.360 --> 01:11:29.530
So we've got it down to one
parameter, the damping ratio,
01:11:29.530 --> 01:11:32.200
to tell us these things.
01:11:32.200 --> 01:11:35.700
Rather than previously
we had to say
01:11:35.700 --> 01:11:37.720
is b squared smaller than 4mk?
01:11:37.720 --> 01:11:40.430
Is it equal to 4mk?
01:11:40.430 --> 01:11:42.510
Is it bigger than 4mk?
01:11:42.510 --> 01:11:47.090
Now we've got those words
down to a single number z.
01:11:47.090 --> 01:11:54.810
And let me just write next to
us here that the z turns out
01:11:54.810 --> 01:11:59.470
to be the ratio of the
damping to-- I think
01:11:59.470 --> 01:12:03.480
it's right-- it's
the damping divided
01:12:03.480 --> 01:12:08.419
by the square root
of 4mk, I think.
01:12:08.419 --> 01:12:09.960
Can I just put a
question mark there.
01:12:12.980 --> 01:12:23.260
You couldn't mess around
with the letters p and z.
01:12:23.260 --> 01:12:30.030
But to get some variation
from some other,
01:12:30.030 --> 01:12:33.110
but the point is, you
see how much cleaner that
01:12:33.110 --> 01:12:34.940
is compared to this?
01:12:34.940 --> 01:12:39.320
You're directly comparing
that number to that number.
01:12:39.320 --> 01:12:42.454
And that ratio is that number.
01:12:42.454 --> 01:12:43.720
Yeah.
01:12:43.720 --> 01:12:47.190
So all the formulas
come out nicely.
01:12:47.190 --> 01:12:49.680
Yeah, the formulas
come out nicely.
01:12:49.680 --> 01:12:55.400
And I guess what we see
here-- final comment-- what
01:12:55.400 --> 01:12:59.770
we see here is what
is the frequency
01:12:59.770 --> 01:13:03.770
of underdamped oscillations.
01:13:03.770 --> 01:13:08.430
So I want to be in
this underdamping case
01:13:08.430 --> 01:13:11.100
where there is oscillation.
01:13:11.100 --> 01:13:16.510
There is an imaginary
number coming out of that.
01:13:16.510 --> 01:13:20.060
But there's also a real number.
01:13:20.060 --> 01:13:23.830
Is the frequency of underdamped
oscillation the same
01:13:23.830 --> 01:13:27.410
as omega m, the
natural frequency?
01:13:27.410 --> 01:13:28.440
No.
01:13:28.440 --> 01:13:35.810
The frequency of that
number-- so final comment,
01:13:35.810 --> 01:13:36.900
let me put it just here.
01:13:42.080 --> 01:13:46.790
I would like this whole thing
to be i, to give me oscillation,
01:13:46.790 --> 01:13:56.720
times omega d, the
damped frequency.
01:13:56.720 --> 01:13:58.870
And let me just
say what that is.
01:13:58.870 --> 01:14:04.500
So omega d, the
damped frequency,
01:14:04.500 --> 01:14:09.180
squared, is this omega
natural frequency
01:14:09.180 --> 01:14:12.750
squared minus the p squared.
01:14:12.750 --> 01:14:20.655
If I had longer and we didn't
have blackboards already full
01:14:20.655 --> 01:14:24.630
of formulas, I
could-- it's the thing
01:14:24.630 --> 01:14:26.510
whose square root
we're taking here.
01:14:26.510 --> 01:14:35.050
So this is minus p plus
or minus i, omega damped.
01:14:35.050 --> 01:14:37.320
Omega damped is
this square root.
01:14:40.340 --> 01:14:51.040
There, we succeeded to
fit in the better ratios,
01:14:51.040 --> 01:14:56.060
the good quantities to look at.
01:14:56.060 --> 01:14:58.100
So again, the good
quantities to look at
01:14:58.100 --> 01:15:05.075
are p, z, the damping ratio,
omega d the damped frequency.
01:15:07.980 --> 01:15:10.490
I think in a first
lecture you could say,
01:15:10.490 --> 01:15:14.080
well, we already had
correct formulas,
01:15:14.080 --> 01:15:17.080
we should just leave it there,
and that's absolutely true,
01:15:17.080 --> 01:15:22.610
but anyway, this is what-- these
are the letters people have
01:15:22.610 --> 01:15:26.020
introduced to make
the formulas easier
01:15:26.020 --> 01:15:29.380
to understand in an
engineering problem.
01:15:29.380 --> 01:15:30.450
OK.
01:15:30.450 --> 01:15:33.110
I'm all done except
for questions.
01:15:36.000 --> 01:15:37.670
Yes?
01:15:37.670 --> 01:15:39.528
Don't ask me about
resonance again.
01:15:39.528 --> 01:15:41.020
Yes, OK.
01:15:41.020 --> 01:15:41.520
Yes?
01:15:41.520 --> 01:15:46.251
STUDENT: In the case of where
we have the delta function, what
01:15:46.251 --> 01:15:48.010
is the velocity [INAUDIBLE]?
01:15:48.010 --> 01:15:49.526
PROFESSOR: What is the what?
01:15:49.526 --> 01:15:51.609
STUDENT: Why is the velocity
equal to [INAUDIBLE]?
01:15:51.609 --> 01:15:53.006
PROFESSOR: A-ha.
01:15:53.006 --> 01:15:55.850
Okay.
01:15:55.850 --> 01:15:57.830
You're right on the ball.
01:15:57.830 --> 01:16:00.990
The question is where
did this come from.
01:16:00.990 --> 01:16:04.060
Where did that come from?
01:16:04.060 --> 01:16:07.360
Can I tell you?
01:16:07.360 --> 01:16:10.250
So if I integrate
everything here,
01:16:10.250 --> 01:16:14.200
if I take the integral
of everything,
01:16:14.200 --> 01:16:20.090
between 0, a little bit left
of 0-- can I call that 0 minus?
01:16:20.090 --> 01:16:22.860
Just a little bit left of 0.
01:16:22.860 --> 01:16:24.310
This is crazy.
01:16:24.310 --> 01:16:27.700
No math professor or
whatever should ever do this.
01:16:27.700 --> 01:16:33.760
To a little bit right of
0, just a real short time.
01:16:33.760 --> 01:16:36.065
So what am I going to call
a little bit right of 0?
01:16:38.920 --> 01:16:41.640
0 plus.
01:16:41.640 --> 01:16:49.960
OK now what is that integral?
01:16:49.960 --> 01:16:52.920
Between a little left of
0 and a little right of 0,
01:16:52.920 --> 01:16:55.525
you know what the integral
of the delta function is.
01:16:55.525 --> 01:16:56.230
It is?
01:16:56.230 --> 01:16:57.850
STUDENT: 1.
01:16:57.850 --> 01:16:59.050
PROFESSOR: 1.
01:16:59.050 --> 01:17:00.290
Good.
01:17:00.290 --> 01:17:03.040
Now, what are these
ridiculous things?
01:17:03.040 --> 01:17:09.870
Well, y is not changing
in this tiny, tiny time.
01:17:09.870 --> 01:17:20.060
So this is something,
it's not getting big.
01:17:20.060 --> 01:17:22.280
I'm integrating it over
this tiny little time.
01:17:22.280 --> 01:17:23.370
It's nothing.
01:17:23.370 --> 01:17:25.660
Forget it.
01:17:25.660 --> 01:17:29.880
Similarly here, y prime,
the velocity is not
01:17:29.880 --> 01:17:31.910
climbing to infinity.
01:17:31.910 --> 01:17:34.540
There's no-- and
I'm just integrating
01:17:34.540 --> 01:17:37.330
over this infinitesimal
little time.
01:17:37.330 --> 01:17:38.930
Nothing here.
01:17:38.930 --> 01:17:45.090
So this term has to be
responsible for the 1.
01:17:45.090 --> 01:17:48.803
And now you can tell me the
integral of m y double prime.
01:17:52.750 --> 01:17:55.220
What's the integral
of m y prime?
01:17:55.220 --> 01:17:56.590
STUDENT: m y prime
01:17:56.590 --> 01:17:58.890
PROFESSOR: m y prime.
01:17:58.890 --> 01:18:09.910
So m y prime plus, at 0 plus,
minus m y prime at 0 minus.
01:18:09.910 --> 01:18:13.600
But at 0 minus, it's 0.
01:18:13.600 --> 01:18:15.080
You see what's happening?
01:18:15.080 --> 01:18:18.980
And on the right
side I'm getting a 1.
01:18:18.980 --> 01:18:25.140
This is-- no person who had any
skill with a blackboard would
01:18:25.140 --> 01:18:26.620
allow this to happen.
01:18:26.620 --> 01:18:28.920
But that happened.
01:18:28.920 --> 01:18:29.630
OK.
01:18:29.630 --> 01:18:33.930
So these lower order
terms are typical of math.
01:18:33.930 --> 01:18:37.330
Lower order terms in
the limit, forget them.
01:18:37.330 --> 01:18:41.920
This is the top term, and it
has to have something there,
01:18:41.920 --> 01:18:44.300
because it has to balance the 1.
01:18:44.300 --> 01:18:47.800
And what it has is
the jump in y prime.
01:18:47.800 --> 01:18:50.330
So this is the instant
jump in y prime
01:18:50.330 --> 01:18:54.580
in velocity, is times m gives 1.
01:18:54.580 --> 01:19:01.980
So the instant jump jumped
us from 0 to 1 over m.
01:19:01.980 --> 01:19:03.670
That's where I came from.
01:19:03.670 --> 01:19:08.630
Well, that was a good question,
and a kind of crazy answer
01:19:08.630 --> 01:19:10.250
but there it is.
01:19:10.250 --> 01:19:14.980
OK, so we've got a mention
of the Laplace transform
01:19:14.980 --> 01:19:18.650
as the algebra tool
that works when you're
01:19:18.650 --> 01:19:21.580
staying with exponentials
and nice functions.
01:19:21.580 --> 01:19:27.172
And you'll see more of that.
01:19:27.172 --> 01:19:31.804
So it's a frequently used tool
to turn problems into algebra.