 Hello and welcome. In continuation to our last lecture, where we introduce the ideas of optimal staging, we will now look at the basic technique of optimal multi-stage design through the Lagrange's method, which provides optimal solution using one extra variable called the Lagrange multiplier. And we will probably also look at the possibilities of alternate ways of arriving at the optimal solution. So, let us begin. So, let us begin our discussion on the optimal staging solution given below is a broad procedure for solving rocket sizing problem in the present context. So, the first step is that we solve all n partial derivative equations for individual pi i's in terms of the Lagrange parameter lambda. So, all the n design variables pi i's are expressed in terms of the Lagrange parameter lambda. Next, all these solutions of pi i's which are in terms of lambda are substituted into the constraint equation, which then becomes an equation in lambda. We can solve all this equation, it is an algebraic equation and the solution of lambda so obtained is then substituted back into the pi i's that we have already expressed in terms of lambda and we obtain all the pi solutions. So, we see that in this procedure, we first have to express all pi i's in terms of lambda which is essentially an algebraic substitution. And then we solve an nth order algebraic equation in lambda which is arrived from the constraint relation. And once the lambda is obtained, we go back to those expressions and simply substitute the value of lambda and evaluate pi i's. So, let us look at this technique through the two options that we have established that is in one case, the V star will be objective function and the pi star will be constraint. And in the other case, pi star will be the objective function and V star will be the constraint. So, let us first look at the case where V star is the objective function. So, we use the augmented function hv as we have seen earlier which is nothing but minus g naught sum i equal to 1 to n ispi ln epsilon i plus 1 minus epsilon i into pi i that is the objective function part. And then we have the constraint part that is ln pi star minus sum i equal to 1 to n ln of pi i multiplied by the Lagrange multiplier lambda. Now, we construct the n partial derivatives of the above augmented objective function by differentiating hv with respect to pi i's. And please note because these are partial derivatives, we use the basic strategy of partial derivative that all terms involving only pi i's will be nonzero, all the terms which involve pi i minus 1 or i plus 1, they all go to 0. The moment we do this, we realize that this partial derivative will contain only terms corresponding to pi i. So, we get this derivative as g naught ispi into 1 minus epsilon i divided by epsilon i plus 1 minus epsilon into pi i plus lambda by pi i equal to 0. And this is an algebraic relation from which we can solve for pi i as minus lambda epsilon i divided by 1 minus epsilon i into lambda plus g naught ispi. So, these are the relations for all the pi i's in terms of the fixed parameters epsilon i and the ispi i and the Lagrange multiplier lambda. Now, the next step is to substitute these solutions of pi i's into the constraint relation. So, we write down the constraint relation as this product that is pi star constraint relation is product i equal to 1 to n minus lambda epsilon i divided by 1 minus epsilon i into lambda plus g naught ispi. We also know that once we obtain the lambda i's, the optimum velocity will be the sum i equal to 1 to n ispi ln epsilon i plus 1 minus epsilon i into pi i. And we immediately realize that a known value of pi star which is the constraint is going to fix the solution of lambda. And because it is a product on the right hand side, it is clearly visible that we will get an algebraic equation of power n in lambda. Let us now look at the counterpart of this particular solution where we would like to maximize m star or in this particular case the pi star the payload fraction with v star as the constraint. So, in this case, we use the augmented objective function h pi where the first term that is sum i equal to 1 to n ln pi i is the objective part coming from the pi star. And then we have the constraint error multiplied by the Lagrange multiplier lambda. Again, we go through the same process of differentiating this augmented function with respect to pi i. And similarly, we get only pi i terms in this. And by solving for pi i, we get an expression for pi i in terms of epsilon i, ispi and lambda i as minus epsilon i divided by 1 minus epsilon i 1 to 1 plus lambda g naught ispi. So, you can see that this expression is different from the expression that we obtain when we use v star as the objective function. We substitute these values of pi i into the constraint relation that is v star constraint. And then once we do that from this constraint relation, again we will get an nth order algebraic equation in lambda whose solution will give us the value of lambda which will fix the solution for all the pi i s. And using those values of pi i s, we can then obtain the optimal value of pi star. Here, the known v star is going to fix the value of lambda. So, we have seen from these two solution procedures that in both the cases the constraint is the one which will fix the solution of the weightage lambda which is the coupling parameter for all the payload ratio pi i s. And then it fixes their values in relation to the constraint that is applied. Now, there are certain special cases which we can examine. So, the first special case that is of interest is that if we had the same structural technology and the same propeller technology to be used in various stages of the rocket, what would happen? So, this is denoted as the equal stages which means all stages have equal epsilon and equal isp. In that case, we assume that epsilon is are all epsilon and isp is all isp. And we substitute these into the expression for pi i s. You will immediately notice from this that all the pi i s are going to be the same because all the pi i s are going to be the same. It is now a simpler algebraic equation for lambda that we get from the constraint. And by putting that equation, we redefine an additional parameter beta as v star by n g naught isp where v star is the velocity constraint. And the pi for every stage is e to the power minus beta minus epsilon divided by 1 minus epsilon. Because all the pi s are the same, the pi star is nothing but pi to the power n. You realize that this particular solution is an extremely simple representation if you have the same structure and the same propellant to be used in all the stages. Of course, if either the structure or the propellant or both are different, then obviously this formula is not applicable and we must use the expressions as given in the previous tool. Derivations. Let us look at the same thing for m star or pi star constraint. So, in this case, because pi star is a constraint, it can be shown that all the pi s will be same. Because all the pi s are same, the pi will be nothing but the nth root of pi square, pi star rather is nth root of pi star. So, directly that is the solution for your stage payload ratio and the v star now can be obtained directly from this value of pi. So, you realize that when we use the simplification of equal stages, the solution simplifies greatly. Let us now demonstrate these expressions through couple of examples. So, let us first look at the case of a two stage sounding rocket having equal stages, that is, it has epsilon 1 equal to epsilon 2 equal to 0.15. Let us try and determine the optimal pi and the lift of mass m naught for a m star of 10 kg. If v star required is 4000 meters per second, while burning a propellant of ISP equal to 240 seconds. So, the solution is as follows. Let us go through the steps one by one. So, let us first calculate beta which is v star by n g naught ISP as it is a two stage, it is 4000 which is the v star divided by 2 into 9.81 into 240. So, we get a beta value of 0.8494. Substituting this into the expression for pi which is e to the power minus beta minus epsilon divided by 1 minus epsilon, we get pi 1 as 0.3267. Now, this is the value which is common for both the stages. So, pi star becomes the square of this which is nothing but 0.1067. So, our payload fraction in this case which is maximizing pi star is 0.1067 and for a payload of 10 kg, the rocket must weigh roughly around 94 kg. So, now we have designed an optimal sounding rocket which has a payload fraction of 0.106 and a 94 kg rocket will be able to launch a 10 kg payload. Let us now flip the problem and look at when we want to put a payload constraint and see what is the solution that we get and what is the velocity that we are going to get. So, in the previous case, the payload fraction that we had got was 0.106. Let us try for a slightly higher payload fraction of 0.15 and let us see what happens to the solution or the same set of structural ratios and same ISP of 240. So, this solution is as follows. Now, we know that both the pi's are same which are nothing but square root of pi star and this 0.387. So, now you can see in the previous case the pi was 0.32, but now the pi has become 0.38. So, the payload ratios are higher because the payload ratios are higher. Now, I substitute these into my V star expression and what I get as V star is slightly lower. Instead of 4000 meters per second, I get only 3466 meters per second and here there is now an important result that we need to note. There is a trade-off between the burnout velocity and the pi star. If you want a higher pi star, you must accept a lower velocity or if you want a higher velocity, you must accept a lower pi star. Let us now go to the general problem where we have stages which are not equal and it is useful to recall the example that we saw in the last lectures about Angara 1.2 and let us say that that is the rocket that we want to redesign so that we get a payload fraction of 0.025 which means I want to use that rocket to achieve a higher payload fraction. So, my pi star has been fixed at 0.025 and let me see what should be the optimal staging and what will be the corresponding optimal velocity which I am going to get. For the first stage, the ISP is given as 310 and the structural ratio is 0.072. For the second stage, the ISP is 342.5 and the structural ratio is 0.089. Let us now try to determine a new stage-wise payload ratios and the corresponding ideal optimal velocity. So, the old parameters if you remember you can go and check the pi1 was 0.188, the pi2 was 0.124 and corresponding to these two, the V star was 9633.9 meters per second. This was the solution that we had obtained when we were looking at the mass configuration of the so basically we are having a overall payload fraction which is not very large. Now, let us formulate this problem in the context of the solution that we have obtained. So, let me just go ahead and substitute the value of epsilon1 and ISP1 into pi1 expression and similarly epsilon2 and ISP2 into pi2 expression and then I say that pi1 into pi2 which is pi star must be equal to 0.025 that is the constraint. So, this is our constraint relation. This results in with some amount of algebraic manipulation, a quadratic equation in lambda for a two-stage whose solution actually results in two values of lambda, lambda1 and lambda2. One is minus 2055.9, other one is minus 0.7140.7. You will pick one of those. In fact, I will leave you to verify which one we should pick because I will give you a hint that the other value will be an invalid value. It will give you an inconsistency in your solution which you could independently verify. So, I am not saying which one of these I have used but using one of those I get two solutions pi1 and pi2 as 0.162 and 0.154 and I get pi star as 0.029. Let me make a comment here. We have started with the specification of a payload fraction of 0.025 but we have ended up with the value of 0.029. Kindly note that this is essentially because of the truncation errors which are part of the solution process that we do not use all the decimal places and particularly when there are large numbers, their manipulations results in smaller numbers and if we ignore the higher digits, it is possible that you will result in little bit of error. You can actually verify this by doing a more accurate calculation and show that your pi star will be close to 0.025 which is the constraint that we have put and for these values of pi1 and pi2, you get V star is 8,337 meters per second and now you make a comparison. Originally, your pi star was smaller, your V star was 9,600 but now because you want a higher pi star, your V star reduces to a smaller value. Let us now look at whether this particular technique has an issue. It is good technique we have already seen but there are certain drawbacks that we must take note of. So, the first thing that it is seen is that we first need to get a solution for lambda before we can get solution for pii at least for the unequal stages. For equal stages, we are in a position to eliminate lambda so that it is a simpler solution but more often than not, we are not going to get equal stage configuration. So obviously, it is going to require a lot more computational effort and then of course, your lambda is an nth order algebraic equation. So, there are two issues involved with it. As you increase the number of stages to 3, 4, 5, the order of the algebraic equation is going to increase. So, you are going to get that many roots for lambda and then you will have to pick the one which is going to give the feasible solution. So, that is going to be an additional effort to pick among the lambda, the value which will give you the correct end. This can only be done by actually checking for all the lambda values. This can become a tedious exercise if all the lambdas are real numbers. If in some cases, some of the lambdas appear as complex conjugate, they can be straight away discarded because lambda has to be a real number that is the original interpretation with which this whole formulation has been done. So, it cannot be complex, but it can be a real number. So, if all the five roots for a fifth stage rocket is a real, then you will have to check for all those lambdas before discarding saying which one of them is consistent solution and remaining are inconsistent solution. So, it becomes lot more computationally intensive. So, is there alternate way in which we can do this? Alternate way should be such that it simplifies the process of solution as compared to the procedure that we have used here, but should not compromise significantly on accuracy, which means in some initial design stages, we may be in a position to sacrifice a bit of accuracy for computational comfort and simplicity of the solution process. So, to summarize, Lagrange multiplier based technique is capable of providing optimal multi-stage solutions that are also in the closed form. That is a great benefit, but we also note that we need to solve a slightly more complicated nth order algebraic equation for the Lagrange multiplier. Hi. So, we have seen in this lecture the mechanization of the basic procedure proposed by the Lagrange for extracting optimal solutions of a constrained optimization problem and we note that it becomes extremely simple in the context of equal stages and we have also seen that in the context of unequal stages, the numerical effort is going to increase almost exponentially as number of stages are increased and that there is a need to look at an alternate methodology that will simplify the process without losing the accuracy, which is what we will look at in the next lecture. So, bye. See you in the next lecture and thank you.