A.1.2.2 pg 1

A.1.2.2 Initial Sizing Function

We began the project by trying to come up with a method for determining the shape of the launch vehicle. The first method we used was to linearly scale the vehicle by payload mass. The dimensions of two rockets were used for data points; the Vanguard rocket and the Purdue Hybrid Launch Vehicle. This method, however, was proven to be ineffective at sizing the vehicle because it yielded unrealistic overall lengths for small payload masses. This method was then abandoned in favor of sizing the vehicle based on the volume of propellant in each stage. This method yielded realistic lengths for every vehicle since the size was based off of how much propellant each stage would need instead of a scaling factor based off of the payload mass. This method involved manually optimizing the length and diameter of the vehicle to obtain the final vehicle dimensions. This proved time consuming so this method was also discontinued due to a similar method being employed in a large optimization code (MAT code).

To begin the initial sizing of the vehicle, a sizing function was needed. We decided on an initial sizing scheme of linearly scaling the Vanguard rocket based on payload mass. The linear relationship was calculated using Vanguard payload mass data along with stage length and diameter data found from an online source for historical rockets.1 For a second set of data points, the payload mass, stage length, and stage diameter data from the Purdue Hybrid Launch Vehicle were used.2 This data was then entered into Excel and a linear relation between length and diameter was found with respect to payload mass for each stage.An example of how the sizing functions were calculated is shown in Figure A.1.2.2.1 below.

Figure A.1.2.2.1: Sizing function regression plot for vehicle second stage.

(Chris Strauss)

Figure A.1.2.2.1 shows the regression plot for the length of the second stage of the launch vehicle along with the sizing function associated with the stage. This was created by entering the data for second stage length of Vanguard and the Purdue Hybrid Launch Vehicle versus the payload mass of each. A linear regression line between the points was then plotted and the equation of the line was used as the sizing function for the stage length where x is the payload mass. A similar method was used for each stage length and diameter until a complete set of dimensions was calculated for each launch vehicle for each different payload mass. The results of this scaling are shown in Table A.1.2.2.1 below for the overall length of the rocket. The results by stage are shown in Tables A.1.2.2.2 through A.1.2.2.4.

Table A.1.2.2.1 Initial Scaling Overall Length vs. Payload Mass
Variable / Value / Units
Payload Mass 1 / .2 / kg
Payload Mass 2 / 1 / kg
Payload Mass 3 / 5 / kg
Overall Length 1 / 0.5112 / m
Overall Length 2 / 2.556 / m
Overall Length 3 / 12.78 / m
Footnotes: 1,2,3 for lengths refer to masses 1,2,3
Table A.1.2.2.2Initial Scaling Stage 1 Dimensions vs. Payload Mass
Payload (kg) / Length (m) / Diameter (m)
.2 / 0.25268 / 0.6953
1 / 1.2634 / 0.7357
5 / 6.317 / 0.9377
Footnotes:
Table A.1.2.2.3Initial Scaling Stage 2 Dimensions vs. Payload Mass
Payload (kg) / Length (m) / Diameter (m)
.2 / 0.11404 / 0.38582
1 / 0.5702 / 0.4271
5 / 2.851 / 0.6335
Footnotes:
Table A.1.2.2.4Initial Scaling Stage 3 Dimensions vs. Payload Mass
Payload (kg) / Length (m) / Diameter (m)
.2 / 0.05532 / 0.11524
1 / 0.2766 / 0.1502
5 / 1.383 / 0.325
Footnotes:

From the data presented, we found that a linear scaling method was not a good method to use. As evidence for this, we looked at the overall length for the 200 gram payload mass and found that it was unreasonably small at 0.5112 meters. The only reasonable dimensions calculated using this method were those for the 5 kilogram payload where the overall length was nearly half that of the Vanguard rocket. This is reasonable for a launch vehicle size considering the smaller payload that will be carried. A new method for sizing the rocket needed to be devised to provide more accurate results reflecting the actual size of the rocket and payload.

After the linear scaling method was proven to be very inaccurate, the new method of vehicle sizing that was devised was based on fuel volume. This method relied on finding the amount of fuel burned for each stage and the densities of the fuel being burned. This information was provided by the propulsion group.Below are tables showing the results of sizing the vehicle based on fuel volume.

Table A.1.2.2.5 5 kg Payload Launch Vehicle Dimensions for Various Fuel Combinations
Fuels / Length (m) / Diameter (m)
LOX/HTPB / 5.035 (stage 1) / 3 (stage 1)
7.634 (stage 2) / 4 (stage 2)
1.884 (stage 3) / 1 (stage 3)
H2O2/RP-1 / 9.250 (stage 1) / 6 (stage 1)
5.569 (stage 2) / 4 (stage 2)
2.330 (stage 3) / 0.75 (stage 3)
AP/HTPB/Al / 16.413 (stage 1) / 6 (stage 1)
9.364 (stage 2) / 4 (stage 2)
2.632 (stage 3) / 0.75 (stage 3)
Footnotes:
Table A.1.2.2.6 1 kg Payload Launch Vehicle Dimensions for Various Fuel Combinations
Fuels / Length (m) / Diameter (m)
LOX/HTPB / 10.352 (stage 1) / 6 (stage 1)
6.279 (stage 2) / 4 (stage 2)
2.753 (stage 3) / 0.75 (stage 3)
H2O2/RP-1 / 10.956 (stage 1) / 5 (stage 1)
4.581 (stage 2) / 4 (stage 2)
1.921 (stage 3) / 0.75 (stage 3)
AP/HTPB/Al / 13.509 (stage 1) / 6 (stage 1)
4.932 (stage 2) / 5 (stage 2)
2.169 (stage 3) / 0.75 (stage 3)
Footnotes:
Table A.1.2.2.7 0.2 kg Payload Launch Vehicle Dimensions for Various Fuel Combinations
Fuels / Length (m) / Diameter (m)
LOX/HTPB / 14.263 (stage 1) / 5 (stage 1)
6.008 (stage 2) / 4 (stage 2)
2.636 (stage 3) / 0.75 (stage 3)
H2O2/RP-1 / 10.483 (stage 1) / 5 (stage 1)
4.383 (stage 2) / 4 (stage 2)
1.833 (stage 3) / 0.75 (stage 3)
AP/HTPB/Al / 12.928 (stage 1) / 6 (stage 1)
4.720 (stage 2) / 5 (stage 2)
2.076 (stage 3) / 0.75 (stage 3)
Footnotes:

From the data shown in Tables A.1.2.2.5 through A.1.2.2.7, it can be seen that the vehicle sizes are all comparable to each other when similar diameters are used. This implies that a single launch vehicle could be used for all three payloads. This conclusion is based on very minimal optimization of each stage diameter, however. This data also shows that the method of sizing the vehicle based on fuel volume is a much better method for sizing than linearly scaling the vehicle based on payload mass. This is based on the vehicle not having unreasonably small lengths for each stage. Since the vehicle has realistic lengths, this method could be used for a more in depth sizing analysis once a particular fuel combination is chosen of each stage. This exact method for determining the size of the vehicle’s stages was not used as the final sizing method, however, since an automatic size optimization routine was included into the MAT code. The MAT code was then used for all sizing problems through the end of the design process.

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