In fact it's pretty straightforward to understand the resonant phenomena now that we understood how to calculate the amplitude of the particular solution.

So the amplitude of a particular solution will for example will be given by this equation we just calculated and we see that we see that this amplitude actually depends on the frequency and we can ask ourselves what's the resonant frequency in other words what's the frequency of the maximum amplitude and then we know since we have the numerator is a constant we just have to minimize the denominator.

This is a straightforward derivative so you try to minimize to make a derivative equal to zero and we find that the maximum of d will be obtained by frequency omega r is equal to omega zero square minus two beta square.

So if you are driving your system at a frequency that's given by the equation here at the bottom of the slide this is when you have the maximum motion, maximum solution, maximum amplitude and motion.

So we call this a resonant frequency omega r omega zero square minus two beta square and then we see that this resonant frequency can be modulated by changing the damping so there's a lot of damping.

I mean if the damping is fairly large not too large but let's say fairly large so that omega zero square minus two beta square is a positive still a positive number then the resonant frequency goes down.

There is no resonance however if two beta square is larger than omega zero square because in that case the resonant frequency is actually a complex number which actually it's an imaginary and then we would have a monotonic decrease.

So that's something that's important to notice.

Okay so just to summarize a little bit what we've done so far we've looked at a number of frequencies we've looked at when we looked at three oscillations no damping no force we found the natural frequency omega zero square equal k over m.

When we look at damping we find that we had an omega one square equal omega zero square one beta square.

This frequency omega one could be either an oscillation like in the under damping or it's no longer an oscillation when we are for example in over damping but in the under damping which is actually the solution that's written on this slide with the envelope function omega one is a frequency not so much of a periodicity of the response since the amplitude goes down so you don't repeat the same solution but as basically the frequency between maxima.

And finally for the driven oscillation we find another important frequency which is the resonant frequency.

So you see when you look at these three frequencies which are typical frequencies for for our driven oscillation that omega zero is always larger than omega one and which is which is itself always larger than omega r.

Now it turns out that driving system at resonant frequency is something that's very important for devices and for to get the best response the maximum response from a system.

And so these are used in many many different situations like for example in loudspeakers or in quantum resonators where we want to have the maximum response.

You see that as well even in NMR actually this is the way it works in the MRI nuclear magnetic resonance.

Every time there's a resonance we would like to maximize the response.

And so for this to maximize the response we call this quality factor and the quality factor Q is defined as the frequency of the resonance divided by twice the damping factor.

Of course the quality factor will be much larger if we have very little damping very little loss right.

So if we have no damping in fact Q is going to be very large but if we have very large damping we can even end up in a situation when there is no more resonance.

And in fact you can see that very easily if you plot the amplitude D on this which is the left hand side.

The resonance always shows as as a spike which is broader and broader as the damping increases.

So a greater as a larger damping means it's and in fact there is a place where beta is so large that there is no resonance as we discussed in the previous slide.

And on the right hand side you have the value of the D phasing delta which goes of course from the maximum at infinite quality.

In other words there's no damping to the flatter situation when there is a very very large damping at Q equals zero.

So as I mentioned oscillators can and resonators can be found in many different circumstances not just mechanical effect.

Each time we have Newton's law where the Hooke's law we find situations like this.

So in mechanical system like a loudspeaker the quality factor will be about 100 but in quantum devices could be up to 10 to power 14.

Electrical circuit is also well described so AC circuit also described as resonators.

And so all those quality factors of course are very important so that we have a sharp response and a very large response.

Now remember we are going to talk about one last thing which is the frequency for the kinetic energy resonance.

Remember you have your system it's actually an oscillator which is damped and driven by an external force and we do not expect the energy to be constant of course because first of all we have an outside force which keeps pumping energy but also damping which keeps taking energy in friction and dissipation.

So we can ask if we were to monitor the kinetic energy resonance, is there a frequency at which there is a maximum kinetic energy.

And so it's pretty easy to calculate because we know that the kinetic x dot is easy to calculate just like this.

And if you calculate the square we obtain an equation like this.

Now the problem is that we don't like equation like this because this is the kinetic energy as a function of time.

So we would like to get rid of the time so what it's typical to do is to calculate the average kinetic energy.

And you calculate the average kinetic energy so basically average kinetic energy will be the average over one between two maxima if you will and so you obtain this by calculating the average of sine square.

Why would we do sine square?

Well because this is the only function that depends on time in the kinetic energy.

And so we end up the over a period of oscillation we find the average kinetic energy will be given by this equation which is of course depends on the frequency.

Now let me reproduce that equation on the next slide.

And so what we see is that in fact if you were to calculate the at what frequency the response the expectation value of kinetic energy is the maximum you will see that it will happen at omega equal omega naught which is the natural frequency of the system for undamped oscillation.

So that's very interesting because this is different.

So it turns out the maximum frequency at which the system has an average kinetic the maximum average kinetic energy is not the same frequency at which the displacement is the largest.

So we can so this is interesting and on top of that we can also look at the other contribution to energy which is the potential energy.

But that one is easy to see where the maximum contribution the potential energy will be because it will be the largest will be when the displacement is the largest and of course this person is the largest at omega r.

So you see energy and potential energy reach the maximum at different time.

By the way this is not completely surprising that they do not happen at the same time since of course we do not have a conservative system at all.

The total energy is not conserved we have damping and then we also have a pumping of energy.

So this is pretty fascinating and it turns out the application of this of this framework is much much broader than than just mechanical systems.

In fact the number of equations that you find in physics that will that will look very much like the ones we looked at today is very very large and it involves quantum system it involves mechanical system electrical system all sort of system where you have where you are pumping energy in the so dissipation of energy.

So I hope it was clear I hope that you enjoyed this screencast and