Let's consider very simple vibratory system that has a mass, and spring constant, k, which is the linear spring constant, and then damping, or linear damping, that has the damping coefficient of c. And then suppose we measure the motion of this mass by using this coordinate. Of course, that is oscillating with respect to time. And then suppose we apply the force, F(t), the general force. Then well-known governing equation In other words, that governs the motion of this kind of vibratory system can be written as Simply means that due to this excitation the single degree of a vibratory system will respond. Responds, and it can be measured by the displacement F(t). Of course, the F(t) can be expressed as the sum of sines and cosine. Sorry. Sum of sine and cosine. Obviously, the sine and cosine has the face difference of 90 degree over pi over two. And note also that the relation between the between F(t) and x(t) is linear. In other words, if I excite a system twice of the force, then the response will be the twice. That's the meaning of a linear system. Therefore if we understand how the system is going to respond due to sine omega1t, or generally sine omega t, then using the principle of superposition, we can generalize how the system will respond due to sine omega1, or omega2, omega nt. Therefore we usually study the following case, that is governed by the following equation. mx double dot + cx dot + kx =, for example, C sine omega t. Then the solution, x(t), can be sine omega t. Because we are exciting the system with the frequency of omega, the response has to be the omega also because we are handling, again, the linear system. Also there will be some component of cosine omega t. Because there is x double dot and x dot, as you can see here, even if you assign a system to the sine omega t, the response can be cosine or sine, or in between. So we can assume that the response x(t) would be some contribution of sine omega t and some contribution of cosine omega t. And say this is A, and contribution of cosine omega t is B. Then, of course, that can be written as square root A squared + B square and the sine omega t + phi. And the phi is the phase angle. Of course, tangent phi would be the ratio between A and B. So even if we excited a system with a sine omega t, the response will be sine omega t + phi. Phi is the phase difference between force and response. In other words, these two terms actually expresses that when we excite a system with a sine omega t, the response will be responded by same omega, but has certain phase difference. Okay, that's the one way to mathematically solve this equation. As you can see here in this manipulation, we have to handle sine and cosine at the same time. What would be the more convenient way to handle this problem? Okay. Observing that C sine omega t can be represent as, imagine C sine omega t can be represented by some magnitude of F exponential j omega t. Imaginary part. Because F exponential, e to the j omega t, is simply the magnitude F multiplied by cosine omega t plus j, and the magnitude F multiplied by sine of omega t. So if I take imaginary part of this, that is the same as F ej, sorry, F cosine. No, sine omega t and magnitude, mj. So if I take the imaginary part, that would be this stuff. In other words, C and F, magnitude is the same. Okay then, suppose I am taking the whole equation mx double dot + cx dot + kx = Fej omega t. I'm taking imaginary part, and this will be true, [COUGH] and [COUGH] this will be somehow related with this equation. Okay, that means if I get a solution that governs by mx double dot + cx dot + kx = Fe + j omega t. Get a solution x(t), then take imaginary part. Then we will get the same solution as we obtained by following this procedure. Let's say, assume that x(t) is composed by the complex amplitude X, this is complex force F, of course, and ej omega t. Okay, and then solve this equation by using this assumed solution. Then what we will get is the following Okay, taking a double derivative with respect to time will give me- omega squared X ej omega t, m, I have to put m over here. And then take the derivative with respect to time, will give me + j omega C and then X ej omega t. And then kx ej omega t, that has to be = F ej omega t. Therefore we can write immediately the following equation. (- omega squared m + j omega c + k)X = F because exponential j omega t is common vector. What does it mean by this equation? This very interesting. First note that X, capital X, and capital F are complex. So, let's invite complex plane. So this is imaginary axis, and this is real axis. Then, because X is complex, that has magnitude and phase. So let's draw this as X. And kx is just follow the same direction but with a different magnification factor, k. So this would be kx. Let's write down over here kx. And what is j omega c X? Okay, the magnitude would be omega c X, but there is a j vector, and a note that j is simply ej pi over two. In other words, it has to rotate 90 degrees from this direction. Okay? So let's draw this j omega c component over here. Maybe I will use a different color. So that is j omega c X. That was essentially over here. And what about- omega squared m and X? And minus one is equal to j multiplied by j, that means this one essentially indicates that this complex magnitude has to rotate again 90 degrees over here and that will be- omega squared m. So I put this one over here, translate. Okay, and I said this is omega square m X, and that has to be equal to F. So, that means, geometrically, that has to be equal to F. Okay? So, that means the phase difference between this excitation force magnitude complex F and then X, phase difference is this. This is a phi. Okay? And, again, by looking at this so-called phasor diagram, we can anticipate, or we can feel, rather, dramatic, freezing cold meaning from this figure. For example, if omega is increasing, then the magnitude of this one will increase linearly. But this one will increase with respect to omega squared. Therefore, if this increased by 10, this will increase by 100. Therefore I can see that the increase like that induced increase like that with the same F. F has to be same. Therefore the phase m will approach to as omega increased to 180 degrees. And what if this one reduced? And then omega squared is reduced by omega. Therefore, this will be drastically reduced. So therefore, the face angle will approach to zero. In other words, that means in low frequency, the phase difference between force and displacement will be approaching to zero, but in very high frequency, the phase difference between force and displacement would be 180 degrees. This is simple. When I excite the spring with a very low frequency, suppose I am pushing some spring with a very low frequency like this. Then there will be no phase difference between the force and the displacement. But when I excite the similar degree of phi through the system, for example, car. When I drive a car in a very rough road, with very high speed, then the excitation frequency will increase very much. Then what you will feel is this kind of response. That means the phase difference between force and displacement is 180 degrees. So using complex function for analyzing this kind of vibratory system rather provide very dramatic and tangible, physical understanding between the force and displacement, in terms of phase. Of course, you using this geometry, you can also get the transfer function rather easily because the square of this magnitude has to be equal to square of this magnitude, and the square of this magnitude. They will give you the so-called, the transfer function between force and displacement. As a conclusion you can see that using this complex function for analyzing vibratory system is convenient. Because we don't have to handle between the sine and cosine, that has 90 degree phase difference. As well as because we can use the phasor, we can see more tangible physics that is associated with the phase difference between force and displacement.
