 Hello and welcome. In the last lecture, we had looked at the basic algebraic strategy to set up the problem of configuration design of a multi-stage rocket and in the process, we defined important parameters for the stage as stage structural ratio and a stage payload ratio. In this lecture, we will now demonstrate those expressions through a simple example and then we will also establish a reasoning for adopting a more rigorous approach for arriving at the launch vehicle mass configuration. So, let us begin. So, let us now mechanize the solution methodology based on the relations that we have derived in the previous lecture. Let us make use of m star epsilon i n pi star to determine the stage configuration which is meant by the MSI and MPI for each stage as well as the total lift of mass in the following manner. So, let us first take the expression for epsilon i which is written as MSI by MSI plus MPI. Now, let me do a little bit of algebraic jugglery and that gives me MSI plus MPI as 1 by epsilon i into MSI. Similarly, I take the expression for pi i, invert it and write it as 1 by pi i as MSI plus MPI plus m naught i plus 1 which is nothing but m naught i divided by m naught i plus 1 and I get another expression for MSI plus MPI as 1 minus pi i by pi i into m naught i plus 1. So, now as you can see, I have two equations for MSI and MPI as unknown in terms of the two known parameters epsilon i and pi i. So, I am assuming that epsilon i and pi i are going to be available to me and based on that I should be in a position to solve for MSI and MPI. Of course, these are not very straightforward equation because if you see the second one m naught i plus 1 appears in the second one. So, it cannot be directly solved and there is a strategy by which we will be in a position to solve these equations. One thing that you might have realized that the relations that we have written are recursive which means they are for each i. So, if I substitute i in the previous relation what I get is the relation in terms of i plus 1. Directly what it means is that my solution for ith stage is going to be possible only if I am able to solve for i plus 1 stage first. Now, let us extend this logic until we reach m star and now we realize that m star is something which is to be specified and it is generally available as a design requirement and now you realize that once m star is specified you should be able to solve for nth stage first. That is you are starting now from the top and once you solve for the nth stage the nth stage solution will drive the n minus 1th stage etc until you reach the last or the first stage and then when you add all this you are going to get the total rocket configuration. So, now this is the mechanization of the equations that we have just now seen. Of course, we can now use this to show that msi for ith stage is a function of m0 i plus 1 and the pi i and the epsilon i. Similarly, mpi is directly driven by the same m0 i plus 1 and now my m0 is going to be m star plus sum of all these. So, this is how I am going to actually solve the problem. You can see that if I have epsilon i and pi i I can solve for the mass configuration of the complete rocket including the lift off mass and because m star is known as I have mentioned the solution proceeds from top downwards. So, this is how the design solution is proceeded for obtaining the overall rocket configuration. Let me demonstrate this through an example of a rocket which is in literature you can look it up called Angara 1.2. It is a two stage rocket which is supposed to launch a 4 ton payload and has the following stage parameter as per our solution methodology. Its first stage has the propellant with ISP of 310 seconds. Of course, for this exercise it is not really required. It has a structural ratio of 0.072 or what I call 7.2 percent structural mass and remaining 92.8 percent propellant mass and a stage payload ratio of 0.188 which means the ratio between the mass above the first stage and the lift off mass of the first stage is 0.188. Similarly, for the second stage the ISP is 342.5 epsilon 2 that is the stage structural ratio is 8.9 percent and the stage payload ratio is 0.124. With these numbers let us try and determine the first the stage wise mass distribution. And then get the total lift off mass and see if it matches with the actual lift off mass of this rocket which is recorded in literature. So, let us start the process. Let us go through it step by step. So, we start from the second stage because there is the last stage and M star is known to us. So, I calculate MS2 as epsilon 2 into 1 minus pi 2 by pi 2 into M star. I perform this simple arithmetic and I get the structural mass of second stage as 2.51 tons. Similarly, I use the MP2 expression and I get the propulsion mass as 25.74 tons. Now, my starting mass for the second stage is MP2 plus MS2 plus M star. So, this is nothing but my M0 2 because this is going to drive the configuration of the first stage. With this M0 2 now I immediately calculate MS1 which turns out to be 10 tons and MP1 which turns out to be 129 tons. I just add all these that is M0 2 plus MP1 plus MS1 which is the M0 1 or the lift off mass it turns out to be 171.6 tons. This I will leave you to verify that the actual lift off mass or the gross lift off mass of Angara 1.2 is around 171 tons which means that the exercise that we have carried out essentially is feasible methodology for arriving at the stage wise distribution of masses for a given rocket which has epsilon i's and pi i's specified and we also note that the configuration is quite practical and realistic. The problem is that this requires the values of epsilons and pi's. Now, we have already said that epsilons are known from structural technologies but pi's are the ones which we do not know and then those are the ones which we are supposed to generate and only after we know the pi i's the methodology that I have demonstrated can be used to arrive at the actual mass configuration. So, even before we can make use of these relations and the solution methodology we have to talk about pi i's as the real unknowns which have to be determined from a different strategy and how do we arrive at such a strategy and that will be nothing but statement of our design problem. So, we have already noted earlier that the mass configuration through this problem is directly a function of pi i's. So, obviously different pi i's all of which satisfying the constraint that product of all pi is equal to pi star will generate different mass configurations. Question is among those many possibilities that we have which is the one which we should choose. How do we ultimately get those pi i's which ultimately are compatible with a mission that we have in mind and in this context we need to realize that from a practical perspective we want a mission cost to be as small as possible. Now, mission cost we normally define as amount of payload per unit lift of mass that we get and higher that value lower is the cost which means our mission is more cost effective if we are able to launch a higher payload with a smaller lift of mass. There is another issue of safety and accuracy that we must do this exercise without affecting the overall safety of the vehicle as well as the accuracy of our orbital mission which is the primary objective. So, the design task in a nutshell is to choose among the many combination of pi i's which all satisfy the constraint and also result in the best rocket for a given mission. This is going to be our overall objective in arriving at the configuration design of a multi-stage rocket. Let me just make a mention that optimal techniques which are available in many forms are commonly employed to achieve this objective. So, to summarize simple algebraic strategy presented here is able to provide a fairly realistic stage wise configuration for multi-stage rockets. However, we see that configuration so obtained strongly depends on the stage wise payload ratios which are generally going to be obtained from an optimization procedure. Hi. So, in this lecture we have seen the working of the formulation that we had given in the last lecture through a simple example of Angara 1.2 and we found that the methodology is workable and gives a realistic estimate of not only the lift of mass but also the mass that is there in each of the stages. Of course, we have also noted the fact that this is subject to availability of values of epsilon 2 and pi 2, epsilon 1, pi 1, etc. which have to be connected to the mission performance because they affect the mission performance itself. Now, among the many possibilities we have kind of given a justification that we would need to use some optimization methodology for arriving at the values of pi i's that will give us a best possible rocket. We will look at some of these ideas in the next lecture. So, bye. See you in the next lecture and thank you.