 Alright, let us talk about the aerodynamic stability of our airships. Now, let us start with pitch stability. Now, earlier in the dynamics lecture, we had talked about the conventions that we have. Now, by that convention, we can define the angle of attack direction. So, we take our airship again. So, we have the x axis about the thumb using the right hand rule and the y axis along the z direction and sorry, the y axis along the lateral direction and the z axis downward. So, alpha would be about the y axis approximately. So, that is why this is kind of like a convention. So, we put the thumb along the y axis and it is upward. So, that is why alpha here would be positive in this case and so will be the pitching moment as well if that would be the direction for positive. Now, let us say we have a pitching moment which is in the opposite direction which is acting in the opposite direction. So, what would happen in that case for an aircraft? So, we have an alpha which is like this and a pitching moment which is acting like this, the airship would basically reduce the alpha to 0. And for this particular thing, what we get is when we plot the pitching moment coefficient Cm versus the angle of attack alpha, what we get is this kind of curve. So, basically a curve which would that is a green line here. So, that is what we would get. And because the aircraft is able to reduce the angle of attack to 0 or to the value that it is required by driving it to 0 by generating a restoring pitching moment, we call this to be a stable system. Now, in the other case when let us say we have this angle of attack and in this case we have the pitching moment which is also in the direction which is also in the positive direction. And in such a case what would happen is this angle of attack a non-zero angle of attack would make it have make it knows up even more. So, the angle of attack and the pitching moment basically just increase. And for this case what we would have is a curve which looks like this. So, the pitching moment curve versus the angle of attack would be in the opposite direction. Now, this basically the directions of the curve is entirely defined by the conventions that we talked about earlier. And in this case what happens is because we have a non-zero angle of attack, we do not generate a restoring moment rather we generate a destabilizing moment. And that is why the moment also increases and thereby the angle of attack also increases and such a system would be unstable. Now, this is for a conventional airplane. Now, what exactly happens in the case of aircrafts? So, typically aircrafts are designed the civilian aircrafts at least are designed to be stable in pitch. So, we have the aircrafts which follow in this line. Now, there is a metric which can be used to determine whether the aircraft is stable or not and that particular metric is the aerodynamic pitching stability derivative or CM alpha. So, CM alpha is basically the slope of the pitching moment to the angle of attack. So, in this case we have the pitching moment versus angle of attack data and when we take the slope of these curves what we get is CM alpha for that particular aircraft. Now, as you can see the green line is stable and the red line is unstable we can say that CM alpha should be negative for aerodynamic pitching stability. Now, this again is because of the conventions that we have defined that is for a positive angle of attack we need a negative pitching moment to stabilize the aircraft. So, by that virtue what we have is CM alpha should be negative for ensuring that there is aerodynamic pitching stability. Now, let us see what happens in the case of airships. So, let us say the airship has some non-zero pitching angle in this case we have some very small angle non-zero and what happens is that it actually moves away and which means the pitching moment is actually increasing it is destabilizing a moment that is being created we do not have a restoring moment. In aircrafts it is positive in conventional aircrafts mainly civilian aircrafts we see that the pitching stability is positive but in airships we cannot ensure that. So, what we have here is that the airships are unstable and typically civilian aircrafts are stable. Now, we further look at the aerodynamic data. So, this is a data of an actual airship which is YZ2A this was an experimental airship and we have the CM versus that is the pitching moment coefficient versus the angle of attack we see that it is an unstable curve because the slope is positive. So, that is the that is the that is the thing with airship. So, they are they are aerodynamically unstable in pitch. Now, as we had talked earlier about the in the dynamics video in the dynamics lecture that we have pendulum stability in airships that is we have the CG which is hanging under the center of volume which kind of looks like a simple pendulum and because of this we have this stability. Now, when the airship is moving we also have this aerodynamic destabilizing moment and because of that there are two things at play here. One is this stabilizing pendulum stability and the other one is the destabilizing aerodynamic moment. Now, the pendulum stability of airship is going to remain same irrespective of what speed the airship is moving. However, the aerodynamic destabilizing moment it is going to increase with the velocity that is because the dynamic pressure of the airship increases. So, typically at low velocities usually within the operating velocity of the airship we find that the airships are stable. But beyond a certain value of this velocity the pendulum stability the aerodynamic destabilizing moment is more than the stabilizing pendulum moment of the airship and at that point the pitching stability the total pitching stability of the airship is lost. So, to summarize in general at low speeds we find that the airships are stable in pitching although they have an unstable aerodynamic pitching moment. Now, let us look at what happens in the lateral plane or as we call it the weather cocks stability. So, let us say the aircraft by virtue of the conventions that we have defined has a positive side slip angle beta and let us say this particular aircraft generates a positive yawing moment N positive aerodynamic yawing moment N. In such an aircraft what would happen is that it would align itself in the direction of the wind. Now, what is basically happening is the aircraft is flying is encountering this crosswind in this direction. Now, this particular angle is positive and the aerodynamic moment yawing moment that is generated is also positive by virtue of the conventions and it aligns itself in the direction of the wind. Now, what this would allow is that the aircraft thrust line or the propulsion engines would be aligned in the direction of the wind and the aircraft would tackle this wind head on. This is also why we want the aircraft to align itself in the direction of the wind and such an aircraft would be called an aerodynamic weather cocks stable or directionally stable aircraft. Now, typically what happens is that when we have a nonzero beta the stable system would drive this beta to 0 and thereby the yawing moment would also go to 0. Now, this is a stable system and when we plot the yawing moment versus the side slip angle what we want is this red line stable which is a directionally stable system. Now, as I mentioned civilian aircrafts are designed to be directionally stable. Now, what would happen in the case of an unstable directionally unstable system? Now, we have this positive side slip angle beta and what we would have in an unstable system is a negative or in the opposite direction. The yawing moment aerodynamic yawing moment N would be acting in the opposite direction and because of that the side slip angle would just increase or blow up. So, this is what happens in an unstable system. Now, because the side slip angle also increases the yawing moment further increases making it even more deviate making it deviate even more from the intended path. Now, this is what the unstable system would look like. Now, again civilian aircrafts are designed to be stable and we do not see this instability in directional mode or weathercock stability in civilian aircrafts. Now, similar to the pitching moment stability derivative CM alpha we can define this directional stability derivative I pardon for the spelling of this which is DIR there should be an R over there it is the aircrafts directional stability derivative CM beta. And what we see here is that we have this metric called CM beta which is basically the slope of the yawing moment coefficient to the side slip angle. So, in the case of directional stability what we want is CM beta to be positive to ensure that we have weathercock stability. So, that is this stable line as we can see should have a it should be positive slope and if it has a negative slope it means that it is not directionally stable or there is no weathercock stability. Now, what happens in the case of airship? So, we have this non-zero side slip angle. So, in case of airships what we see is that it is actually directionally unstable or it lacks weathercock stability. So, airships are basically directionally unstable. So, it does not align itself in the direction of the wind and this is also the reason why we have seen many disasters because of bad weather or bad wind conditions. And this is how the slope this is how the line or directional yawing moment versus side slip angle would look like. Further when we look at the experimental data again for YZ to airship we have this yawing moment coefficient CM on the Y axis and on the X axis we have the side slip angle we see that there is a positive there is a negative slope for yawing moment coefficient that is C and beta is negative meaning this airship is unstable. And most of the airships that we see around they are actually unstable. Now, as I mentioned civilian aircrafts are designed to be stable and airships are typically directionally unstable. Now, why is this to answer that question let us look at the normal force distribution about the hull of the airship. Now, we see that in the first figure what we have is the force normal force distribution along the length of the airship and it is normalized with respect to the dynamic pressure Q naught that is half rho V squared. Now, what we see here is that there is this uneven distribution of the forces about the longitudinal axis. Now, because of this what we see is a kind of resultant moment that happens at non-zero angle of attack. Now, this particular graph is for the angle of attack at 12 degrees and what we see here is that there is some kind of positive pitching moment that is basically the nose would be going upward. So, that is what happens here. Now, as I mentioned that is this destabilizing aerodynamic moment as a resultant of this uneven normal force distribution about the longitudinal axis of the airship. And this is what is known as the monk moment the one that we had mentioned that I had mentioned earlier in the dynamics lecture. So, this is what is known as the monk moment it is the destabilizing aerodynamic moment that occurs at non-zero angle of incidences. And it is primarily because of the uneven normal force distribution across the length of the airship. Now, monk moment itself for this destabilizing aerodynamic moment can be explained solely using potential flow theory. However, there are many more models which include the viscosity, the interaction effects between the hull and the fin as well. Now, this monk moment or the destabilizing aerodynamic moment it happened it is by virtue of the shape of the particular airship or the aircraft. Now, even aircrafts the conventional aircrafts experience this monk moment. So, the fuselage is the part which is similar to the envelope of the airship. So, as you can see from this model that the airship is actually the envelope of the airship is a body of revolution. The fuselage of the aircraft is also a body of revolution and both of these they experience this monk moment. However, in the case of aircrafts we have fins or tails conventional aircrafts have tail which are pretty huge to stabilize this. But in the case of airship we see that we have smaller fins. Now, let us go ahead and answer the question why. So, as I mentioned earlier it is because of the undersized fins. Now, for comparison let us look at the airship here and look at the relative sizes of the fin. So, as compared to the envelope the fins are smaller as compared to a conventional airliner. So, in this case we have this fuselage which is analogous to the envelope of the airship because both of these are destabilizing aerodynamic components. But just look at the tail size. When we look at this tail size we see that there is a clear distinction. Now, why do not airships have bigger fins? Why are they undersized? Well, obviously there are some restrictions. Now, one of the restriction is that the weight and the location of the fins. Now, in the case of airships we have the center of buoyancy which is about let us say here about where there is the maximum where all the aerostatic force is acting and then we have this fins. Now, fins are placed at the back and if we put a very large fin here what would happen is that the CG of the airship would be shifted tailward. Now, you could argue that that is the same case with aircrafts. But in aircrafts everything is not light, but in airships everything that you have everything that you see is actually lightweight. So, the component the tail would actually be a considerable object which is placed from away from the CG and because of that the CG would be shifted towards the tail and that is something that we do not want. That is the reason that is what this weight and location of the fins mean. And the fin size also pose some constraints such as the ground handling. So, we know from earlier lectures by Professor Panth we know that ground handling is a real issue for LTA vehicles because they are huge and they need to be handled by multiple people and that is one of the issues. The other reason is the storage space or the hanger size. So, if we have larger fin, larger tail or larger fins we would require a much larger hanger size. So, that also imposes a constraint on the fin. And one of the and the other important thing or the other important restriction is that it needs to be mounted on the envelope. Now, remember this envelope is actually a flexible material which is inflated it is basically a balloon and the fins have to be mounted on top of it which makes it even more difficult. And that is the reason why we are not able to mount a very big fin on an airship. Now, as a penalty what we have is a steady rate of turn in the direction of the disturbance. Now, we will talk about this in a little detail in some time. So, what it means is that it would basically, so if there is a disturbance what the airship would do is it is basically going to turn in that particular direction forever. So, that is what it means. And so, we also have a solution for this and the solution is that we always we need a active feedback control law that would correct the directional heading. So, basically the moment it deviates from the path you have this corrective action which you sense that there is this deviation and you manually give some control inputs maybe the elevator or the yaw thrust and then you align itself in the direction of the wind. So, that is how you would be able to stabilize an airship. Now, this is a simulation. So, here at this point the airship is moving straight as you saw and for 5 seconds at this particular point there is a wind disturbance for just 5 seconds. So, it is basically the wind appears for 5 seconds and the wind disappears. Let us see what happens after the wind disturbance disappears. So, this is the response. Now, there is no wind there was a wind disturbance earlier but now there is no wind what the airship is doing is basically the steady rate of turn. This is the direct resultant of the airship being directionally unstable. So, it basically just turns in this circle forever unless and until a corrective control action is taken. So, this is what happens. So, to summarize airship stability in general, so not just aerodynamic stability the entire airship stability we know that the airship is stable in pitch at low speeds. Now, this is because of the low hanging CG which is being suspended by the center of volume which kind of creates a pendulum effect and at low velocities we see that this is stable and at higher velocities as the speed increases it is quite possible that the aerodynamic unstable aerodynamic pitching moment might increase and it would overpower this overcome it would be much larger than the stabilizing pendulum moment provided by the low hanging CG. And as I mentioned, yes the fin damping as well. So, stable in roll again it is also because of the pendulum effect and the fins also provide some damping for roll and we have seen that they are directionally unstable and similar to what we have in pitch because airships are stable pitching moment a yawing moment the equations if you look at them they are actually quite the same because if you take out the gondola from the airship it is basically symmetric about the axis about the longitudinal axis. So, what we have here is the yawing moment and the pitching moment they are basically one and the same because in terms of calculations and equations. However, in case of pitch we have this pendulum effect because of gravity but in case of the yawing moment there is no gravity because in the lateral direction there is no gravity acting and because of that there is no restoring force from gravity and that is the reason why there is no corrective forces or elements which would stabilize it directionally and therefore airships are directionally unstable. It is also because of the undersized fins and to counter this problem what we do require is a active control system which would basically take the feedback whenever the airship is off course you basically take the feedback and continuously a computer would generate a corrective action for stabilization.