WEBVTT
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I just recalling some of the
notation we are going to need
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for today, and a couple of the
facts that we're going to use,
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plus trying to clear up a
couple of confusions that the
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recitations report.
This can be thought of two
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ways.
It's a formal polynomial in D,
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in the letter D.
It just has the shape of the
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polynomial, D squared plus AD
plus B.
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A and B are constant
coefficients.
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But, it's also,
at the same time,
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if you think what it does,
it's a linear operator on
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functions.
It's a linear operator on
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functions like y of t.
You think of it both ways:
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formal polynomial because we
want to do things like factoring
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it, substituting two for D and
things like that.
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Those are things you do with
polynomials.
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You do them algebraically.
You can take the formal
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derivative of the polynomial
because it's just sums of
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powers.
On the other hand,
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as a linear operator,
it does something to functions.
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It differentiates them,
multiplies them by constants or
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something like that.
So it's, so to speak,
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has a dual aspect this way.
And, that's one of the things
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we are exploiting what we use
operator methods to solve
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differential equations.
Now, let me remind you of the
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key thing we were interested in.
f of t:
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not any old function,
we'll get to that next time,
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but f of t,
exponentials.
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So, it should be an exponential
or something like an
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exponential, or pretty close to
it, for example,
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something with sine t
and cosine t,
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or e to the,
that could be thought of as
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part of the real or imaginary
part of a complex exponential.
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And, maybe by the end of today,
we will have generalized that
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even little more.
But basically,
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I'm interested in exponentials.
Let's make it alpha complex.
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That will at least take care of
the cases, e to the ax times
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cosine bx, sin bx,
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which are the main cases.
Those are the main cases.
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Then, remember the little table
we made.
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I simply gave you the formula
for the particular solution.
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So, what we're looking for is
we already know how to solve the
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homogeneous equation.
What we want is that particular
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solution.
And then, the recipe for it I
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gave you, these things were
proved by the substitution rules
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and exponential shift rules.
The recipe was that if f of t
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was, let's make a
little table.
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f of t is, well,
it's always e to the a t.
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So, in other words,
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it's e to the a t.
The cases are,
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so yp, what is the yp?
Well, it is the normal case is
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yp equals e to that alpha t
divided by the
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polynomial where you substitute,
you take that polynomial,
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and wherever you see a D,
you substitute the complex
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number, alpha.
There, I'm thinking of it as a
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formal polynomial.
I'm not thinking of it as an
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operator.
Now, this breaks down.
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So, that's the formula for the
particular solution.
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The only trouble is,
it breaks down if p of alpha is
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zero.
So, we have to assume that it's
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not.
Now, if p of alpha is zero,
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that means alpha is a root of
the polynomial,
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a zero of the polynomial is a
better word.
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So, in that case,
it will be e to the alpha t
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divided by p prime of alpha.
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Differentiate formally the
polynomials, --
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-- and you will get 2D plus A.
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And now, substitute in the
alpha.
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And, this will be okay provided
p prime of alpha
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is not zero.
That means that alpha is the
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simple root, simple zero of p.
And then, there's one more
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case, which, since I won't need
today, I won't write on the
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board.
But, you'll need it for
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homework.
So, make sure you know it.
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Another words,
if this is zero,
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then you've got a double root.
And, there is still a different
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formula.
And, this is wrong because I
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forgot the t.
Yes?
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I could tell on your faces.
That was before,
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and now we are up to today.
What we are interested in
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talking about today is what this
has to do with the phenomenon of
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resonance.
Everybody knows at least one
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case of resonance,
I hope.
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A little kid is on his swing,
right?
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Back and forth,
and they are very,
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very little,
so they want a push.
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Okay, well, everybody knows
that to make the swing go,
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a swing has a certain natural
frequency.
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It swings back and forth like
that.
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It's a simple pendulum.
It's actually damped,
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but let's pretend that it
isn't.
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Everybody knows you want to
push a kid on a swing so that
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they go high.
You have to push with
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essentially the same frequency
that the natural frequency of
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the spring, of the swing is.
It's automatic,
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because when you come back
here, it gets to there,
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and that's where you push.
So, automatically,
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you time your pushes.
But if you want the kid to
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stop, you just do the opposite.
Push at the wrong time.
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So anyway, that's resonance.
Of course, there are more
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serious applications of it.
It's what made the Tacoma
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Bridge fall down,
and I think movies of that are
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now being shown not merely on
television, but in elementary
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school.
Resonance is what made,
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okay, more resonance stories
later.
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So, my aim is,
what is this physical
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phenomenon, that to get a big
amplitude you should have it
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match the frequency?
What does that have to do with
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a differential equation?
Well, the differential equation
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for that simple pendulum,
let's assume it's undamped,
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will be of the type y double
prime plus,
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I'm using t now since t is
time.
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That will be our new
independent variable,
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plus omega nought squared
is the natural
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frequency of the pendulum or of
the spring, or whatever it is
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that's doing the vibrating.
Yeah, any questions?
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What we're doing is driving
that with the cosine,
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with something of a different
frequency.
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So, this is the input,
or the driving term as it's
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often called,
or it's sometimes called the
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forcing term.
And, the point is I'm going to
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assume that the frequency is
different.
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The driving frequency is
different from the natural
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frequency.
So, this is the input
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frequency.
Okay, and now let's simply
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solve the equation and see what
we get.
00:08:03.000 --> 00:08:09.000
So, it's if I write it using
the operator,
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it's D squared plus omega
nought squared applied to y
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is equal to cosine.
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It's a good idea to do this
because the formulas are going
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to ask you to substitute into a
polynomial.
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So, it's good to have the
polynomial right in front of you
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to avoid the possibility of
error.
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Well, really what I want is the
particular solution.
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It's the particular solution
that's going to give me a pure
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oscillation.
And, the thing to do is,
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of course, since this cosine,
you want to make it complex.
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So, we are going to complexify
the equation in order to be able
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to solve it more easily,
and in order to be able to use
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those formulas.
So, the complex equation is
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going to be D squared plus omega
nought squared.
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Well, it's going to be a
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complex, particular solution.
So, I'll call it y tilde.
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And, on the right-hand side,
that's going to be e to the i
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omega1 t.
Cosine is the real part of
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this.
So, when we get our answer,
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we want to be sure to take the
real part of the answer.
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I don't want the complex
answer, I want its real part.
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I want the real answer,
in other words,
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the really real answer,
the real real answer.
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So, now without further ado,
because of those beautiful,
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the problem has been solved
once and for all by using the
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substitution rule.
I did that for you on Monday.
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The answer is simply e to the i
omega1 t
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divided by what?
This polynomial with omega one
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substituted in for D.
So, sorry, i omega one,
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the complex
coefficient of t.
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So, it is substitute i omega
for D, I omega one for D,
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and you get (i omega one)
squared plus omega nought
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squared.
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Well, let's make that look a
little bit better.
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This should be e to the (i
omega one t)
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divided by, now,
what's this?
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This is simply omega nought
squared minus omega
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one squared.
But, I want the real part of
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it.
So, as one final,
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last step, the real part of
that is what we call just the
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real particular solution,
so, yp without the tilde
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anymore.
And, the real part of this,
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well, this cosine plus i sine.
And, the denominator,
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luckily, turns out to be real.
So, it's simply going to be
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cosine omega one t.
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That's the top,
divided by this thing,
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omega nought squared minus
omega one squared.
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In other words,
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that's the response.
This is the input,
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and that's what came out.
Well, in other words,
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what one sees is,
regardless of what natural
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frequency this system wanted to
use for itself,
00:11:20.000 --> 00:11:26.000
at least for this solution,
what it responds to is the
00:11:25.000 --> 00:11:31.000
driving frequency,
the input frequency.
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The only thing is that the
amplitude has changed,
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and in a rather dramatic way,
if omega1, depending on the
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relative sizes of omega1 and
omega2.
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Now, the interesting case is
when omega one is very close to
00:11:48.000 --> 00:11:54.000
omega, the natural frequency.
When you push it with
00:11:53.000 --> 00:11:59.000
approximately it's natural
frequency, then the solution is
00:11:59.000 --> 00:12:05.000
big amplitude.
The amplitude is large.
00:12:04.000 --> 00:12:10.000
So, the solution looks like the
frequency.
00:12:07.000 --> 00:12:13.000
The input might have looked
like this.
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Well, it's cosine,
so it ought to start up here.
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The input might have looked
like this, but the response will
00:12:18.000 --> 00:12:24.000
be a curve with the same
frequency and still a pure
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oscillation.
But, it will have much,
00:12:24.000 --> 00:12:30.000
much bigger amplitude.
And, it's because the
00:12:28.000 --> 00:12:34.000
denominator, omega nought
squared minus omega
00:12:32.000 --> 00:12:38.000
one squared,
is always zero.
00:12:35.000 --> 00:12:41.000
So, the response will,
instead, look like this.
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Now, to all intents and
purposes, that's resonance.
00:12:44.000 --> 00:12:50.000
You are pushing something with
approximately the same
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frequency, something that wants
to oscillate.
00:12:52.000 --> 00:12:58.000
And, you are pushing it with
approximately the same frequency
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that it would like to oscillate
by itself.
00:13:00.000 --> 00:13:06.000
And, what that does is it
builds up the amplitude
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Well, what happens if omega one
is actually equal
00:13:10.000 --> 00:13:16.000
to omega zero?
So, that's the case I'd like to
00:13:14.000 --> 00:13:20.000
analyze for you now.
Suppose the two are equal,
00:13:18.000 --> 00:13:24.000
in other words.
Well, the problem is,
00:13:21.000 --> 00:13:27.000
of course, I can't use that
same solution.
00:13:24.000 --> 00:13:30.000
It isn't applicable.
But that's why I gave you,
00:13:28.000 --> 00:13:34.000
derived for you using the
exponential shift law last time,
00:13:33.000 --> 00:13:39.000
the second version,
when it is a root.
00:13:38.000 --> 00:13:44.000
So, if omega one equals omega
nought,
00:13:42.000 --> 00:13:48.000
so now our equation looks like
D squared plus omega nought
00:13:47.000 --> 00:13:53.000
squared, the natural frequency,
y.
00:13:51.000 --> 00:13:57.000
But this time,
the driving frequency,
00:13:54.000 --> 00:14:00.000
the input frequency,
is omega nought itself.
00:13:57.000 --> 00:14:03.000
Then, the same analysis,
a lot of it is,
00:14:00.000 --> 00:14:06.000
well, I'd better be careful.
I'd better be careful.
00:14:04.000 --> 00:14:10.000
Let's go through the analysis
again very rapidly.
00:14:10.000 --> 00:14:16.000
What we want to do is first
complexify it,
00:14:13.000 --> 00:14:19.000
and then solve.
So, the complex equation will
00:14:17.000 --> 00:14:23.000
be D squared plus omega nought
squared times y tilde equals e
00:14:22.000 --> 00:14:28.000
to the i omega nought t,
this time.
00:14:29.000 --> 00:14:35.000
But now, i omega is zero of
this polynomial.
00:14:32.000 --> 00:14:38.000
That's why I picked it,
right?
00:14:35.000 --> 00:14:41.000
If I plug in i omega zero,
I get i omega zero
00:14:40.000 --> 00:14:46.000
quantity squared plus omega
nought squared.
00:14:46.000 --> 00:14:52.000
That's zero.
00:14:48.000 --> 00:14:54.000
So, I'm in the second case.
So, i omega nought is a simple
00:14:53.000 --> 00:14:59.000
root, simple zero,
of D squared plus
00:14:58.000 --> 00:15:04.000
omega nought,
that polynomial squared.
00:15:04.000 --> 00:15:10.000
Therefore, the complex
particular solution is now t e
00:15:08.000 --> 00:15:14.000
to the i omega nought t
divided by p prime,
00:15:13.000 --> 00:15:19.000
where you plug in that root,
the i omega nought.
00:15:17.000 --> 00:15:23.000
Now, what's p prime?
00:15:19.000 --> 00:15:25.000
p prime is 2D, right?
00:15:22.000 --> 00:15:28.000
If I differentiate this
formally, as if D were a
00:15:26.000 --> 00:15:32.000
variable, the way you
differentiate polynomials,
00:15:29.000 --> 00:15:35.000
the derivative,
this is a constant,
00:15:32.000 --> 00:15:38.000
and the derivative is 2D.
So, the denominator should have
00:15:38.000 --> 00:15:44.000
two times for D.
You are going to plug in i
00:15:42.000 --> 00:15:48.000
omega zero.
So, it's 2 i omega zero.
00:15:47.000 --> 00:15:53.000
And now, I want the real part
00:15:51.000 --> 00:15:57.000
of that, which is what?
Well, think about it.
00:15:55.000 --> 00:16:01.000
The top is cosine plus i sine.
The real part is now going to
00:16:00.000 --> 00:16:06.000
come from the sine,
right, because it's cosine plus
00:16:05.000 --> 00:16:11.000
i sine.
But this i is going to divide
00:16:09.000 --> 00:16:15.000
out the i that goes with this
sine.
00:16:11.000 --> 00:16:17.000
And, therefore,
the real part is going to be t
00:16:15.000 --> 00:16:21.000
times the sine,
this time, of omega nought t.
00:16:18.000 --> 00:16:24.000
And, that's going to be divided
00:16:22.000 --> 00:16:28.000
by, well, the i canceled out the
i that was in front of the sine
00:16:27.000 --> 00:16:33.000
function.
And therefore,
00:16:28.000 --> 00:16:34.000
what's left is two omega nought
down below.
00:16:34.000 --> 00:16:40.000
So, that's our particular
solution now.
00:16:36.000 --> 00:16:42.000
Well, it looks different from
that guy.
00:16:39.000 --> 00:16:45.000
It doesn't look like that
anymore.
00:16:42.000 --> 00:16:48.000
What does it look like?
Well, it shows the way to plot
00:16:46.000 --> 00:16:52.000
such things is basically it's an
oscillation of frequency omega
00:16:50.000 --> 00:16:56.000
nought.
But, its amplitude is changing.
00:16:54.000 --> 00:17:00.000
So, the way to do it is,
as always, if you have a basic
00:16:58.000 --> 00:17:04.000
oscillation which is neither too
fast nor too slow,
00:17:02.000 --> 00:17:08.000
think of that as the thing,
and the other stuff multiplying
00:17:06.000 --> 00:17:12.000
it, think of it as changing the
amplitude of that oscillation
00:17:10.000 --> 00:17:16.000
with time.
So, the amplitude is that
00:17:14.000 --> 00:17:20.000
function, t divided by two omega
zero.
00:17:18.000 --> 00:17:24.000
So, just as we did when we
talked about damping,
00:17:21.000 --> 00:17:27.000
you plot that and it's negative
on the picture.
00:17:25.000 --> 00:17:31.000
So, this is the function whose
graph is t divided by two omega
00:17:29.000 --> 00:17:35.000
nought.
That's the changing amplitude,
00:17:34.000 --> 00:17:40.000
as it were.
And then, the function itself
00:17:37.000 --> 00:17:43.000
does what oscillation it can,
but it has to stay within those
00:17:41.000 --> 00:17:47.000
lines.
So, the thing that's
00:17:43.000 --> 00:17:49.000
oscillating is sine omega nought
t,
00:17:47.000 --> 00:17:53.000
which would like to be a pure
oscillation, but can't because
00:17:52.000 --> 00:17:58.000
its amplitude is being changed
by that thing.
00:17:55.000 --> 00:18:01.000
So, it's doing this,
and now the rest I have to
00:17:58.000 --> 00:18:04.000
leave to your imagination.
In other words,
00:18:03.000 --> 00:18:09.000
what happens when omega nought
is equal to, when the driving
00:18:07.000 --> 00:18:13.000
frequency is actually equal to
omega nought,
00:18:11.000 --> 00:18:17.000
mathematically this turns into
a different looking solution,
00:18:15.000 --> 00:18:21.000
one with steadily increasing
amplitude.
00:18:18.000 --> 00:18:24.000
The amplitude increases
linearly like the function t
00:18:22.000 --> 00:18:28.000
divided by two omega nought.
00:18:25.000 --> 00:18:31.000
Well, many people are upset by
this, slightly,
00:18:29.000 --> 00:18:35.000
in the sense that there is a
funny feeling.
00:18:32.000 --> 00:18:38.000
How is it that that solution
can turn into this one?
00:18:38.000 --> 00:18:44.000
If I simply let omega one go to
omega zero, what happens?
00:18:43.000 --> 00:18:49.000
Well, the pink curve just gets
taller and taller,
00:18:47.000 --> 00:18:53.000
and after a while all you see
of it is just a bunch of
00:18:52.000 --> 00:18:58.000
vertical lines which seem to be
spaced at whatever the right
00:18:57.000 --> 00:19:03.000
period is for that function.
It's sort of like being in a
00:19:03.000 --> 00:19:09.000
first story window and watching
a giraffe go by.
00:19:08.000 --> 00:19:14.000
All you see is that.
Okay.
00:19:20.000 --> 00:19:26.000
So, my concern is how does that
function turn into this one?
00:19:24.000 --> 00:19:30.000
I have something in mind to
remind you of,
00:19:27.000 --> 00:19:33.000
and that's why we'll go through
this little exercise.
00:19:31.000 --> 00:19:37.000
It's a simple exercise.
But the function of it is,
00:19:35.000 --> 00:19:41.000
of course that as omega one
goes to omega zero cannot
00:19:39.000 --> 00:19:45.000
possibly turn into this.
It's doing the wrong thing near
00:19:43.000 --> 00:19:49.000
zero.
It's already zooming up.
00:19:45.000 --> 00:19:51.000
But, the point is,
this is not the only particular
00:19:49.000 --> 00:19:55.000
solution on the block.
Any solution whatsoever of the
00:19:52.000 --> 00:19:58.000
differential equation,
the inhomogeneous equation,
00:19:56.000 --> 00:20:02.000
is a particular solution.
It's like Fred Rogers:
00:20:01.000 --> 00:20:07.000
everybody is special.
Okay, so all solutions are
00:20:05.000 --> 00:20:11.000
special.
We don't have to use that one.
00:20:08.000 --> 00:20:14.000
So, I will use,
where are all the other
00:20:11.000 --> 00:20:17.000
solutions?
So, I'm going back to the
00:20:14.000 --> 00:20:20.000
equation D squared plus omega
zero squared,
00:20:18.000 --> 00:20:24.000
applied to y,
00:20:20.000 --> 00:20:26.000
is equal to cosine
omega one t.
00:20:24.000 --> 00:20:30.000
Now, the particular solution we
found was that one,
00:20:27.000 --> 00:20:33.000
cosine omega one t divided by
that omega nought squared minus
00:20:32.000 --> 00:20:38.000
omega one squared.
00:20:39.000 --> 00:20:45.000
What do the other particular
solutions look like?
00:20:44.000 --> 00:20:50.000
Well, in general,
any particular solution will
00:20:49.000 --> 00:20:55.000
look like that one we found,
what is it, omega nought
00:20:54.000 --> 00:21:00.000
squared minus omega
one squared,
00:21:01.000 --> 00:21:07.000
plus I'm allowed to add to it
any piece of the complementary
00:21:07.000 --> 00:21:13.000
solution.
Equally particular,
00:21:11.000 --> 00:21:17.000
and equally good,
as a particular solution is
00:21:14.000 --> 00:21:20.000
this plus anything which solved
the homogeneous equation.
00:21:18.000 --> 00:21:24.000
Now, all I'm going to do is
pick out one good function which
00:21:23.000 --> 00:21:29.000
solves the homogeneous equation,
and here it is.
00:21:26.000 --> 00:21:32.000
It's the function minus cosine.
In fact, what does solve the
00:21:32.000 --> 00:21:38.000
homogeneous equation?
Well, it's solved by sine omega
00:21:36.000 --> 00:21:42.000
nought t,
cosine omega nought t,
00:21:40.000 --> 00:21:46.000
and any linear combination of
00:21:44.000 --> 00:21:50.000
those.
So, out of all those functions,
00:21:47.000 --> 00:21:53.000
the one I'm going to pick is
cosine omega nought t.
00:21:51.000 --> 00:21:57.000
And, I'm going to divide it by
this same guy.
00:21:55.000 --> 00:22:01.000
So, this is part of the
complementary solution.
00:22:00.000 --> 00:22:06.000
That's what we call the
complementary solution,
00:22:02.000 --> 00:22:08.000
the solution to the associated
homogeneous equation,
00:22:06.000 --> 00:22:12.000
to the reduced equation.
Call it what you like.
00:22:08.000 --> 00:22:14.000
So, this is one of the guys in
there, and it's still a
00:22:12.000 --> 00:22:18.000
particular solution to take the
one I first found,
00:22:15.000 --> 00:22:21.000
and add to it anything which
solves the homogeneous equation.
00:22:19.000 --> 00:22:25.000
I showed you that when we first
set out to solve the
00:22:22.000 --> 00:22:28.000
inhomogeneous equation in
general.
00:22:24.000 --> 00:22:30.000
Now, why do I pick that?
Well, I'm going to now
00:22:27.000 --> 00:22:33.000
calculate, what's the limit?
So, these guys are also good
00:22:31.000 --> 00:22:37.000
solutions to that.
This is a good solution to that
00:22:35.000 --> 00:22:41.000
equation, this equation.
All I'm going to do now is
00:22:38.000 --> 00:22:44.000
calculate the limit as omega one
approaches omega zero of this
00:22:42.000 --> 00:22:48.000
function.
Well, what is that?
00:22:46.000 --> 00:22:52.000
It's cosine omega one t minus
cosine omega zero t divided by
00:22:50.000 --> 00:22:56.000
omega nought squared minus omega
one squared.
00:22:57.000 --> 00:23:03.000
Now, you see why I did that.
If I let just this guy,
00:23:02.000 --> 00:23:08.000
omega one approaches
omega zero,
00:23:07.000 --> 00:23:13.000
I get infinity.
I don't get anything.
00:23:10.000 --> 00:23:16.000
But, this is different here
because I fixed it up,
00:23:14.000 --> 00:23:20.000
now.
The denominator becomes zero,
00:23:17.000 --> 00:23:23.000
but so does the numerator.
In other words,
00:23:21.000 --> 00:23:27.000
I've put myself in position to
use L'Hopital rule.
00:23:26.000 --> 00:23:32.000
So, let's L'Hopital it.
It's the limit.
00:23:29.000 --> 00:23:35.000
As omega one approaches omega
zero, and what do you do?
00:23:34.000 --> 00:23:40.000
You differentiate the top and
the bottom with respect to what?
00:23:42.000 --> 00:23:48.000
Right, with respect to omega
one.
00:23:44.000 --> 00:23:50.000
Omega one is the variable.
That's what's changing.
00:23:47.000 --> 00:23:53.000
The t that I'm thinking of is,
I'm thinking,
00:23:50.000 --> 00:23:56.000
for the temporary fixed.
This has a fixed value.
00:23:53.000 --> 00:23:59.000
Omega nought is fixed.
All that's changing in this
00:23:57.000 --> 00:24:03.000
limit operation is omega one.
And therefore,
00:24:01.000 --> 00:24:07.000
it's with respect to omega one
that I differentiate it.
00:24:05.000 --> 00:24:11.000
You got that?
Well, you are in no position to
00:24:08.000 --> 00:24:14.000
say yes or no,
so I shouldn't even ask the
00:24:11.000 --> 00:24:17.000
question, but okay,
rhetorical question.
00:24:14.000 --> 00:24:20.000
All right, let's differentiate
this expression,
00:24:17.000 --> 00:24:23.000
the top and bottom with respect
to omega one.
00:24:20.000 --> 00:24:26.000
So, the derivative of the top
with respect to omega one is
00:24:24.000 --> 00:24:30.000
negative sine omega one t.
00:24:28.000 --> 00:24:34.000
But, I have to use the chain
rule.
00:24:32.000 --> 00:24:38.000
That's differentiating with
respect to this argument,
00:24:35.000 --> 00:24:41.000
this variable.
But now, I must take times the
00:24:38.000 --> 00:24:44.000
derivative of this thing with
respect to omega one.
00:24:43.000 --> 00:24:49.000
And that is t is the constant,
so times t.
00:24:46.000 --> 00:24:52.000
And, how about the bottom?
The derivative of the bottom
00:24:49.000 --> 00:24:55.000
with respect to omega one is,
well, that's a constant.
00:24:53.000 --> 00:24:59.000
So, it becomes zero.
And, this becomes negative two
00:24:57.000 --> 00:25:03.000
omega one.
So, it's the limit of this
00:25:01.000 --> 00:25:07.000
expression as omega one
approaches omega zero.
00:25:05.000 --> 00:25:11.000
And now it's not indeterminate
00:25:09.000 --> 00:25:15.000
anymore.
The answer is,
00:25:10.000 --> 00:25:16.000
the negative signs cancel.
It's simply t sine omega nought
00:25:15.000 --> 00:25:21.000
t divided by two omega nought.
00:25:19.000 --> 00:25:25.000
So, that's how we get that
00:25:21.000 --> 00:25:27.000
solution.
It is a limit as omega one,
00:25:24.000 --> 00:25:30.000
but not of the
particular solution we found
00:25:28.000 --> 00:25:34.000
first, but of this other one.
Now, it's still too much
00:25:34.000 --> 00:25:40.000
algebra.
I mean, what's going on here?
00:25:37.000 --> 00:25:43.000
Well, that's something else you
should know.
00:25:41.000 --> 00:25:47.000
Okay, so my question is,
therefore, what does this mean?
00:25:46.000 --> 00:25:52.000
What's the geometric meaning of
all this?
00:25:50.000 --> 00:25:56.000
In other words,
what does that function look
00:25:54.000 --> 00:26:00.000
like?
Well, that's another
00:25:56.000 --> 00:26:02.000
trigonometric identity,
which in your book is just
00:26:01.000 --> 00:26:07.000
buried as half of one line sort
of casual as if everybody knows
00:26:07.000 --> 00:26:13.000
it, and I know that virtually no
one knows it.
00:26:13.000 --> 00:26:19.000
But, here's your chance.
So, the cosine of B minus the
00:26:17.000 --> 00:26:23.000
cosine of A can be expressed
as a product of
00:26:22.000 --> 00:26:28.000
signs.
It's the sine of (A minus B)
00:26:25.000 --> 00:26:31.000
over two times the sine of (A
plus B) over two,
00:26:29.000 --> 00:26:35.000
I believe.
00:26:33.000 --> 00:26:39.000
My only uncertainty:
is there a two in front of
00:26:37.000 --> 00:26:43.000
that?
I think there has to be.
00:26:40.000 --> 00:26:46.000
Let me check.
Sorry.
00:26:42.000 --> 00:26:48.000
Is there a two?
I wouldn't trust my memory
00:26:46.000 --> 00:26:52.000
anyway.
I'd look it up.
00:26:49.000 --> 00:26:55.000
I did look it up,
two, yes.
00:26:51.000 --> 00:26:57.000
If you had to prove that,
you could use the sine formula
00:26:57.000 --> 00:27:03.000
to expand this out.
That would be a bad way to do
00:27:03.000 --> 00:27:09.000
it.
The best way is to use complex
00:27:05.000 --> 00:27:11.000
numbers.
Express the sign in terms of
00:27:08.000 --> 00:27:14.000
complex numbers,
exponentials,
00:27:10.000 --> 00:27:16.000
you know, the backwards Euler
formula.
00:27:13.000 --> 00:27:19.000
Then do it here,
and then just multiply those
00:27:17.000 --> 00:27:23.000
two expressions involving
exponentials together,
00:27:20.000 --> 00:27:26.000
and cancel, cancel,
cancel, cancel,
00:27:23.000 --> 00:27:29.000
cancel, and this is what you
will end up with.
00:27:26.000 --> 00:27:32.000
You see why I did this.
It's because this has that
00:27:31.000 --> 00:27:37.000
form.
So, let's apply that formula to
00:27:33.000 --> 00:27:39.000
it.
So, what's the left-hand side?
00:27:36.000 --> 00:27:42.000
B is omega one t, and A is
omega nought t.
00:27:39.000 --> 00:27:45.000
So, this is omega one t,
00:27:42.000 --> 00:27:48.000
and this is omega nought t
00:27:44.000 --> 00:27:50.000
All right, so what we get is
00:27:47.000 --> 00:27:53.000
that the cosine of omega one t
minus the cosine of omega nought
00:27:51.000 --> 00:27:57.000
t, which is exactly the
00:27:54.000 --> 00:28:00.000
numerator of this function that
I'm trying to get a handle on.
00:28:00.000 --> 00:28:06.000
Then we will divide it by its
amplitude.
00:28:02.000 --> 00:28:08.000
So, that's this constant factor
that's real.
00:28:05.000 --> 00:28:11.000
It's a small number because I'm
thinking of omega one
00:28:10.000 --> 00:28:16.000
as being rather close to omega
zero,
00:28:13.000 --> 00:28:19.000
and getting closer and closer.
What does this tell us about
00:28:17.000 --> 00:28:23.000
the right-hand side?
Well, the right-hand side is
00:28:21.000 --> 00:28:27.000
twice the sine of A minus B.
00:28:24.000 --> 00:28:30.000
Now, that's good because these
guys sort of resemble each
00:28:28.000 --> 00:28:34.000
other.
So, that's (omega nought minus
00:28:32.000 --> 00:28:38.000
omega one) times t.
00:28:35.000 --> 00:28:41.000
That's A minus B,
and I'm supposed to divide that
00:28:39.000 --> 00:28:45.000
by two.
And then, the other one will be
00:28:42.000 --> 00:28:48.000
the same thing with plus:
sine omega nought plus omega
00:28:46.000 --> 00:28:52.000
one over two times t.
00:28:50.000 --> 00:28:56.000
Now, how big is this,
approximately?
00:28:52.000 --> 00:28:58.000
Remember, think of omega one
as close to omega zero.
00:28:56.000 --> 00:29:02.000
Then, this is approximately
00:28:59.000 --> 00:29:05.000
omega zero.
So this part is approximately
00:29:03.000 --> 00:29:09.000
sine of omega zero t.
00:29:06.000 --> 00:29:12.000
This part, on the other hand,
that's a very small thing.
00:29:09.000 --> 00:29:15.000
Okay, now what I want to know
is what does this function look
00:29:13.000 --> 00:29:19.000
like?
The interest in knowing what
00:29:15.000 --> 00:29:21.000
the function looks like it is
because we want to be able to
00:29:19.000 --> 00:29:25.000
see that it's limited is that
thing.
00:29:21.000 --> 00:29:27.000
You can't tell what's what its
limit is, geometrically,
00:29:25.000 --> 00:29:31.000
unless you know it looks like.
So, what does it look like?
00:29:30.000 --> 00:29:36.000
Well, again,
the way to analyze it is the
00:29:35.000 --> 00:29:41.000
thing, that thing.
What you think of is,
00:29:41.000 --> 00:29:47.000
yeah, of course you cannot
divide one side of equality
00:29:49.000 --> 00:29:55.000
without dividing the equation by
the other side.
00:29:56.000 --> 00:30:02.000
So, that's got to be there,
too.
00:30:02.000 --> 00:30:08.000
Now, what does that look like?
Well, the way to think of it
00:30:06.000 --> 00:30:12.000
is, here is something with a
normal sort of frequency,
00:30:10.000 --> 00:30:16.000
omega nought.
It's doing its thing.
00:30:14.000 --> 00:30:20.000
It's a sine curve.
It's doing that.
00:30:16.000 --> 00:30:22.000
What's this?
Think of all this part as
00:30:19.000 --> 00:30:25.000
varying amplitude.
It's just another example of
00:30:23.000 --> 00:30:29.000
what I gave you before.
Here is a basic,
00:30:26.000 --> 00:30:32.000
pure oscillation,
and now, think of everything
00:30:29.000 --> 00:30:35.000
else that's multiplying it as
varying its amplitude.
00:30:35.000 --> 00:30:41.000
All right, so what does that
thing look like?
00:30:38.000 --> 00:30:44.000
Well, first what we want to do
is plot the amplitude lines.
00:30:44.000 --> 00:30:50.000
Now, what will they be?
This is sine of an extremely
00:30:48.000 --> 00:30:54.000
small number times t.
The frequency is small.
00:30:52.000 --> 00:30:58.000
How does the sine curve look if
its frequency is very low,
00:30:57.000 --> 00:31:03.000
very close to zero?
Well, that must mean its period
00:31:02.000 --> 00:31:08.000
is very large.
Here's something with a big
00:31:05.000 --> 00:31:11.000
frequency.
Here's something with a very,
00:31:08.000 --> 00:31:14.000
very low frequency.
Now, with a low frequency,
00:31:11.000 --> 00:31:17.000
it would hardly get off the
ground and get up to one here,
00:31:15.000 --> 00:31:21.000
and it would do that.
But, it's made to look a little
00:31:19.000 --> 00:31:25.000
more presentable because of this
coefficient in front,
00:31:23.000 --> 00:31:29.000
which is rather large.
And so, what this thing looks
00:31:27.000 --> 00:31:33.000
like, I won't pause to analyze
it more exactly.
00:31:32.000 --> 00:31:38.000
It's something which goes up at
a reasonable rate for quite a
00:31:36.000 --> 00:31:42.000
while, and let's say that's
quite awhile.
00:31:39.000 --> 00:31:45.000
And then it comes down,
and then it goes,
00:31:42.000 --> 00:31:48.000
and so on.
Of course, in figuring out its
00:31:45.000 --> 00:31:51.000
amplitude, we have to be willing
to draw its negative,
00:31:49.000 --> 00:31:55.000
too.
And since I didn't figure
00:31:51.000 --> 00:31:57.000
things out right,
I can at least make it cross,
00:31:54.000 --> 00:32:00.000
right?
Okay.
00:31:55.000 --> 00:32:01.000
So, this is a picture of this
slowly varying amplitude.
00:32:01.000 --> 00:32:07.000
And in between,
this is the function which is
00:32:05.000 --> 00:32:11.000
doing the oscillation,
as well as it can.
00:32:08.000 --> 00:32:14.000
But, it has to stay within that
amplitude.
00:32:12.000 --> 00:32:18.000
So, it's doing this.
Now, what happens?
00:32:15.000 --> 00:32:21.000
As omega one approaches omega
zero,
00:32:21.000 --> 00:32:27.000
this frequency gets closer and
closer to zero,
00:32:25.000 --> 00:32:31.000
which means the period of that
dotted line gets further and
00:32:30.000 --> 00:32:36.000
further out, goes to infinity,
and you never do ultimately get
00:32:35.000 --> 00:32:41.000
a chance to come down again.
All you can see is the initial
00:32:42.000 --> 00:32:48.000
part, where it's rising and
rising.
00:32:45.000 --> 00:32:51.000
And, that's how this curve
turns into that one.
00:32:49.000 --> 00:32:55.000
Now, of course,
this curve is enormously
00:32:52.000 --> 00:32:58.000
interesting.
You must have had this
00:32:55.000 --> 00:33:01.000
somewhere.
That's the phenomenon of what
00:32:59.000 --> 00:33:05.000
are called beats.
Too frequencies--
00:33:03.000 --> 00:33:09.000
Your book has half a page
explaining this.
00:33:05.000 --> 00:33:11.000
That's the half a page where he
gives you this identity,
00:33:09.000 --> 00:33:15.000
except it gives it in a wrong
form, so that it's hard to
00:33:13.000 --> 00:33:19.000
figure out.
But anyway, the beats are two
00:33:16.000 --> 00:33:22.000
frequencies when you combine
them, the two frequencies being
00:33:20.000 --> 00:33:26.000
two combined pure oscillations
where the frequencies are very
00:33:24.000 --> 00:33:30.000
close to each other.
What you get is a curve which
00:33:27.000 --> 00:33:33.000
looks like that.
And, of course,
00:33:31.000 --> 00:33:37.000
what you hear is the envelope
of the curve.
00:33:34.000 --> 00:33:40.000
You hear the dotted lines.
Well, you hear this.
00:33:37.000 --> 00:33:43.000
You hear that,
too.
00:33:39.000 --> 00:33:45.000
But, what you hear is-- And,
that's how good violinists and
00:33:43.000 --> 00:33:49.000
cellists, and so on,
tune their instruments.
00:33:46.000 --> 00:33:52.000
They get one string right,
and then the other strings are
00:33:51.000 --> 00:33:57.000
tuned by listening.
They don't actually listen for
00:33:54.000 --> 00:34:00.000
the sound of the note.
They listened just for the
00:33:58.000 --> 00:34:04.000
beats, wah, wah,
wah, wah, and they turn the peg
00:34:02.000 --> 00:34:08.000
and it goes wah,
wah, wah, wah,
00:34:04.000 --> 00:34:10.000
and then finally as soon as the
wahs disappear,
00:34:07.000 --> 00:34:13.000
they know that the two strings
are in tune.
00:34:13.000 --> 00:34:19.000
A piano tuner does the same
thing.
00:34:16.000 --> 00:34:22.000
Of course, I,
being a very bad cellist,
00:34:19.000 --> 00:34:25.000
use a tuner.
That's another solution,
00:34:22.000 --> 00:34:28.000
a more modern solution.
Okay.
00:34:49.000 --> 00:34:55.000
Oh well.
Let's give it a try.
00:34:51.000 --> 00:34:57.000
The bad news is that problem
six in your problem set,
00:34:55.000 --> 00:35:01.000
I didn't ask you about the
undamped case.
00:34:58.000 --> 00:35:04.000
I thought, since you are mature
citizens, you could be asked
00:35:03.000 --> 00:35:09.000
about the damped case.
00:35:21.000 --> 00:35:27.000
I warn you, first of all you
have to get the notation.
00:35:26.000 --> 00:35:32.000
This is probably the most
important thing I'll do with
00:35:31.000 --> 00:35:37.000
this.
Your book uses this, resonance.
00:36:02.000 --> 00:36:08.000
I'm optimistic.
[LAUGHTER] Let's say zero or f
00:36:06.000 --> 00:36:12.000
of t.
It doesn't matter.
00:36:09.000 --> 00:36:15.000
In other words,
the constants,
00:36:12.000 --> 00:36:18.000
the book uses two sets of
constants to describe these
00:36:17.000 --> 00:36:23.000
equations.
If it's a spring,
00:36:20.000 --> 00:36:26.000
and not even talking about RLC
circuits, the spring mass,
00:36:26.000 --> 00:36:32.000
damping, k, spring constant.
Then you divide out by m and
00:36:33.000 --> 00:36:39.000
you get this.
You're familiar with that.
00:36:36.000 --> 00:36:42.000
And, it's only after you
divided out by the m that you're
00:36:41.000 --> 00:36:47.000
allowed to call this the square
of the natural frequency.
00:36:46.000 --> 00:36:52.000
So, omega naught is the natural
frequency, the natural undamped
00:36:51.000 --> 00:36:57.000
frequency.
If this term were not there,
00:36:54.000 --> 00:37:00.000
that omega nought
would give the frequency with
00:36:59.000 --> 00:37:05.000
which the system,
the little spring would like to
00:37:03.000 --> 00:37:09.000
vibrate by itself.
Now, further complication is
00:37:08.000 --> 00:37:14.000
that the visual uses neither of
these.
00:37:11.000 --> 00:37:17.000
The visual uses x double dot
plus b times x prime,
00:37:16.000 --> 00:37:22.000
I think we will have to fix
this in the future,
00:37:20.000 --> 00:37:26.000
but for now,
just live with it,
00:37:22.000 --> 00:37:28.000
plus kx,
and that's some function,
00:37:27.000 --> 00:37:33.000
again, a function.
So, in other words,
00:37:30.000 --> 00:37:36.000
the problem is that b is okay,
can't be confused with c.
00:37:37.000 --> 00:37:43.000
On the other hand,
this is not the same k as that.
00:37:42.000 --> 00:37:48.000
What I'm trying to say is,
don't automatically go to a
00:37:48.000 --> 00:37:54.000
formula one place,
and assume it's the same
00:37:53.000 --> 00:37:59.000
formula in another place.
You have to use these
00:37:58.000 --> 00:38:04.000
equivalences.
You have to look and see how
00:38:03.000 --> 00:38:09.000
the basic equation was written,
and then figure out what the
00:38:08.000 --> 00:38:14.000
constant should be.
Now, there was something
00:38:12.000 --> 00:38:18.000
called, when we analyzed this
before, and this has happened in
00:38:17.000 --> 00:38:23.000
recitation, there was the
natural, damped frequency.
00:38:22.000 --> 00:38:28.000
I'll call it the natural,
damped frequency.
00:38:25.000 --> 00:38:31.000
The book calls it the
pseudo-frequency.
00:38:30.000 --> 00:38:36.000
It's called pseudo-frequency
because the function,
00:38:33.000 --> 00:38:39.000
if you have zero on the right
hand side, but have damping,
00:38:38.000 --> 00:38:44.000
the function isn't periodic.
It decays.
00:38:41.000 --> 00:38:47.000
It does this.
Nonetheless,
00:38:43.000 --> 00:38:49.000
it still crosses the t-axis at
regular intervals,
00:38:47.000 --> 00:38:53.000
and therefore,
almost everybody just casually
00:38:50.000 --> 00:38:56.000
refers to it as the frequency,
and understands it's the
00:38:54.000 --> 00:39:00.000
natural damped frequency.
Now, the relation between them
00:38:59.000 --> 00:39:05.000
is given by the little picture I
drew you once.
00:39:02.000 --> 00:39:08.000
But, I didn't emphasize it
enough.
00:39:07.000 --> 00:39:13.000
Here is omega nought.
00:39:09.000 --> 00:39:15.000
Here is the right angle.
The side is omega one,
00:39:12.000 --> 00:39:18.000
and this side is the
damping.
00:39:15.000 --> 00:39:21.000
So, in other words,
this is fixed because it's
00:39:19.000 --> 00:39:25.000
fixed by the spring.
That's the natural frequency of
00:39:23.000 --> 00:39:29.000
the spring, by itself.
If you are damping near the
00:39:27.000 --> 00:39:33.000
motion, then the more you damped
it, the bigger this side gets,
00:39:31.000 --> 00:39:37.000
and therefore the smaller omega
one is, the bigger the damping,
00:39:36.000 --> 00:39:42.000
then the smaller the frequency
with which the damped thing
00:39:40.000 --> 00:39:46.000
vibrates.
That sort of intuitive,
00:39:44.000 --> 00:39:50.000
and vice versa.
If you decrease the damping to
00:39:47.000 --> 00:39:53.000
almost zero, well,
then you'll make omega one
00:39:49.000 --> 00:39:55.000
almost the same size as omega
zero.
00:39:51.000 --> 00:39:57.000
This must be a right angle,
and therefore,
00:39:54.000 --> 00:40:00.000
if there's very little damping,
the natural damped frequency
00:39:57.000 --> 00:40:03.000
will be almost the same as the
original frequency,
00:40:00.000 --> 00:40:06.000
the natural frequency.
So, the relation between them
00:40:05.000 --> 00:40:11.000
is that omega one squared is
equal to omega nought squared
00:40:11.000 --> 00:40:17.000
minus p squared,
00:40:15.000 --> 00:40:21.000
and this comes from the
characteristic roots from the
00:40:20.000 --> 00:40:26.000
characteristic roots of the
damped equation.
00:40:26.000 --> 00:40:32.000
So, we did that before.
I'm just reminding you of it.
00:40:31.000 --> 00:40:37.000
Now, the third frequency which
now enters, and that I'm asking
00:40:37.000 --> 00:40:43.000
you about on the problem set is
if you've got a damped spring,
00:40:43.000 --> 00:40:49.000
okay, what happens when you
impose a motion on it with yet a
00:40:49.000 --> 00:40:55.000
third frequency?
In other words,
00:40:53.000 --> 00:40:59.000
drive the damped spring.
I don't care.
00:40:56.000 --> 00:41:02.000
I switched to y,
since I'm in y mode.
00:41:02.000 --> 00:41:08.000
So, our equation looks like
this, just as it did before,
00:41:06.000 --> 00:41:12.000
except now going to drive that
with an undetermined frequency,
00:41:10.000 --> 00:41:16.000
cosine omega t.
00:41:13.000 --> 00:41:19.000
And, my question,
now, is, see,
00:41:15.000 --> 00:41:21.000
it's not going to be able to
resonate in the correct-- you
00:41:19.000 --> 00:41:25.000
really only get true resonance
when you don't have damping.
00:41:24.000 --> 00:41:30.000
That's the only time where the
amplitude can build up
00:41:27.000 --> 00:41:33.000
indefinitely.
But nonetheless,
00:41:31.000 --> 00:41:37.000
for all practical purposes,
and there's always some damping
00:41:37.000 --> 00:41:43.000
unless you are a perfect vacuum
or something,
00:41:42.000 --> 00:41:48.000
there's almost always some
damping.
00:41:46.000 --> 00:41:52.000
So, p isn't zero,
can't be exactly zero.
00:41:50.000 --> 00:41:56.000
So, the problem is,
which omega gives,
00:41:54.000 --> 00:42:00.000
which frequency in the input,
which input frequency gives the
00:42:00.000 --> 00:42:06.000
maximal amplitude for the
response?
00:42:20.000 --> 00:42:26.000
We solved that problem when it
was undamped,
00:42:22.000 --> 00:42:28.000
and the answer was easy.
Omega should equal omega zero.
00:42:26.000 --> 00:42:32.000
But, when it's damped,
the answer is different.
00:42:30.000 --> 00:42:36.000
And, I'm not asking you to do
it in general.
00:42:33.000 --> 00:42:39.000
I'm giving you some numbers.
But nonetheless,
00:42:37.000 --> 00:42:43.000
it still must be the case.
So, I'm giving you,
00:42:40.000 --> 00:42:46.000
I give you specific values of p
and omega zero.
00:42:45.000 --> 00:42:51.000
That's on the problem set.
Of course, one of them is tied
00:42:50.000 --> 00:42:56.000
to your recitation.
But, the answer is,
00:42:53.000 --> 00:42:59.000
I'm going to give you the
general formula for the answer
00:42:58.000 --> 00:43:04.000
to make sure that you don't get
wildly astray.
00:43:03.000 --> 00:43:09.000
Let's call that omega r,
00:43:05.000 --> 00:43:11.000
the resonant omega.
This isn't true resonance.
00:43:09.000 --> 00:43:15.000
Your book calls it practical
resonance.
00:43:11.000 --> 00:43:17.000
Again, most people just call it
resonance.
00:43:15.000 --> 00:43:21.000
So, you know what I mean,
type of thing.
00:43:18.000 --> 00:43:24.000
It is omega r is very much like
that.
00:43:20.000 --> 00:43:26.000
Maybe I should have written
this one down in the same form.
00:43:25.000 --> 00:43:31.000
Omega one is the square root of
omega nought squared minus p
00:43:29.000 --> 00:43:35.000
squared.
00:43:34.000 --> 00:43:40.000
What would you expect?
Well, what I would expect is
00:43:37.000 --> 00:43:43.000
that omega r should be omega
one.
00:43:40.000 --> 00:43:46.000
The damped system has a natural
frequency.
00:43:43.000 --> 00:43:49.000
The resonant frequency should
be the same as that natural
00:43:47.000 --> 00:43:53.000
frequency with which the damped
system wants to do its thing.
00:43:52.000 --> 00:43:58.000
And the answer is,
that's not right.
00:43:54.000 --> 00:44:00.000
It is the square root.
It's a little lower.
00:43:57.000 --> 00:44:03.000
It's a little lower.
It is omega nought squared
00:44:01.000 --> 00:44:07.000
minus two p squared.