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

事實上，既然我們已經瞭解瞭如何計算特定解的振幅，那麼要理解共振現象就非常簡單了。

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.

我們可以問自己共振頻率是多少，換句話說，最大振幅的頻率是多少，然後我們知道，由於分子是一個常數，我們只需將分母最小化。

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.

這是一個直接的導數，所以你要儘量使導數等於零，我們發現 d 的最大值將通過頻率 omega r 等於 omega 零平方減去兩個 beta 平方得到。

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.

是以，如果以幻燈片底部等式給出的頻率驅動系統，就會產生最大運動、最大解、最大振幅和運動。

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.

是以，我們稱其為共振頻率歐米茄 r 歐米茄零平方減去兩個貝塔平方，然後我們可以看到，這個共振頻率可以通過改變阻尼來調節，是以阻尼很大。

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.

我的意思是，如果阻尼相當大，不是太大，但也可以說相當大，使歐米茄零平方減去兩個貝塔平方是一個正數，仍然是一個正數，那麼共振頻率就會下降。

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.

不過，如果兩個貝塔平方大於歐米茄零平方，就不會產生共振，因為在這種情況下，共振頻率實際上是一個複數，實際上是一個虛數，然後我們就會看到單調的下降。

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

是以，這一點很重要。

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.

好了，總結一下我們目前所做的工作，我們研究了一些頻率，當我們研究三個振盪時，沒有阻尼，沒有力，我們發現固有頻率Ω零平方等於k大於m。

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

當我們查看阻尼時，會發現我們有一個歐米茄 1 平方等於歐米茄 0 平方一個貝塔平方。

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.

這個頻率 omega 1 可以是一個振盪，比如在阻尼不足的情況下，或者當我們處於過度阻尼時，它就不再是一個振盪，但在阻尼不足的情況下，實際上就是這個幻燈片上所寫的包絡函數的解決方案，Ω 1 是一個頻率，而不是響應的週期性，因為振幅會下降，所以你不會重複相同的解決方案，但基本上是最大值之間的頻率。

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

最後，我們發現了另一個重要的驅動振盪頻率，即諧振頻率。

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.

是以，當你觀察這三個頻率時就會發現，它們是我們驅動振盪的典型頻率，Ω 0 總是大於Ω 1，而Ω 1 本身總是大於Ω 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.

事實證明，在共振頻率下驅動系統對設備和從系統中獲得最佳響應和最大響應非常重要。

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.

是以，它們被用於許多不同的場合，例如，在揚聲器或量子諧振器中，我們希望獲得最大的響應。

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

即使在核磁共振中也能看到這一點，實際上這就是核磁共振的工作原理。

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

每次產生共鳴時，我們都希望能得到最大程度的響應。

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.

為了使響應最大化，我們稱之為品質因數，品質因數 Q 的定義是共振頻率除以阻尼係數的兩倍。

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

當然，如果阻尼很小，損耗也很小，品質因數就會大得多。

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.

是以，如果我們沒有阻尼，實際上 Q 值會非常大，但如果我們有非常大的阻尼，我們甚至會出現不再有共振的情況。

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

事實上，如果在左側繪製振幅 D，就可以很容易地看到這一點。

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

共振總是表現為一個尖峰，隨著阻尼的增加，尖峰的範圍也越來越大。

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.

是以，阻尼越大，就意味著貝塔越大，事實上，正如我們在上一張幻燈片中討論的那樣，有一個地方的貝塔非常大，以至於沒有共振。

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.

右側是 D 相位三角的值，當然，它是從無限品質的最大值開始的。

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

換句話說，當 Q 等於零時，有一個非常非常大的阻尼，在平坦的情況下沒有阻尼。

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

是以，正如我所提到的，振盪器和諧振器可以出現在許多不同的環境中，而不僅僅是機械效應。

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

每次我們在牛頓定律和胡克定律之間發現這樣的情況。

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.

是以，在揚聲器等機械系統中，品質因數約為 100，但在量子設備中，品質因數可高達 10 到功率 14。

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

電路也很好描述，是以交流電路也被描述為諧振器。

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

是以，所有這些品質因素當然都非常重要，這樣我們才能有敏銳的反應和巨大的響應。

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

現在，請記住，我們要討論的最後一件事就是動能共振的頻率。

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.

請記住，你的系統實際上是一個振盪器，它受到阻尼並由外力驅動，我們當然不會期望能量是恆定的，因為首先我們有一個外力，它不斷抽取能量，但也有阻尼，它在摩擦和耗散中不斷吸收能量。

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.

是以，我們可以問，如果要監測動能共振，是否存在一個動能最大的頻率。

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

是以計算起來非常容易，因為我們知道動能 x 點的計算就像這樣簡單。

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

如果計算平方，就會得到這樣一個等式。

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

現在的問題是，我們不喜歡這樣的方程，因為這是動能與時間的函數關係。

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

是以，我們希望擺脫時間的影響，典型的做法是計算平均動能。

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.

然後計算平均動能，基本上，平均動能就是兩個最大值之間的平均值。

Why would we do sine square?

為什麼要做正弦平方？

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

因為這是動能中唯一與時間有關的函數。

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.

是以，在振盪週期內，我們發現平均動能將由這個等式給出，當然這取決於頻率。

Now let me reproduce that equation on the next slide.

現在讓我在下一張幻燈片上再現這個等式。

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.

是以，我們可以看到，事實上，如果計算動能期望值最大的響應頻率，就會發現它將發生在歐米茄等於歐米茄無的頻率，也就是系統無阻尼振盪的固有頻率。

So that's very interesting because this is different.

這很有趣，因為這與眾不同。

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.

原來，系統平均動能最大的頻率與位移最大的頻率並不相同。

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.

是以，我們可以看到這很有趣，除此之外，我們還可以看到能量的另一個貢獻，那就是勢能。

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.

但很容易看出，當位移最大時，勢能的貢獻最大，因為位移最大時，勢能的貢獻也最大，當然，這個人在歐米茄 R 處的位移最大。

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

是以，你會看到能量和勢能在不同的時間達到最大值。

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.

順便說一句，這兩種情況不同時發生並不完全令人驚訝，因為我們當然根本沒有保守的制度。

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

總能量是不守恆的，我們有阻尼，然後我們也有能量泵。

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.

是以，這非常吸引人，而且事實證明，這個框架的應用範圍比機械系統要廣泛得多。

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.

事實上，物理學中與我們今天所研究的方程式非常相似的方程式數量非常多，其中涉及量子系統、機械系統、電氣系統等所有系統，在這些系統中，你都可以抽取能量，從而耗散能量。

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

所以，我希望你能明白，我希望你能喜歡這個截屏，並且