 not Professor Panth, I am Sohan Swarna, I am a PhD research scholar at IIT Bombay and Monash University. I am also a project research engineer at the LTA lab in the department of aerospace engineering at IIT Bombay. So I work on the design and development of autonomous airships and during that study I have also done dynamics modeling and also studied the aerodynamics of airships. So in this talk I am going to talk about the dynamics of airships. So let us start. So let us look at a few obvious facts that we already know about airships. We know that they are low speed vehicles, they operate around less than 160 kilometers per hour and that corresponds to a marked range of about 0.2. They also have large surface area and we also know that they are buoyant vehicles. So therefore, the weight of the displaced air and is it nearly equal to the weight of the airship. So you could in some sense say that the total weight of the airship is 0. The net weight, the net resultant weight of the airship is approximately 0. And we also know that they are susceptible to wind disturbances that is they are very sensitive to wind and weather conditions. So these are some of the things that we are going to look in these series of talks. So let us start. So let us look at some historical background as to all the airship disasters or accidents that have happened from 1896 to 2016. So we see that the major chunk is related to lifting gas that is 33 percent. Now let me tell you this 33 percent of the accidents most of them have happened before 1950s because after that there were several restrictions on the lifting gas being used and also several safety precautions were taken even when hydrogen was used. So that is the reason why most of the early accidents were because of lifting gas related. But something that has pertained for a long time is weather or wind related that contributes about 30 percent of the total airship accidents that have happened since 1896 to 2016. And this kind of suggests that airships are susceptible to bad weather and windy conditions. Another major component is the loss of control or airship colliding into surrounding objects. Now this could also be because they are huge and they are difficult to maneuver. And it is also because there is this loss of control and airships are very tricky to control. So that is the reason why you have this major chunk which has caused disasters in airships. Now in this talk we are going to try and understand why it is so difficult to control an airship and also try to decode a little bit of the motion of airship. Now what do we mean when we say we are doing a dynamic modeling of an airship? Well dynamics is nothing but the study of the motion. So we are basically studying how the airship moves when we give certain control inputs to it may be some amount of thrust input or some control surface deflection something like that. Or maybe even study the influence of the wind. When wind acts on the surface of the airship it is going to show some behavior, some changes in the motion that it is already traversing. So that is what we are going to study. And we are also going to, we also use dynamics modeling in the control in the design of control algorithms for autonomous airships. So that is why it is important to learn about the dynamics of airships. Now dynamics of airships have been mostly derived from submarines. It is primarily because both of these are buoyant vehicles and they have a lot of similarities. For instance the aerodynamics of airships are very much similar to the hydrodynamics of the submarines. So that is the reason why most of the things have been adapted from submarine dynamics modeling. Now just let us look at some of the conventions that we use when we have to do the dynamics modeling of an airship. Now the first major thing that is different as compared to conventional aircraft is that we center the dynamics model and about the center of volume. Typically in aircrafts we center it about the center of gravity, we do not consider the center of volume at all. Now the reason that we do this in airships is because the center of gravity keeps on changing and that is due to the fact that we have ballonets inside which can change the distribution of air inside the airship and thereby that would also change the center of gravity. So that is one of the major differences between a conventional aircraft and an airship that we center the dynamics about the center of volume. Now similar to other aircrafts and any object which moves in 3D space, airships also have 6 degrees of freedom which is 3 translation and 3 rotation. So the 3 translation motion would be these along the x-axis and this along the lateral axis that is the y-axis and then we have the upward motion that is the z-axis and similarly we also have rotation. So about the x-axis we have roll and about the y-axis we have pitch that is this motion and about the z-axis we have yaw. So that would be about this. So we have 3 translation and 3 rotational motion and this corresponds to 12 state variables. So we have 3 position coordinates as I explained. So we have 3 x, y and z. So these are the 3 position coordinates that would be required to locate and tell where in 3D space the airship is and then we also have 3 Euler angles or basically these are the orientations. So for instance this would be the pitch angle. So the rotation about the y-axis would be the pitch angle and then the rotation about the x-axis that is this axis would be the roll. So when the airship does this it is basically rolling and then about the z-axis we have yaw. So it basically moves like this. Now about these 3 position coordinates the x, y, z and the 3 Euler angles that is roll, pitch and yaw we have 3 velocities corresponding to it. So it would be the linear velocity, lateral velocity and the vertical velocity. Similarly we have the pitch velocity, the roll velocity and the yaw velocity. Now how do we define what the x-axis are for this? So when I say x, y and z so we use the right hand rule. So we have the x-axis along the thumb. So we place the airship like this. So about the thumb we have the x-axis, about the index finger we have the y-axis and about the middle finger we have the z-axis. So this basically forms the x, y, z frame of reference and we center this above the center of volume. So that forms the x, y, z-axis. Now we also have this angular velocities and momentum corresponding to this particular coordinate system. Now that comes from the right hand thumb rule. So for instance when we have to do, when we have to look for the roll direction, which direction is positive and negative, we simply align the thumb about the axis. So roll is rotation about the x-axis. So about the x-axis we have this and the rotation is this way. So which means this is a positive direction. So if the airship is rotating in this direction, it means it is positive, positive roll. Similarly about the z-axis so which is downward. So we have this from the right hand rule, z is downward. So we have, we place the thumb, we use the right hand thumb rule which is downward and the rotation is this way. So basically the airship if it turns in this direction, it means that it has a positive yawing moment. And the last thing is about the pitch which is rotation about the y-axis. Now as I mentioned the index finger represents the y-axis. So we place the thumb, right hand thumb about the y-axis and upward direction. So basically when the airship pitches up, it is positive pitching moment. So these are the conventions that we follow for airships. Moving ahead, we have also do some assumptions to simplify our dynamics model. So some of the assumptions I am going to talk about. So one we assume that the airship mass remains fixed. We also assume that there is no error elasticity that is basically there is no fluid structure interaction. So we assume the airship to be rigid body which means there is no deformations when the airship is in a flight. And we also assume that the airship is symmetric along the vertical exit plane. Now as I explained the x is along the thumb and the z is along the middle finger. So about the exit plane it is actually symmetric. So this is a fair enough assumption that about the exit plane we have the airship to be symmetric. And the last assumption is that the CG and the CV line the same vertical exit plane. Now this basically means that we have this plane the exit plane and within that in the same plane we have the CG and the CV. Now this is not a bad assumption as well because we know that airship is symmetric about the vertical exit plane. So that is why the CG and CV will also lie in the vertical exit plane. Now two of these assumptions which is the first one airship mass remains fixed. We could say that some conditions apply to this. Now this is mainly because the mass of the airship changes. Now because by virtue of this assumption that we make the dynamics model that we will discuss is applicable when the airship is not is flying in the same at the same multitude without the bolognese changing their distribution of mass. And also this is an active research topic which is the fluid structure interaction. How does the airship deform or how does aeroelasticity of the airship envelope affect its dynamics? So that is an active research topic. So as I mentioned the mass of the airship changes. So you can see from this figure about the x axis you have the mass of the airship. So the airship is the heaviest at sea level and it increases the altitude. The air in the bologna gets plused out and they deflate. And at the pressure height what we see is the completely deflated bologna and that would be the lightest of the airship configuration. So this is what we have. So basically what I want to say here is that the mass of the airship does not remain constant. So whenever we have to apply the dynamics model we will make we will assume that the airship is not changing altitude and the bologna air distribution in bologna do not change. And the second one as I mentioned which is an active research topic is flow structure interaction. So people have been doing wind tunnel test to see how a inflated airship envelope would behave in when there is a wind flow around it. And this could also be done using computational methods using FEM and CFT and this also have been reported in the research. Now let us look at the airship dynamics model. So how do we actually model? So we are going to use in this case a Newton Euler approach. So which means we are going to take up all the forces, add it and then that would be the total force acting on the airship. So what we have here is that all the forces we take them all together, add them up and that would be the total force acting on the airship. Similarly for the moment, so all the moments acting on the airship. Now when I say all the moments or all the forces it includes everything. So it would be the forces due to the rigid body coupling or it could be aerodynamics, it could be buoyancy, it could be gravity. So we basically cover everything, we add all the forces and moment and what we get is the total force acting on the airship and the total moment acting on the airship. Now once we know what the total force of the force is acting on the airship and the total moment that is acting on the airship, we can find the linear acceleration. So that is basically given by Newton's second law that is force equal to mass into acceleration. So about each axis we have a certain force and we divide that by the mass of the airship what we would get is linear acceleration. Now because we have three axis we will also get three linear acceleration. Similarly we would get the angular acceleration when we divide the total moment by the total the inertia of the airship. Now the inertia of the airship is going to be different about the x, y and z axis. Now so that is something that needs to be kept in mind and that is how you would be able to find the three angular accelerations associated with the axis of the airship x, y and z. Now once we have the acceleration we could find the velocities by integrating it. So basically when we solve the above two ordinary differential equations now this linear acceleration and angular acceleration equations would be basically in the form of ordinary differential equations when we solve this what we would get is the linear velocities and when we further solve it by integrating it what we would get is the position values as well. So that is how we come up with the solution for the motion of the airship and that is how we basically decode. If these are the inputs or if these are the wind disturbances this is how the airship is going to behave. Now let us look at all the forces and moments that act on the airship. So what are the components that go into airship dynamics? So the one component that is primary is the mass and the inertia of the airship. So that is basically what we require to get the accelerations. Then as I mentioned we also have rigid body dynamics. Now this is because of the coupling between the linear velocities and angular velocities this has nothing to do with aerodynamics or any other thing. This is entirely because there is a coupling between the linear accelerations and angular velocities. The second one is the gravity. Now this is pretty obvious because airship has some weight it is also going to have some gravity which would always be normal to the surface and acting towards the center of the earth. Aerostatics is basically the buoyancy force acting on the airship. Now this can be determined using the volume of the airship, the density of the lifting gas and the density of the ambient air. So that basically gives us the total aerostatic or the buoyant force acting on the airship and this would be opposite to gravity. So this would be normal to the surface of the earth but it would be acting in the opposite direction of gravity that is away from the center of earth. Now there is another component which is propulsion. So basically whatever propulsion unit you have you could have propulsive motors for pushing the airship forward or you may have vertical thrusters which basically gives vertical thrust. So all those components all those forces and moments that are generated by propulsion that comes under this segment and then we have something very very peculiar that we only consider in buoyant vehicles like airships and submarines that is added mass. I am going to talk about it more in the next part of this lecture and the other one is aerodynamics which is basically because of the motion of the airship. Now added mass and aerodynamics both of these are by virtue of the motion with respect to the air mass. Now both of these are because the airship is moving through the fluid mass that is the air mass that is the reason why we have added mass and aerodynamics. We are going to talk more about it in the next lecture. Now let us talk about how do we trim an airship? By trimming I mean how do you put an airship in a steady level flight basically which is what is required of an aircraft when in normal conditions. So basically cruising an airship. So the first thing we are going to do is we are going to assume neutral buoyancy. Well this is not a very necessary assumption because if let us say you are statically heavy that is your weight is a little the weight of the airship is a little higher than the total buoyancy that resultant the total heaviness of the airship could be addressed by giving the extra propulsion units vertically or can even be compensated by aerodynamic lift. Now in this particular case let us assume that we have neutral buoyancy which means the weight of the airship would be equal to the lift of the airship. The second thing that we require to ensure for trimming an airship is that thrust would be equal to drag. So that is basically the thrust which is provided by the propulsion units and the drag which is by virtue of the aerodynamics of the airship. Now these should be equal. Now this is a simplified model and they may not be aligned as shown in this figure so which basically means that we would have to consider the components. But just for the sake of this lecture let us consider that thrust and drag are acting along the same line and that is why we are going to assume that thrust would be equal to drag. The second thing also acting on the airship would be the moment and we are in this particular case we are going to consider the pitching moment and that also needs to be zero. So basically if the pitching moment is not zero what would happen is the airship would either go nose up. So let us say we have a resultant nose up pitching moment so the airship would do this upward or let us say we have a nose down moment so basically the airship would do this. We do not want that kind of moment so that is why we say that the total moment on the airship should be equal to zero. Now for the first condition that is thrust equal to drag we need to identify what this thrust would be. So for that first we have to fix the airspeed at which the airship is going to operate or cruise and then we will determine the drag. Now the way to determine the drag we are going to discuss this in the next lecture and once we know what the drag is we know that that would be equal to the thrust and that is how you determine what thrust is required to maintain a certain cruise speed. Now about the moment it is a little bit trickier as compared to the first one that is thrust equal to drag. So the first thing we have is to determine the destabilizing moment. Now what is this destabilizing moment? There are several destabilizing moments that could happen in an airship. The first one being monk moment. Now this again is a topic for a next lectures video and the other reason why we would have a destabilizing moment is the CG and CV may not be aligned about the visor plane. So what I mean by that is let us say your CG is about here, your center of volume is about here and for some reason your CG is towards the back. Now because your CG is here and center of volume is here the buoyancy force is going to act upward in this direction here at this point and the center of gravity is here. So the weight is going to act downward along this direction. So what you have is this destabilizing moment. So basically the airship is not going to pitch angle itself in total 0 pitch angle. So you are going to have a non-zero pitch angle. So that is the reason why airship would do this. Now that is something that we do not desire and the other one is the thrust offset or vectoring. So typically we always have this because the motors, the propellers are typically put on the gondola which is at the bottom half here. Now because the thrust acts along this direction from here so we are going to have this upward moment. So whenever the airship is flying because the forward facing motors are here we always have this moment which would make the airship pitch nose up. So that is another thing and this could also happen because of vectoring the thrust. Now how do we stabilize this? So obviously we have to create a stabilizing moment but how do we do that? So the first one is elevated deflection. So this is using the aerodynamic features of an airship. So basically you have elevators on the fins of the airship. So you basically deflect it such that in during the motion you have a total net 0 or equivalent nearly 0 pitch angle while flying and the other one other thing that you could do is offsetting the horizontal fin. However nobody does this. So what offsetting the horizontal fin means that you have these fins the horizontal fins. So these fins could be offset it could be set at an angle. Now offsetting the horizontal fin has its repercussions. So basically if you offset it you will always have a fixed aerodynamic normal force and moment because it is offset and that is something a little bit tricky to do and that is why people do not go for this option. The other one is adjusting the air in the bologna. Now this is something very very practical because just by adjusting the air distribution between the two bologna you can adjust where the CG of the airship would lie. Now the elevated deflection one is a very good option when you have monk moment or the destabilizing aerodynamic moment and when you have a CG and CV offset or even for that matter the thrust offset adjusting the air in the bologna is a better option to go for. Now there is something other than this that we need to discuss which is the pendulum stability of the airship. Now as I mentioned earlier in the assumptions that the CV and the CG they both lie on the same XZ plane. So basically you have this XZ plane and they both lie in that particular plane. Now CV is always related to the envelope because that is the part which generates buoyancy. So the center of volume of the envelope is somewhere in the middle of this so it would always be about the envelope somewhere inside the envelope. So in this case in the figure you can see there is this yellow dot so that is the center of volume. Now CG is typically downward and that is because the gondola is placed there which is the heaviest component apart from the envelope that is the next heaviest component and that is why the CG is typically brought down because of because rest everything if you look at the airship mass distribution in the airship everything is symmetric except for the gondola. So if you look at this it is basically axisymmetric so you have four fins or you may have three fins. So if you take the central axis it would be axis it would always be axisymmetric except for the part except for the gondola. So that is why the CG of the airship is always under the center of volume. Now because of that and also we also have ballonet inside. Now because of the ballonet the air distribution the air is concentrated at the bottom of the envelope the CG is even further pushed down. Now because of these two reasons that is the gondola and the air distribution in the ballonet we always have the CG lying under the center of volume. Now let us just look at this. At the center of volume we have the buoyancy force which is acting upward so it is like this from the center of volume it is going upward and at the CG we have the gravitational force which is acting downward. Now this is something which is very very similar to a simple pendulum. So basically what we have is the airship is suspended about the center of volume upward direction so that is like a reaction and we have the CG which is downward and which it basically does this. Now from our basic understanding of dynamics we know that a simple pendulum is kind of stable because there is damping air damping and also damping provided by the fins. So it would eventually stop oscillating and you would it would go into its normal state of zero oscillation. Now this is the case for two motions which is one is pitch so basically the airship would do this and eventually it would basically the oscillations would die out and the second one where it would act is in the roll. So whenever you have this roll kind of motion it is going to basically oscillate and you are going to have this oscillation dampen out. Now we do not have this for yaw but we have this for pitch and roll. Now this is something that we need to look at later on we are going to discuss this a little more in the aerodynamics part of this lecture but for now just keep in mind that we have this pendulum stability of airship mainly because the center of gravity lies a center of gravity of the airship lies under the center of volume of the envelope of the airship. Now that concludes the dynamics part of the this lecture and in the next lecture we are going to talk about the aerodynamics of airship. Thank you.