[MUSIC] So from previous lecture, To get a motor share we need to get mass matrix and stiffness matrix. How to get for the case of general structure? Okay, general structure does not really, Looks like a mass spring, mass springtime. General structure may look like this. For example, if I want to, Get a board shape of this general structure, and how could I get it? Looking at this, this one is composed by, kij stiffness matrix, matrix. And this is composed by mij element. How to get mij? To get mij and kij of this kind of structure, For example I can divide the structure, Like this. And like that. And I say this is M11. Okay, this is another mass, this another mass, another mass, another mass. Okay, and, obviously, there is a stiffness-like force acting on this mass. In other words, when I push this unit displacement, there is some restoring force acting on opposing to my force, that is K11. And a K12, K13, stuff like that. So in matrix form I can write, here is K11 K, 21 K31 blah, blah, blah, and K21 K22, To K31, big matrix you can make. But, as I demonstrated before, The procedure to get both shape would be exactly same as what we demonstrated for using two degrees of freedom system. But having mass metrics it looks like a simple we can you can measure the mass, but how to get this? There are many ways to do that in practice. But principle is as I said before, if I push unit force over here and the major the restoring force. By element to, by the other element, by the other element. That's essentially what modal analysis is doing. I will go on more detail about this subject. Let me summarize what we've done in this lecture. We start with the very simple single degree of 503 system. Which is experienced by the force function of time and major displacement, x(t). And using Free Body Diagram and Newton's Law we obtain the governing equation. And [COUGH] Investigating how this single degree of freedom system, Is behaving, looking at governing equation. Okay, there are many ways to solve a governing equation. In time domain approach, We can solve the governing equation as the initial value problem. What is the initial value problem. For example if you give the initial displacement like that, it will oxalate like this, and the solution will look like some amplitude multiply by cosine omega t. And what is omega? Omega is the natural frequency of this system or the characteristic equations solutions that essentially provider us natural frequency. And then we show the physical, I mean the way to see the physics of the system. That, We found that the transfer function, Essentially shows the displacement with respect to force in terms of frequency. In other words, we are seeing that how we got the displacement for each frequency like an omega is equal to omega one and omega to an omega three stuff like that. And we found that the magnitude of this look like omega 0 is natural frequency of the single degree of freedom system follows Like this curve. In other words, when the excitation frequency is the same as natural frequency, we have a very high peak. In other words, amplification factor is very big, that will called the, Vibration behavior at natural frequency that's really, practically important. And we also explain how it behaves as the high frequency and low frequency. And then we move on to the two degree of freedom system. To understand the essential physics, in other words, what is the difference between single degree of freedom system and two degree of freedom system? We first found that because it has a two degree of freedom system we have two different mass of frequency. And if this mass is the same, and stiffness is the same. We found that the first natural frequency corresponding to the motion of this system, two mass move with the same phase. A second natural frequency corresponds to the motion of this two mass move out of phase. So, The difference between the single degree of freedom system and two degree of freedom system is that, this two degree of freedom system has two natural frequency. Also, the vibration shape on both in other words, they're distinctly two different vibration mode. And that vibration mode corresponding to each natural frequency, Can be, Used as the essential component of vibration for general vibration of two degree of freedom system. Okay, that generally corresponding to the model analysis we explained very briefly. For example for, we have a string, string has this vibration mode for the second natural frequency and first vibration mode look like that. And each corresponding to have natural frequency of omega 1, omega 2, and omega 3. General vibration of string, Will composed by, that is to be equal to first natural frequency vibration and second natural frequency vibration, so on, so on. Same concept can be applied to any general structure. So for example as we saw in the very first lecture if we have a car. Okay, car is moving off the road, we want to see how this car is vibrating, and what we can do is, we can sub divide a whole structure, like that. And then we measure the, say this is element 1, this is element 2. Then we measure mass metrics and governing equation will look like that. And CX dot + KX. That has to be excited by some, Excitation. We follow the same procedure as we did before. And then we can have this matrix can be expressed by super pollution of a mode that is the solution of this matrix equation.
