For a base excitation case, in other words, I have base and I have a spring, and I have a mass. But, because we understand the contribution of damping, so we include a damping over here and this is cx dot, and that is kx, and I have a mass over here. I am measuring the response of a mass as x, and I have another coordinate that expresses the base excitation, displacement excitation. Okay. In this case, the displacement would be relative so that could not be x, that would be y minus x. So, that has to be changed. So, I express this as ky minus x because when I move this y amount of this and move like that, then the displacement would be y minus x. That is the force applied to this mass. Similarly that has to be y dot minus x dot. So, that gives me the equation mx double dot has to be equal to ky minus x plus cy dot minus cx dot. Because free body diagram upon m has to be, there will be a force acting on this, and that is ky minus x, that is cy dot minus x dot is equal to mx double dot, because that is the acceleration experienced by the mass. So, Newton's expression of Newton's second law, graphical expression of Newton's second law. So, this is the equation we obtained for this excitation case. When this five rotary system has damping. So, what is the next step? You want to understand this and physical domain using representative measure, which is usually in this case a transfer function. So, let's try to understand this in physical domain, and also we want to understand this practical or application domain. The equation we obtained as the stiffness force is ky minus x and damping force, which is proportional to the velocity, in this case, relative velocity and relative displacement has to be balanced by inertia according to Newton's second law. Rearranging this, I will put all the response related time, in other words x on this left-hand side, that has to be mx double dot, plus cx dot plus kx equal to cy dot plus ky, which is somehow different. So, in this case, this is response related time, and this is excitation related time. Again, assuming that response x is complex amplitude exponential j Omega t, assume. Then, y also has to be complex amplitude and steady state time oscillation. In other words, this one is oscillating, real part is oscillating like that, and imaginary part oscillating like this. Now, this assumption lead us to rewrite this governing equation into the following form. This governing equation minus Omega square m, plus jc Omega, plus k, complex amplitude x. Let's omit this exponential j Omega t term. That has to be equal to jc Omega, plus ky. From this expression, it is obvious that this is excitation term. In the previous lecture, we expressed this complex amplitude f. Now, let's attempt to express this relation, using complex domain geometrical expression. Okay. Now, I have real part and imaginary part. Now, expressing this is rather straightforward because we had before. This is kx. Put a 90 degree rotation and then move and we have j Omega cx. Then, I have another 90 degree rotation, that is j multiplied by j. So, that is minus Omega square minus. So, I have minus Omega square mx. That has to be balanced by this complex expression but that has to be balanced by this. That has to be j Omega c plus k, not x, y. So, this is interesting. So, resulting complex expression over has two components. One is this and the one is that. So, I have ky that does not necessarily have to be same as this kyx. So, I put, this is ky. Then, a 90 degree rotation. This has to be j Omega cy, resulting complex expression is this. So, interesting observation. This is the phase difference between the base and response. Similar physical observation can be observed and Omega is increasing. This part will increase in proportional to the square, very rapidly increasing, slowly increasing. Also, you can observe the excitation parts is influenced by Omega linearly. So, this part is dominating the response all over. So, when this part is dominating, in other words, when mass is controlling the vibration, the phase this will go that. Therefore, this has to go that. Therefore, it's not possible to have a y. So, y in this direction. So, it has to be going this. Therefore, the phase difference between y and x tend to be going to 180 degree. If Omega is getting small and small, then this one go there. As you can see the y and x phase difference approach to 0, as you can see easily in this phasor diagram. Usually, in physics, we call this is a phasor diagram. Okay. That is interesting. Also, you can convert this picture into the famous magnitude and phase diagram. One of interesting things you can observe from here is, compared to when we have no damping, we have like this. In the phase we have very quick change of phase, from 0 to Pi, but in here, you will see the Pi over 2 phase difference at resonance. So, again I can demonstrate here, 0 phase difference at low frequency and a Pi over 2 phase difference at high frequency but I have Pi over two phase difference over here. In other words, at resonance base excitation difference and the response of a mass has to be Pi over 2. That's very interesting observation. In other words, when at resonance you are providing your force or base excitation energy to the system, through velocity, because there is a Pi over 2 phase difference.
