 Hello and welcome with the background that we have created in the last lecture with regard to the multi-stage rocket configurations. Let us begin our discussion on how to set up the problem that will give us a configuration for a multi-stage rocket along with the issues involved in arriving at the solution. So let us begin our discussion on the multi-stage configuration of rockets with some ideas of how to set up the problem. In this context, let us consider the following configuration that captures the spirit of a multi-stage rocket that we may have. In this picture, you note that the complete rocket has been divided into segments with the subscript 1 and 2 called the stages and a final payload module given a subscript m star. Now within the each of the stages, it is further subdivided into two parts, the propulsion and the remaining structure. So we have mp1 and ms1, similarly we have mp2 and ms2. Let us now introduce additional symbology that we are going to use in our solution procedure. So we introduce m01 which is defined as the starting mass for stage 1 operation. In the present case, when the stage 1 starts operating, it is almost like the complete rocket and in most cases, this will be same as your lift off mass. In some special cases, it may not be so that we shall see later. Now as the stage starts operating, the propulsion gets burnt until you finish all the propulsion and once you finish all the propellant, what you are left with is mf1 which is the final mass for stage 1 operation and that contains the residual mt shell or the inert mass of stage 1 and then of course, the remaining part of the rocket which is the stage 2, the stage 3 etc and the final payload stage called m star. Let us now come to m02 definition and this is where we implement the idea that we have discussed in the previous lecture that before starting the second stage, we get rid of the structural mass or the inert mass of the first stage ms1 so that we now have a smaller m02 compared to mf1. So m02 is nothing but mf1 minus ms1. And then of course, the cycle continues that when second stage completes the operation, it would have burnt the propellant mp2 and then what would be left will be mf2 and then of course, the subsequent stages. We can now write these m01, mf1 etc in terms of the applicable parameters as shown here. So we know that the lift of mass m0 is going to be sum of propulsion mass for all the stages, the structural mass for all the stages and the payload stage mass. And now we introduce an important definition, the stage-wise starting mass as the mass of that particular stage given by mpi plus msi and the mass of everything that is above it given by m0 i plus 1. Similarly, stage-wise ending mass would be m0 i minus mpi. So you can see that in this expression of mfi, mpi does not appear. And then of course, we have the stage-wise specific impulse ispi. So when we say that an i-th stage of n-stage rocket is operating, then for that i-th stage, whatever is above it is like a load on it, which means the i-th stage is carrying everything starting from i plus 1 stage till the payload end. And that becomes a kind of a payload for the i-th stage and that brings us to the definition of what we call the stage payload. So m0 i plus 1 becomes the payload for i-th stage because that is the amount of mass that the i-th stage will have to push along with its own mass of course and it is appropriate that we use the same interpretation. And then of course, we have already seen that the inert mass of the i-th stage is separated after its burnout before we start the i-th one stage and that becomes m0 2 which is the starting mass for the second stage for which m0 3 will become the payload and this cycle will continue. Based on this, let us now define the applicable design variables for a multi-stage configuration. So let me introduce a parameter called pi i named stage-wise payload ratio which is nothing but the ratio of the payload of the i-th stage and the starting mass of the i-th stage. You may immediately connect this with the basic definition of the payload ratio that we have already used for the mission which is defined as m star by m0. It is the same strategy that we are using here to say that for the rocket as a complete unit the payload ratio is m star by m0. So in the same light for each stage the payload ratio is what it is pushing to its starting mass. We will find that it is an important parameter which will help us to set up the configuration design of multi-stage vehicles. We also introduce another ratio called the stage-wise structural ratio and denoted by symbol epsilon i which is nothing but an indication of how much of the structure is used in a given stage. So if the stage has 100 kgs, how much kg is the structure and the remaining will all be treated as the propellant. In some way it also tells you how efficient is your stage that how much propellant you can carry for a given stage and lastly of course ISPI is as we have seen earlier. I think it is also worth noting here that epsilon i's also indicate the status of the structural technologies, for example whether you are talking about metallic structures, composite structures or even frame structures or honeycomb structures the epsilon i's will be different for each of those. Similarly ISPI captures the status of propulsion technologies which indicates whether you are talking about a solid propellant, liquid propellant, cryogenics or any other hybrid combination like air breathing engines while the PI i's are the primary parameters which indicate how the lift of mass is distributed between different stages. So how do we formulate this problem? We formulate this problem by defining mission payload ratio pi star which is m star by m naught and there is a constraint that this particular ratio has to obey in terms of the stage payload ratios that we have defined in the following manner. So we define pi star as m star by m naught and then it is not very difficult to see that we can define a recursive process as shown through this multiplication until we arrive at m01 and as you can see each of these represents a stage payload ratio. For example m star by m naught n is the payload ratio for nth stage and not n by m naught n minus 1 is the stage payload ratio for n minus 1 stage etc. So finally we come to the last stage that this m02 by m01 is the payload ratio for the first stage and this in a compact form we can represent as product of all the individual stage payload ratio and that becomes now the main constraint that this rocket must obey that the product of all the individual stages which is the actual mass configuration of the vehicle is not unconstrained it cannot be done in any arbitrary manner it has to be done in such a manner that their product is always equal to pi star of course even within that constraint there are infinite possibilities of pi i s which will result in the same in pi star and that will result in multiple design solutions for the same payload ratio specification and this is something that we will have to deal with when we are setting up the formal design process. Let me also make a mention of a related issue which is going to be discussed in some detail later is that there are strap on stages or what we call the boosters which in most rocket are used then there will be one more payload ratio which will get into this definition called pi naught which will be the payload ratio for the zeroth stage. So sometime the booster stage or the strap on stage is also called the zeroth stage to differentiate from the first second stage etcetera we will talk a little bit more about it when we talk about the parallel staging. Now let us understand how we are going to solve this problem and then we note that in most cases for most space agencies the propulsion and the structural technologies are generally available as a set of options that they have developed over a period of time and because of that the ISPs and epsilon IS are available as a set of discrete values depending upon the number of technological options that a particular agency has. You will note that in such a case epsilon IS and ISPs are not really directly designed per say but are only selected from available set of discrete values in a given context and the only unknown for the design problem is the stage payload ratios pi IS under the constraint that their product must be equal to pi star and that will be our statement of the design problem. Apart from m star or pi star as the primary driver for the design we have also seen that a rocket has a mission objective in terms of orbit and we have also established in our earlier lecture that the ideal burnout velocity is a good indicator of the nature of the mission that the rocket is required to perform. So, in that sense design specifications that are commonly used to drive the design or the solution for pi IS which are m star or pi star and the v star the burnout velocity. You will find that this is generally adequate for initial sizing and initial conceptual or even primary design of launch vehicles because most of the other parameters are now going to be related to these parameters. For example, if I know the ideal burnout velocity and if I know the mass configuration then by appropriately implementing the burn profile I can find out what are going to be the actual velocity and altitude profiles and I am going to get the trajectory. So, which means that once I get a configuration of rocket in terms of pi IS for a given v star and pi star or m star as the case may be we have the whole rocket configuration and we also have the trajectory. So, how do we set up this problem? Ideal burnout velocity for a multi-stage rocket is nothing but the sum of the velocity increments given by each of those stages. Each of those stages add an ideal delta v and if I add all of these the total is nothing but the ideal burnout velocity that I will get from this multi-stage rocket. So, I can express this in terms of the parameters that I have defined as follows. So, you already know from our earlier discussion that the ideal burnout velocity from an IF stage is nothing but g0 ISPI ln of m0 minus mpi which is nothing but mfi divided by m0i. Of course, we just use the minus sign here. I could easily invert the logarithm and the minus sign would disappear. So, it does not really matter. I going from 1 to n indicates that this velocity increment is summed up over n stages that gives us v not or v star that both symbols are used interchangeably. Now, we have already defined the two mass parameters pi IS and epsilon IS as non-dimensional parameters and as the ratio inside the ln is also non-dimensional it is now appropriate that we introduce those two parameters as part of our velocity expression. This is shown next. So, v0 in terms of the three parameters ISPI epsilon i and pi i can be written as follows. So, I am just showing you the longhand derivation of this. So, m0 minus mpi by m0i. So, m0i is nothing but msi plus mpi plus m0i plus 1. I know it from my definition, but now mpi gets cancelled. So, what is left is msi plus m0i plus 1 divided by m0i. I separate the two terms and the first term that is m0i plus 1 by m0i is nothing but my pi i by definition. Similarly, the second term msi and m0i I multiplied by msi plus mpi and divide by the same. So, the first term with the division becomes my epsilon i. In the second term which is msi plus mpi in the numerator, I can rewrite in the form as m0i minus m0i plus 1. This is going to be which means the difference between the starting mass of the ith stage and the starting mass of i plus 1th stage is nothing but the mass of the ith stage msi plus mpi. The reason for this is quite obvious because I want to get this in terms of the parameters that I have defined. Again, I see that m0i by m0i is nothing but 1 and m0i plus 1 by m0i is pi i and I get an expression called pi i plus epsilon i into 1 minus pi i and this can also be written as epsilon i plus pi i into 1 minus epsilon i. Substituting this back into our velocity expression, we now have these two forms of the velocity expressions in terms of the non-dimensional parameters pi i and epsilon i and the dimensional value G0 ISP. G0 into ISP is nothing but the velocity units. So, to summarize, the multi-stage configuration design essentially is a simple algebraic methodology that makes use of the basic staging philosophy that we have discussed. The staging philosophy that we have discussed is typically called the serial or the series staging and in some context it is also called restricted staging. You also note that by defining stage payload and structural ratios, it is possible to create a feasible design methodology. We will look at the solution for these in the next lecture. Hi. So, in this lecture, we have seen a simple algebraic formulation for a multi-stage rocket configuration based on the fundamental ideas of stage operations as has been discussed earlier and we have noted that by defining stage payload ratios, the end the stage structural ratios, it is possible for us to set up equations which when solved in an appropriate manner will generate a mass configuration that should meet the specification. In the next lecture, we will look at the basic concept solving these through a simple example and then understand the possibilities of arriving at the solution through different techniques. So, bye and see you in the next lecture. Thank you.