Okay, let's talk about two degree of freedom system. In short, many people call two DOF. Okay, why we need to study two degree of freedom system? Suppose we are riding a car, look like that. So feel. And this is passing this kind of road. Then we can expect that there will be a lot of motion in this direction or this direction. Some of the rotation, things like that. Normally we can have six different kinds of motion. But suppose we have only interest in the motion of this direction, then we can regard this system that has some rolling element that is connected by the some spring which I call k_1. That represent the stiffness of tyre in this direction. Of course tyre has a some mass. So I have a mass over here. Everybody knows that the tyre is connected to the car with a suspension system that we can say the suspension system's stiffness is, can be represented by k_2. Then we have certain another mass which will be sort of effective mass of car in moving that direction. So, I have another mass over here, and the passenger riding over this mass, so we have interests about the motion of this mass. So our objective would be to try to find the best way that minimize the motion of m_2. Of course we do have interests about the motion of m_2 and m_1. So we have to employ the coordinate x_1t and the coordinate x_2t. So, normally recall this kind of process is modeling process. That is of course very essential and important to get the right answer or meaningful answer. The next step is to find out how to achieve our objectives. In other words how to achieve to minimize vibration of m_2. To get this objective we'd need to import some good means to analyze the system. So, we need to analysis. What would be the best way to look at the physics related to whether this motion, of course that is Newton's second law. Okay, let's apply the Newton's second law upon this mass m_2 and m_1. How to do it. Okay, first we have to look at the, how the force is acting on m_1 for example. Then lets find out what kind of force is acting on m_1, provided that we have this two coordinate system. Okay, suppose x_1 one is moving upwards by x_1 and x_2 is moving upward x by amount x_2, then the force acting on m_1 to the elongation of this stiff, this spring k_1 would be, the magnitude would be k_1, x_1. The direction is, because it is moving up, the direction would be this way. How about the force acting on this side due to the motion of x_2 and x_1? Okay if x_1 is larger than x_2, okay, x_1 moving this amount, x_2 is moving amount, moving this amount, then the relative displacement will be x_1 minus x_2. In this case, the spring k_2 will be compressed by this amount, so that will be k_2. The direction of this force would be this way. Okay, that has to be balanced by according to Newton's second law, m_1, x_1 double that. Because the reason why my inertia back to m_1, x_1 double that is toward in this direction is because I used the coordinate x_1 like this. Okay, this will give me the equation; m_1, x_1, double that is equal to k_2, x_1 minus x_2. The design has to be over here is minus. Also I have minus k_1, x_1. Okay, this is forced equation. What about applying Newton's second law upon mass m_2? Okay, here is mass m_2. First thing I have to find out to apply Newton's second law is to find out all the forces acting on this mass two. That has to be balanced by m_2, x_2 double that. Okay, the force acting on m_2, obeying these two coordinate system. If x_2 is moving this amount and x_1 is moving this amount. If x_2 is larger than x_1, then relative displacement acting on this DFS will be x_2 minus x_1. Then x2 is moving this amount, x1 is moving this amount. Therefore, there is some elongation. Therefore, the force will, moving down that has to be k2. I will check whether this is right. Their relation or. Okay. One thing we omitted in this derivation is the displacement due to the rolling of this massless sphere and I have to employ another coordinate y. Then, I have to correct this equation. That has to apply over here k1 x1. Instead of k1 x1, I have to minus y over here. k1 x1 minus y. That should be correct. So, next step is to write down equation according to this two picturely expressed Newton's Second Law. We already did one equation over here. I want to write down the equation according to this picturely expressed in Newton's equation. Now, I can simply write down m2 x 2 double dot is equal to, because I'm using the coordinate to add in this direction as positive. So, m2 x2 double dot. Then, this one has to be minus k2 x2 minus x1. Okay. Then, the first equation can be written as m1 x1 double dot and then minus k2 has to be plus. So, I have k2 x1 minus x2. Then I have plus k1 x1 has to be equal to k1y, that has to be minus. We have to check whether this expression is correct or not by checking the limit cases. The second picturely expressed the Newton's second law will give me m2 x2 double dot plus k2 x2 minus x1 is equal to 0. Okay. This looks very complicated to have a very compact expression of this mathematical expression. We can employ matrix expression that express like this, where I have x1 double dot, x2 double dot and I have some matrix. I have x1 x2 and that has to be the equalized with some other vector form that will be expressing the force. Okay. What kind of term? What kind of expression I can get from here? This term multiply x1 double dot. Okay. I have to find over here that should be m1. Then what about over here? That has to be m2 because I have m2 x2 double dot over here. This term has to multiply with, no, no. What about over here? This term has to multiply x1. So, find out x1 contribution over here that should be k1 plus k2, k1 plus k2. This term has to be multiply x2, and the contribution of kx2 will be minus k2. Moving over here, the contribution of x1 over here will be minus k2, and the contribution x2 over here would be, no, x1 minus k. Okay. Contribution of kx2 would be k2. That has to be over here, one is minus k1y and the other one is 0. So, this is matrix expression that certainly governs the motion of this two degree of freedom system. Okay. What does it mean by two degree of freedom system? Simply meaning that the least number of coordinate that requires to completely describe this vibratory system is two, x1, x2. Okay. Analysis tool, in this case, we use Newton Second Law provide us mathematical expression over here and over there. Then, we invite matrix expression to have a compact expression that represents the dynamics of this car in the [inaudible] direction, [inaudible] motion. But when you followed my lecture but you feel very boring because you can not see the physics, so coming from this mathematical matrix expression. Therefore, we need to find some good way to examine or to explore the meaning of this equation. Of course, this term is still be 0 and that term also has to be 0. To understand this two degree of freedom system, of course, we can solve this problem step by step but I'd rather suggest to you to go into a rather simple case. So, let's start with a simplest case. Simplest two degree of freedom system. Then using this simplest two degree of freedom system, let's try to understand the physics behind, of the mathematical expression in matrix form or two degree of freedom system and then go back to the problem we generated in this lecture. Little mistake I made in this derivation is there is a minus k1 x1 and minus and minus that has to be plus k1y. So, this one has to be plus and this one has to be plus.
