[MUSIC] So as I mentioned earlier, let's invite simplest, 2 degrees of freedom, System [COUGH]. Okay, Suppose I have a mass over here, And it is connected by the spring. And say this is k. And I have the same mass over here. And this is rolling, On the frictionless floor. Very ideal case. And also invite another spring over here that is ended by the wall. And another spring over here that is ended by wall over here, and say this is k and this is k. How this very simple two degree of freedom system will move? Okay, to understand the motion, we have to employ two coordinate system, denoted by x1 and x2. So our question is, how this one is moving? Then we need, as I talked to you earlier, we need a tool that can, That can lead us to understand the motion of this 2 degree of freedom vibratory system. What is the law that governs this motion? Of course Newton's second law. So invite Newton's second law again. On each mass, on each mass. So first the mass, I have a mass And I have to force this thing to do it, applying the Newton's second law is to find out all the forces acting on this mass. So if I move this mass, the amount of x1, and there is a spring, so this will push over here, and that will be kx1. And the force acting over there will be the relative displacement between x1 and x2. So if x1 moves this and x2 moves less than x1, then the force will acting over there, that will be k(x1- x2). Okay, that has to be balanced b, mx1 over that. Okay, what about applying Newton's second law upon this mass. I have a mass over here. Suppose we are moving this mass with x2 amount, then there will be a spring force due to the compression of this k. And that will be kx2. And if x2 was this amount and x1 is a little bit less than x2 amount, then this spring will be compressed or elongated. It moves x to this amount, then is elongated by x2 amount. But shortened it by x1 amount by x1, therefore the force acting on this mass due to the presence of this k, would be k(x2-x1). And that has to be balanced by m, Mx2 double dot. This is interesting pictorially expressed Newton's second law. Now let's try to express this equation of motion into matrix form. Where I use x1 double dot, x2 double dot, over here, and I have a K matrix that has x1 and x2. Of course, suppose that there is a force acting on this mass, would be F1, and the force acting on this mass, F2. Then, in matrix form, I can say here, the excitation force, F1, F2. Let's look at what kind of term has to be filled out over here. So this one multiply x1, has to come out over here, x1 double dot, that is m. And this one multiplied by x2 double dot has to be over here. And in this case, there is no contribution in terms of x2 double dot, therefore, that is 0. And that over here of the contribution of x1 has two terms k and k, and this direction is minus, so if I move this term over here that has to be 2k. And what about the contribution of x2? That is -k and x2. If I move this one, Over here, that has to be, Plus k. x2. Is it correct? Okay, let's check it later. From this express the Newton's second law has to somehow compile the matrix form over here. So mx2, therefore, this is 0 because there is no inertia force expressed in terms of x1 double dot, so m. And over here, I can get a contribution of x1, that would be minus kx1. So minus kx1, move over here, that is plus k. And I have kx2 and a kx2, that is, there's 2kx2, therefore, that is 2k. Okay, I found what kind of mistake I made. Okay, if I write over here that according to the coordinate system over here, mathematically, that is 2kx1, minus, because the direction of this vector in this direction, and plus, And- kx2. And according to this coordinate system, that has to be minus again, so that is plus. So I have plus 2k, too, and this one has to be minus. Similarly, this one has to be minus. Okay, this has very interesting matrix form, first, This two term is the same. And these two terms are also same. And these two terms are also same. This is symmetric. What it means by having symmetric mass term and the stiffness term? Okay, let's put that question in your mind, and then, Think about this system physically by doing some experiment, mentally experiment. I have two mass, same mass over here, that is vibrating to these two force. So remember, as we remember in the previous, single degree of freedom system, I have single degree of freedom system over here. I have a mass, frictionless floor, and I have a stiffness k. And using coordinate over here, x and the force is acting over here, F. What is the governing equation? Governing equation in this case you can now automatically write mx double dot + kx = F. Okay, you remember that, If you release this vibratory system by using, Initial displacement or initial impact or velocity, then this system will vibrate with a natural frequency. And the natural frequency, in this case, is omega squared is famous k over m. So applying same concept to this two degree of freedom system. Now, suppose I am giving initial displacement, like the one centimeter, and then let it go. Then this will vibrate. How will it vibrate? Think carefully, then you can find out, Ultimately, you can find out there's two possible way. One possible way is moving these two mass with the same phase, like this. Another possible way of moving this mass would be like this, opposite direction, opposite direction. That we will call eigen. Let's find out how this eigen can be obtained from this compact mathematical expression. Mathematical, I mean, in matrix, we express mathematical expression in matrix form and we invited the very simplest two degree of freedom system, why? Because we want to find out, to explore the meaning of a two degree of vibratory system. From this matrix, the equation expressed it in matrix form. Okay, this is the two degree of freedom system we are studying now. So as I talked to you before, the equation that represent the motion of this system would be in matrix form x1 double dot, x2 double dot, plus, and k matrix, x1, x2. And that has to be equal to excitation vector, F1, F2. Okay, contribution of this would be m, 0, 0, m. But we can find out the inertia term and multiply x1, can be found immediately from this picture. So inertia term acting over here is obviously m multiplied by x1 double dot. That is acceleration of this mass. There's no way to have a nicer term. Some mass multiplied by x2. Right? And what about this term over here? That term, if I say, in matrix term, k 1 1, meaning that the force acting on mass m1, that's m over here, due to the motion of x1. So if I move this mass amount x1, and the last stop of the motion of this mass, then the force acting on this mass would be one elongation by amount of x1 and one elongation by amount x1 over here. So therefore, that is 2k. What about over here? That is force acting on this mass. So as I said before, we have to envisage the compression of this stiffness by amount x1. But that will be the compressed force in this direction. That has to be k. The direction is minus. Similarly, we can obtain here -k and 2k. And this matrix, k 1 1, and k 1 2, and k 2 1, and k 2 2. Similar notation we can apply over here, m 1 1, m 1 2, m 2 1, m 2 2. Now, you can expand this notation to many degrees of freedom by virtue of system. You cannot analyze like a three degree or four degree of freedom system. On your paper, you have to go to the computer program. But at least you have to understand physically what does it mean by each element. That's why we are using simplest, two degree of freedom over here. As we did before for signal degree of freedom system, suppose, I'm exciting the system harmonically. Then we can say x1, x2, Can be expressed by, Complex amplitude x1, complex amplitude x2, the same excitation frequency, omega. Because we are handling linear bivariate system, we have to assume F1, F2. Similarly, complex amplitude F1 and complex amplitude F2, explanation j wt, very straightforward. And putting this assumed response and assumed excitation upon this mathematical expression will give me the following term. Differentiate twice with respect to time, will give me minus omega squared. Then there is a mass, so I have omega squared m 0, sorry. And I have 2k over here. So I put + 2k. This term, there is a 0 of minus omega k multiplied by 0 is 0. And I have a -k over here, -k. Similarly, I have a -k over here. I have minus omega squared m plus 2k. Then I have, let's use for simplicity, beta form over here, that represent x1 and x2. That has to be balanced by what else? F of x. Okay? So I have a 2 by 2 matrix, column vector and column vector. What is the solution? From linear algebra understanding, the solution would be x has to be equal to F divided by the determinant of this matrix. What is the determinant of this matrix? Okay, the determinant of this matrix is simply, Determined by notation of this as a determinant would be equal to multiply this term and multiply that term would be minus omega squared m + 2k, Squared, and minus k squared. Okay, what happen if this terminal approach to 0? Then the solution will blow up. What? What does it mean by physically? Now this resonance of this system, let's calculate the resonance of this system, putting this term as equal to zero. Then we can obtain c- omega squared m + 2k squared = k squared. Right, and that has to be, +-k. For +k case, I have, I'm using space over here, omega square is equal to and this +k, if I move this one over here, then that has to be -k. I have a minus over here, so that [COUGH] would be [COUGH]. Sorry, k over m. Another frequency I can obtain when this is minus k. So if I move this one over here, over there, that is 3k. So another omega is 3k over m. Wow, so let's denote this is first natural frequency and that is second natural frequency. So first natural frequency is k over m, how to make first natural frequency k over m? Suppose I am moving this mass this way. Okay, then this stiffness cannot influence the motion, right? Therefore, there is a 2n and the stiffness will be k + k. So 2k divided by 2m is k over m. So this natural frequency corresponding to this motion. So first natural frequency. And this one, suppose I'm moving this way, then the stiffness acting up on this system will be how much? There is a k, and there is k, there's another k, that is 3k, so that is a second natural frequency.