Скачать 202.35 Kb.

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 0.51 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 launch vehicle needed to be devised to provide more accurate results reflecting the actual size of the launch vehicle and payload. After the linear scaling method was proven to be very inaccurate, we based our next attempt at sizing the vehicle 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.
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 provides us with better results than linearly scaling the vehicle based on payload mass. 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 for each stage. This exact method for determining the size of the vehicle’s stages is not used as the final sizing method; however, since an automatic size optimization routine is included into the MAT code. The MAT code is then used for all sizing problems through the end of the design process. References ^{1}Wade, M., “Vanguard”, 19972007. [http://www.astronautix.com/lvs/vanguard.htm] ^{2}Tsohas, J., “AAE 450 Spacecraft Design Spring 2008: Guest Lecture Space Launch Vehicle Design”, 2008 A.1.2.3 Aerodynamic Coefficients A.1.2.3.1 Drag Coefficient The coefficient of drag C_{D} is one of the most important aspects of launch vehicle aerodynamics. This small, nondimensional number impacts many features of the overall launch vehicle design. A few examples include the amount of thrust needed for an appropriate thrust to weight ratio, the overall ΔV required to reach orbit, and the ability to control the launch vehicle. The C_{D} is essentially a means of representing the impact a launch vehicle’s shape will have on the amount of drag incurred as the launch vehicle speeds through the atmosphere. The manner in which the C_{D }achieves this impact can be seen through Eq. (A.1.2.3.1.1)
where D is the total drag, q is the dynamic pressure, and A is the area. One of the first goals of the aerothermodynamics group was to further understand the impact of launch vehicle geometry, Mach number, and angle of attack on the C_{D}. Doing so would allow us to put some preliminary limits on certain aspects of the launch vehicle design, such as diameter, and maximum tolerable angle of attack. The C_{D }is highly dependent upon Mach number. In the subsonic regime C_{D} is relatively low. In the transonic regime it raises to its highest value, and in the supersonic regime it reduces back to a lower value. An example of this trend is shown in Fig. A.1.2.3.1.1. Fig. A.1.2.3.1.1: Impact of Mach number on C_{D} for V2 rocket.^{1} (Jayme Zott) Not only is the C_{D} defined by the speed of the launch vehicle, it is also defined by aspects of the geometry such as diameter, number of fins, and length. Referencing data from the Vanguard and other historically successful launch vehicles, we realized that as the rocket diameter increased, so did the C_{D. } To get an idea of exactly how much the C_{D }increased with respect to diameter, we referenced established model rocket programs. Table A.1.2.3.1.1 shows outputs from the Aerolab^{3} model rocket program using Vanguard geometry with varying diameter.
(Jayme Zott) From this information we were able to determine that our final launch vehicle diameter should not exceed 2.00 meters in length. Doing so would lead to undesirable C_{D} values in the transonic regime. The angle of attack also has a noticeable impact on the C_{D}. In order to deduce the magnitude of this impact, we referenced historical data from various launch vehicles.^{1,2} Using this historical data, we created general trends for the subsonic, transonic, and supersonic regimes shown in Eqs. (A.1.2.3.1.2), (A.1.2.3.1.3), and (A.1.2.3.1.4) respectively.
where C_{D} is the coefficient of drag, C_{D0} is the initial coefficient of drag, and M is the Mach number, and α is the angle of attack. By extrapolating these empirical results we were able to show the impact of a wide variety of angles of attack on C_{D}. Fig. A.1.2.3.1.2: Impact of Angle of Attack on C_{D}.^{3} (Jayme Zott) Knowing historical trends for the impact of Mach number, launch vehicle geometry, and angle of attack on the C_{D} is of great use in preliminary analysis. When the team began work on creating a final design configuration, it was necessary to solve for the C_{D} in a much more refined manner. In order to take into consideration all elements of the launch vehicle geometry, angle of attack, and Mach number for the final design analysis, linear perturbation theory was used. Linear perturbation theory is the method in which the pressure over the top and bottom surfaces of the launch vehicle is integrated to solve for axial and normal force coefficients acting on the launch vehicle. From these axial and normal force coefficients, we are then able to use Eq. (A.1.2.3.1.5) to solve for the C_{D}.
where C_{D} is the coefficient of drag, C_{N} is the normal force coefficient, C_{A} is the axial force coefficient, and α is the angle of attack. An explanation of how linear perturbation theory is implemented can be found in the following sections on aerodynamic forces, A.1.2.3.2A.1.2.3.7. 