MAE 4262: Notes on Gravity Loss and Optimum Acceleration

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, 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 often called 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 reasonable estimates are shown in the table below:

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

Now we can put these estimates into equation 8 to yield:

/ 10

Finally, putting equation 10 into the rocket equation to yields:

/ 11

For this simple model we hav 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 equation 11 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 . This is shown in the figure below:

Figure 1: Optimization of ao/ge: Effects of Gravity Loss and Engine Weight Included

Now, if we specify at 1.0, we can see that there is indeed a best at about 3, but it is a very flat 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. We can modify equation 11 to read:

/ 12

The result of this analysis is shown in Figure 2:

Figure 2:

We can combine these plots, as shown in Figure 3 to see how the locus of maxima have shifted with the added assumption.

Figure 3:

The optimum shifts to lower values. For the limiting case with no payload, it suggests that we would want to launch at an initial acceleration that is exactly equal and opposite to the earth’s gravitation.

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