Let's review what we learned in the last lecture, okay. The last lecture I asked one of the students, what would be the word that can represent last lecture? And then he answered mass law. Then let me ask you, what is mass law? [LAUGH] Mass law simply says, the transmission laws or sometimes writes T.L. is 20 log 10, omega m, 2Z0 or z0? >> [INAUDIBLE] >> Hm, 2Z0, okay, 2Z0. Remember that. We have a strange number 2 over there, okay? If you look at carefully, this is a valid, This is a valid when omega is sufficiently large. Okay, when omega is sufficiently large, the vibration system normally follow, so called, mass law. Okay, I think everybody remember the famous phase diagram that graphically represents dynamic behavior of spring mass system that can be expressed symbolically as many of you already saw in many times And if you measured displacement from here. Let me use r instead of c which is often used in classical vibration system, but the reason why we do not use c and use r because we use c as the speed of sound propagation. So this is dumping talk, strictly, or precisely speaking that is [INAUDIBLE]. Then equation of motion that describes this single vibratory system is, If I say there is external force, F of t. I'm trying to relate the mass law with the well-known single vibratory system, okay. Then let's assume that the force applied to this in the vibratory system has some complex magnitude to the F of exponential minus j omega t. Maybe strictly speaking, F of t would be the imaginary part of this or the real part of this. Then, x of t can be written as some complex amplitude, exponential minus j omega t, okay. This physically means that if we excite a system with a frequency of omega, Then the force and displacement could have some phase difference, okay. And this gives us the expression of ordinary differential constant equation into very well known form that is minus omega skill m plus, and that should be -j omega r+K x=F. Okay, that's very well known. That's not very interesting, but if you look this complex expression. In terms of real part and imaginary part in other words I'm trying to look at this equality in complex play. Okay, then suppose this is x. X is complex therefore it has a real part as well as imaginary part, okay. And then exponential, because I'm multiplying this with exponential j omega t, actually this x has to rotate with omega. Okay, then. Kx could be this. Okay? Then I express kx over here. And I'm trying to express kx multiplied by minus j omega r. Okay, minus j has to rotate to this in this direction. Okay, so I will say that is minus j omega rx. That's simply the magnitude of this is r times the magnitude of that. Multiply omega and then I rotate the complex with minus j, that is 90 degree of course. Because minus j is simply cosine pi over 2 minus j sine pi over 2, or exponential minus j pi over 2. That means I'm rotating the complex in this direction. And then let's look at this term, this minus omega square m. Minus 1 is minus j multiply minus j, that means I have to rotate again, this arrow 90 degree in this direction again, and then that is minus omega square m. That makes sense, because this one is kx. This one is minus omega square mx. And if this minus of this, that means in complex domain, it has to be 180 degree face difference. Well first, understanding this will help you to understand this Mass Law. This Mass Law as I mentioned in the last lecture is essential concept that you can. Understand anything related with the partition design, okay. Okay, then. This equation says this one has to be equal to F. Right, what it means in complex domain? Graphically that means if I translate this over there, minus j omega rX, and then move this over here in parallel. That is minus omega scale mX. That, this mathematical expression says that this one plus that one, plus that one, has to be equal to complex magnitude F. So that is F. And then what is this? This angle simply says, the phase difference between force and displacement so that is very useful, okay? Then let's see when omega Is getting smaller and smaller. Then this one shrink linearly over there. And this one shrinks proportional to scale. Because I'm reducing omega so this one shrinks very rapidly, and this one shrinks this amount. Therefore, what I get would be something like that. So in this case, I have to draw F in circles, right, but approximately say this is correct. That is there is no phase difference between force and displacement. So that's physically sensible because when I have a spring, because this is the case where spring force is dominant compared with the other force. So that means when I push free the displacement of a vibratory system is just to follow the force. So then I have a spring that I've pushed, the spring compress and I release the spring will follow me. Therefore there's a very small face difference. Okay. When omega is getting larger and larger as this mass low says. What's going to happen? What's going to happen is this will increase rapidly as a square and this will increase linear, and therefore, the diagram we will have looks like that. Therefore, the force and displacement approach to 180 degree difference. So when you ride a car on a very rough road with high speed what's going to happen is because excitation frequency is getting larger and large the behavior of your car follow the Maslow. Follow behavior of your car is dominated by omega squared m over here. That is why when you ride a car in a rough road it’s not like this. Drawn slowly in that case it follows over the edge therefore the force and displacement has the same phase, but when you drive your car in rough road then omega is increasing. Therefore you feel like that is dominated by that slope. And that is explained very well by using this face of diagram. So, that's why there is only omega m over here. Because, in this case, we are considering when the frequency is sufficiently large. Therefore, it follows the mass law. Okay? And another point, this is due to the observation of this, what is 2z0? Why there is a 2? Okay, let's leave that question later on to be answered. And the message I'm trying to give you is that, this is the formula. That I say this is the formula that expresses Mass Law. And then each term of the formula is talking to you. The physics. The secret behind of this formula. Right? I mean, if you just look at the formula and say, that is a 20 log over omega m divided by two z's and go, okay, that's the Mass Law. Then the formula does not tell anything secret behind that formula. But if you look at the formula very carefully, then the formula open the door. Of a secret garden. And there is a hint to enjoy science and engineering. So I strongly suggest to you to look at any formula, key formula, essential formula, representative, representor formula very carefully maybe you look at it for whole one day or whole one week. Then you may find another view which I did not introduce to you. You'll own another view, your own view. So why we have a 20. Why? Because actually it was proportionate to 10 log 1 plus omega over 2z 0 square. And because omega is sufficiently large, or strictly speaking omega m, is sufficiently large. In other words, how sufficiently large? Omega m over 2z 0 has to be much, much larger than 1. Then this mass law force, and using practical Guideline. In other words, to practically use this mass law. I suggest to use the concept of block frequency that is related with again very strange number, 123 divided by mass per the area. That's the starting frequency to estimate how much transmission loss you will obtain by using this mass law. And the u just count of frequency, then you can obtain how much noise reduction you can have. This is kind of review, I'll introduce the concept of phase diagram that visualize very easily the mass law in vibratory system. Then let's move another concept. In the last lecture, We found that on this limpwall We'll assume the instance where normal reflected away and transmitted away. And apply the boundary condition over here that expresses like that. The pressure reflect incidents plus pressure magnitude complex reflected minus pressure transmitted. That is the net force acting on the limpwall that has to be equal to mass times exeration. So you complex notation that would be minus omega square mY. We assume that this wall will move Y exponation minus j omega t. And also we found that the velocity on this side will be Pi divided by z0, and the Pr divided by z0, and this is reflected with there. So therefore, I put a minus over there. And that has to be equal to the values of this wall, that would be minus j omega Y. And also develops it on this side and that has to be Pt over z0, has to be equal to minus j omega Y. And using these three equations, we obtain the mass law. Everybody remember that, right? But let's look at this equation again, carefully in rather different way, and trying to find out whether or not there is another meaning that we could find. Anything wrong over there? If you find something wrong, please tell me because I often make a mistake. Sorry this has to be small i. One could have some curiosity. Well, how about the pressure Pi + Pr? That is the pressure acting on limpwall on the left hand side. What is this? Of course, if you look at this equation, you can say that Pi + Pr- omega square mY + Pt. Any other relation I can find? If we can Pi + Pr in relation with the Pt. As well as why that would be very interesting. So from here, what I can see from these two equations, what I can see is, simply Pi say, this is equation one, this is equation two, this is equation three. From over here I say, equation. 2, I can write Pi-Pr is equal to Pt. Then I can write simply -Pi + Pr = -Pt. Okay. Then I want to have this, Pi plus Pr. What is this? And what is Pi+Pr? That has to be 2Pi-Pt. Can anybody follow how I can get this interesting formula? Over here. Over here, ok. Pr is equal to Pr minus Pt. Therefore, Pi plus Pr is 2Pi minus Pt. Simple, right? Ok. What does it mean? That means many things. Ok, I have a wall. And this is the pressure. Acting on this on left-hand side. And that is equal to, according to this equality 2P_i. What is 2P_i? Okay, 2P_i, Is the pressure that can be sensed by non-moving Okay, so I call this block frequency or blocked pressure. Plus minus Pt. What is minus Pt? - Pt is this. What is that? That is -j omega y Z0. That is the pressure in use to by the oxalation of this panel, this is a velocity and z0 is a characteristic impedance that produces pressure because z0 is p0 over z0. And then this is velocity. Velocity multiplied by the impedance is pressure. So this the pressure induced by the oscillation of this panel. The velocity J omega one. So what it essentially says, interestingly, is that Okay. When we have four incidents reflected, transmitted, that is equivalent with we have 2 Pi or Pblock frequency plus Oscillation of work with the velocity j omega Y that will produce The Pt. Okay? So actually, this is a very interesting result for this oscillation to produce the radiation. Okay, now so everything associated with this kind of problem can be Thought equivalently as the super position of two problems, one is simply blood pressure plus radiation as if the panel is moving. And this is so simple because that is just the two times of the instance away. And this is the radiation. So if you think carefully for those control purpose, you do not want to have big radiation. Due to the panel.
