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So far, we analyzed...
We calculated the periods of lots of oscillators: pendulums, springs, rulers, hula hoops.
We gave them a kick, moved them off equilibrium, and then they were oscillating at their own preferred frequency.
Today, I want to discuss with you what happens if I force upon a system a frequency of my own.
So we call that forced oscillations.
I can take a spring system, as we have before.
This is x equals zero, this is x, and we have the spring force, very familiar, minus kx.
But now this object here, which is mass m, I'm going to add a force to it, F zero, which is the amplitude of the force, times the cosine of omega t.
So I'm going to force it in a sinusoidal fashion with a frequency that I choose.
This frequency is not the frequency with which the system wants to oscillate.
It is the one that I choose, and I can vary that.
And the question, now, is what will the object do? Well, we have Newton's Second Law--
ma equals minus kx plus that force, F zero cosine omega t.
a is x double dot, so I get x double dot, plus-- I bring this in--
k over m times x equals F zero divided by m times the cosine omega t.
Now, the question is what is the solution to this differential equation? It's very different from what we saw before, because before, we had a zero here.
Now we have here a driving force.
It's clear that if you wait long enough that sooner or later that system will have to start oscillating at that frequency.
In the beginning, it may be a little different.
In the beginning, it may want to do its own thing, but ultimately, if I take you by your arms and I shake you back and forth, in the beginning you may object, but sooner or later, you will have to go with the frequency that I force myself upon you.
And when we reach that stage, we call that the steady state as opposed to the beginning, when things are a little bit confused, which we call the transient phase.
So in the steady state, the object somehow must have a frequency which is the same as the driver, and it has some amplitude A.
And I want to evaluate with you that amplitude A.
So this is my trial function that I'm going to put into this differential equation.
x dot equals minus A omega sine omega t.
x double dot equals minus A omega squared cosine omega t.
And so now I'm going to substitute that in here, so I'm going to get minus A omega squared cosine omega t plus k over m times A cosine omega t, and that equals F zero divided by m times the cosine of omega t.
And that must always hold.
So therefore I can divide out my cosine omega t.
I can bring the A's together, so I get A times k over m minus omega squared equals F zero divided by m.
Now, this k over m is something that we are familiar with.
If we let the system do its own thing--
we bring it away from equilibrium and we don't drive it--
then we know that omega squared, which I will give the zero, equals k over m.
This is the frequency that we have dealt with before.
This is the driving frequency--
it's very different.
And so I'm going to substitute in here for k over m omega zero squared, and so I find, then, that the amplitude of this object here at the end of the spring will be F zero divided by m divided by omega zero squared minus omega squared.
And this amplitude has very remarkable characteristics.
First of all, if I drive the system at a very low frequency so that omega is much, much smaller than omega zero, we call omega zero often the natural frequency.
It is the one that it likes.
If you have omega much, much less than omega zero, this goes--
omega zero squared is k over m--
so you get an amplitude A which is F zero divided by k.
If you go omega way above the natural frequency or, let's say, omega goes to infinity--
it becomes very, very large--
then downstairs becomes very, very large, so A goes to zero.
But now, what happens when omega is exactly omega zero? Then the system goes wacky.
Look what happens.
The downstairs becomes zero and the amplitude goes to infinity.
And that's what we call resonance.
So if we drive it at that frequency, the system goes completely berserk.
I can make a plot of A as a function of frequency.
When I say "omega," you can obviously always change to hertz, if you prefer that, because omega is two pi times F, so you can do it either in hertz or you can do it in radians per second, of course.
So if I make a plot of the amplitude versus frequency omega, then at low values-- I have here F zero divided by k--
is the amplitude.
When I hit the resonant frequency, the natural frequency of the system, it goes out of hand, it goes to infinity.
The moment that omega is larger than omega zero, notice that the amplitude becomes negative.
A negative amplitude simply means that you get all of a sudden a phase change of 180 degrees, so the object is 180 degrees out of phase with the driver.
I will not expand on that too much today, but it is negative, and so it comes up here, and then it goes here to zero for very high frequencies of omega.
So something very spectacular is going to happen at the resonant frequency of the system.
In practice, of course, the amplitude will not go to infinity, and the reason for that is that there is always friction of some kind.
There is always damping, but you get a very high amplitude but not infinitely high.
So if I make you a more realistic plot of the amplitude, and I will take now the absolute value, the magnitude, so we don't have to worry about it getting negative--
we don't have to worry about the 180-degree phase shift--
then you would get a curve that looks like this.
And here, then, if this is frequency F, then here you would get the natural frequency when things go out of hand.
And depending upon how much damping there is, this curve would look either very narrow and very spiky--
it goes very, very high, then there is very little damping--
or if there is a lot of damping in the system, it's more like this.
So the narrower this...
We call this the resonance curve.
The narrower that is, the less damping there is.
I have here a system on the air track which is an object that I can drive with a frequency that I can choose.
Here is a spring...
object mass m, and here is another spring.
It's fixed on this side, right there, and here I'm going to drive it, so I have here this variable force that you see there.
And what I want to show you now is that first I will drive it at a frequency which is way below the resonant frequency.
Then you will see an amplitude, not very large.
I will then drive it way above the resonant frequency.
Again, you will see an amplitude which is very low.
If I can go very high frequency, you will see that it almost stands still, and then I will try to hit the resonant frequency, and that will be...
It's about one hertz, the resonant frequency, if I just let this object do its own thing, just this.
Now you see the natural frequency--
it's about one hertz--
but now I'm going to drive it here with a system, and we are going to pull on this spring with a frequency that we control.
So let me start.
You see here the frequency.
You see an indicator, very low, way lower than one hertz.
And when you look at the way that the system responds, if you wait long enough when the transients are died out, you will see that they go hand in hand, that the amplitude is in phase with the driver.
That's why we have plus amplitude here.
But the phase is not so important today.
See, they go hand in hand.
Very small amplitude, roughly at zero divided by k, which is the spring constant of the spring.
Now I'll go way above resonance, and this system will slip here, so don't pay attention to the arrow anymore.
Way above resonance.
You see, it starts to slip.
Look at the amplitude--
very modest, very small.
But we went over this curve, so first we probed it here and now we're probing it here.
Look, it's almost not moving at all, almost standing still, and I'm driving it at a high frequency now.
And now I'm going to find you this resonance, which is near one hertz...
which is somewhere here.
And look-- very high amplitude.
If we're not careful, then we can actually break the system.
Very high amplitude--
I'm trying to scan over it now, go a little bit off resonance.
Now I'm back on resonance.
See what a huge amplitude! Better turn it off.
So you see here the response when I drive a system.
When my system is a little bit more complicated--
for instance, if I had two masses here so I would add one here, spring constant k, spring constant k--
I could repeat this experiment, and if I did that, I would find two resonant frequencies.
And if I do it with three objects,
I would find three resonant frequencies.
If I did it with five, I would find five resonant frequencies.
And when I make, then, this curve of A amplitude as a function of frequency--
either hertz or in radians per second, whichever you prefer--
then if I had three objects there in a row, you would see something like this.
And depending upon how many of these objects you have, you get more and more resonances.
And these resonances can all be found by driving the system and searching for them.
If I go to a system whereby I have an infinite number of these masses...
We call them coupled oscillators; these oscillators are coupled through the springs.
An infinite number of coupled oscillators would be a violin string.
Here's a violin string.
And the reason why I call it "infinitely" number of oscillators is that I can think of each atom or each molecule as being driven, as being connected by springs to the neighbor.
And so it's an infinite number of coupled oscillators.
And so when I start to shake this system, I would expect a lot of resonances, and that's what I want to explore with you now.
In the case here, that the objects move in the same direction of the spring...
I call this the y direction and I call this the x direction, so the spring is in the x direction, the objects, the beats are in the x direction, and the oscillations are in the x direction.
We call those longitudinal oscillations.
There is also a way that you can have transverse oscillations, transverse...
whereby the motion is in the y direction, whereas the beats are in the x direction.
I could even do that with this system--
I could make them oscillate like this, because the springs will obviously also work if I do this with the system.
And that's the way I want this violin string or piano string to oscillate now, because that's the only meaningful way that I can make it oscillate.
And so I wonder--
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