We have discussed free oscillations of harmonic
oscillators without damping, and then we introduced the
damping. But in each of those cases,
we let the simple harmonic oscillator do its own thing.
We did not interfere with it. Today, that's going to change.
Today we're going to impose our will on to the simple harmonic
oscillator. And, we can impose our will on
to it by driving it with a force, and then see what the net
result is. And let's start with a simple
example that I have here. [NOISE OBSCURES] K,
and an object of mass M. Let this be equilibrium
position X equals zero. That will be damping.
I will introduce, again, B over M equals gamma,
and omega zero squared equals K over M.
We've seen this before. There's the shorthand notation.
So now, in addition to the fact that when the object is away
from equilibrium, that there is here a spring
force, I am now going to apply on that object a force.
It may not be easy to do. We will get back to that how we
do that. But I can apply a force on that
maybe through magnetic fields, maybe through electric fields,
and I'm going to, this force is not going to have
the character F zero times cosine omega T.
I impose on a system, now, that frequency omega.
I can choose anything I want to.
So now I can write down the differential equation of motion,
Newton's second Law, MX double dot equals minus KX,
nothing new. That's the spring force minus
BX dot, nothing new. That's the damping.
But now comes this external force at zero cosine omega T.
What I'm going to do now: I'm going to move this to the
complex plane, not that is absolutely
necessary, but I'm so used to that.
So, I'm going to write this now in terms of Z,
and then we'll take the real part of Z later.
That goes back to X. So, I'm going to write this now
in terms of Z double dot. I divide M out.
We get plus gamma times Z dot plus omega zero squared times Z.
And, that now becomes F zero divided by M.
Remember, I divided M out. And then we get cosine omega T,
for which I will write E to the power J omega T because I work
now in the complex plane. And, through Euler,
I can always convert that back to cosine.
My trial function for Z, which is a complex notation,
is some amplitude times E to the power J omega T minus delta.
Now, crucial is that you understand why this omega and
this omega are the same. This is the omega of my driving
decision that is why well that I impose on that system.
Clearly, given enough time in the beginning,
the system may be unhappy, and it may do all kinds of
nasty things, which we will discuss next
lecture. But, ultimately,
I will come out to be the winner, and ultimately that
system is bound to start oscillating with the frequency
that I impose on it. If I start shaking you in the
beginning, you may not like that, and you may oppose to
that. But ultimately I will be the
winner, and I will make you shake with that frequency,
omega. So, clearly,
the ultimate solution must have the same omega as the driver.
What is the meaning of this delta?
Well, it is not at all obvious that the object will have been
in the same phase as the driver. It is possible that when the
force is pointing in the direction that the object may be
going in the other direction. And you will see that,
indeed, can happen. And so, this delta is a phase
angle which takes into account the possibility that the driver
and object in motion are not exactly in phase.
We call this solution a steady state solution,
a steady state that you must wait long enough for the system
not to fight you any longer. That will be part of my next
lecture, the fighting issue. This is when I ultimately come
out to be the winner and when the system follows my will.
So now, I'm going to take the second derivative.
So I get minus omega squared. J omega comes out twice.
So I get minus omega squared. Then I get plus gamma times J
omega. Then I get plus omega zero
squared. And, that whole thing,
multiplied by A to the power E J omega T minus delta,
that now equals F zero divided by M times E to the power J
omega T. The whole thing is now in the
complex plane. And, you see that E to the
power J omega T cancels on both sides.
So, I lose my E to the power J omega T, and I'm going to
multiply both sides by E to the power J plus delta.
So, I lose my delta here, but it appears then here.
And so, if I make that simple change, algebraic change,
we are going to get minus omega squared plus gamma J omega plus
omega zero squared. That multiplied by A must now
be equal to F zero divided by M times E to the power J delta.
Omega T is gone, and I have moved the delta to
the right side. Can we live as that?
And this can be written F zero divided by M times the cosine of
delta plus J times the sine of delta.
I'm still in the complex plane, but that's Euler.
Now, let's compare the apples with apples, and oranges with
oranges. This is an apple,
and this is an apple. That means it's real.
And that means this is an apple.
But there are also oranges. This is an orange as a J,
and this is an orange that has a J.
And so, for this equation, to always hold at all moments
in time, the apples must be equal to the apples on this
side. And oranges on this side must
be equal to the oranges on that side.
So, it looks like one equation, but it really is two equations.
So, we now get that minus omega squared plus omega zero squared
times A must be equal to F zero divided by M times the cosine of
delta apples on both sides are equal, and now we get the
oranges on both sides. Gamma omega must be equal to
times A. I still have this A here.
So, gamma omega times A equals F zero divided by M times the
sine of delta. Two equations with two
unknowns: A is unknown and delta is unknown.
And, they are easy to solve. If you square them,
then you get sine squared delta and cosine squared delta.
You add them up, that's one.
And so, that immediately gives you, then, what A is.
So, A is going to be F zero divided by M upstairs,
and downstairs you're going to get omega zero squared minus
omega squared, squared, plus omega gamma
squared. So, this is the amplitude of
the object. We'll massage that.
We'll talk about this for at least the next ten minutes.
It's a very complicated function.
We want to see through that equation what that actually
means. And, the tangent of delta is
easy to find because you do find this equation by that one.
You get immediately the tensions of delta.
A's disappear, and F zero over M disappear,
and so you will get that the tangent over that angle,
delta, is omega gamma divided by omega zero squared minus
omega squared. We can now return to the real
world. And if we return to the real
world, we have to change Z back into X.
And so, our final solution, which I will put in color,
would then be that X as a function of T is this amplitude
A times the cosine of omega T minus delta.
And, that is the omega. It's my will that I imposed on
the system. Notice that there are no two
adjustable constants, which we were so used to in the
past. In the past we said,
well, you can start the system at T equals zero.
You can define the position. You can give it a certain
velocity. So, you always expect that in
your solution, there are two adjustables in
order to meet the initial conditions.
There are none here. And, the reason for that is
that this is a steady state solution, which means the system
doesn't even remember anymore what the situation was at T
equals zero. It has lost all its memory.
And so, A, which is the amplitude of that object,
is not something that you may choose.
A follows immediately from this equation, which is a complex
function: omega zero, F zero, and so on.
And delta is always nonnegotiable.
Delta has nothing to do with your initial conditions.
Delta follows from gamma, omega, and omega zero.
So now we're going to look at A, and try to understand the
complexity of that amplitude. For one thing,
it is pleasing that F zero is upstairs.
It is an intuitive pleasing that if the force that you apply
becomes larger, then the amplitude will become
larger. That's reasonable.
It is also pleasing to see that there is a gamma here
downstairs. That means if there is a huge
amount of damping, you don't expect A to be very
large. So, that's also pleasing that
you see a gamma downstairs. Now, I want to evaluate in
detail what's hidden in this very difficult equation.
And let me try out your intuition.
Common sense, without looking at my
solutions, without looking at differential equations,
without looking at the equation, A, just common sense
now, suppose I apply a force here on the subject with a
frequency which is near zero. So, it takes 100 million years
for it to reach a maximum. And then, it takes another
hundred million years for the force to go to zero and so on.
When that force has a value, F zero, what do you think will
be the position of that object? If you know that position,
that may tell you what A is, what the amplitude is of that
object without any differential equations.
Any one of you able to immediately say,
of course A has to be this? Maybe that is a little tougher.
I see some hands there now. Or you're just doing your hair?
Yeah?
Of course. If that force goes so slowly,
then at all moments in time there must be equilibrium
between the spring force, which is, of course,
KX, and the force that you apply, which is UF.
And so, if you do it extremely slowly, the two must always
cancel each other. And so, I make the prediction
now that when omega goes to zero, that A should become F
zero divided by K. That is that X.
Now, let's look at that equation.
Let's see whether that is true. We make omega zero.
So, this equation tells us that it is F zero divided by M.
And then, downstairs we have omega zero squared.
But, omega zero squared is K over M.
And you see, indeed, that is exactly what
you get. Now, without looking at
equations, can you guess what the phase difference is between
the driver and the follower? If it takes 100 million years
for that force to slowly reach its maximum at 100 million years
to go back again, what do you think will be your
phase difference between the two?
It will be zero, of course.
There's plenty of time for that object to follow.
So, you expect that delta becomes zero.
Well, if omega becomes zero, this zero is omega squared.
So, this goes away. He has a zero upstairs,
so you see the tangent of delta is zero, and that indeed is what
you see. So, the two follow each other.
It's extremely boring, the whole thing,
to watch, and the amplitude is exactly what you predict.
Let's now do something more interesting, and let us drive it
at what we call the resonance frequency.
We give it that word. That is the frequency that the
system really would love to oscillate in the absence of any
damping, and in the absence of my doing this silly thing by
driving it. So, now we are at what we call
residents. So, this term goes away,
and this term now becomes omega zero gamma.
So you now get that A becomes F zero divided by M,
and then downstairs you have F omega zero gamma.
Well, if you remember that we introduced a quality factor
omega zero divided by gamma, which is a dimensionless
number, then you can also write this as F zero divided by K
times Q. So, that's nice to remember
that at resonance, if you define this as
resonance, the amplitude of the object is Q times higher than
what it would be at extremely low frequencies.
Interesting to remember, so, this is the amplitude at a
very low frequency that when you drive it at resonance,
it is Q times higher. And then, I will put that here.
When omega goes to infinity, everything goes so fast that
the object has no time to follow the driver.
The object goes nuts because of this high frequency.
It can do anything. And so, A would then go to
zero. And, let's check that.
If omega goes to infinity, you see the downstairs here
goes to infinity so, A goes in D2 zero.
So, you have no amplitude at all.
What is not so obvious that delta here is pi,
and what is not so obvious here either, that delta goes to pi
over two in case of resonance. In other words,
at resonance, the driver and the follower are
90° out of phase. The follower is 90° behind:
very hard to imagine what that is like.
But, I will demonstrate it. You will be able to see it.
What it means here that at very high frequencies,
the amplitude of the object goes to zero,
but what I will be able to show you, that if the driver goes in
this direction, that the object goes in this
direction. So, they are 180° out of phase
that I could show you. That's what it means when delta
equals pi. So, now we can make a graph,
a plot of A as a function of omega --
So, here is omega, and here is A.
And, let this be the resonance frequency omega zero.
I'll make it a little straighter.
So, you start at very low frequencies if this is zero.
You start here with F zero divided by K.
We all agree that that was obvious.
And then, the amplitude will build up, go through a maximum,
goes down, and ultimately goes to zero.
And at this value, omega zero, this value is Q
times F zero divided by K. Now, for those of you who look
very carefully, you may have noticed that the
maximum here that I have drawn is not at omega equals omega
zero, which may go against your instinct.
This maximum occurs at a frequency, which we will call
omega max, which is always a little bit below omega zero,
but for high Q systems, as I will show you shortly,
it is effectively the same. I will come back to this.
The delta, the phase as a function of omega,
if this is pi and this is pi over two, and if this is omega
zero, then that delta will change in the following way.
It is way harder to imagine than what A is doing you are in
phase at very low frequencies, at resonance precisely at omega
zero. You hit the pi over two,
90° out of phase, and at very high frequencies
you will see that the two are out of phase.
And I will be able to demonstrate that to you.
Coming back to this mysterious maximum, is also mysterious
actually, where is this? At what frequency do we have
really the maximum amplitude? Well, to calculate that,
you would have to take the derivative of that monstrous
equation. You will have to take DAD
omega, and you [ask?] that to be zero.
So, that's when the maximum occurs and I will leave you with
that exercise. It may take you a few minutes
to do that, and you will find, then, that omega max,
so, where the real maximum is located, the maximum in terms of
amplitude is omega zero squared minus gamma squared over two,
not four, but two to the power of one half.
Not so intuitive that it is there.
And, if you'd like to write that in terms of Q,
which is often done, then omega x,
so that is the frequency at which the amplitude reaches a
maximum is omega zero times one minus one over two Q squared.
And then, you square root of that whole thing.
And the reason why this is nice, you can immediately,
if you know Q, you can immediately evaluate
what the difference is percentage-wise between omega
max and omega zero. If you want to know what the
maximum amplitude itself is, so what A max is,
so that's really this value. It must be very close to Q
times F zero over Q. But, it's a little higher.
Then, you can write that in the following form.
And that's just a matter of algebraic manipulation.
And you get a Q here, what you'd expect,
and then downstairs you get something like this,
one minus one over four. And then you get a Q squared
through the power of one half. And so, now let's put in some
numbers so that you get some feeling for the answers that we
have. Suppose we have an example by Q
equals five. It's a modest value for Q.
Most pendulums that we have, the Q is way higher than five.
So, I take a modest number for Q.
If I go to this equation here, Q squared is 25.
Two times 25 is 50. That's 2% that I have to take
the square root. So, it's only 1% off.
So, omega zero, omega max divided by omega zero
is 0.99. It's only 1% lower.
It's only 1% below omega zero. And then, if you want to know,
now, what A max is, so you would think that A max
is very close to Q times F zero over K, but it is not.
Q times, it is a little larger. And so, if we defined A max
divided by A zero, A zero now is meant to be the
amplitude when omega equals zero.
That is a shorthand notation. This number is not Q.
It's a little higher. It's now 5.03.
And, you can see it if Q is higher, then of course these
numbers become even closer than omega max becomes even closer to
omega zero, and then this maximum A becomes even closer to
Q times F zero over K. Rarely ever will we be bothered
too much with the fact that the resonance frequency,
which we call omega zero, is not exactly the frequency
whereby the response of the object is A maximum.
Very rarely ever will that become an issue.
I want to show you, now, the transparency from your
own book. So, don't take notes.
This is from French. You see here that a function A,
that it is divided by A zero, which is that F zero divided by
K. So, that is the amplitude for
zero frequency. And so, when you start off,
that ratio is one by definition, right,
because it's A omega divided by A zero.
And, horizontally, you see of omega divided by
omega zero. So, by definition,
right here at the one sign is that point that I put omega zero
there. And, you see here these various
curves for different values of Q.
And, the one that I made green is Q equals ten.
And, no surprise that the height is at plus ten because we
predicted that it is Q times higher than the amplitude when
we have low frequency. And you see,
indeed, that the red one is very close to ten.
If you look at the one that a mark Q equals three,
which is this one, if you look very carefully you
may see that the maximum A shifted slightly below the value
one, which is omega zero. But even for Q equals three,
the differences in significantly small.
And at the bottom, you see the delta function,
the phase delay. The object follows the driver
at very low frequencies. Precisely, delta is zero.
At resonance, it is precisely pi over two,
90°, very hard to imagine that I will try to show it to you.
And, at very high frequencies, they go like this.
They are 180° out of phase. So, now comes the question,
how do you apply a force on a system?
It's nice to say there's a force, but you have to think of
a way that you can actually do that.
And, I will just discuss one case with you,
and then I'll try to demonstrate it also.
If I have a pendulum and I want a force on this object,
then I can do that as you will see in an indirect way by
starting to move my hand here. You'll see how that translates
into a force on this object by moving my hand.
I am now the driver. My displacement,
now, is in inches, not [a force?],
but is in inches. Here is the pendulum,
length L mass M. And, this is the equilibrium
position of my hand. And here's the object.
But, going to move my hand in a way, [eta?] equals eta zero
times cosine omega T. That is the frequency that I
decide. I impose that frequency on the
top of the pendulum. And, the amplitude of my hand,
in terms of inches or miles or light-years, that is linear
scale, eta zero. It's not a force.
It's a displacement.
Well, I take a picture at one moment in time and what do I
see? I see that this is what the
pendulum looks like. This angle is theta.
The pendulum is displaced over a distance, X,
from equilibrium. And the top is displaced over a
distance eta. This is Walter Lewin.
I am doing that. I am there with my hand.
I can't help it. This is where I am.
And this is where the object is.
So, now I want to put in all the forces that are at work.
We'll put it up a little bit because I want to have a little
bit of room for my forces. So, we'll make a little
shorter. So here's the object,
and here's the object. There are only two forces on
this object, and that is gravity, which is MG,
and that is the tangent. There is nothing else.
I call this X equals zero. And I call this displacement X
away from equilibrium. For small angles,
I want to argue that T is very close to MG.
For one thing, if you hold them vertically and
he did nothing and there is no motion, it's obvious that T is
MG. The two forces have to cancel
each other out. That's clear.
But I can show you that even if the angles are modest,
that that should also be the case.
Suppose I decompose T in two directions, vertical direction.
So, this is T times the cosine of theta.
And in the horizontal direction, so this is T times
the sine of theta. If the angles are very small,
the object is hardly moving at all in this direction.
The motion is almost exclusively in this direction.
So, there is no acceleration in the Y direction,
or I should say the acceleration in the Y direction
is negligibly small. So, that means to a high degree
of accuracy, T cosine theta is always the same as MG:
high degree of accuracy. But for small angles,
cosine theta itself is one. Therefore, T equals MG.
And so, the force that is driving this object back to
equilibrium is T sine theta. And so, that force is MG sine
theta to a high degree of accuracy.
I'm going to introduce, again, that gamma is B over N.
So, it is damping. And, I'm going to introduce
that omega squared equals G over L.
Omega zero squared is G over L, the square root of G over L
being the resonance frequency of a pendulum length L independent
of the mass of the object as we've seen before.
So, now I'm going to write down Newton's Second Law.
So, I get MX double dot. And then I get minus BX dot.
That is the damping minus BX dot.
And now comes this force which is the only one that wants to
drive it back to equilibrium. It's the restoring force.
And so, that's force, if you set my T being MG.
That is MG times the sine of theta.
That's the differential equation that I now have to
solve. That is a driven system.
Now, here in the driven system, and boy, I saw force here.
I don't see anything like that there.
Where on Earth does Walter Lewin come in to this picture?
Who's doing something? Have I overlooked myself,
perhaps?
Excuse me? I'd change nothing.
Would I change anything? I've changed nothing that I
don't see myself anymore. So, what's wrong?
Is there anything wrong with this?
Where do I show up in this equation?
Yeah? So where in that equation do I
show up? What is the sine of theta?
What is the sine of this angle?
X minus [eta?]. That's Walter Lewin.
X minus eta divided by L. There I am.
And so, I'm going to substitute that in here,
and I'm going to divide by M. No, let's not divide by M yet.
Let's just say X double dot minus BX dot,
and now we get minus MG times X over L.
And, now we bring Walter Lewin to the other side.
And so, we get plus MG times eta divided by L.
And, eta is eta zero times cosine omega T because I am
moving my hand that eta is a function of time.
Let me write down MG in here, and then we'll check this.
So, MG times sine theta has two terms.
It has an MG times X over L, but it has also an MG times eta
over L. And, I bring that eta L over on
this side, but I know that eta is changing in time.
And so, you see now Walter Lewin is right there.
And now divide by M, and I substitute omega zero
squared over here. Oh, I have an M here,
too. You should have screamed there
was an M there. I decided not to divide by M,
remember? Now I'm going to divide by M,
so I get X double dot. There is an equal sign here.
You should not be sleeping. You're not supposed to sleep.
This is an equal sign minus BX dot, right, minus B,
yeah? We're in business now.
And so, you are all sleeping. All of you are sleeping,
my goodness. MX double dot minus BX dot
minus MG KX over L, and then the minus and the
minus becomes plus. Right, try not to sleep.
So, X double not, now we get plus gamma times X
dot. And now we get plus omega zero
squared times, because G over L is omega zero
squared times X. So, I divide the M out,
and now I get equals omega zero squared times eta zero times
cosine omega T. I will move this L up a teeny
little bit, and on going to look now at that equation at the
bottom. And I am now overjoyed,
happiness, because this one looks almost like a carbon copy
of the one that I had here was an F zero cosine omega T.
And now, instead of an F zero divided by M cosine omega T,
I now have this. So, this takes the place of my
earlier F zero divided by M. F zero divided by M is an
acceleration, by the way.
Now, it better be an acceleration,
because this is an acceleration.
And apples have to be apples. So, this is an acceleration.
This is an exhilaration, and this is also an
acceleration. Multiply omega squared by
distance. Then you get distance divided
by time squared. So, you see now how the
connection between the two go, where originally I got an F
zero over M times cosine omega T, now because of Walter Lewin's
motion, I'm going to get an omega squared times eta zero.
And so, you see now how this motion of my hand indeed
translates into a force on the object.
Well, I have the solution. I don't have to do anything.
All I have to do is change this by omega zero squared times eta
zero, and I'm done. Differential equations are
identical. I don't even have to change the
tension of Delta. Nothing changes.
This is the only thing that changes.
So, we are done. We can now make some
predictions. The prediction is that if I'm
going to shake this pendulum, and are going to do that very
slowly, taking one hour to go to the left and taking one are to
the right. And, if my amplitude is eta
zero, what do you think that the amplitude, A,
the solution of the differential equation,
will be? In other words,
I'm going to shake very slowly. What do you think A will be
without looking at the differential equations?
So, I just go with my hand like this: amplitude eta zero,
amplitude eta zero, and I do it very slowly.
What would be the amplitude of A?
Eta zero. So, you expect that this goes
to eta zero. Well, if you don't believe it,
go to this equation, substitute in here zero,
in here zero. You get omega zero squared.
Each of us: omega zero squared, and you see eta zero.
It's exactly what that equation predicts.
But, your common sense says the same thing.
Now, what do you think delta is if I'm going to move this
pendulum very slowly to the left and to the right?
Of course the object [UNINTELLIGIBLE] will be
ridiculous if I take one week to go from here to here.
The object would be there, right?
The object is always here. So, you also predict that delta
is zero. And so, now we can make
[UNINTELLIGIBLE] prediction that at resonance,
you probably get Q times eta zero, and then delta would
become pi over two. And, when you go to very high
frequency, then A would go to zero.
And then delta would go to pi. And this is what I want to
demonstrate to you. So, the final solution of this
pendulum, which I will write down in red is going to be that
X equals A times the cosine of omega T minus delta just as we
had before. A is nonnegotiable.
It has nothing to do with the initial conditions.
Delta is nonnegotiable. It has nothing to do with the
initial conditions. This is the steady state
solution. All right, let me take my shoes
off because then you can see it better.
All right, here's the pendulum. Here's a pendulum.
It's going to be very exciting. I'm going to tell you.
I'm going to move this with omega very close to zero.
Very exciting. I'm doing it right now.
Aren't you thrilled? No, you're not thrilled?
[You're not?] moving. And now I'm going to go back.
Do we agree that A, the amplitude of that object
it's exactly the same as the amplitude of my hand?
Do we agree? Do you see that?
That is why that A is eta zero, and that follows from that
rather complicated equation. Did you see that delta was
zero? Did you see that we went hand
in hand, so to speak, no pun implied?
We're going hand in hand. That one follows exactly my
hand. So, delta is zero.
Let's now go to high frequency, very high frequency,
way above resonance. And, what you see now is that
the object is not moving very much.
But, if you look very carefully, you'll see when my
hand as here, the object tends to go there.
When my hand is here, the object tends to go there.
That is that pi. Ready?
You see that there's almost no motion?
A is near zero, but can you really see that the
phase difference is pi? Can you see the 180°?
You see, if A is exactly zero, of course you cannot tell.
So, I try not to go infinitely fast.
I go a little slower than [OVERLAPPING VOICES].
Can you see it? OK, now comes resonance.
And now it will be very difficult to see this pi over
two. That's almost impossible.
That's not my objective. My objective is to show you
that enormously small, very small eta zero here will
give an amplitude there are which is Q times higher.
So, you get a huge swing when my hand is hardly moving at all.
That's the power of Q. So, there we go.
First [carried?] into it; there it is.
Now, this is resonance. Would you agree?
This is resonance. Now, look at my hand.
My hand is moving, probably no more there is an
amplitude of 3 mm, no more.
And yet, I see an amplitude there of 60 cm.
That would mean that very roughly, this pendulum has a Q
of 200, namely, 60 cm divided by 3 mm.
So, this is even a way to make an extremely rough guess,
admittedly very rough, of the Q value.
You cannot even see my hand move.
Be honest. You can't even see my hand
move, but I know I am moving it a little.
You're lying. [LAUGHTER] No you were not.
No you were not. OK, so you see all the goodies
that we have calculated actually can be demonstrated and show up
quite dramatically. Suppose I had a spring system
like this, and I wanted the force on that object here.
Well, what I can do is just shake it here [NOISE OBSCURES].
And, when I shake it there, we can make certain
predictions. We can make predictions now
based on the knowledge that we have.
Suppose I shake it with an amplitude eta zero.
No differential equations, no nothing for now.
But I know that somehow it will come out in terms of the force
at the object. So, I know that when I write
down the differential equation, of course it shows up exactly
this way again in omega zero squared times eta zero except
the omega zero squared is now K over M.
So, what do you think if I shake it at omega equals zero?
What is, then, the amplitude that this object
will have relative to my motion eta zero?
If I move my hands, eta zero infinitely won't,
what will this object do? It will just follow it.
So, you get this answer. What will be the delta?
It will be zero. When I get resonance,
what will be the amplitude of that object hanging from this
spring? It will be Q times higher than
my eta zero. What will be the phase
difference? 90°.
When I shake like crazy, A will go to zero.
So, there is a spring. If you shake it like this,
which is part of your problem set, you will see exactly the
same results that we have done for a pendulum.
This, now, I want to independently demonstrate to
you. I have here in air track.
I can blow out air so that the object here starts floating.
So, we can make the damping very slow by making it float.
But, if we lower the airflow, the damping becomes a little
higher. I have a spring here with
[spring?] constant K. And I have another spring here
with spring constant, K.
They both have spring constant, K.
And now I'm going to drive this here at extremely low
frequencies over a distance eta zero at maximum.
Eta zero, cosine omega T. What do you think the amplitude
of this object will be at that very low, yeah?
Very, very good. Not eta zero,
but why is it half? Because we have two springs.
So, effectively, the spring constant is twice
that, exactly. So, if I go very slowly,
you'll see that this displacement here will be twice
as high as this displacement. But what I really want to show
you is they are in phase. This one will go to the right
and this one goes to the right. Now comes the catch.
I showed you earlier there is a steady state solution.
In the beginning, the system doesn't like me.
It hates me. It fights me.
It doesn't like that omega. It wants to do something
different, which is part of next week's lecture.
And, you'll see that in the beginning and so,
we have to be a little patient before my will survives.
Ready for that? So, I'm going to start now to
drive the system at a frequency which is below resonance.
I want you to see two things, that they go hand-in-hand.
And, I want you to see that. See, you're going to a very low
frequency. Here, this is twice the
amplitude of the driver. Now it is here,
the spring, and now the spring is here.
So, only this much is two eta zero.
So, eta zero is no more than three quarters of an inch.
And now we're going to let that object the exposed to this
driver. And, we'll give it a little bit
of time to recognize me. It takes a little bit of time
to reach the steady state solution.
And, next time we will learn how much time it actually takes.
So, if you want to be a little bit patient, then you'll see --
-- if we give it too much damping, too little air,
then of course it starts to get stuck.
Yeah, we're close. We are close,
for me close enough. Now, look at it.
They're going both to the left for me, and both to the right
for me. For you, they're going now both
to the right, and they're going both to the
left. They're going both to the right
and both to the left. Now, this was the amplitude,
twice the amplitude of the driver.
And then, you look very carefully here,
it's less. It is that eta zero over two
that this gentleman immediately noticed because we have two
springs. So, you see here apart from the
factor of two, you see the delta zero,
and you see that the amplitude indeed is half of the amplitude
of the driver because of the two springs.
Now, we're going to resonance, omega zero.
And now, nasty things may happen.
It may break. We have to give it time.
See what funny things it's doing?
Not in steady-state yet. You have to wait.
Just a little patient.
Give it more time.
Notice, also, that remember this is only
moving this much. Look how much this is moving.
I may not even be exactly at resonance.
We can only do the best we can here.
It may not be exactly at resonance.
Oh boy, I'm close to resonance now.
Oh yeah. Oh, man, look at that.
Am I at resonance! I think I got it.
You see, they are neither in phase nor out of phase.
Now you see the 90°. And, look at this teeny-weeny
little displacement here. And, look what this man is
doing. That is resonance.
Remarkable. Where's Marcos?
We hit it. Right on!
[LAUGHTER] Now, I will oscillate it way over
resonance, not way, but over resonance.
The system must first calm down.
And now I will change the frequency of a resonance so that
now you will see the phenomenon that I discussed earlier that
the amplitude is very small. Again, we have to wait a
little. Look how fast it's going.
And, then I will go 180° out of phase.
Now, look, this is going this much back and forth.
And this one is not doing very much.
Now, you can't see it. I can.
It looks like this. Can you see it?
That's what 180° out of phase is, very clear.
Five minute break, see you back here in exactly
five minutes. [SOUND OFF/THEN ON]
So, we have discussed today some simple systems:
pendulum, one object, springs, one object,
one resonance frequency, but soon in 8.03 will discuss
systems with more than one object.
For instance, if I put three cars on here
with four springs, three resonance frequencies,
if I have a trouble pendulum, which I will demonstrate next
week, when pendulum below the other below the other,
three resonance frequencies. Five cars on there,
five resonance frequencies. So, simple objects like a
dinner plate or just a regular glass have an enormous number of
resonance frequencies. It can oscillate in many,
many different ways. If you drive your car,
your wheels turn around, that's certain oscillation,
a certain period underlying. And, you may notice at certain
speeds that something in your car begins to rattle:
very annoying. All you have to do is go
slower, go a little faster, and it stops.
You go off resonance for that object.
Now, you may go on resonance for another object,
of course, and some cars rattle at any speed.
You have a radiator in your room which rotates that is also
an underlying oscillation and period.
That may start to cause residents in the frame.
You may hear some awful noise sometimes.
Unfortunately, these fans, you cannot change
the speed so easily. But you can go from state
three, to two, to one, and then this terrible
noise will go away. You take your washing machine
or dryer. They used to remember a friend
of mine in the Netherlands had a dryer.
And, when he started the dryer at a very early phase,
when it was a certain frequency, the whole dryer would
start to walk through the room. [LAUGHTER] It would.
And then, at higher frequencies, it of course would
stop. Resonances are everywhere,
and they often occur when you don't expect them to.
You open a faucet; you think it's a steady stream
of water which I'm sure it is. But sometimes you hear
[SCREAMS] an unbelievable sound that drives you almost nuts.
I'm sure all of you have heard that sometime.
If it isn't in your dormitory, maybe at hotels or at home.
All you have to do is open the faucet a little more or a little
less and it goes away. And, it's really an extremely
loud and annoying resonance. If you take something as simple
as a wine glass, which has a tremendous number
of resonances, then I can make you listen to a
well-known resonance, which is by rubbing the rim of
the glass. When I rub the rim of the
glass, I'm not exciting it at one particular frequency,
and certainly not at a resonance frequency.
I am exciting it at lots and lots of frequencies.
I dump on it a whole spectrum of frequencies.
But the glass is mean. It just picks out the one which
is its resonance. That's where it builds up a
large value for [A?]. It ignores all the others.
And that's why I can make it resonate at that particular
frequency. Listen to it.
This is not one frequency what I'm doing.
It has nothing to do with the time for me to go around.
It's a very high pitch. It's about 420 Hz.
So, the rubbing is like dumping a spectrum of frequencies on it.
And it selects what it likes the most.
When I was a student I remember we often had an after dinner
speaker. We had dinners at the
fraternity, and we had an after dinner speaker.
And more often than not we didn't like after dinner
speaker. We didn't like the speech.
And so, we made that very clear.
And the way we do that is all our wine glasses,
an enormous sound in that dining hall.
And the speaker very quickly got the message,
of course. That is an enormous sound that
you can generate. And most of these wineglasses
were roughly the same. So, it was always a tone that
was loud and clear and almost one frequency.
You've seen footage lately of the storms, three storms in a
row. And you must remember sometime
that you saw a traffic sign. You see a pole and then the
traffic sign. And then even though there was
some kind of a crazy when going, the traffic sign goes like
this: all resonance frequencies. It can break.
Even though the wind appears to be relatively steady,
the wind then generates in a way the whole spectrum of
frequencies and this traffic sign picks out the one that
likes the most. And then it goes nuts at a
frequency that is a resonance frequency.
And, resonances can become destructive, of course,
if these amplitudes are too high.
Then things can break down. You may have noticed I was
worried here when amplitude became so large that the spring
might even break or the car might jump off that track.
And, of course, a classic example of this
district of resonance is the destruction of the very famous
bridge in this country in 1940. The [common arrow is bridge?],
Washington State, was destroyed by wind.
And this bridge had many different resonant frequencies:
won't like this and some like this.
And depending upon the wind strengths, different resonances
were excited at different moments in time,
but that ultimately led to the destruction of the bridge.
Now, most of you have seen this movie, but just for the few that
haven't I really want you to see this movie.
You cannot go to MIT and not have seen the destruction of the
[Coma Bridge?] movie. [SOUND OFF/THEN ON]
There are countries where soldiers are not allowed to
cross a bridge when they're marching in step.
That's the case in the Netherlands, my home country.
It's also the case in many European countries.
The story has it that in England I think more than a
hundred years ago when soldiers went over the bridge and step
that the bridge collapsed. Whether that was the result of
the soldiers, we will never know.
But in any case, from those days can order that
soldiers have to go out of step before they cross the bridge.
There is a rumor that most of you have heard that there are
women who are capable of singing with such a loud voice they can
break a wine glass. They have to tune exactly at
the resonance frequency of the wineglass, and they come out at
a huge volume and the glass breaks.
I don't believe it, but it's a rumor.
There was a commercial many years ago, some of you may never
have seen it, for Memorex.
Memorex was a special tape for audio tape recorders this guy is
going to a concert. And there is his woman singing,
loud voice, beautiful frequency: bingo.
The glass breaks. He comes home and he tells his
wife the story. Well, his wife was smart
enough, of course, not to believe it.
And he says, well, it just so happens that I
recorded it on my Memorex tape. So, he plays the tape at home,
and at the moment that the woman's voice goes crack,
what happens? The glasses break at home in
his cabinet. So much for the physics of
Memorex, because you can imagine that the resonance frequency of
the glasses at home were very different than the glasses in
the orchestra. So, it's all a swindle,
but that's, of course, what commercials are all about.
[LAUGHTER] What is now the bottom line?
The bottom line was that if you buy Memorex, these tapes,
then the reproduction is so perfect that you can even take
it home and you see the glasses break.
[SOUND OFF/THEN ON]
This brings up, now, the $64 million question,
and that is, can it be done or can it not be
done? So, any women in my audience
you want to give it a try? That would be wonderful.
[LAUGHTER] We've asked ourselves that question.
Can this be done or can this not be done by person?
And I think we came to the conclusion that a person alone
without all kinds of electronic equipment would not do that.
And I still believe that today. But, Professor Michael Felt
[SP?] here at MIT with one of his graduate students many years
ago developed some powerful equipment which you see here
which was designed to make an attempt to break a wineglass.
It doesn't always work, but it works often,
the idea being then that here is the wineglass.
It's almost a carbon copy of this one when they finally made
this to work at near 440 Hz. He went to Crate & Barrel and
asked, how many of these one glasses do you have?
They said we have 5,000. And they bought all 5,000
[LAUGHTER] because it's not obvious if you have to [check?]
the glasses that it will ever work again.
And so, here's one of those glasses.
Here is a loudspeaker. The sound comes from this side,
and what is nice about this arrangement: that we can make
you see the distortion of the glass in this resonance mode.
The glass oscillates like this, goes from oval,
to circular, to oval, to circular.
And, the way we can make you see that it is by strobing the
glass at a frequency which is a little bit different from the
frequency of the sound. Think about it:
two frequencies almost the same give you a beat phenomenon.
So, what that comes down to is you see the motion of the glass
very slowly. And then, if we had that
resonance frequency, we will increase the volume.
And then maybe it will break. Now, I have to warn you,
the sound will be unbearably high, and so for some of you
here in the front row, you may even want to cover your
ears. Or, you may want to move back.
But, be careful, this is really an enormously
strong signal that you're going to hear.
So, I'll be careful. You can move back if you want
to. I have hearing aids,
as you have noticed, and I have the option that I
can turn them off. But in spite of that,
I'm not deaf without them. I will still cover my ears.
So, let me first give you the light setting that we want,
so we make it a little dark. And then I will show you the
glass. I have to change the setting
here. There is the glass.
There's no sounds. It's not oscillating.
And now I turn on the speaker. So, I am now going to turn my
hearing aids off and put this on.
And I'm going to increase the volume.
And if it doesn't want to break, I will change the sound
frequency little bit to sweep over that resonance curve so
that I get onto the maximum because of it's only off,
the cue is so high as the system that if I'm a little bit
off in frequency, then the amplitude will below.
So, let's first see what happens when I increase the
volume. Did I do something wrong?
Yes, I changed the frequency.
Getting very close.
[APPLAUSE] I think I've convinced you that a woman
cannot do this. I have some last words of
wisdom for you, and that is falling in love is
also a form of resonance. And the two can be destructive
because it can break your heart. So try to remember that next
time. Have a good weekend.