Let me review what we learned about the vibration of a string beam extending to membrane and the plate. We argue that generally, for example, vibration of a beam y is the displacement of vibration of, sorry, string, and x denotes the spatial coordinate, t as the time. We actually attempted to see what is related to space and time, mathematically speaking by separation of variable. Then we come up with this part. This part can be expressed as some modal function, Phi one, Phi two, Phi three, that expresses the how generally this vibration can be expressed in space. For example, string case first mode, and Phi two is second mode, and the Phi three is third mode and so on so on. Then later on, we showed that the general solution of this will be the summing up of all this contribution. Then somehow, find out the weighting factor. How much each modal or the mode contribute to the essential response. The time function, of course we can do it in time domain but if we see this characteristics and frequency domain. Say, I excite this or I am looking at this, each frequency Omega. Then, the each Omega, this contribution would be different. So in the last lecture, I argued that, there will be some amplification factor due to the Omega. For example, distribution of force of natural frequency, second natural frequency and so on so on. Summing up. Then, there will be a contribution due to the modal function, or you might put the amplification factor over here, and depending on Phi i. In other words, mode shape, this contribution will be different. So Phi i can be for a string case. If this is L, this string case, mathematical expression will be sine, Pi, divided by L, x. Because when x equal L, that is a Pi. So that is correct. The second one it will be sine two Pi x, three Pi x, things like that. So generally I can say, Phi n, for a string case that has many many many and that will be sine, n Pi, L, over x. That certainly satisfy the boundary condition. Also I just have briefly mentioned that, what if the physical object we're handling does not really correctly, or precisely satisfy these kind of boundary condition. For example, the one end of string is connected by some hard, not rigid, but has some mass spring damper system. But I argue that, in the area far away from the boundary. For example, one Lambda apart from the boundary, the vibration characteristics of this area will be quite similar with or approximately similar with what we obtained using this homogeneous boundary condition based Eigenfunction. Okay for bending case, as I said before only difference is the mode shape at the boundary is different, because bending has to satisfy the boundary condition over here asking the derivative. Derivative has to be zero to satisfy the momentum boundary condition. So in this case, instead of sin the function would be sinh hyperbolic. Hyperbolic sin, some depending on the boundary condition that can go to the hyperbolic cosine. Expanding this to the membrane and plate can be conceptually easily implemented. Okay, this is very briefly explained. What I meant by the what I observed, the vibration of stream beam so on and so on, then mathematical domain. Okay, try to impose as a physical meaning. But not yet very much enough to see the mathematical expression in terms of physical meaning. Let's explore the results that we obtained mathematically, to get more physical meaning. For example, suppose I have a string over here. This point, this is a very interesting point that DOT the boundary. Also consider this point at the center. This geometrical difference is essentially say that if I excite at the center then, for example, this kind of odd mode, cannot be excited. However, even modes, for instance, look like that as a maximum amplitude over here while the excited of course this kind of even mode would be well excited. So depending on the position, some of the modes can be excited and some of the modes cannot be excited. Okay, suppose that we have to put some instrument. I mean, I have to mount instrument on an uncertain structure for simplicity and a string. If I install my instrument over here and if I do know that the excitation frequency only excites odd mode then this instrument has the least vibration. Right? So that is somehow related with practical application or implementation. So all I'm saying is looking at or investigating what we observed mathematically. What physical insight that bring us to get some practical application. Many be coal has exhausted system? Yeah, you have exhaust system coming from the engine. So more flow and there is exhaust. Then you have to mount or hand this exhaust system to the body, car body how where you hand your exhaust system to the car body. Then what you require is the transmissibility of the vibration due to the vibration of this muffler system to the body has to be minimized, how to do it? There are many ways. But starting with this kind of concept, if you know the position where you get many more modal contribution then that is most likely possible position to hang up your muffler system. Another extreme case over here where you can sense every possible boat, because over here every modes contribute. So for example, if you want to measure the every vibrational mode then this is the position that is likely possible position to put your measurement instrument. So when you measure vibration and you pick up the accelerometer, this would be the best position to put the accelerometer to measure the vibration frequency and so on and so on. What I am saying there are many many practical application they come up with your precise and physical observation which is based on mathematical expression.