[MUSIC] F(t) can be, Composed by sin and cosine. Say cosine, 2 pi ft. Its magnitude is A, and there will be some sin component too. All right. You can use this model, And the plug into governing equation which is, mx double dot + kx = F(t). You could do that. But what if I write F(t) =, because it does have cosine and sine components, I say inviting exponential of, Complex function, because it does possess this, cosine and sine. Because as you know this is cosine 2 pi, cosine 2 pi f(t), + sine 2 pi. Ft, but has phase difference j. Okay? This is rather compact form than this. And noting that this is a linear system. So if I excite the system with a cosine, this will give me, in this case, cosine response. And if I excited this system with a sine, this will give me sine response due to the sine. And this is linear system, therefore, we can express this general excitation F of t, using complex notation, all right? And I put over here, magnitude of F, in complex. So this is rather compactful. So our details related to this formulation, we will provide some supplementary material, which is available in this course. Okay, if I use this excitation, then I can write, Mathematical expression of the simplest vibratory system is mx double dot + kx. Is equal to F complex magnitude, exponential j 2 pi ft, and because it is complex, I would say the response is also complex. Right? So now I have this mathematical expression in complex notation. That made us easier, To understand what's going to happen in singular degree of vibratory system. I mean, you may not agree that this is easier, because if you do not know of the power and the convenience of using complex notation. But as I said, if you look at it carefully, this supplementary material that explains in some detail about this notation, then you will observe that it is very convenient. Now, the response of x can be obtained, Mathematically. By saying that x of t, Can be also expressed by exponential j 2 pi f of t, and to say this magnitude is complex. Maybe I use notation hat over here. To be consistent, I use hat over here, too. Okay? Then plug these things over here, will give me minus, Minus, 2 pi, (f) square m + k, x hat = f. Right? That's very interesting. So if I see the response to complex excitation, that mathematically expressed like x hat over F hat. Is equal to, 1 over (2 pi f) squared m + k, and I have to put minus over here. So the response with respect to excitation, is expressed by system parameter m and k, and excitation frequency f. That is interesting. That is interesting. And inviting what we obtained from the previous lecture, The natural frequency is k over m. Okay, that is 1 over 2 pi f, 2 pi, no, no. 1 over 2 pi, 2 pi, sorry. Natural frequency is 2 pi f0 squared, okay? So plug this one over here. Has to divide this by k and that is 1. And I'm dividing this by k therefore I have to divide this by k too. So k And I have -2 pi f square m over k + 1. That's very interesting expression. Then for complex rest pause, with respect to complex excitation. Excitation look like this. And using this as the inversion of this, I come up with 1 over k, Okay, -(2 pi f) squared. And that is (2 pi f 0) squared. And I have a +1 over here, and that gives me 1 over k. F over f0 squared- +1. Wow, this is nice, because the frequency component, excitation frequency is seen with respect to natural frequency. So if f is greater than f0, this value is bigger than 1. So bigger than 1, +- and +1 would be negative. Okay, so complex magnitude, with respect to frequency, would be negative. The f natural, f excitation forces are smaller than natural frequency. This is smaller than 1, therefore this one is a positive. That's very interesting. Another interesting point, of course. If excitation frequency f is the same as natural frequency, then this will be -1. So -1 + 1 is 0, therefore this will go to infinity? That is very interesting observation, so let's look at what it means physical in this. So we obtained the simplest single degree of vibration system in terms of complex amplitude, with respect to complex force. Actually this measures how much it will vibrate for unit force. And we found that these expressed in terms of vibration systems parameters that has stiffness of a spring. And, interestingly, it has the relation between natural frequency and excitation frequency, squared. I have to, okay, squared, and- 1. So, in this expression, it is very interesting that how much it accelerates with respect to unit excitation force is very much related with 1 over k, and this interesting frequency ratio. So response depends very much on frequency ratio. In other words, the excitation frequency has to be always measured with respect to natural frequency. Where the natural frequency f0 = 1/2 pi square root k/m, m is the mass of, single degree of freedom by system. And look at very carefully again what it means, because we obtained the result by analyzing single degree of freedom system. Using Newton's Second Law, we obtained mathematical expression. Now, what do you have to do is, we have to look at this expression, and translate into how we can use in application. So transforming this expression to the application domain, you have to look at very carefully of this expression. And f 0 certainly controls the whole expression, okay? So one thing we can immediately see, that this expression, this has to be plus, sorry. So this expression, I will write down this one the other way around. 1- f over f 0 squared. So first case, if f over f0, in other words, if excitation frequency is smaller than f0, then this denominator has to be positive. So, the whole magnitude Complex magnitude, Has to be positive. And second case, if frequency ratio is greater than 1, then this 1, Will be negative, sorry, this has to be [LAUGH] greater than 0, okay? So if I draw this, x hat over f hat, in graph to realize what it really means. Then it look like. Okay, I am seeing, Complex magnitudes with respect to the complex force, in terms of frequency ratio. So obviously, when this is 1, The, so this is one of very important singular points. In this case, this one has to be positive, and in this case, this answer has to be negative. When this one is 0, then this is 1 over k. Essentially expressing that f excitation frequency is general, in other words, a static case. The response has to be inversely proportional to. That makes sense. And over here, it goes to infinity, therefore, the graph has to be look like that. And over here if f over f0 over here is very, very big, very, very, big, compared with 1, then this denominator tends to be going to minus infinity. Therefore, it goes to some here, over here, over here, that has to behave like that. It's interesting. So this is what we obtained. Okay, note that, The ratio between two complex value can be always expressed as magnitude, And the phase term. In this case, plus or minus. Okay, then let's try to see this magnitude from this expression. It's very straightforward, okay, I'm seeing its magnitude. Absolute value. In terms of frequency ratio. Of course, there is a singular point when natural frequency and excitation frequency is the same, so absolute value of a plus is plus. Absolute value of minus, It has to be upside down, so it look like that. Okay, that's looks okay. And very reasonable. And we realize that we don't need this domain. Okay, we need to see how this phase term will behave. So generally, we express this phase term mathematically, exponential J theta. And, obviously, in this case, because that is cosine theta plus J sine theta, To express this to plus 1, would simply mean theta has to be, theta has to be what? Pi over 2, 90 degree? Or 180, or 0, okay? If theta is 0, cosine 0 has to be 1, so in this case, 0. And what about when theta is pi or 180 degree? Then cosine theta would go to -1. So that meets our requirement expressing that in this case, it has to go, -1, and over here in this region, theta has to go 0. So the behavior of our phase has to be, starting from 0, and then it has to be -1 from here, and at this singular point, actually, we do not know at the present time, but later on, we will see what's going to happen over here when we have a damping. So physically, it means, over this region, when the excitation frequency is smaller than the natural frequency, the vibration of the system is in phase with the excitation force. So when I excite the system, when I push the system over this way, the amplitude follow. So like that. Over here, when I push the system this way, the mass will move like that, okay? So the phase difference, there is a phase difference between excitation force and the response.
