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Fig. A.1.2.3.4.1, Variation of pitching moment coefficient with Mach number at 0º aoa (Alex Woods) Fig. A.1.2.3.4.2, Variation of pitching moment coefficient with angle of attack at Mach 3 (Alex Woods) We can see here that once again, C_{M} has a linear progression with angle of attack and a nice curve with changing Mach number. There are several characteristics that we must note about these plots. First, the slope of the plot in Fig. A.1.2.3.4.2 is slightly steeper than that in the normal load. This is expected, as it produces a changing CP with changing angle of attack. Also, in reality the plot in Fig. A.1.2.3.4.2 would have some nonlinearity, but in a less pronounced fashion than what one would find in the normal coefficient.^{3} Finally, we note once again that this data is not reasonable for the transonic or hypersonic regions. A.1.2.3.5 Center of Pressure Since we calculate both normal and moment coefficients, we can produce a reasonable location for center of pressure using Eq. (A.1.2.3.2.3). For the purposes of this analysis, we found it more useful to use the following modification, which outputs the CP location as a fraction of the body length from the tip of the nosecone: This equation was used directly to produce a CP location that changes with angle of attack, as the Vanguard report suggests it should.^{2} A visualization of this variance can be found in the following figure. Fig. A.1.2.3.5.1, X_{CP} vs. angle of attack for a 3 stage vehicle at Mach 3 (Alex Woods) Figure A.1.2.3.5.1 shows that the center of pressure will move aft along the LV body as angle of attack changes, which is what we expect for a launch vehicle.^{3} We have some issues with the validity of the results however. We found that the CP values being output by the code tend to begin lower and higher than real world data, by as much as 30% of the actual value. The change of CP from minimum to maximum also occurs faster than the Vanguard data would suggest.^{2} Finally, the location of the CP does not vary with Mach number in our results. While this is consistent with linear theory, it does not agree with information found in the Vanguard report. Vanguard has wind tunnel data showing an aft CP in the subsonic region, and a spike even farther aft in the transonic region. In the subsonic region this difference can probably be attributed to viscous effects. Since the location of the CP is determined by an integral of forces acting along the vehicle surface, it seems reasonable that if viscous effects were included, they would heighten the effect of long cylindrical stages present on the launch vehicle. This would be particularly true if flow separation occurred on the aft surfaces, which is also something not modeled by the aerodynamic codes. We note the same characteristics in the transonic region, with the addition of possible shocks as flow accelerates over the vehicle surfaces. A.1.2.3.6 Axial Force Coefficient We find axial force along the launch vehicle is the prime component of drag for low angles of attack. As such, deriving an axial force coefficient (C_{A}) based upon vehicle geometry is a top priority for the design team. Once again we turn to linear perturbation theory for a solution. Using the pressure distribution as described in A.1.2.3.2, we integrate with respect to body thickness as shown below: where dy denotes that we are integrating with respect to thickness, lengthwise along the LV. Figure A.1.2.3.6.1 provides a visual example. Fig. A.1.2.3.6.1, Axial force acting along the launch vehicle body (Alex Woods) CP_Linear_two.m calculates the C_{A} value in a similar fashion to C_{N}, with the major exception being that we left out the integration around the launch vehicle (leaving the analysis in two dimensions). We did this in order to more accurately fit our results to historical data, which was larger than we were predicting. These differences may have been in part due to viscous or separation effects along the vehicle body. Fig. A.1.2.3.6.2, Variation of axial force coefficient vs. angle of attack for a 3 stage LV (Alex Woods) Fig. A.1.2.3.6.3, Variation of C_{A} with Mach number for a 3 stage LV (Alex Woods) The model axial force coefficient does not change with angle of attack. This is consistent with historical and experimental data.^{3} These results should be reasonable up to at least 10º angle of attack, and the nonlinearity experienced afterwards is not very significant. We find that the axial force coefficient for a range of Mach numbers is fairly accurate when compared to historical data. The exception to this is that the Vanguard results have a decrease in drag for the middle of the subsonic region, while our model predicts a small increase. Recall as well that the transonic results from our model are not to be trusted. The axial force coefficient can be used to find a simple drag value by using the equation: where C_{D} is the drag coefficient for the LV. For 0º angle of attack the drag coefficient is equal to the axial force coefficient. Eq. (A.1.2.3.6.2) predicts an increase in drag as angle of attack increases, as we expect. Once our model progresses past approximately 14º it no longer predicts an accurate drag, because sizable flow separations will occur on the leeward side of the LV. A.1.2.3.7 Shear Coefficient We find that the derivation of normal forces and pitching moments allows us to derive some of the forces working within the launch vehicle. Shear stresses and bending moments are important considerations for the structures personnel to factor in to their analysis. To provide a solution, the aerothermodynamics group developed a code called CP_Structures.m. This code analyzes the lowest “connection point” on the LV at any given time. We define this as the point where the skirt meets the lowest stage; for our final models this was always the top of the first stage. We first derived the theory behind the shear stress on the LV, and ran the method by our structures contacts to promote accuracy. We defined shear stress as the force of one stage acting on another in a horizontal fashion. Fig. A.1.2.3.7.1, Normal coefficient along a LV surface (Alex Woods) The shear stress is the differential between the normal forces acting on the LV on either side of the shearing point. This means that if the sections of the launch vehicle are causing different amounts of aerodynamic force, the differences between those sections is going to manifest as shear forces within the vehicle structure. Or more bluntly, where x is the shear point. If this value emerges negative it means that the forces acting on the lower portion of the vehicle (the first stage) are greater than those on the rest of the vehicle. CP_Structures.m is designed to ignore shoulder sections such that the shear output is for each stage along the vehicle. The maximum loads experienced are at the junction between the first stage and the upper stages. A.1.2.3.8 Bending Moment The bending moment is slightly more complicated to compute than the shear stresses. In principle it once again uses the normal force. Since moment equals force multiplied by distance, and each section of the vehicle geometry has a local center of pressure, we can say that the normal force acting over a section of the LV will cause a moment acting about a point from the local CP. This can be visualized in Fig. A.1.2.3.8.1 below. Fig. A.1.2.3.8.1, Bending moments caused by normal forces acting at local centers of pressure (Alex Woods) If we take a point within the vehicle geometry to sum the moments about, we have an opposing moment pair causing the structure to fold in on itself. If we take nose up to be a positive pitching moment, the value for the first moment will be: where C_{bend,1} is the portion of the bending moment caused by the upper stages, C_{N,1} is the normal force coefficient acting on the same section and X_{CP,1} is the local center of pressure. X_{CP,1} is negative because we take the nose tip to base direction to be positive. We then take a moment about the point caused by forces acting on the other side of the launch vehicle. Please note that X_{CP,2} will be positive because it is on the opposite side of the summing point: where all the variables are identical to those in Eq. (A.1.2.3.8.1), but for the opposite side. These two moments can then be summed to create the overall bending moment: where C_{bending} is the bending moment. Since C_{N,2} is causing a nose down moment, this will be subtracted from the first moment, causing us to sum moments, just like what common sense would dictate. Because of the way we defined the unit vectors, the moment being output by CP_Structure.m is negative, but the magnitude is the same as if it were a positive moment, and just as important to the design process. References ^{1} Barrowman, James and Barrowman, Judith, "The Theoretical Prediction of the Center of Pressure" A NARAM 8, August 18, 1966. www.ApogeeLVs.com ^{2} Anderson, John D., Fundamentals of Aerodynamics, McgrawHill Higher Education, 2001 ^{3} Ashley, Holt, Engineering Analysis of Flight Vehicles, Dover Publications Inc., New York, 1974, pp. 303312 ^{4} Klawans, B. and Baughards, J. "The Vanguard Satellite Launching Vehicle  an engineering summary" Report No. 11022, April 1960 ^{5} Steven Collicott, Ph.D., In personal communication regarding linearized perturbation theory, 2:002:30 at his office in Armstrong Hall, Purdue University on Feb. 6^{th} 2008. ^{6} Marc Williams, Ph.D, Personal communication regarding pressure distribution and derivation of aerodynamic forces, 2:303:30 at his office in Armstrong Hall, Purdue University on Feb. 19^{ }2008 A.1.2.4 Lift and Lifting Bodies Though lifting bodies are not implemented on the final design, they are still researched in order to determine a cost effective means of launch. Lifting bodies, such as a wing, are beneficial for an aircraft launch. We discuss in detail the aerodynamic coefficients which include lift, drag, and moment that are created with the addition of lifting bodies. Lifting bodies create additional nose up pitching moments that would allow for the launch vehicle to pitch from an initial horizontal configuration, which is assumed to be angle of attack zero degrees, to a final vertical configuration, which is assumed to be an angle of attack of 90 degrees. This extra nose up pitching moment is needed if an aircraft launch configuration is considered. To help us better visualize this configuration, refer to Fig. A.1.2.4.0 below. Fig. A.1.2.4.0 Launch Vehicle with a Delta Wing Configuration (Kyle Donahue) A.1.2.4.1 Drag and Drag Coefficient Though the pitching moment is a known benefit of the wing, induced drag is not. Induced drag is defined as a drag force which occurs whenever a lifting body or a finite wing generates lift. If all other parameters are held constant, the induced drag will increase with increasing angle of attack. Let us look deeper into this subject. The induced drag is calculated using (A.1.2.4.1.1) where D is the induced drag, ρ is the air density, V is the true airspeed, S is the reference area, and C_{D} is the coefficient of drag.^{1} It was previously noted that induced drag increases with increasing angle of attack. But this is not apparent from Eq. (A.1.2.4.1.1). Therefore, in order to see this relation we must further dissect Eq. (A.1.2.4.1.1). The variable that changes with angle of attack is the coefficient of drag. This is shown using (A.1.2.4.1.2) where C_{D} is the coefficient of drag, C_{N} is the normal force coefficient, α is the angle of attack, and C_{A} is the axial force coefficient.^{1} The normal force and the axial force coefficients can then be computed for a lifting body. The derivation of the coefficients follow three basic steps: first we must determine the geometric shape of the body, next we must integrate the theoretical pressure coefficients over the body and evaluate the basic force coefficients, and finally we must determine the appropriate moment coefficients from the vehicle center of mass. All of the extensive integrations necessary to derive the aerodynamic force coefficients are omitted and only the results are presented. For this analysis, we assume an aircraft launch, being that an aircraft launch is the only launch configuration that requires a wing. In order to determine the normal and axial force coefficients we make several assumptions. We implement the Newtonian Model; this assumption is made because the launch vehicle is traveling at supersonic and hypersonic speeds throughout most of the trajectory. We assume turbulent flow; once again this is a valid assumption due to the high velocities. Finally a delta wing configuration is employed. A.1.2.4.2 Normal and Axial Force Coefficients With the assumptions stated, we can now determine the axial and normal force coefficients. In order to determine the total axial and normal force coefficients we must divide the wing surface up into two separate parts, the leading edge and the lower surface. The leading edge and the lower surface are chosen because they are the two portions of the wing that are exposed to the relative wind given an angle of attack. The normal and axial force coefficients from the leading edge are found using (A.1.2.4.2.1a) (A.1.2.4.2.1b) where C_{N} is the normal force coefficient, R_{LE} is the radius of the leading edge, l_{LE} is the length of the leading edge, S is the reference area, k_{LE} is the correction factor for the leading edge, Λ is the wing sweep, Λ_{e} is the effective wing sweep, α is the angle of attack, and C_{A} is the axial force coefficient.^{1} Next we must look at the lower surface of the wing. The normal and axial force coefficients from the lower surface can be found using (A.1.2.4.2.2a) (A.1.2.4.2.2b) (Laminar Flow) (A.1.2.4.2.2c) (Turbulent Flow) where k_{LS }is the lower surface correction factor, S_{LS }is the lower surface area, S is the reference area, α is the angle of attack, S_{w} is the wing area, V_{∞} is the relative velocity, c is the chord length, μ_{∞ }is the relative air viscosity, , n = 0.5 laminar, n = 0.8 turbulent, and m is the planform taper ratio.^{1} Once we find the normal and axial force coefficients for the leading edge and the lower surface, the total normal and axial force coefficients are determined by summing the two.^{1} (A.1.2.4.2.3a) (A.1.2.4.2.3b) Now that the axial and normal coefficients are known, they can be substituted back into Eq. (A.1.2.4.1.2) to solve for the coefficient of drag. Prior to doing that though, let us first look at the behavior of the normal and axial force coefficients against angle of attack. Logically the normal force should be the greatest when the launch vehicle is at a high angle of attack. Therefore, as the angle of attack is increased, the normal force should also increase. This can be shown through Fig. A.1.2.4.2.1. 