 So, in the last part we discussed about the aerodynamics of airship, so I have to be honest that we did not discuss the entire story of aerodynamics, there are things, there are other forces, there are an extended part of aerodynamics which are known as added mass effects and the it is not that they are not present in conventional aircrafts, they are however these the effects of this particular component of aerodynamics is actually quite less in conventional aircrafts and that is the reason why we only consider added mass effects in buoyant vehicles such as submarines, hot air balloons, aerostats and airships. Now, what exactly are added mass effects? So, before we answer that question let us look at this bluff body which is moving in fluid. Now, when this red colored ball is moving in the fluid we see that it is moving from the pink colored section to the blue colored section. So, those pink you can think of it as particles, fluid particles. So, let us look at it again. So, when it is moving it is actually displacing some of the particles. So, you see that the pink particles have actually entered this blue region. Now, that is a complete displacement of particles from one portion to the other. Now, let us see why and what exactly it is. Now, this can be explained using potential flow. Now, potential flow is simply that the object and the surrounding fluid cannot occupy the same physical space simultaneously. Now, this makes total sense because obviously you there are no two objects can occupy the same physical space. So, that is the underlying concept behind potential flow. And this is also what we see in this particular animation. We also see that there is a permanent displacement of fluid particles along with the body that is the yellow region here. The pink particles have moved to permanently displaced and move to the blue region. And this particular volume the yellow region that we saw here that particular volume of fluid is known as the Darwin drift volume because the elements the particles have drifted from one region to another permanently. So, this particular thing is the Darwin drift volume and the added mass effect or the added mass forces or moments they are basically the product of Darwin drift volume into the fluid density. So, that is what Darwin added mass is. So, now let us have another perspective of what exactly is happening around ice ships. Now, first thing is that this happens by displacement of surrounding fluid as we saw in the bluff body animation. And they happen only when the object is accelerating or decelerating through the fluid. And we also have this relation which is basically the added mass forces they are directly proportional to the acceleration or F equal to K a. Now, let me remind you of an equation which is very similar to this F equal to MA or F equal to mass into acceleration that is Newton's second law. And that is also where we get this term added mass. Now, the reason why we have this name added mass is because it is directly proportional to acceleration. Now, this K term will have the units of mass because force is equal to mass times acceleration K here in this case which is a factor for added mass would have the dimensions of mass and that is the reason why we get this name added mass effects. Now, this acts like a inertia addition to the system as you can see from this relation. Again, this is an aerodynamic effect and it happens only when the object is moving through a fluid. Now, it is because it is directly proportional to the acceleration it has the aerodynamic coefficients have the K this added mass coefficient K has the units of mass and that is why it looks like it appears as though it is the inertia addition to the system. Now, another way of looking at it is something like this. So, let us say you have this bluff body this gray bluff body which is going through fluids. Now, I beg your pardon because this is a very poorly animated slide. Now, what we see here is that when this bluff body moves there are these particles which move away as we saw in the bluff body animation and there are some particles which this object carries along with it. Now, these objects that move away they can be modeled using the conventional aerodynamics model that is the one that we saw in the previous talk but the one that is carried along with the bluff body it looks like it is the inertia addition because this much extra mass is being attached to the bluff body itself. Now, this can be modeled using several model using empirical relation which was given by LAMP in 1918. Now, again as I mentioned these are predominantly used in buoyant vehicles like submarines, hot air balloons, aerostats where basically the displaced mass of the air or the fluid is comparable to the total weight of the system. Now, as you can imagine they are going to be different coefficients about different axes. So, let us say the airship is moving along the x axis like this in that case we would have something which is about the x axis. So, the added mass about the x axis let us call it mx would be given by k1 into m where k1 is the added mass coefficient and m here would be the displaced volume of fluid in this case displaced volume of the air. So, x would be in one direction now the airship could also have added mass along the y direction or the lateral direction. So, basically when the airship does this so that would be my so here as I mentioned m is the mass of the fluid displaced by the body and mz could be in this direction laterally. Now, it is mz because if you look at the airship like this or if you look it from the top it is because it is axis symmetric you are going to have the same. So, my would be equal to mz. So, that is given by k2 into m. So, you have two different added mass coefficients with different values about the x axis y axis y axis would be equal to z axis in this case m y would be equal to mz there is another thing which is also acting. So, basically when the airship is turning let us say yawing or pitching again both of this would be equal the added inertia terms corresponding to these. Now, it is not just only for the masses it is also for the inertia because when you are turning even then you are displacing this fluid mass. So, that is why in the yawing motion and also in the pitching motion in both of these cases we have this added inertia to the system and that is given by j dash equal to k dash into j where j is the inertia of the mass of the inertia of the fluid displaced by the body. Now, it is given by these curves which is taken from this book called fundamentals of aircraft and airship design volume 2 by Karichner and Nikolai. So, we have k1 here and so we basically see that as the fineness ratio increases the k1 value decreases. Now, it makes sense because what we are doing is we are elongating the airship you are basically reducing the frontal area. So, therefore the inertia and addition along the x direction would be less because you are basically reducing the area here projective area and you are elongating the airship. And in this case in the YZ case as the fineness ratio increases we also increase the added mass effects. So, as it increases we would see that more and more fluid mass would be displaced and that is why we have more higher values as we increase the fineness ratio. The same thing also goes with k dash which is corresponding to the pitching and yawing moment added mass inertias. We see the same effect as k2 and k3 that is as fineness ratio increases it increases as well and for sphere we see that it is actually constant it is the value is 1. All right. So, let us do a small comparison between a lighter than aircraft and the heavier than air aircraft. So, for this exercise I am going to pick these two aircrafts. So, we have skyship 600 and Boeing 777-200. Now, the reason for picking or choosing these two aircrafts is because they are of similar size. So, you can look at the length of the aircraft they are very similar even the volume is in a similar range. Now, note that volume does not really make sense for Boeing 777 because it is a heavier than air aircraft. So, that is why we only use the fuselage volume we have the total empty max takeoff weight. So, let us only talk about the maximum takeoff weight MTO WMTO which is 5200 kgs for skyship 600 and about 247200 kgs for Boeing 777-200. So, let us calculate the added mass for skyship 600. So, we have the volume. So, this is also the displaced volume. Now, we have we can calculate the fineness ratio that is the length over the diameter which is 3.88. Now, that corresponds to a K1 value. So, we are only going to talk about the x direction added mass. So, it is about 0.1 approximately. So, MA which is the added mass about the x direction is volume that is this 666 into rho which is the density of air 1.225 and K1 which is 0.1 for skyship 600. Now, the value that we get is about 816.5 kg. Now, the maximum takeoff weight as we saw earlier is 5200 kgs. And what we see here as a percentage is that it is about 15% of the maximum takeoff weight. The added mass about the x axis for skyship 600 is about 15% of the maximum takeoff weight. Now, let us do the same exercise for a heavier than aircraft which is Boeing 777-200. Now, the length of this is 63.73, 6.2 as the diameter and volume as we saw is 7692 meter cube. Now, I have not considered the volume of the fins or the wing. So, it is only the fuselage. So, what we get here would be kind of an underestimate. But just let us for the sake of comparison, let us just do it. So, we have the L by D ratio fineness ratio to be 10.27. Now, although in this graph, we do not see beyond 10. So, we are going to assume that it is about 10. And what we get is K1 equal to 0.04. So, let us calculate the added mass which corresponds to these numbers. So, we multiply volume displaced that 7692 meter cube into rho 1.225 and K1 which is 0.04. Now, the added mass in this case is about 376 kgs. The maximum takeoff weight is 247200 kgs. Now, when we take this as a ratio, what we see here is about 0.15% of the maximum takeoff weight that is the added mass of Boeing 777-200 about the longitudinal axis is 0.15% of the maximum takeoff weight of the aircraft. So, this is the reason why we do not consider added mass in conventional aircrafts or heavier than air aircrafts. And it makes a lot of sense to consider in lighter than air vehicles because it is about 15% of the maximum takeoff weight. And that is the reason why it is important to consider added mass effects in heavier than in lighter than air systems. With that, I thank you for listening to me patiently. If you have any questions, please do post them on the forum and I will answer them to my best understanding. Thank you.