Let me summarize what we have studied. First, we derived the governing equation that actually governs the motion vibration of a string that is expressed as this QY. The X square has to be equal to one over C square, this QY the t squared. That is actually homogeneous governing equation which means that there is no steady state excitation. In other words, the string that has this rigid boundary condition either plucked, in other words, I push up this and then let it go then it vibrate, or I can struck the string then it will vibrate. If I demonstrate this initial excitation condition, but my assistant it may look like this. Okay, here is the string. The first case, I gave initial displacement then it will vibrate. If I give initial velocity, then it'll vibrate like this. Okay. All other thing I would like to you observe from this simple demonstration is that, does it depend on the location of excitation? This may look like that. The response may look like that. If I pluck this position, may look like that. Obviously, you can see there is some difference. Okay. If I excite this string over here for the initial velocity condition look like that. But if I excite this string at the position over here, it may look like obviously the response is different. Okay. What if I excite this string at this position with a certain frequency of omega like this. Then, it may oxalate like that. If I oxalate the string over here with a certain velocity, you may see the vibration at this region and over this region would be different. So that means depending on the excitation position, the response would be different. How this simple demonstration could be related with the theoretical approach. I mean, how this simple demonstration is mathematically related with what we want to study. If there is some excitation, then this governing equation will look like this or over the skill y, the X square minus one over C square. This QY, the t square, what I may have some excitation which is a function of position as well as time. We argue that the solution of this inhomogeneous governing equation can be composed by superposition of some function phi i, that is only depending on space and its contribution ai. If I excite this with a certain frequency omega, then we could say it would be exponential j omega t, or we can say in frequency domain. If I convert this in frequency domain, then the solution of string with respect to certain frequency omega, or I may write this is y x as certain frequency omega. Then, I can argue that the solution is composed by superposition of phi i x. Actually, this is mode shape, and the contribution due to the excitation of omega. Okay. It must contain the amplification due to natural frequency. So I may write over here amplification including or representing natural frequency omega i. Okay. That is conceptually written solution. If you go back to the homogeneous equation, this approach essentially argues that the solution of inhomogeneous equation of in physical term, the solution due to excitation can be expressed by the solution of a homogeneous equation. In other words, this string whether this boundary condition may vibrate with this, I may denote this is first mode, and second mode, and then there will be third mode that look like that [inaudible]. So, I can argue that any response due to excitation could be expressed by this mode shape. They make sense as we demand demonstrated before. Depending on the position of excitation, special distribution of excitation, the response would be different. So generally, we can say that the solution of the string with respect to space and frequency would be composed via some amplification factor, and multiply by the contribution of each mode. That makes sense. Then phi i. In this case, the first mode, and the second mode, and the third mode, and so on, that can be obtained either by analytically, or later on, we will show that it can be obtained by experimentally. Expanding this idea to more general case for example, vibration of a beam for example. The mode shape, model contribution, of the beam case would be this Phi i shape mode shape one, two, three has to be what is governed by the vibration of beam. So what is beam? What the beam actually say in terms of mathematical expression. Mode shape of a beam would be very similar with the mode shape of a string but different in this region boundary, because the beam can transmit normal. Therefore, in this region, beam has to be to satisfy the rigid boundary condition. The beam has to get zero slope over here. Therefore, the beams mode shape would look like that if I exaggerate it a bit. Other than this region, the mode the shape of a beam will be very similar way the mode shape of a string. Second mode will look like this and satisfying the boundary condition over there. So next, the question is, what if the boundary condition is different? Okay, as I mentioned before, what if we have different boundary condition? For example, for string case, if I have a string but not rigid boundary condition. For example, I have over here, mass, spring, and a dashpot. Over here, I have mass, spring, and dashpot. A general approach would be quite similar with what we have tried. First, given boundary condition which can be expressed by the mass M1 over here, K1 over here, C1 over there, mass M2 over here, K2 over here, C2 over here. We can get the mode shape. For example, that would be not same as the mode shape I obtain but there is some component over here. The mode shape will be somehow look like that or may look like that. Depending on this boundary condition, mode shape would be slightly different with what we described for the rigid boundary condition. But one can argue that if we have interest about the vibration in this region considerably far away from the boundary, then it is very likely to use the mode shape that has this boundary condition to this case, would not generate a lot of errors. So in engineering sense, if you have an interest of obtaining the response in this region away from the boundary then we can use the mode shape that is obtained for more simple boundary condition case. Of course, we have to use the exact boundary condition that represents this boundary condition if you want to get the response over here. But practically, that is not very likely to happen in general. A similar approach, similar argue can be applied to the vibration of a beam. What if we want to study about the vibration of membrane, or plate, or shell, or more general case? Approach will be quite similar with what we have tried for the case of a string. For membrane, it may look like that or if you have interest about vibration of a membrane that has a circular shape that will be different look. For the membrane case, first the mode everything go up and second mode will be, if I write over here this is first mode everything move up and down. Second mode would be, this part move up, this part move down. Third mode, this part move up, this part move down, and the fourth mode, this part move up, down, up, down, and so on. So if I denote this is first, second, third, fourth. Then, as we said before the vibration of membrane will be the superposition of this first, the second and third mode so on and so on and the contribution of each mode depending on where you exert and which frequency you exert. Also, there is a magnification factor. Those magnification factor and in other words, exact expression of membrane and the string and the beam, you can find in the supplementary material of this course. Similarly, for the plate case, plate is a two-dimensional version of beam. Similar mode shape can be obtained, as I said before, the only difference is the shape along the boundary. For the shell case, we can apply the similar approach one, two, three, four mode shape but for general case, we have to obtain the mode shape experimentally that I will explain later on.