[MUSIC] The mathematical expression we learned is like this, m x double dot + k x = F(t). We learned the physics that is expressed by this mathematical expression in terms of transfer function. Transfer function is mathematical expression, That expresses the relation between force and motion in terms of magnitude and phase. Why it has magnitude and phase relation? This is simply because we regard F(t), That has complex amplitude F and exponential j omega f or if you like you can write this F exponential j2 pi ft. For convenience we will use omega instead of j2 pi f because it is more convenient. And x(t) we assume that x also has a complex amplitude and j omega t. Reason why we're using same omega t is because this is linear system or we could say this is expressed in terms of constant coefficient, second-order linear differential equations. And in the last lecture, we showed that its magnitude, Looks like this, We have an f over f0, or omega over omega 0, this's 1, in other words, when system is excited by the resonant frequency, we have certainly infinite magnitude. And we also look at its phase, and the phase is changing from 0 degree to very abrupt change at the resonance frequency to pi or 180 degrees. In some case, you could say there is the -180 degree, depending on which expression you use. If you use a plus j omega t, if you use a -j omega theta phase change would be a little different. Okay, this approach to understand magnitude and phase trend is rather complicated. To solve this complicated approach, we can try to follow a sense. For example, noting that this is complex amplitude, therefore we can invite complex domain over here. Say this is real part, this is imaginary part. And because amplitude x is complex, therefore we could say amplitude x has a real part and an imaginary part. Actually, x is rotating because we have exponential j omega t, if it is +j omega t, it is rotating this way. If it is -j omega t, it is rotating that way. But at certain time, the complex amplitude is over here. Then, if you express this mathematically, the expression in terms of this complex domain, then kx might look like this. And mx double dot, as you know, mx double dot is equal to minus omega square x, then this mx double dot has a magnitude omega square x, but there's a minus, therefore, it should be like this. Depending on the scale of omega, it could be much, much bigger than this, or like this. So this is minus omega square x. For convenience, if I move this magnitude and direction over here, this will look like minus omega square x. And then noting that this equation says, minus omega squared x + kx has to be equal to f, therefore, complex amplitude of f has to be like this, right? Kx minus semi vascular x has to be f. So this is very clear. Now you can see very vividly the relation between kx and minus omega square. Okay, if omega is getting large and large, then this minus omega square x will move, if I draw over here again. Okay, this is the case when omega square is small, but omega square is large, this is kx, then what's going to happen is this omega square x minus omega square x. This master will increase as omega square getting larger and larger like this, omega square x. Okay, now you can see that the phase difference between this kx and minus omega square is pi, 180 degree, right? 180 degree. This case minus omega square x, and kx and x, what's the phase difference? In this case -182, but if omega is getting small and small, If omega is getting small and small, and getting very small, then the stiffness and so let me review this. When omega square is getting smaller and small, the phase difference between stiffness and minus omega square x is getting smaller and smaller, in other words, the whole picture is dominated by kx, therefore, stiffness control. In this case, whole picture is dominated by minus omega square x, therefore mass controlled region. Another interesting point in this case if f has to be this, therefore, for mass dominated case, the phase between exciting force and mass is the same. But in this case, phase of the excitation force and mass is different, 180 degree difference.