[MUSIC] Okay. Today, we'd like to review what we learned in the past few weeks. We start with the single degree of freedom, 1 degree of freedom vibratory system to understand general vibration. Okay, one example that can represent the single degree of vibration system can be depicted like that has spring and mass that will oscillate due to the inertia force and the interaction or mutual toss back and forth of kinetic energy and the energy that can be stored in the spring. And the motion has to be measured by coordinate. In this case, we use coordinate that measure the motion from this equilibrium position. And then we denote that x(t), where t is time, which means that the motion of x is varying with respect to time. And stiffness, that is force required to displace the spring by unit length that is k. Okay, we attempt to explain the vibration of this with respect to two measures, two expressions. One is using transfer function, That has magnitude and phase that look like this, if you have a damper over here. And a damper we usually denote as c, That represent viscous damping, in other words, the damping force proportional to the velocity. And the governing equation, we can obtain the using Newton's second law up on this mass. We obtain that governing equation look like mx double dot cx dot + kx = F(t). F(t) is the excitation force applied to this single degree of vibratory system. Then we can see how the physics is going on on this system by looking at transfer function that has big amplitude over here that corresponding to the natural frequency, which if I say omega 0. In this region, we call mass controlled region. This region, we call a stiffness, Controlled region. And over here, this region, This is controlled by damping, so we call it damping controlled, Region. Today, we will talk a little bit more about damping and what is critical damping also. Another measure we'll look at, this behavior of this is using phasor diagram. Simply expressing the force and the response and the mass dominated response and the stiffness dominated response and damping dominated response in complex domain. All right, realizing that x(t) for harmonic excitation, we can express that as complex amplitude and exponential j omega t. And F(t) can also be expressed by complex excitation force, exponential j omega t. Then this mx double dot cx dot + kx can be expressed like kx. This axis complex amplitude. And the kx is the force exerted by the stiffness upon the system. And then this will be j omega cx. The reason why we have a 90 degree phase difference is because the kx and the j omega cx has phase difference of j. And a phase difference of j corresponds to 90 degree. And then we can see that there will be- omega squared mx component, because, again, this one and that one has phase of j difference because j multiply j is minus 1. And then that has to be balanced by, Complex force over here. That essentially expresses this mathematical expression in terms of complex domain. So I demonstrate that if omega is becoming small and small, the phase between F and x will approach to zero. Because if omega is small, it's going to be like that. And because this omega is proportional to omega squared, that will be like that. And therefore, phase approach to zero. And that corresponds to this region. In other words, if If omega getting larger and larger than this omega increase linearly like that. And then but this omega will increase, omega's in terms of omega scale like that. Therefore, the phase difference between the force and displacement, will approach to 180 degree. And that is already observed from this magnitude and phase diagram. Phase diagram look like that, this is 180 degree over here. And that is 0 degree. This tells a lot, and that is corresponding to over here. Let's certainly summarize this, what we learned the vibration of a single degree of freedom system. It also possesses a lot of implications that can be applied to have a practical obligation. One immediate practical application you can think about is to reduce the vibration, we have to use the concept that we learned from this transfer diagram. Transfer function that is in a mass controlled reason, the vibration is very small, okay? Here is the my vibration singular degree of freedom system and mass controlled reason, I am accelerating this so it will very large frequency compare with natural frequency. I have a very small vibration. But in this stiffness controlled reason, I'm oscillating this with that low frequency then I can have the amplitude that is proportional, that is inversely proportional to the stiffness. But when damp, when I oscillated this in natural frequency, they'll oscillate it like that and the amplitude is totally determined by the damping coefficient of c. One other thing I didn't really go into the detail but that is and has to be rather, rather important, important The important one is the concept of critical damping, okay When I oscillated this system by giving it an initial displacement like that, it will oscillate. And then decay as you can see. If I plot this initial displacement, and then go up and down, and then decay. Okay, we can see that there could be some damping they exponentially decay, the response like that. So it is one of very strange or extreme vibration case because does not, doesn't look like vibrate but exponentially decay. And we call, this is the case when we call, in this case we call this single degree of vibratory system has critical damping. What is critical damping? How does critical damping is related with other system parameters, like m, c, k? That's our question, right, how to find it? Mathematically, if you express this kind of physical behavior or say the wave is exponentially decaying, in mathematical form we can find the relation of this. Okay, that can be done like that I have a question and I'm giving initial displacement then it oscillates and decay. The solution possible solution could be expressed as x(t) will be proportional some amplitude and exponential decay lambda t. Lambda could be real and imaginary part. So plot this assumed solution over there, then I will get lambda square m, No, + lambda square. If I differentiate once, then I have lambda c + k = 0. We often call, this is characteristics equation of the linear differential equation. This gives me immediately, lambda has to be Denominator should be 2m. And then 2b or I mean plus minus square root Square root of C squared- 4 m k. That has to be 2c, 2c, right, 2c, thanks My assistance.So that I can write this is minus C over N plus minus square root Z Square minus 4 MK. So this means that this term, this term, represents exponentially decayed term. And this term, depending on the magnitude, this term can be either imaginary or real. But it is rather interesting if c squared equal to four mk. Then there is no chance to have oscillation, right. That corresponding to this case. And if that happens we call this is. Critical damping, in this case c = cc. Okay, I think this has to be, I think this has to be -c, just -c. So this is the concept of critical damping. And the damping ratio, zeta, is defined as the ratio between damping coefficient c and critical damping, which is usually very small, very small. For general steel pipe or steel case, this is very much smaller than 0.01. If zeta equals to 1, then this is critical damping, then the vibration is exponentially decay, there's no oscillation. Let's think about another vibration system. We talked about single-degree-of-freedom system that oscillates mass in one direction. Therefore, we need to have one single coordinate. But if you have a two-mass over here, we need coordinate to measure the motion of this mass, as well as that mass. Therefore, we call it two-degree-of-freedom system. In previous lecture, we talked about the various spatial two-degree-of-freedom system that has the same mass, not like this. And it oscillates symmetrical way. And then we found that the possible modes are one is oscillating like that and the other is oscillating like that. And each oscillation has it's own frequency, natural frequency. So the first natural frequency corresponds to the motion of this. Second natural frequency corresponds to the motion of that. And each natural frequency produce the unique motion of vibration that we called natural, that you call the mode shape. Okay, what if I have a big mass and a small mass over here? If I oscillate with this frequency, look like that. If I increase a little bit the frequency, and you will see that, there is motion, that is dominated by this mass, okay. If I increase natural frequency a little bit more, We can find that depending on the frequency, the motion of little mass oscillates a lot more than the mass that has a bigger mass. Okay, that is one very interesting work. So suppose I have a small mass over here, and I have a spring, and I have a big mass, and I have a spring over there. And I say this is k1 and this is k2. And I have a damper over here, c2, and I have a damper over here, c1. And I use the coordinate x1 over here and the coordinate x2. Okay, If I want to make x2 approach to 0, but x1 getting bigger and bigger, This case we call, this is dynamic absorber. Because that case I have mass over here. Sorry, it takes some time. If there is excitation, but if this mass does not move very much, but this mass moves a lot, that is the case I have. As you can see here, I cannot very well demonstrate. But we can find a certain frequency that oscillated this a lot and oscillated this very little. That is called the dynamic absorber case. And we have many, many application that corresponds to dynamic absorber. You can find typical approach or mathematical approach or solution in many textbooks. Okay, another interesting example would be, there's a small mass and a stiffness and a damping, and a big mass, and a stiffness, and a damping. But I have a field, massless field, and that is moving through this rough surface, that corresponding to, as I mentioned earlier, I have a car over here and here. And there is a mass stiffness, and this is moving around. If I have interest on having some stiffness corresponding to the suspension system, the stiffness is corresponding to the stiffness that has to do with the seat, and so on and so on. But my objective is to have the motion of this, x(t) to 0, I mean in any kind of vibration that is induced by a rough surface. We can also solve that problem using similar approach, as I mentioned earlier, looking at the vibration of this model in terms of Newton's second law and then solving the system assuming harmony excitation. Then we can see the solution in terms of its mode shape, as well as two natural frequencies. Okay, we can expand, what I said up to now, to many degree of freedom system, say, I have mass. Stiffness and the mass and stiffness, so on, so on. We can approach very similarly what we did in two degree of freedom system. The equation is simple. So mass matrix and, Acceleration vector plus damping matrix plus stiffness matrix. That has to be the last of our force matrix. And the solution is very well known, okay? And you can use MATLAB and so on to find the solution. Okay, one thing I didn't mention in this course is about the measurement. Okay, we are living in the very high technology. Therefore, measuring vibration is no longer belong to the people who are very much trained in vibration. In other words, everybody can measure vibration using, for example, smartphone. Okay, because smartphone does have triaxial accelerometer. So you can use this very handy instrument to measure vibration. Even you can buy some app that can show the vibration graphically, free of charge. Of course, there is some software requires some significant amount of money, but you can start with the free app that supplies the frequency response information of the vibration.
