 Now, some idea, some gyan about the envelope shapes. First question is why is the envelope shape important? What does it affect? One is a drag. So, what would you like to have? A shape with low drag, a better aerodynamically shaped envelope, where are you? What else does a shape affect? Correct. So, a shape that you choose decides the storage space, the hanger and the clear area required to operate. If you go for spherical, you will have the least height and the least width for a given volume. You cannot play with the volume because volume decides the buoyant force, density difference in volume. So, the best shape from the point of view of size is spherical, but very poor drag characteristics for sphere. Anything else is affected by the envelope shape, location of CG, how does it get affected? So, the location from the nose would not be at the center of the length, it will be maybe slightly forward. Okay, center of buoyancy will be at the center of volume, which will also be slightly ahead of the half-way mark because it will be more towards the front as you can see in this shape. Agree. So, that affects the trimming, but you can locate your tail appropriately, something more fundamental. Remember that, sorry, surface area, that is the killer. Sphere is preferred because sphere has the least surface area for a given volume. And the more slender you make it, you may get better aerodynamics, but you need more area, not only you store it, but it also weighs more because it has more surface area. Anything else? Volume is fixed, shape is changing between sphere and a long one. What else will change? Dynamic lift, yes, will change because that depends upon the CL and CL will be different for spherical and what else. But dynamic lift, I really do not bother too much because at mass it is 15% of the total and I need to be able to cater to operate without dynamic lift anyway. So that is a bonus, but yes, it will matter, anything else will matter. How about stress in the envelope? What will that depend on? Yes, delta P inside and outside, let us say I fix that number. Why should I have more pressure for sphere and less for a oblique, you can have the same. So yes, pressure inside, how much it is more than outside will determine the stress other than that. So let us say that P is fixed. I know that we do not have sharp corners. In an inflatable structure, you cannot have sharp corners, yes, but if the envelope is inflatable and if you start filling it, it will automatically acquire a smooth curvature. You cannot say that it will be very difficult for you to make a very sharp change in the curvature if it is inflatable, it will not allow you. It will get more stressed at that area. So you have to put more material there to strengthen it, which will of course increase the weight. How about hoop stress? It depends mainly on the pressure inside which is constant. Then maximum diameter is Pd by 2t, thickness will be the same because it is the same material. So diameter. So if you use maximum diameter, if it is reduced, you can reduce the hoop stress. So the shape optimization of the envelope, one more thing is the weight of the fins of the control system. If it is slender, as I said, you can put a very small fin far away. A smaller fin far away will give you the same moment as compared to a larger fin on a less slender shape. So there is flight dynamics or stability, there is structure, there is weight, there is aerodynamics. That is why we have done an MTech project on shape optimization of envelopes, keeping in mind all these multiple concentrations. So GNVR shape is one standard shape. This was suggested by a professor of IISC Bangor called as GNV Rao. That is why it is called as GNVR shape. And this shape has been suggested by Professor GNV Rao for aerostats used by ADRD in Agra. All the aerostats use this particular shape. And NAL has done a lot of wind tunnel testing, CFD analysis about this shape. So the data is kind of available. It is not a very good shape for airships, it is a very good shape for aerostats. So now my question to you is, what do you think is the principal difference in the aerodynamic considerations between airships and aerostats or why would a shape like this be good for aerostats but bad for airships or not so good for airships. Remember aerostats are designed to be to remain stationary, airships are designed to move. So this is a question for Moodle. A question for Moodle, think about it, you will not probably get the answer on some website by searching, you will have to apply your mind slightly. Interestingly, there is a paper by Professor GNV Rao on modified GNVR shape suitable for airships. It is a very small paper, 6-7 pages in which he argues that in the same GNVR shape, now what you do is, if you give a simple Google search on modified GNVR shape, I am sure you will hit into one of our papers, there is a dual degree student called Rabindra Zoshi. He did his dual degree project on an interesting topic on knowledge based engineering design. And for that as a sample he used this, so he has described the modified GNVR shape in his paper. Basically, what GNVR Rao said is, put a constant diameter portion if possible in the center of this shape, extend the L by D to 4.5, it will become a good shape for airships. But what we did is, we said that for demo airship we will stick to this shape just because we had aerodynamic data for it, just because we had its coordinates. And just because it is a very tested shape, it may not be super efficient for airships, but what the hell, let us go with it. Now what is so special about this shape? So first I want you to just look at this particular picture and try to get a hang. See this is the genius of a person who has experience picked up by years of aerodynamic analysis and experiment. Try to understand the contribution of the GNVR Rao in giving this particular shape. First thing that comes out is that this shape consists of 3 standard geometrical constructs. From the max diameter forward it is an ellipse with the semi-minor axis equal to 1.25 times diameter. So if you know that it is an ellipse and if you know that the semi-minor axis or semi-major axis sorry is 1.25 times diameter, you can easily generate the coordinates analytically from nose to the max diameter portion. On the rear somewhere here the last 17.5% times of the diameter it is a parabola. Now why a parabola? Because he was told by the scientists who make aerostat envelopes that a parabolic shape is a good shape to attach the fins. This is just a input from a user saying that you can give me any shape on the back, you can make it straight line but it would not remain straight line, it is an infratable structure. You can give me high curvature but that will be bad as Amoeba pointed out. So they said that if the tail cone is parabolic it kind of remains rigid and we are able to mount fins easily. So he gave the last portion as parabolic and he gave the equation for that. In between these 2 he fitted a circle or arc of a circle. So 1.25 times dia from nose is ellipsoid shape, mid portion is circle with radius r equal to 4 times diameter and the rear part is the tail cone parabola which is 0.1.5 times diameter length and these numbers 1.625 etc they have come by intersection. So where will geometrical construct, where will it intersect if it is 4 times diameter radius. Interestingly behind the maximum diameter the slope of this particular shape is something like 1 is to 7 roughly you can make out or you can cross check it. So it is 1.625 times d plus 0.175 times d, so it is what 1.8 times diameter. So through wind tunnel testing and through experience of aerodynamics of bodies Prof Rao knew that if you give almost 1 is to 7 times you know if you reduce the diameter 1 meter in 7 meters you get a nice shape after max diameter from aerodynamics point of view from the flow control point of view, from the flow behaviour point of view or separation point of view. Interestingly the complete coordinates of this shape can be obtained analytically in terms of diameter. If I say g and v are shape of diameter equal to 1 meter I do not have to say anything else. All of you individually can get the exact coordinates of the shape from nose to tail. So that is the beauty not only that you can analytically calculate the volume and area in terms of diameter. So how easy it becomes just multiply the diameter cube by 1.4784 you get the exact volume of this shape. You multiply diameter square with 7.4481 you get the exact and this number will not come from top of the head or from any numerical integration or this has come from analytical information so they are exact. So this is for you on the model page to upload. Take this shape do an integration analytical and confirm that you can actually get. So what will be the equation of the ellipse it will be you know x square by a square plus y square by b square equal to 1 with a and b related with diameter. Similarly parabolic equation is given similarly the arc of a circle r equal to 4 it will be very interesting for you to do it. And just do an integration I did it myself and I confirm okay and when I see sufficient interest I am going to upload the solution also for your information but not right away I do not want to spoon feed you do it yourself struggle a little bit and you upload your calculations or your results in eventually I will give you the correct calculation. So now the length upon diameter ratio of this shape is 3.05 it is a very popular shape and we have also used this shape for aerostats as well as small airships. In fact the first airship that we made the one which I showed you called as a micro airship it had this particular shape admittedly it is not good for airships but it is a convenient shape to work with when you work for the first time you always work with known and established geometrical information. The first uo that you make is somebody else's design you do not start sizing and doing the calculations every time you do something. And do it once get it right get the hang of it then you can say ah I can improve it now that is what we also did.