4262: Rockets and Mission Analysis Assigned: September 9, 2014

Homework #2 Due: September 23, 2014

Purpose of this homework assignment:

  1. Review the use of the rocket equation
  2. Consider the impact of initial acceleration on launch vehicle performance
  3. Understand the benefits and drawbacks of rocket staging, using a slightly different approach than that examined in lecture.

Question 1:

Our discussion of staging in class suggested it as a solution if a single-stage rocket was not capable of reaching the desired velocity. Assume a rocket of total mass 100 tons, carrying a spacecraft payload of 1 ton. The engines develop a constant exhaust velocity of 3,000 m/s. The structural mass is assumed to be 10% of the fuel mass.

1)  Determine the velocity of this configuration as a single stage rocket

2)  If the rocket is divided into two smaller stages, each with half the fuel, and the structural mass also shared equally, and the payload being the same, determine the total velocity increment for the two stage configuration.

3)  Repeat part (2), assuming 3 stages. What do you notice about the total velocity increment as you add more and more stages? As an engineer, how would you determine how many stages to use?

Question 2:

Our discussion of staging in class suggested it as a solution if a single-stage rocket was not capable of reaching the desired velocity. It may also be desired to use a staged rocket even when the desired final velocity can be reached in a single stage. In fact, one of the most contentious issues about launch systems is whether single-stage-to-orbit is the way of the future. To help your thinking on this important matter, it is suggested that you compare launch systems with 1, 2 and 3 stages, using as a Figure of Merit the ratio of Payload Mass / Initial Mass.

To make the comparison simple, assume the following:

Final velocity: 8,000 m/s

Exhaust velocity, constant for all stages: 4,000 m/s

Structural weight fraction (Structure Mass / Initial Mass) for all stages, including the engine mass: 0.07

Initial acceleration fixed for all stages at ao/g=2.


Question 3:

The purpose of this question is to address the issue of whether or not there is an optimum acceleration for an earth launched rocket (without drag).

Recall that the rocket equation can be written as:

/ 1

In this equation we have neglected the effects of drag and have assumed that the rocket is traveling in vertical flight. Vo is the initial velocity, which is zero for vehicles being launched from the surface of the earth, and Ve is the exit velocity of the propellant relative to the rocket. In the case where Pe≠Pa, we can replace Ve in equation 1 with Veq, which is defined as:

/ 2

If we make the assumption that that is constant (which is a pretty good assumption), then we can write:

/ 3

In equation 3, ao is the initial acceleration, and we can use this result to eliminate t in equation 1:

/ 4

The last term is the gravity loss, since it is the reduction in velocity increment that the vehicle suffers due to the acceleration of gravity. This is an implicit equation for in terms of , with the initial acceleration ao as a parameter. As ao increases, the gravity loss decreases, since the time over which gravity acts is reduced. So why not use a very large ao to get the largest possible for a given? To answer this question, we can divide the mass of the rocket into parts:

/ 5

When all the propellant has been expended at time tb, we have:

/ 6

We can write:

/ 7

Examining equation 4, we can see that the smaller or the closer is to 1, the larger the can be achieved. But the limit is set by:

/ 8

We want to design components of the rocket so that and are as small as possible, given the loads and available technology. Some estimates are shown in the table below:

Component or Component Ratio Relative to the Initial Mass Estimates
/ 9.a
/ 9.b
/ 9.c

Substitute these estimates into equation 8 and then into the rocket equation at time tb. Doing so, you should have 3 design parameters: , , . Ordinarily is set by the mission requirement, for example to get to LEO, and clearly we would like to have as large as possible. The key question is: What is the best? It is a bit hard to see the answer by looking at the rocket equation and trying to solve for. To find an optimum is analytically messy. The best way to do this is to plot as a function of for various values of . Plot these results for =0, 0.01, 0.05, 0.1, 0.2, and 0.5.

Now, say we specify at 1.0, which is the maximum efficiency condition. Is there a best ?. How would you describe the optimum?

The optimization has considered only the effects of the gravity loss and the engine weight on the choice of . If we include other effects, we will get a different answer. This is shown in the next section.

The above example determines the initial acceleration that minimizes the initial mass, accounting for the gravity loss and the engine mass. In doing so, we assumed that the structural mass fraction was independent of the acceleration. In fact, it is not, because the structure must withstand larger forces if the vehicle is accelerated more rapidly. We can repeat this analysis, with the assumption that the structural mass fraction is proportional to the initial acceleration. The constant of proportionality is such that when =2. From the rocket equation we showed:

/ 4

Again the quantity which is equal to is given by equation 7. Modify the rocket equation for this new assumption criteria and re-plot the results. How has the locus of optima shifted? How would you explain this result? How would you expect these results to differ if the effects of drag were included?

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