Integration of Grid Fins for the Optimal Design of Missile Systems by Timothy Wayne Ledlow II A thesis submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Master of Science Auburn, Alabama August 2, 2014 Keywords: grid ns, optimization, missile design Copyright 2014 by Timothy Wayne Ledlow II Approved by Roy Hart eld, Chair, Walt and Virginia Waltosz Professor of Aerospace Engineering John Burkhalter, Professor Emeritus of Aerospace Engineering Andrew Shelton, Assistant Professor of Aerospace Engineering Abstract Grid ns are unconventional missile control and stabilization devices that produce unique aerodynamic characteristics that are vastly di erent from that of the conventional planar n. History has shown that grid ns are able to achieve much higher angles of attack than planar ns without experiencing any e ects of stall. They are also able to produce much lower hinge moments than planar ns, which allows for the use of smaller actuators for n control. However, the major drawback of grid ns that has prevented them from seeing more applications in missile control is the high drag that is associated with the lattice structure, which is substantially larger than that of a comparable planar n. Despite the high drag produced by grid ns, there are still several applications where the grid n is an ideal candidate for missile control. One such application is the maximization of the target strike capability of a missile that is released from an airplane at a designated altitude. The goal of this work is to integrate a set of grid n aerodynamic prediction algorithms into a missile system preliminary design code in an e ort to maximize the target strike area of a missile using both planar ns and grid ns as aerodynamic control devices. It was found that a missile system using grid ns for aerodynamic control is able to strike a larger target area with a higher degree of accuracy than a similar missile system using equivalent planar ns for aerodynamic control. ii Acknowledgments The author would like to thank Dr. Roy Hart eld for providing the opportunity, en- couragement, and support for conducting this research. Special thanks are also due to Dr. John Burkhalter for providing his assistance and expertise regarding grid ns and the missile trajectory optimization program, as well as allowing the use of his subsonic and supersonic grid n prediction programs. The author would also like to thank his parents for their years of support and encouragement, without which none of this would be possible. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Background: Grid Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 Subsonic Grid Fin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Linear Analysis: Subsonic . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.2 Nonlinear Analysis: Subsonic . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Transonic Grid Fin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Supersonic Grid Fin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.1 Linear Analysis: Supersonic . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.2 Nonlinear Analysis: Supersonic . . . . . . . . . . . . . . . . . . . . . 19 3.4 Fins in the Vertical Position . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Fin-Body Carry-Over Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Validation of Grid Fin Prediction Algorithm . . . . . . . . . . . . . . . . . . . . 23 5 Algorithm Description and Integration . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 Standalone AERODSN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Missile System Preliminary Design Tool . . . . . . . . . . . . . . . . . . . . 36 5.2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2.2 Flight Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 iv 5.2.3 Program Modi cations . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Target Strike Envelope Maximization . . . . . . . . . . . . . . . . . . . . . . . . 44 6.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 56 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Standalone AERODSN Input File . . . . . . . . . . . . . . . . . . . . . . . . . . 62 B Standalone AERODSN Output File . . . . . . . . . . . . . . . . . . . . . . . . . 64 C Grid Fin Geometry Output File . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 D Best Fitness vs. Number of Function Calls Output File . . . . . . . . . . . . . . 67 E Best Fit Member Output File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 F Target Fitness Output File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 v List of Figures 2.1 Comparison of Grid Fins vs. Planar Fins . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Grid Fin Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Wind Tunnel Results Showing the High Angle of Attack Capability of Grid Fins (Mach 0.35 Flow) [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Examples of Uses of Grid Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1 Di erent Possible Shock Structures for a Grid Fin [24] . . . . . . . . . . . . . . 9 3.2 Vortex Lattice on a Single Grid Fin Panel [9] . . . . . . . . . . . . . . . . . . . 10 3.3 Resulting Flow eld from a Freestream Doublet [9] . . . . . . . . . . . . . . . . . 11 3.4 Grid Fin Normal Force Coe cient Transonic Bucket [1] . . . . . . . . . . . . . . 15 3.5 Classical Evvard?s Theory [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.6 Modi ed Evvard?s Theory [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.7 Imaging Scheme for Fin-Body Carry-Over Load Calculation [9] . . . . . . . . . 21 4.1 Grid Fin Geometries Used for Validation . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Parameters De ning Missile Geometry [12] . . . . . . . . . . . . . . . . . . . . . 24 4.3 Sign Convention for Orientation Angles [9] . . . . . . . . . . . . . . . . . . . . . 25 vi 4.4 Subsonic Mach Numbers, Including Fin-Body Carry-Over Loads . . . . . . . . . 28 4.5 Varying Incidence and Roll Angles at Mach 0.5, Including Fin-Body Carry-Over Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.6 Varying Roll Angle at Mach 0.7, Including Fin-Body Carry-Over Loads . . . . . 30 4.7 Varying Incidence Angle at Mach 2.51, Including Fin-Body Carry-Over Loads . 31 4.8 Single Grid Fin, Not Including Fin-Body Carry-Over Loads . . . . . . . . . . . 32 4.9 Single Grid Fin, Subsonic Speeds, Not Including Fin-Body Carry-Over Loads . 33 4.10 Single Grid Fin, Supersonic Speeds, Not Including Fin-Body Carry-Over Loads 34 5.1 Standalone AERODSN Flow Chart . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Missile System Preliminary Design Tool Flow Chart . . . . . . . . . . . . . . . 37 5.3 Illustration of Line-of-Sight Guidance . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4 Grid Fin Parameter Optimization Constraints . . . . . . . . . . . . . . . . . . . 41 5.5 Grid Fin Parameters [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.1 Illustration of the Missile Drop Problem . . . . . . . . . . . . . . . . . . . . . . 44 6.2 Illustration of a Target Grid for Optimization . . . . . . . . . . . . . . . . . . . 45 6.3 Unoptimized Target Strike Envelopes . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4 Target Strike Envelope for Optimized Grid Fin Con guration . . . . . . . . . . 48 6.5 Target Strike Envelope for Optimized Planar Fin Con guration with Limited Hinge Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 vii 6.6 Target Strike Envelope for Optimized Planar Fin Con guration with Unlimited Hinge Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.7 Grid Fin Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.8 Side View of Optimized Missile Geometry . . . . . . . . . . . . . . . . . . . . . 54 6.9 Front View of Optimized Missile Geometry . . . . . . . . . . . . . . . . . . . . 55 viii List of Tables 3.1 Assumed Grid Fin Flow Structure in Di erent Mach Number Regimes . . . . . 8 4.1 Missile Body Con guration Parameter Values . . . . . . . . . . . . . . . . . . . 24 6.1 Optimized Missile Con guration Data . . . . . . . . . . . . . . . . . . . . . . . 47 6.2 Optimized Grid Fin Geometry Parameters . . . . . . . . . . . . . . . . . . . . . 52 ix Nomenclature Angle of Attack Compressibility Factor Cp Di erential Pressure Coe cient Fin Incidence Angle th Displacement Thickness of Boundary Layer Vortex Strength Speci c Heat Ratio Fin Leading Edge Sweep Angle Mach Angle Roll Angle Density al Density of Aluminum A Area A Throat Area A th Reduction in Fin Capture Area Aref Reference Area B2 Fin Span from Body Centerline x C Chord Length CA Axial Force Coe cient CN Normal Force Coe cient CS Side Force Coe cient CMmc Pitching Moment Coe cient about Moment Center CNi Imaged Normal Force Coe cient CSi Imaged Side Force Coe cient CF Transonic Correction Factor DB Diameter of the Missile Body Dref Reference Length ft Feet H Height of Grid Fin HB Height of Grid Fin Support Structure Iyy Mass Moment of Inertia About the Y Axis ibase Number of Cells in Fin Base Corner in: Inches itip Number of Cells in Fin Tip Corner Lptot Total Length of the Grid Fin Panels M Mach Number mGF Mass of a Single Grid Fin xi mph Miles per Hour ndy Number of Cells in Horizontal Direction ndz Number of Cells in Vertical Direction np Number of Fin Element Intersection Points p Pressure RB Radius of the Missile Body Re Reynolds Number Sabi Span-wise Length of the Imaged Panel Sab Span-wise Length of the Actual Panel THT Distance from Nose of Body to Tail Location TXCG Distance from Nose of Body to Center of Gravity Location thk Average Fin Element Thickness ttle Total Length of the Grid Fin Elements in the Plane Perpendicular to the Freestream XL Length of Missile Body XLN Length of the Nose xii Chapter 1 Introduction The grid n (also known as a lattice control surface) is an unconventional aerodynamic control device that consists of an outer frame which supports an inner lattice of intersecting planar surfaces of small chord length. Unlike the conventional planar n, the grid n is positioned perpendicular to the freestream direction, which allows the oncoming ow to pass through the inner lattice structure. This design provides unique aerodynamic performance characteristics that are vastly di erent from that of the planar n. Extensive research has been performed on grid ns since their development in the 1970s in an e ort to better un- derstand these unique aerodynamic performance characteristics. This research has included wind tunnel testing of di erent grid n con gurations [1{5], Computational Fluid Dynam- ics (CFD) analysis [6{8], and the development of di erent theoretical formulations for the prediction of grid n aerodynamics [9{14]. The U.S. Army Aviation and Missile Command in particular has performed extensive research on various grid n con gurations [1{4], including a suite of grid n performance prediction codes that was developed for the U.S. Army Aviation and Missile Command by Burkhalter in the mid 1990s [9{11]. Di erent theoretical formulations were used for the subsonic, transonic, and supersonic ow regimes due to the drastically di erent ow elds that the grid n experiences in each Mach regime. These codes use a vortex lattice approach in the subsonic and transonic ow regimes, with a correction factor that is applied in the transonic ow regime to account for mass ow spillage due to the choking of the ow within each individual cell of the grid n. The formulation for the supersonic ow regime uses a modi ed version of Evvard?s Theory to determine the di erential pressure coe cient for each panel of the grid n. The results produced by these theoretical formulations have 1 been compared with experimental data obtained from wind tunnel test results of di erent grid n geometries at various Mach numbers, incidence angles, and roll angles. It has been determined that these theoretical formulations are able to accurately and e ciently predict the aerodynamics of a wide range of grid n geometries, and is a suitable tool for application in the preliminary design analysis of missile systems. The purpose of this work is to incorporate the grid n aerodynamic prediction capabil- ity that was developed by Burkhalter into an existing preliminary design analysis tool that has been developed at Auburn University for the optimal design of missile systems. This program uses an aerodynamic prediction tool known as AERODSN to predict the aerody- namic characteristics of typical missile con gurations in ight. AERODSN was developed by Sanders and Washington for the U.S. Army Missile Command in 1982 to provide an e cient and reliable tool to predict the aerodynamics for a typical cylindrical missile body con guration with wings or planar tail ns or both [15]. AERODSN has proven to be an e ective aerodynamic prediction tool, and has been successfully applied to a multitude of aerospace design problems [16{19]. Optimization is a very important tool that provides the ability to nd good solutions for highly complex problems where the best solution is not readily apparent and cannot be solved for directly. There are several di erent optimization schemes that can be used to intelligently search a given solution space for global optimal solutions. The missile system preliminary design tool used in this work has four di erent optimization schemes incorporated in the code that have been successfully applied to aerospace design problems in the past [16{22]. These optimization schemes include a modi ed ant colony optimizer, a particle swarm optimizer, a binary-encoded genetic algorithm, and a real-encoded genetic algorithm. The modi ed ant colony optimization scheme was selected for use in this work due to its proven e ectiveness at solving complex aerospace design problems. It has been shown that the modi ed ant colony has the ability to be more e ective than many established optimization methods, as 2 it is able to converge more quickly and nd better solutions than competing optimization algorithms [20,21]. The incorporation of the grid n aerodynamics into AERODSN allowed for a preliminary analysis that compared the performance of planar ns versus grid ns as the aerodynamic control device for an unpowered missile system. The modi ed ant colony optimization scheme was used to nd the optimal planar n and grid n designs for a given missile con guration that maximized the target strike area for an unpowered missile dropped from an airplane ying with a horizontal velocity of 492.8 ft/sec at an altitude of 23,000 ft. 3 Chapter 2 Background: Grid Fins Grid ns were initially developed in the 1970s by the Soviet Union. The rst ow eld analysis of grid ns was performed by Russian researchers, who were able to provide a basic understanding of the unique aerodynamics associated with the grid n [23]. Grid ns can be used as either an aerodynamic stabilizer or a control surface for a missile or munition con guration. The unconventional geometry of the grid n is what really separates it from the conventional planar n. Planar ns can generally be characterized by four geometrical parameters: root chord, tip chord, span, and thickness. The grid n, however, adds an extra dimension, requiring ve geometrical parameters: element thickness, cell spacing, span, height, and chord length [2]. Figure 2.1 below shows a comparison between an example of a general missile con guration with grid ns (2.1a) versus an example of a general missile con guration with planar ns (2.1b). (a) Missile Con guration with Grid Fins (b) Missile Con guration with Planar Fins Figure 2.1: Comparison of Grid Fins vs. Planar Fins 4 (a) Foldable for Compact Storage [24] (b) Generate Low Hinge Moments [25] Figure 2.2: Grid Fin Features There are several distinct advantages to using grid ns as an aerodynamic stabilizer or as a control surface instead of planar ns: 1) Grid ns are able to be folded down against the body of the missile for compact storage (Figure 2.2a), which can be particularly helpful when there are size limitations for the missile, such as if it is a tube-launched device. 2) Grid ns generate much lower hinge moments than planar ns (Figure 2.2b), and are therefore able to use smaller actuators for n de ection than their planar n counter- parts would require [8]. 3) The multiple cell arrangement of the grid n makes it less prone to stall at higher angles of attack than the traditional planar n. A typical grid n can reach angles of attack near 40 - 50 degrees before experiencing any loss in lift, as seen by the wind tunnel test results of a single grid n in Mach 0.35 ow in Figure 2.3. 4) The truss structure of the grid n is inherently strong, which allows the lattice walls to be extremely thin, thus reducing the weight of the n. 5) Grid ns provide greater control e ectiveness in the supersonic ight regime than a comparable planar n [2]. 5 Figure 2.3: Wind Tunnel Results Showing the High Angle of Attack Capability of Grid Fins (Mach 0.35 Flow) [2] 6) A grid n that is mounted in the vertical position will still produce a normal force at any nite body angle of attack, which is a unique characteristic compared to any other lifting surface system that is currently in use on missile systems [10]. There are also several disadvantages to the use of grid ns as an aerodynamic stabilizer or as a control surface: 1) Grid ns produce higher drag than planar ns, especially in the transonic ight regime. 2) Grid ns perform very poorly in the transonic ight regime due to the choking of the ow within the individual grid n cells and the shocks that are present in the ow eld. 3) Grid ns have a high manufacturing cost due to their complex geometry [8]. The large amount of drag that is produced by the grid n is undesirable in most ap- plications. There have been several di erent e orts that have been conducted in an e ort to reduce the drag of the grid n [4, 8]. Miller and Washington found that it is possible to considerably lower the drag of a grid n without resulting in a major impact on other aerodynamic properties by adjusting the cross-section shape of each panel within the lattice 6 (a) American Massive Ordnance Penetrator (MOP) [26] (b) Russian Vympel R-77 [25] Figure 2.4: Examples of Uses of Grid Fins structure [4]. Each advantage and disadvantage of the grid n concept must be thoroughly analyzed within the constraints of the given problem before a de nitive decision can be made on their use [1]. Since their inception in the 1970s, grid ns have found rather limited use compared to their planar n counterparts. The majority of the application of grid ns has been on Russian ballistic missile designs such as the SS-12, SS-20, SS-21, SS-23, and the SS-25. Grid ns have also been used on some launch vehicle designs, most notably as emergency drag brakes on the launch escape system for the Russian Soyuz spacecraft [24]. Grid ns have also found use on conventional bombs such as the American Massive Ordnance Penetrator (Figure 2.4a), and the Russian Vympel R-77 (Figure 2.4b). Another recent application of grid ns is the Quick Material Express Delivery System (Quick-MEDS), which is a precision-guided supply pod that is designed to deliver small, critically needed packages from Unmanned Aircraft Systems (UAS) in the air to troops on the ground [27]. 7 Chapter 3 Theoretical Analysis A set of robust theoretical analysis tools capable of quickly predicting the aerodynamic coe cients associated with a cruciform grid n con guration on a missile body was developed for the U.S. Army Aviation and Missile Command by Burkhalter in the mid 1990s [9{11]. Separate theoretical formulations were developed for the di erent Mach regimes in order to correctly capture the ow structure of the grid n for any given ight condition. Table 3.1 gives a brief description of the assumed ow structure for a grid n for each Mach regime. For a freestream Mach numberM < 0:8, the ow is assumed to be compressible subsonic ow, and a vortex lattice solution is used to calculate the loading on each individual element of the grid n [9]. In the transonic regime (0:8 1:9 Supersonic Unre ected shocks 8 Figure 3.1: Di erent Possible Shock Structures for a Grid Fin [24] being applied to the normal force coe cient to account for the mass ow spillage. This bow shock is \swallowed" by the grid n at higher supersonic Mach numbers (1:4 < M < 1:9), leading to the formation of attached oblique shocks that are re ected within the grid n structure (Figure 3.1). At higher supersonic Mach numbers (M > 1:9), the ow structure for the grid n is assumed to consist of attached, unre ected oblique shocks (Figure 3.1). A modi ed version of Evvard?s Theory is used to determine the loading produced by the grid n in the supersonic Mach regime. The grid n aerodynamic prediction algorithms as developed by Burkhalter consisted of two standalone programs, one for the subsonic ow regime and the other for the super- sonic ow regime. Each program contained a modi ed form of slender body theory that was combined with Jorgensen?s theory for the prediction of the body alone aerodynamic coe - cients [10]. The integration of the grid n aerodynamic prediction programs into AERODSN required the combination of the subsonic and supersonic prediction programs as well as the removal of the body alone aerodynamic coe cient prediction method. Another modi cation that was required for the integration of the grid n aerodynamics into AERODSN was the addition of the normal shock equations to account for the e ects of the bow shock that forms in front of the grid n in the low supersonic Mach regime (1:0 < M < 1:4). This addition 9 allowed for the grid n aerodynamic prediction algorithms to be successfully used for any Mach number ranging from approximately Mach 0.1 up to Mach 3.5. 3.1 Subsonic Grid Fin Analysis 3.1.1 Linear Analysis: Subsonic A vortex lattice solution was used as the linear subsonic formulation for predicting the loading on each individual element of the grid n lattice structure. A horseshoe vortex is placed on each individual element of the grid n, as illustrated in Figure 3.2. This vortex is de ned using ten node points. Points 1 and 6 are placed at the quarter chord location of the grid n element, while points 2 and 7 are located at the trailing edge of the element. The remaining points de ne the vortex trailing legs, which extend aft of the panel in the direction of the freestream ow [9]. A control point and a unit normal vector are then placed at the three quarter chord location of the panel. A boundary condition that requires the ow at the control point location of a panel to be tangent to the surface of that panel is then applied so that the strength of each vortex within the lattice structure can be determined. The velocity at each control point is composed of three di erent components: the freestream velocity, the cross- ow velocity (body up-wash term), and the induced velocity that takes into account the vortex strengths of the surrounding vortices [9]. Figure 3.2: Vortex Lattice on a Single Grid Fin Panel [9] 10 Figure 3.3: Resulting Flow eld from a Freestream Doublet [9] The cross- ow velocity term is determined using a potential ow solution of an in nite doublet in the freestream, an illustration of which can be seen in Figure 3.3. The induced velocity from the doublet and freestream can be written in vector form as: V 0 = V1cos( )^{+ ( V sin( ) +Vrcos( )) ^|+ (V cos( ) +Vrsin( ) +V1sin( )) ^k (3.1) The compressible form of the Biot-Savart law gives the velocity induced by a vortex lament segment at a control point [9]: v = 2 4 Z rxdl jr j3 (3.2) where is the compressibility factor: =p1 M2 (3.3) The dot product of the velocity vector from Equation 3.1 and the unit normal for a panel gives the velocity component normal to the surface of the panel. This component of velocity must be equal to zero in order to satisfy the ow tangency boundary condition mentioned previously. The application of this boundary condition results in the following 11 expression: v n = 2 4 2 4 Z rxdl jr j3 ! x nx + 2 4 Z rxdl jr j3 ! y ny + 2 4 Z rxdl jr j3 ! z nz 3 5 = B (3.4) which can be rearranged to solve for the unknown vortex lament strength , where the \A" matrix is the inverse of the bracketed term from Equation 3.4 and the \B" matrix is the known velocities induced at the panel control point by the freestream doublet combination [9]: [ ] = [A] 1 [B] (3.5) An iterative procedure is used to nd the vortex strengths in order to avoid taking the inverse of a large \A" matrix [28], and is as follows. First, the vortex strengths associated with the rst n are found as if there are no other ns present in the ow. Second, the vortex strengths for the second n are found as if ns 1 and 2 are the only ns present in the ow, and the vortex strengths of n 1 are known. This process continues in a similar fashion for the remaining two grid ns, each time including the known strengths from the previous ns [28]. Once the vortex strengths of the fourth and nal n are known, the entire process is repeated for several iterations until there is no signi cant di erence in vortex strength values from one iteration to the next. It has been found that a large number of iterations is not necessary to achieve accurate results, and therefore a total of six iterations are used in this work. Once the iterative process is complete and the vortex strengths for all four grid ns are known, the Kutta-Joukowski theorem is used to compute the aerodynamic loads on each individual element of each grid n: C^F = 2 SS ref (3.6) 12 where S is the slant length of the individual grid n element that is being analyzed. This force coe cient can then be turned into di erent components by multiplying by the unit normal vector for that grid n element: C^N = C^F nz (3.7) C^S = C^F ny (3.8) C^A = C^F nx (3.9) The total axial force for the grid n is assumed to consist of four di erent components: induced drag, skin friction drag, pressure drag, and interference drag from the n element intersection points [9]. The induced drag is the drag produced due to n de ection angle, and is given by: CAi = CNtan( ) (3.10) The skin friction drag is determined by calculating the wetted area of the n and con- verting it to a at plate area with an assumed laminar or turbulent boundary layer, as a function of Reynolds number. The drag contribution due to pressure is a function of the frontal area of the grid n and the local dynamic pressure [9]: CAdp = Swetft2 S refC (3.11) An empirical formulation that was derived using experimental data is used to determine the interference drag due to the n element intersection points: CAxp = 2 0:000547 (np+ 2) (3.12) 13 where (np + 2) is the total number of n intersection points, including the base support structure. The total grid n axial force coe cient is simply the sum of these four contribu- tions: CAx = CAxi +CAxf +CAxdp +CAxp (3.13) It was observed that the n axial force changes very little with angle of attack, and is therefore assumed to be independent of angle of attack [9]. 3.1.2 Nonlinear Analysis: Subsonic Unlike planar ns, the linear aerodynamics region of the grid n begins to break down at an angle of attack around 5 to 8 , resulting in the need for a nonlinear theoretical formulation to more accurately model the aerodynamic coe cients. Experimental results obtained through wind tunnel testing of di erent grid n geometries were used to develop a semi-empirical formulation for the nonlinear aerodynamic region at higher angles of attack: CN = 2 66 4 0 B@ CN sin( ) 1 +k2 B g H 2 C H sin2( ) 1 CA 0 BB @k6 k3 k4 qB g H CN sin( ) 1 +k5 B g H 2 2 1 CC A+ 0 B@ CN sin( ) 1 +k1 B g H 2 C H sin2( ) 1 CA 3 75cos2 (2 ) (3.14) This subsonic semi-empirical formulation uses the initial lift-curve slope from the linear vortex lattice theory and also attempts to incorporate the in uence from the major grid n geometric properties, such as the n span to height ratio (Bg=H) and the n chord length to height ratio (C=H). A more in-depth analysis of the nonlinear aerodynamics of grid ns can be found in Reference [12]. 14 3.2 Transonic Grid Fin Analysis Grid ns exhibit unique aerodynamic characteristics in the transonic ight regime com- pared to the traditional planar n. Planar ns experience their maximum normal force coe cients in the transonic region, while grid ns experience what is known as a \transonic bucket". An illustration of the transonic bucket can be seen in Figure 3.4. This phenomenon is a result of the choking of the individual cells of the grid n, which causes mass ow spillage around the edges of the grid n. In order to capture this characteristic, the grid n aero- dynamic prediction tool applies a correction factor to the normal force coe cient when operating in the transonic ow regime. The rst step to calculating the correction factor is to determine the thickness of the boundary layer within the cells of the grid n. Blasius? theorem is used to calculate the displacement thickness of the boundary layer: th = 1:7208 CpRe c (3.15) Figure 3.4: Grid Fin Normal Force Coe cient Transonic Bucket [1] 15 The presence of the boundary layer results in an area reduction within the individual grid n cells, which can be determined by: A th = 2 th +thk ttle (3.16) The area required to choke the ow at the given Mach number is then determined using the following isentropic relationship: A = Acap M 2(1+ 12 M2) +1 ( +12( 1)) (3.17) If the calculated exit area is less than or equal to the value calculated using Equa- tion 3.17, the ow is considered to be choked and a correction factor is then determined by calculating the reduction in mass ow rate between the choked and unchoked conditions: CF = Aex r P0 0 2 +1 ( +1 1) Acap 1V1 (3.18) which is then applied as a multiplier to the calculated n forces and moments, thus capturing the reduction in grid n performance in the transonic ow regime. For any transonic Mach number above Mach 1.0 (Table 3.1), a normal shock is assumed to be present in front of the n, which results in a subsonic ow- eld behind the shock. The well-known normal shock equations (found in Reference [29]) are used to determine the resulting Mach number and freestream pressure behind the shock: M22 = 1 + h 1 2 i M21 M21 12 (3.19) p2 p1 = 1 + 2 + 1 M21 1 (3.20) 16 The subsonic theoretical formulation is then used to determine the loads on the grid ns, including the transonic correction factor if the ow within the cells of the grid n is determined to be choked. 3.3 Supersonic Grid Fin Analysis 3.3.1 Linear Analysis: Supersonic A modi ed version of Evvard?s theory is used as the linear supersonic theoretical formu- lation in the grid n aerodynamic prediction tool. The original version of Evvard?s theory determines the di erential pressure coe cient distribution over a swept wing with a super- sonic leading edge [30]. Figure 3.5 shows an illustration of the original version of Evvard?s theory, showing the various supersonic regions associated with a typical planar n. Similar to the subsonic formulation, the supersonic formulation is applied to the grid n on a panel by panel basis, resulting in the need for a modi ed version of Evvard?s theory that takes account of end-plate e ects. These end-plate e ects are a result of the unique lattice structure of the grid n, and are not accounted for in the original form of Evvard?s theory. Figure 3.6 shows the two possible cases for the ow over the individual grid n elements: without crossing Mach lines (Figure 3.6a) and with crossing Mach lines (Figure 3.6b). Figure 3.5: Classical Evvard?s Theory [9] 17 (a) Without Crossing Mach Lines (b) With Crossing Mach Lines Figure 3.6: Modi ed Evvard?s Theory [9] It can be seen in Figure 3.6 that only the di erential pressure calculations for regions 1, 2, and 4 are required for the determination of the loading on a grid n element. The di erential pressure coe cients for each region were derived by Evvard and can be seen by the following equations: [ Cp]1 = 4 q B2 tan2( ) (3.21) [ Cp]2 = 1 [ Cp]1 " cos 1 x tan( ) B2y B(x+y tan( )) ! +cos 1 x tan( ) B2y B(x y tan( )) !# (3.22) [ Cp]4 = [ Cp]2 1 [ Cp]1 " cos 1 x a +ya (2B +tan( )) xa +yatan( ) !# (3.23) where B2 = M2 1 (3.24) The orientation of the grid n is extremely important in the supersonic theoretical formulation, as that is what dictates the size of the di erent pressure regions on the grid n panels. Each element of the grid n must be oriented at some dihedral angle , pitched at some de ection angle , rolled to some angle , and nally pitched to some angle of attack [11]. The grid n elements are terminated at each end by end plates that are not 18 necessarily perpendicular to the lifting surface, making it very important to understand the geometric angles involved with each element lifting surface. Once each point of each element of the grid n has been pitched and rolled to its nal position, the leading edge sweep and the e ective angle of attack can be determined. This process is described in further detail in References [11] and [31]. Once the di erential pressure coe cients for the di erent regions on the grid n element are known, the aerodynamic coe cients for normal force, side force, and axial force can be determined by subdividing the element into a series of small rectangles with area \A" [9]. The loading for each subelement can be seen by: CN = Cp A UNZS ref (3.25) CS = Cp A UNYS ref (3.26) CA = Cp A UNXS ref (3.27) The total force coe cients for the grid n can then be found by simply summing the loading on each individual subelement. The total n axial force coe cient is calculated using the same component contributions as those that are described for the subsonic theoretical formulation. 3.3.2 Nonlinear Analysis: Supersonic Similar to the linear subsonic grid n aerodynamic formulation, the linear supersonic grid n aerodynamic formulation is only accurate for low angles of attack, and typically begins to break down at angles of attack around 5 to 8 . A semi-empirical formulation for the supersonic nonlinear aerodynamic region at higher angles of attack can be seen as: CN = CN 1 + max =0 + CN 1 + max =0 0 B@1 CN 1 + max 2 =0 1 CA (3.28) 19 where CN and CN are the initial lift-curve slopes with respect to angle of attack and n de ection, respectively, that are calculated from the modi ed Evvard?s theory. A more in-depth discussion of the development of this formulation can be found in Reference [31]. 3.4 Fins in the Vertical Position As was mentioned previously, grid ns possess a unique characteristic that no other lifting surface system that is currently in use on missile systems possesses, as they are able to produce a normal force at any nite body angle of attack when mounted in the vertical position [10]. The normal force coe cient for a grid n in the vertical position (either the top or bottom of the missile body) is still found in the same manner as discussed in the previous sections, but the values of CN and CN are considerably smaller than those for the ns in the horizontal position [11,28]. The lift-curve slopes for a n in the vertical position are still determined via the vortex lattice theory (subsonic) or Evvard?s theory (supersonic), although an error is introduced that is not present in the linear analysis. This error arises due to the streamline ow near the top and bottom of the body surface and from the body vortices emanating from the nose of the missile [11]. The grid ns on the top and bottom of the missile body do not experience oncoming ow that is the same as the freestream direction, and are therefore assumed to be immersed in a stream tube that is nearly parallel to the body [11]. Burkhalter determined from an analysis of available experimental data that the average angle of attack of the top and bottom n due to body alteration of the incident streamlines should be assumed to be approximately =2 [28]. 3.5 Fin-Body Carry-Over Loads In traditional airplane design, the e ects of the wing-body carry-over loads can be ac- counted for by viewing the aerodynamic characteristics as being dominated by the wing such that no body is present in the ow [32]. This traditional assumption is valid for cases where the wing span is large compared to the body diameter, but a di erent approach is required 20 in the case of very small wings in comparison to the body diameter, as is characteristic of many missile designs [32]. The n-body carry-over loads are modeled in the grid n aerodynamic prediction tool through an imaging scheme in which each panel element is imaged inside the missile body, as illustrated by Figure 3.7. The basic assumption is that each imaged element inside the body carries the same load per unit span as its corresponding element outside the body [10]. The geometry of the imaged element is de ned by imaging the endpoints of the \real" panel outside the body along radial lines to the center of the body, using the following equations: ya = y1R 2 B r2a za = z1R2B r2a (3.29) yb = y2R 2 B r2b zb = z2R2B r2b (3.30) The chord length of the imaged element is assumed to be the same as that of the correspond- ing \real" panel. The normal force coe cients and side force coe cients due to carry-over loads are both determined by this method, while the axial force coe cients are not im- aged [10]. The equations for the determination of the imaged normal force coe cient and the imaged side force coe cient can be seen by: Figure 3.7: Imaging Scheme for Fin-Body Carry-Over Load Calculation [9] 21 CNi = CN S abi Sab (3.31) CSi = CS S abi Sab (3.32) where Sab and Sabi are the span-wise lengths of the \real" and imaged panels, respectively. These values are determined by: Sab = q (y1 y2)2 + (z1 z2)2 (3.33) Sabi = q (ya yb)2 + (za zb)2 (3.34) This imaging technique is used to predict the n-body carry-over loads for each theo- retical formulation described in this section. 22 Chapter 4 Validation of Grid Fin Prediction Algorithm The validation of the subsonic, transonic, and supersonic grid n aerodynamic theo- retical formulations was performed by conducting several tests using three di erent grid n designs (as seen in Figure 4.1) for multiple Mach numbers, n incidence angles, and con guration roll angles. The theoretical results obtained from the grid n prediction al- gorithm were compared with experimental wind tunnel data that was extracted from Ref- erences [1, 2, 4, 9{11]. Experimental data was available for two general cases: four grid ns mounted on a cylindrical missile body in a cruciform con guration (allowing for the capture of the n-body carry-over loads), and a single grid n mounted on a n balance (eliminating the n-body carry-over loads). The missile body that was used for these wind tunnel tests consisted of a 15 in: tangent ogive nose section followed by a 37 in: cylindrical main body section [2]. Table 4.1 shows the Figure 4.1: Grid Fin Geometries Used for Validation 23 Table 4.1: Missile Body Con guration Parameter Values XLN 15 in: TXCG 26 in: THT 46 in: XL 52 in: RB 2.5 in: Aref 19.63 in:2 Dref 5 in: values for the missile body con guration parameters in inches, which are de ned in Figure 4.2. A consistent reference length (Dref = 5 in:) and reference area (Aref = 19:63 in:2) was used for all experimental test data. The values of the reference length and reference area correspond to the missile base diameter and the missile base area, respectively. The three grid n geometries that were analyzed for this validation e ort can be seen in Figure 4.1. The same base support structure is used for each grid n, as well as the same chord length (0:384 in:). The average element thickness of the inner lattice structure is 0:008 in: and the average element thickness of the outer support elements is 0:03 in: for each grid n design. The validation results presented in Figures 4.4 - 4.7 in the following pages represent the normal force coe cient, axial force coe cient, and pitching moment coe cient versus angle Figure 4.2: Parameters De ning Missile Geometry [12] 24 Figure 4.3: Sign Convention for Orientation Angles [9] of attack for four grid ns mounted on a missile body in a cruciform con guration. The n-body carry-over loads are included in the results in each of these plots. Figures 4.8 - 4.10 show the normal force coe cient and axial force coe cient (when available) for a single grid n, allowing for a comparison of theoretical and experimental data without the inclusion of n-body carry-over loads. Figure 4.4 shows CN, CA, and CMmc for grid n \A" mounted in a cruciform con gura- tion at Mach numbers ranging from 0.5 to 0.9. The grid ns are at an incidence angle of 0 and a roll angle of 0 . The theoretical results were found to closely match the wind tunnel data overall, with slight discrepancies at high angles of attack. It is clear that the correction for the transonic e ects is accurate for this particular con guration, as the theoretical data matches the experimental data very well in both the Mach 0.8 case and the Mach 0.9 case. Figure 4.5 shows CN, CA, and CMmc for grid n \A" in a cruciform con guration at Mach 0.5 at di erent n incidence angles (ranging from = 0 to = 20 ) and di erent roll angles (0 and 45 roll angles). The n orientation sign convention is de ned in Figure 4.3. It can be seen that the theoretical results have excellent agreement with the experimental data for these cases. The theoretical model was able to accurately predict the aerodynamic coe cients for each n de ection case as well as the roll case for this particular con guration. It should also be noted that the aerodynamic coe cients are symmetric about = 0 for all cases where there is no n de ection, and is not symmetric about = 0 in the cases with n de ection, as would be expected. 25 Figure 4.6 shows CN, CA, and CMmc for grid n \B" in a cruciform con guration at Mach 0.7 at several di erent roll angles, ranging from 0 to 67:5 . It can be seen that the theory matches the experimental results well in the low angle of attack region, but tends to under-predict the normal force coe cient and the pitching moment coe cient at high angles of attack. Figure 4.7 shows CN, CA, and CMmc for grid n \A" in a cruciform con guration at Mach 2.51 at several di erent n incidence angles, ranging from 0 to 20 . It can be seen that Evvard?s theory provides an accurate match in the linear aerodynamics region, but the nonlinear theoretical model tends to over-predict the normal force coe cient and pitching moment coe cient at high angles of attack. Figure 4.8 shows the n balance normal force coe cient and axial force coe cient for grid n \B" at various Mach numbers and roll angles. It can be seen that the theoretical grid n formulation provides good agreement with experimental data in the single grid n case. A comparison of Figures 4.6a and 4.8c and Figures 4.6e and 4.8d show that the imaging scheme for the n-body carry-over loads is accurately predicting the aerodynamics associated with the n-body interaction. Figure 4.9 shows several di erent subsonic n balance cases, including varying the n de ection angle. A comparison of the normal force coe cient for grid n \A" at Mach 0.5 both with and without n-body carry-over loads (Figures 4.5 and 4.9, respectively) show that the imaging scheme is once again able to accurately capture the e ects of the n-body interactions. Figure 4.10 shows several supersonic n balance cases, including the low supersonic cases where there is a normal shock in front of the grid n. It can be seen that the theoretical formulations are able to match the aerodynamic characteristics of the grid n very well in each of these cases for a wide range of angles of attack. It can be concluded from this comparison of theoretical versus wind tunnel test results that the grid n aerodynamic prediction algorithms are suitable for use as a preliminary 26 design tool for missile systems. The grid n prediction algorithms were able to match the aerodynamic characteristics of three di erent grid n geometries for a wide range of Mach numbers ranging from Mach 0.25 to Mach 3.5, for several di erent n incidence angles and con guration roll angles. It was shown that both the subsonic and supersonic linear aerodynamic theoretical models are able to provide very good matches with the initial lift curve slope, and that the nonlinear semi-empirical formulations are su cient for preliminary- level engineering design analysis. It was also shown that the imaging scheme used to capture the n-body carry-over loads for the grid ns is able to accurately capture the e ects of the interaction between the grid ns and the missile body. 27 (a) Grid Fin \A", Mach 0.5, CN and CA (b) Grid Fin \A", Mach 0.5, CMmc (c) Grid Fin \A", Mach 0.8, CN and CA (d) Grid Fin \A", Mach 0.8, CMmc (e) Grid Fin \A", Mach 0.9, CN and CA (f) Grid Fin \A", Mach 0.9, CMmc Figure 4.4: Subsonic Mach Numbers, Including Fin-Body Carry-Over Loads 28 (a) Grid Fin \A", 0 Incidence Angle, CN and CA (b) Grid Fin \A", 0 Incidence Angle, CMmc (c) Grid Fin \A", 10 Incidence Angle, CN and CA (d) Grid Fin \A", 10 Incidence Angle, CMmc (e) Grid Fin \A", 20 Incidence Angle, CN and CA (f) Grid Fin \A", 20 Incidence Angle, CMmc (g) Grid Fin \A", 45 Roll Angle, CN and CA (h) Grid Fin \A", 45 Roll Angle, CMmc Figure 4.5: Varying Incidence and Roll Angles at Mach 0.5, Including Fin-Body Carry-Over Loads 29 (a) Grid Fin \B", 0 Roll Angle, CN and CA (b) Grid Fin \B", 0 Roll Angle, CMmc (c) Grid Fin \B", 22:5 Roll Angle, CN and CA (d) Grid Fin \B", 22:5 Roll Angle, CMmc (e) Grid Fin \B", 45 Roll Angle, CN and CA (f) Grid Fin \B", 45 Roll Angle, CMmc (g) Grid Fin \B", 67:5 Roll Angle, CN and CA (h) Grid Fin \B", 67:5 Roll Angle, CMmc Figure 4.6: Varying Roll Angle at Mach 0.7, Including Fin-Body Carry-Over Loads 30 (a) Grid Fin \A", 0 Incidence Angle, CN and CA (b) Grid Fin \A", 0 Incidence Angle, CMmc (c) Grid Fin \A", 10 Incidence Angle, CN and CA (d) Grid Fin \A", 10 Incidence Angle, CMmc (e) Grid Fin \A", 20 Incidence Angle, CN and CA (f) Grid Fin \A", 20 Incidence Angle, CMmc Figure 4.7: Varying Incidence Angle at Mach 2.51, Including Fin-Body Carry-Over Loads 31 (a) Grid Fin \B", Mach 0.25, 0 Roll Angle (b) Grid Fin \B", Mach 0.5, 0 Roll Angle (c) Grid Fin \B", Mach 0.7, 0 Roll Angle (d) Grid Fin \B", Mach 0.7, 45 Roll Angle (e) Grid Fin \B", Mach 2.5, 0 Roll Angle (f) Grid Fin \B", Mach 2.5, 45 Roll Angle Figure 4.8: Single Grid Fin, Not Including Fin-Body Carry-Over Loads 32 (a) Grid Fin \A", Mach 0.5, 0 Incidence Angle (b) Grid Fin \A", Mach 0.5, 10 Incidence Angle (c) Grid Fin \A", Mach 0.5, 20 Incidence Angle (d) Grid Fin \C", Mach 0.5, 0 Incidence Angle (e) Grid Fin \A", Mach 0.7, 0 Incidence Angle (f) Grid Fin \C", Mach 0.7, 0 Incidence Angle Figure 4.9: Single Grid Fin, Subsonic Speeds, Not Including Fin-Body Carry-Over Loads 33 (a) Grid Fin \A", Mach 1.1, 0 Incidence Angle (b) Grid Fin \A", Mach 1.8, 0 Incidence Angle (c) Grid Fin \A", Mach 2.5, 0 Incidence Angle (d) Grid Fin \C", Mach 2.5, 0 Incidence Angle (e) Grid Fin \A", Mach 3.5, 0 Incidence Angle (f) Grid Fin \A", Mach 3.5, 15 Incidence Angle Figure 4.10: Single Grid Fin, Supersonic Speeds, Not Including Fin-Body Carry-Over Loads 34 Chapter 5 Algorithm Description and Integration The grid n aerodynamic prediction programs were integrated into two di erent existing codes: a standalone version of AERODSN and a missile system preliminary design tool. The standalone version of AERODSN was used to conduct the validation e orts presented in the previous chapter, while the preliminary design tool was used to conduct the target strike envelope maximization problem. 5.1 Standalone AERODSN The grid n aerodynamic prediction algorithm was integrated with a standalone version of AERODSN for the purpose of the validation e orts shown in the previous chapter. Fig- ure 5.1 shows a ow diagram of the program. The program begins by loading the required initial parameters from an input le that has been modi ed to include the information nec- essary for the grid n aerodynamic prediction tool. An example of this modi ed input le can be seen in Appendix A. The program then begins a sweep of the speci ed Mach num- bers, and subsequently calculates the aerodynamic coe cient derivatives for the low angle of attack region. A sweep of the speci ed angles of attack is then performed within the Mach number loop, where the aerodynamic coe cients for the missile con guration are calculated. This process is repeated for every angle of attack at each Mach number. The resulting aerodynamic coe cients are then written to the output le, in either a long or short format, as speci ed by the user. An example of the short format output le can be seen in Appendix B. The short format shows the resulting normal force coe cient for the tail, normal force coe cient for a single grid n, axial force coe cient for the tail, and pitching moment coe cient for each angle of attack at each Mach number. The long 35 Figure 5.1: Standalone AERODSN Flow Chart format includes each of these parameters, as well as the values for the body alone and the total con guration. An additional output le is generated (Appendix C) that de nes the grid n geometry, including the (x;y;z) coordinates of each intersection point of the grid n and a list of each panel and the corresponding intersection points that de ne the endpoints of that panel. 5.2 Missile System Preliminary Design Tool Once the grid n aerodynamic prediction tool had been validated in conjunction with AERODSN, the program was integrated with the missile system preliminary design tool. The missile system preliminary design tool consists of a suite of optimizers that drive a full six-degree-of-freedom (6-DOF) model capable of designing single-stage missile systems to y given trajectories or to hit speci ed targets. This code consists of an aerodynamics model (AERODSN), a mass properties model, and a solid propellant propulsion model. This code 36 has proven to be a reliable tool for aerospace design applications and has been successfully used in many previous optimization studies [20{22]. Figure 5.2 shows the ow diagram for this program. The program begins by allowing the user to select the desired optimizer and ight case and set the maximum and minimum bounds for the optimization parameters. Once this is done, the optimizer lls the initial population with feasible solutions and then begins the generational loop. The ight charac- teristics (mass properties, aerodynamics, and propulsion) are determined for each member of the population, and a 6-DOF y-out is generated for each member. The tness of each member is calculated based on the speci ed objective function, and a new set of solutions are then created based on the tness of the previous population. This process is repeated until the maximum number of generations has been reached. Figure 5.2: Missile System Preliminary Design Tool Flow Chart 37 5.2.1 Optimization As was discussed previously, a modi ed ant colony optimization scheme was selected for use in this work due to its proven e ectiveness at solving complex aerospace design problems [20, 21]. The ant colony is an example of a swarm intelligence algorithm, and is based on the foraging behavior of ants. The ants communicate by depositing a trail of pheromone, which allows them to determine the optimal paths to sources of food over time. The original ant colony algorithm is very e ective at solving complex combinatorial problems, but is ine ective at solving complex problems in the continuous domain. The modi ed ant colony that is in the missile system preliminary design tool has been extended to the continuous domain by replacing the discrete pheromone links with Gaussian probability density functions. The solutions are ranked in order by their respective tness values and a pheromone model is created, where the pheromone amount is determined by rank. The new ants are created by choosing an existing ant with a probability that is proportional to their assigned pheromone strengths. The ants sample the Gaussian distributions around each variable and combine the variables to form the new solution. The methodology behind the modi ed ant colony optimization algorithm is fully detailed in Reference [20]. The objective function that was used for the target strike envelope maximization prob- lem was de ned to simply be the sum of the tness values for each individual target within the speci ed target grid. The tness for each individual target was de ned to be the miss distance in feet between the (x;y) location of the target and the (x;y) location where the missile actually landed, as seen by: Error = q (xtarget xactual)2 + (ytarget yactual)2 (5.1) In order to ensure that the ant colony was optimizing to increase the strike capability area rather than just minimizing the miss distance, a \maximum" miss distance of 35 feet was set so that if the missile missed a target by more than the designated distance, the tness for 38 that particular target location would be set to 35. This value was chosen for the maximum miss distance because it adequately captured the zone where the target is considered to be hit while still providing room for error. If no \maximum" miss distance is set, a missile could actually miss every target in the grid and have a better tness value than a second missile that hits 25% of the grid but misses the remaining targets by a greater amount than the rst missile. 5.2.2 Flight Characteristics The mass properties of the missile con gurations are determined using a variety of empirical formulations for the di erent components of the missile. Included in the mass calculations are: the nose of the missile, the solid rocket motor case and liner, the warhead, the sensors and wiring, the servo actuators, the igniter, the nozzle, the wing assembly, the tail assembly, the rail, and the fuel grain. In addition to the mass of the individual components, the mass moments of inertia and the x-location of the center of gravity of the missile are calculated in this section of the code. The aerodynamic properties for the conventional planar n and for the cylindrical missile body are determined via AERODSN in the missile system preliminary design tool. The grid n aerodynamic prediction algorithms were incorporated into the code so that a wide variety of grid n designs could also be evaluated. The propulsion properties are determined through the geometric analysis of the burning of a solid rocket motor grain. The typical grain geometry used in this code is the star grain. A parabolic nozzle design is also used in this code. 5.2.3 Program Modi cations Several modi cations had to be made to the missile system preliminary design code in order to implement the target strike envelope optimization problem. The missile system preliminary design code was originally set up so that a single-stage, solid propellant missile 39 Figure 5.3: Illustration of Line-of-Sight Guidance could launch from sea level and follow a given trajectory. For this work, all of the propulsion properties were removed, including the solid rocket motor grain modeling subroutines and the nozzle subroutines. The missile guidance algorithm was also modi ed to better t the current problem. Since the purpose of the code was to match speci ed trajectories, the guidance algorithm was set up to follow predetermined points along the trajectory for the duration of the missile ight. For the target strike envelope maximization problem, the guidance algorithm was modi ed to a line-of-sight guidance system that seeks to minimize the rotation of the line-of-sight vector between the location of the missile and the location of the target. This approach is similar to proportional navigation (Pro-Nav), except that the acceleration terms are not considered in this work. Figure 5.3 shows an illustration of the line-of-sight guidance system concept for this application. A total of ten grid n parameters were added to the missile system preliminary design tool, which increased the total number of optimization parameters from 35 to 45. However, with the removal of the propulsion properties, the total number of parameters was reduced to 34. A total of 20 parameters were used for the planar n optimization cases while a total of 40 25 parameters were used for the grid n optimization cases. The ten grid n parameters that were added can be seen in Figure 5.4, along with their respective maximum and minimum bounds that were used for this work. The de ning geometrical parameters for the grid n can be seen in Figure 5.5, and are: 1) Body centerline to the base of the grid n (Y0) 2) Body centerline to the tip of the grid n (B2) 3) Height of the n support base (HB) 4) Total height of the grid n (H) 5) Chord length of the grid n (C) 6) Average n element thickness (thk) 7) Number of cells in base corner (ibase) 8) Number of cells in tip corner (itip) 9) Number of cells in span-wise direction (ndy) Figure 5.4: Grid Fin Parameter Optimization Constraints 41 Figure 5.5: Grid Fin Parameters [12] 10) Number of cells in vertical direction (ndz) Another modi cation made to the missile system preliminary design code was adding the ability to hold any desired optimization parameter constant. Since a direct comparison of the performance of planar ns and grid ns is desired, it is imperative to be able to hold the missile body geometry constant for each run so that any variation in performance can be attributed directly to the ns. Check boxes were added to the user interface that allow the user to mark each individual parameter that is to be held constant for that run, an example of which can be seen in Figure 5.4. If the \hold variable constant" box is selected for a parameter, the corresponding data from the most recent single run case is used for each subsequent call to the objective function. A nal modi cation that was required in the missile system preliminary design code was the addition of a method for the determination of the mass properties for any given grid n geometry. In order to calculate the mass of a given grid n geometry, a routine was 42 added to determine the total length of the panels of the grid n (Lptot). This value was then multiplied by the chord length (C) and the average thickness (thk) of the elements to obtain an e ective grid n volume. Under the assumption that the grid n is made of aluminum, the e ective grid n volume was then multiplied by the density of aluminum ( al) in order to obtain the mass of the grid n: mGF = C thk Lptot al (5.2) The x-location of the center of gravity for the tail con guration was assumed to simply be at the half chord location of the grid n. For the mass moment of inertia calculation, the grid n was assumed to be a point mass, the equation for which can be seen by: Ixx = mGFi r2i (5.3) where ri is the distance from the centerline of the missile body to the half span location of the n. The mass moments of inertia about the y and z axes are assumed to be negligible in this work. 43 Chapter 6 Target Strike Envelope Maximization 6.1 Problem Description The goal of this problem is to compare the performance of an optimized missile con g- uration using both planar ns and grid ns as aerodynamic control devices in an e ort to maximize the target strike envelope of an unpowered missile. An illustration of this problem can be seen in Figure 6.1 below. For each case, the missile was dropped from the (x;y) location of (0;0) at an altitude of 23,000 ft with a freestream (x-component) velocity of 492.8 ft/sec (336 mph). A stationary target was placed directly in front of the missile drop point at sea level at a range of 20,000 ft downstream, and a [21x21] grid of targets was then constructed around this speci ed central target location, as seen in Figure 6.2. The [21x21] grid size was chosen for the optimization runs in an e ort to nd a balance between the number of function calls required for each missile con guration that was analyzed and the Figure 6.1: Illustration of the Missile Drop Problem 44 Figure 6.2: Illustration of a Target Grid for Optimization resolution of the target grid area. Values of dxt = 39;000 ft and dyt = 40;000 ft were used to construct the target grid for the optimization runs so that the entire vicinity in front of the aircraft was captured. A population size of 35 members was used for each optimization run for a total of 25 generations. This resulted in the evaluation of 875 solutions at 441 di erent target locations each, for a total of 385,875 function calls per optimization run. A maximum n de ection of 15 was allowed for the planar n cases, while a maximum n de ection of 30 was allowed for the grid n case. This problem was approached by rst conducting the optimization of a missile con g- uration with grid ns so that it could strike the largest area of the target grid structure as possible. Once the optimal grid n con guration had been found, another optimization run was conducted in which the grid ns were replaced by planar ns but the missile body pa- rameters were held constant. In an e ort to produce comparable results between the grid n and planar n con gurations, several di erent constraints were applied to the problem. The rst constraint was to ensure that the missile geometry had approximately the same static margin regardless of the aerodynamic control device used. This resulted in the placement 45 of the grid ns closer to the nose of the missile compared to the planar ns. The second constraint that was used ensured that the semi-span of the planar n and the semi-span of the grid n would be nearly identical. This was satis ed by using identical maximum and minimum bounds for the optimization runs for both the grid n and planar n cases. The third and nal constraint limited the maximum hinge moment possible for the planar n con guration. Larger hinge moments require a larger control actuator to move the n, which requires more control power and a larger internal volume of the missile. For the purposes of this work, the maximum allowable hinge moment coe cient for the planar n case was set to be two times the maximum hinge moment coe cient from the grid n analysis. 6.2 Results To show the importance of optimization in complex aerospace design problems, two unoptimized cases were run: one for a generic grid n missile con guration and one for a generic planar n missile con guration. The resulting target strike envelopes for these two cases can be seen in Figure 6.3 below. In the target strike envelope plots, the missile is (a) Unoptimized Grid Fin Con guration (b) Unoptimized Planar Fin Con guration Figure 6.3: Unoptimized Target Strike Envelopes 46 dropped from the (x;y) location of (0;0) and the color represents the miss distance in feet, as de ned by the colorbar beside each plot. The total target strike area for the unoptimized grid n case was found to be 2.60 square miles, while the total target strike area for the unoptimized planar n case was found to be 0.51 square miles. Figure 6.4 shows the strike area for the optimized grid n missile con guration. It can be seen that the ant colony optimizer was able to design a missile con guration that drastically improved the target strike area, improving it from 2.60 square miles in the unoptimized case to 13.21 square miles in the optimized case. Figure 6.5 shows the strike area for the optimized planar n missile con guration with limited hinge moment coe cient. Similar to the grid n case, the optimizer was able to drastically improve the performance of the planar n missile case. The target strike area was increased from 0.51 square miles in the unoptimized case to 8.65 square miles in the optimized case. For the optimized planar n missile con guration with unlimited hinge moment coe cient that is shown in Figure 6.6, the target strike area was found to be 19.02 square miles. Table 6.1 shows a comparison between the optimized grid n case, the optimized planar n case with limited hinge moment coe cient, and the optimized planar n case with unlim- ited hinge moment coe cient. It can be seen that the grid n resulted in a substantial weight reduction, as it weighs approximately 85% less than either of the planar n con gurations. It can also be seen in Table 6.1 that the average ight time of the grid n con guration is Table 6.1: Optimized Missile Con guration Data Parameter Grid Fin Case Limited PlanarFin Case UnlimitedPlanar Fin Case Target Strike Area 13:21 mi2 8:65 mi2 19:02 mi2 Mass of Single Fin 20:66 lbs 139:70 lbs 115:30 lbs Maximum Hinge Moment Coe cient 0:0862 0:1538 1:0516 Average Flight Time 59:4 sec 47:7 sec 49:5 sec 47 Figure 6.4: Target Strike Envelope for Optimized Grid Fin Con guration 48 Figure 6.5: Target Strike Envelope for Optimized Planar Fin Con guration with Limited Hinge Moment 49 Figure 6.6: Target Strike Envelope for Optimized Planar Fin Con guration with Unlimited Hinge Moment 50 substantially higher than that of the planar n cases. This is due to the higher drag that is produced by the grid ns compared to the planar ns. A comparison of the target strike envelope of the grid n con guration in Figure 6.4 and the planar n con guration with limited hinge moment coe cient in Figure 6.5 shows that the missile with the grid ns is able to hit a larger range of targets than a comparable missile with planar ns. In addition to being able to hit a larger area than the planar n con guration, the grid n con guration is also able to hit the targets with greater precision. To show this, the average miss distance within the target strike zone was calculated for each of these cases. It was found that for the region where the missile con guration is considered to hit the target, the average miss distance for the grid n case is 2.42 feet, while the same value for the planar n case with limited hinge moment is 5.30 feet. This calculation was also done for the planar n case with unlimited hinge moment coe cient in Figure 6.6, and the average miss distance was found to be 6.50 feet. Figure 6.7 shows a comparison between the optimized grid n geometry found in this work and a classical grid n geometry (Grid Fin \B" from Figure 4.1). It can be seen that the cells of the optimized grid n have been stretched in the span-wise direction so that Figure 6.7: Grid Fin Comparison 51 the panels are not at a 45 angle. This seems to suggest that a missile con guration with classical grid ns is more e ective at some nite roll angle rather than in the cruciform con guration, which is supported by the ndings of Kless and Aftosmis in Reference [6]. In addition, it was found that the design parameters for the optimized grid n geometry did not reach any of the limits that were set for the optimization runs, which indicates that the bounds used in this work were su cient for this particular problem. The values for the optimized grid n geometry parameters as well as their respective maximum and minimum bounds can be seen in Table 6.2. It was also noted that the initial velocity and altitude used in the target strike optimization problem resulted in strictly subsonic and transonic ow conditions for the missile con gurations, meaning that the supersonic grid n aerodynamic prediction capabilities were not used for this particular problem. Figures 6.8 and 6.9 show the optimized missile con gurations for each of these three cases. As expected, the missile body geometry is identical for all three cases. In addition, it can be seen that the grid ns are located at approximately 80% of the missile body length, while the planar ns are located closer to the tail of the missile. This placement was chosen by the optimizer to satisfy the equivalent static margin constraint discussed previously. It Table 6.2: Optimized Grid Fin Geometry Parameters Parameter Minimum Limit Optimized Value Maximum Limit Y0=DB 0:5 0:6276 1:0 B2=DB 1:56258 1:6414 1:79871 HB=DB 0:1 0:3489 0:6 H=DB 0:3 0:4652 1:0 C=DB 0:05 0:0583 0:2 thk=DB 0:0008 0:0014 0:0036 ibase 0 1 3 itip 0 1 3 ndy 2 2 10 ndz 2 4 10 52 can also be seen that the ns in each case have approximately the same semi-span, as expected. Another interesting observation from Figure 6.8 is the optimized geometry of the planar ns in the limited and unlimited hinge moment coe cient cases. Since the missile is in completely subsonic and transonic ow, the best planar n con guration should have an un-swept leading edge similar to that of the missile geometry for the unlimited hinge moment coe cient case. However, this design results in a hinge moment coe cient that is over twelve times higher than that of the grid n case. In order to have lower hinge moment coe cients for the planar n, the leading edge of the n must be more swept, similar to the geometry found for the limited hinge moment coe cient case. 53 (a) Optimized Grid Fin Con guration (b) Optimized Planar Fin Con guration with Limited Hinge Moment (c) Optimized Planar Fin Con guration with Unlimited Hinge Moment Figure 6.8: Side View of Optimized Missile Geometry 54 (a) Optimized Grid Fin Con guration (b) Optimized Planar Fin Con guration with Limited Hinge Moment (c) Optimized Planar Fin Con guration with Unlim- ited Hinge Moment Figure 6.9: Front View of Optimized Missile Geometry 55 Chapter 7 Conclusions and Recommendations The subsonic, transonic, and supersonic grid n aerodynamic prediction algorithms were successfully integrated into two di erent codes: a standalone version of AERODSN and a missile system preliminary design tool. The transonic grid n aerodynamic prediction method was altered to account for the bow shock that forms in front of the grid n at low supersonic Mach numbers, and was shown to provide accurate estimations of grid n aerodynamics in that region. A validation of the grid n aerodynamic prediction capability was performed using the standalone version of AERODSN for several di erent grid n designs for multiple Mach numbers, con guration roll angles, and n de ection angles. It was found that the theoretical formulations provide accurate estimations for the normal force, axial force, and pitching moment coe cients for a wide range of Mach numbers and angles of attack, and are su cient for the prediction of grid n aerodynamics in a preliminary-level engineering design tool. It was also shown that the imaging scheme used to model the n- body carry-over loads is able to accurately capture the interference e ects of the grid n with the missile body. The target strike envelope maximization problem was then conducted using the missile system preliminary design tool, where it was found that an optimized grid n con guration is able to outperform a comparable optimized planar n con guration. Several constraints were set in order to ensure the grid n and planar n cases were comparable. The rst constraint ensured that the grid n and planar n cases both had the same n semi-span. The second constraint ensured the planar n missile con guration had approximately the same static margin as that of the grid n missile con guration. The third and nal constraint ensured that the planar n could not have a maximum hinge moment coe cient that was 56 more than two times larger than that of the grid n. With these constraints, the grid n missile was able to hit a larger target area and was able to hit those targets with greater accuracy than the planar n missile. The grid ns produced increased performance while substantially reducing the mass of the ns and the size of the control actuator required for n control. This research shows that, despite the high amounts of drag associated with grid ns, there are some applications where the grid n should be seriously considered for use as a control and stability device. Additional research that could be conducted to supplement and enhance the results achieved in this work include: 1) The inclusion of wing-tail interference e ects with the grid n aerodynamics so that wings can be added to the missile con guration to see how the target strike envelope is a ected by the additional lifting surfaces. 2) Investigation of di erent missile body geometries, including a multitude of di erent diameters and neness ratios. 3) Expansion of the limits of the n design parameters so that the optimizer is able to consider a wider range of planar n and grid n designs for the di erent missile body geometries. 4) Testing the planar n and grid n missile con gurations at di erent roll angles to nd the optimal orientation of the missile. 5) Performing additional wind tunnel testing on a more diverse set of grid n geometries for further validation of the subsonic, transonic, and supersonic grid n aerodynamic prediction codes. 6) Performing a supersonic grid n analysis similar to the target strike problem that was done in this work. 57 Bibliography [1] William D. 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Burkhalter, \Ramjet Powered Missile Design Using a Genetic Algorithm," Journal of Computing and Information Science in Engineering, Vol. 7, No. 2, June 2007. [19] Roy J. Hart eld, Rhonald M. Jenkins, and John E. Burkhalter, \Optimizing a Solid Rocket Motor Boosted Ramjet Powered Missile Using a Genetic Algorithm," Applied Mathematics and Computation, Vol. 181, No. 2, 2006, pp. 1720-1736. [20] Zachary J. Kiyak, \Ant Colony Optimization: An Alternative Heuristic for Aerospace Design Applications," Master?s Thesis, Auburn University, December 2013. [21] Zachary J. Kiyak, Roy J. Hart eld, and Timothy W. Ledlow, \Missile Trajectory Opti- mization Using a Modi ed Ant Colony Algorithm," 2014 IEEE Aerospace Conference, AIAA Paper 2014-2185, March 2014. [22] Timothy W. Ledlow, Zachary J. Kiyak, and Roy J. Hart eld, \Missile System Design Using a Hybrid Evolving Swarm Algorithm," 2014 IEEE Aerospace Conference, AIAA Paper 2014-2225, March 2014. [23] Sergei M. 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Anderson, Jr., \Fundamentals of Aerodynamics," McGraw-Hill Publishing, Fifth Edition, 2011, pp. 549-598. [30] John C. Evvard, \Use of Source Distributions for Evaluating Theoretical Aerodynamics of Thin Finite Wings at Supersonic Speeds," NACA Report 951, 1950. [31] John E. Burkhalter, \Grid Fins in Supersonic Flow," Contractor Final report, Con- tract No. DAAH01-92-D-R002, DO Number NRC 0211, AMSMI-RD-SS-AT, Redstone Arsenal, AL, September 1994. [32] Jack N. Nielsen, \Missile Aerodynamics," Nielsen Engineering & Research, Inc., 1988, pp. 112-138. 60 Appendices 61 Appendix A Standalone AERODSN Input File Optimizer Type||||||||||||||| |{(1=RealGA, 2=BinaryGA, 3=PSO, 4=ASO, 5=SingleRun)|{ 5 ;Choose Desired Optimizer ...ioptimizertype 2 ;Tail Fin Flag (0=no ns, 1=planar ns, 2=grid ns) ...itail n 0 ;Wing Flag (0=no wing, 1=planar n wing) ...iwing n Flight Conditions|||||||||||||| 1 ;Number of Freestream Mach Numbers ...NFMA 0.5 ;Table of Freestream Mach Numbers ...TFMA 0.0 ;Altitude for Each Mach Number (ft) ...TALT Missile Body|||||||||||||||{ 1 ;Number of Freestream Mach Numbers ...NFMA 0.4167 ;Reference Length (ft) ...DREF 0.136354 ;Reference Area (ft2) ...AREF 1.25 ;Nose Length (ft) ...XLN 4.333 ;Total body Length (ft) ...XL 0.2083 ;Radius Body at Wing (ft) ...RBW 0.2083 ;Radius Body at Tail (ft) ...RBT 1.75 ;Nose to Wing Hinge Line (ft) ...THINGW 3.833 ;Nose to Tail Hinge Line (ft) ...THINGT 1 ;Nose Type 1-Ogive 2-Cone ...NOSE 4 ;Total Number of Fins on Tail ...numb nsT 2 ;Total Number of Fins on Wing ...numb nsW 0 ;Add Boattail (0=NO, 1=YES) ...NBTL 1.0 ;Boattail Diameter/Cylinder Diameter ...DBOD 0.001 ;Boattail Length/Cylinder Diameter ...XLBOD Grid Fin Parameters||{(G12)||||||||| 0.28025 ;Body CL to Base of Grid n (ft) ...yzro 0.280167 ;Min Radius for Grid Points (ft) ...r1 0.508583 ;Body CL to Grid n Tip (ft) ...b2 0.09 ;Height of Fin Support Base (ft) ...hb 0.07192 ;Span of Fin support Base (ft) ...ylb 0.18167 ;Total Height of Fin (ft) ...h 0.032 ;Chord Length of Fin (ft) ...chord 62 0.000667 ;Average Fin Element Thickness (ft) ...thk 2 ;Fin Base corner type; No Cells in Base Corner ...ibase 2 ;Fin Tip corner type; No Cells in Tip corner ...itip 5.0 ;No Cells in Spanwise Direction ...ndy 4.0 ;No Cells in Vertical Direction ...ndz 1 ;No vortices per element chordwise ...nvc 1 ;No vortices per element spanwise ...nvs 00.00 ;Roll Angle for Con guration ...phii 1, 90.0, 0.0 ;Fin No, Angle Phi, Incidence Angle ...i n,phi,delta 2, 0.0, 0.0 ;Fin No, Angle Phi, Incidence Angle ...i n,phi,delta 3, 270.0, 0.0 ;Fin No, Angle Phi, Incidence Angle ...i n,phi,delta 4, 180.0, 0.0 ;Fin No, Angle Phi, Incidence Angle ...i n,phi,delta 16 ;No Alphas (Angles Listed Below) ...nalpa -10.0,-8.0,-6.0,-4.0,-2.0,0.0,2.0,4.0,6.0,8.0,10.0,12.0,14.0,16.0,18.0,20.0 Planar Tail Fin Parameters||||||||||| 2.0 ;Tail Exposed Semispan (B/2) ...TBOT 4.00 ;Tail Root Chord (Croot) ...TCRT 0.5 ;Tail Taper Ratio (Ctip/Croot) ...TTRT 0.0 ;Tail Trailing Edge Sweep Angle (deg) ...TSWTET 20.0 ;Tail Position (Measured from nose) ...TXTAIL 0.0 ;Tail De ection (deg) ...TDELT Wing Parameters||||||||||||||{ 0.001 ;Wing Exposed Semispan (B/2) ...TBOW 0.001 ;Wing Root Chord (Croot) ...TCRW 1.0 ;Wing Taper Ratio (Ctip/Croot) ...TTRW 0.0 ;Wing Trailing Edge Sweep Angle (deg) ...TSWTEW 5.0 ;Wing Station (Measured from nose) ...TXWING 0.0 ;Wing De ection (deg) ...TDELW AERODSN Inputs||||||||||||||| 1 ;Wing Station Flag ...NWPOS 1 ;Tail Station Flag ...NTPOS 1 ;Use NACA Report 1253 ...MCDVT 3 ;Alpha and Trim Output ...NOUT 1 ;Initial Run Number ...NRUN Miscellaneous Inputs||||||||||||| 0.5 ;Mach Number for XCG ...TMF 5.1996 ;C.o.G. (calibers from nose) ...TXCG 6000.0 ;Weight (lbs) ...TWEIGH 6 ;No Iteration Loops < 10 ...nloops 1 ;Short Output (=1) or Long Output (=0) ...ishortoutput 63 Appendix B Standalone AERODSN Output File ALP CNT CNTbal CDT CMCG -10.0000 -0.76221 -0.29774 0.45077 1.49997 -8.00000 -0.67739 -0.26119 0.45077 1.46332 -6.00000 -0.55770 -0.21148 0.45077 1.29273 -4.00000 -0.40153 -0.14936 0.45077 0.97893 -2.00000 -0.21171 -0.07741 0.45077 0.53225 0.00000 0.00000 0.00000 0.45077 0.00000 2.00000 0.21171 0.07741 0.45077 -0.53225 4.00000 0.40153 0.14936 0.45077 -0.97893 6.00000 0.55770 0.21148 0.45077 -1.29273 8.00000 0.67739 0.26119 0.45077 -1.46332 10.0000 0.76221 0.29774 0.45077 -1.49997 12.0000 0.81590 0.32182 0.45077 -1.42802 14.0000 0.84327 0.33502 0.45077 -1.26682 16.0000 0.84945 0.33930 0.45077 -1.04070 18.0000 0.83929 0.33661 0.45077 -0.77262 20.0000 0.81702 0.32870 0.45077 -0.48718 64 Appendix C Grid Fin Geometry Output File Panel Coordinates Point Number x y z 1 45.8040 2.2104 0.5352 2 45.8040 2.2104 -0.5352 3 45.8040 3.1584 0.5250 4 45.8040 3.1584 -0.5250 5 45.8040 3.6834 1.0500 6 45.8040 3.6834 0.0000 7 45.8040 3.6834 -1.0500 8 45.8040 4.2084 0.5250 9 45.8040 4.2084 -0.5250 10 45.8040 4.7334 1.0500 11 45.8040 4.7334 0.0000 12 45.8040 4.7334 -1.0500 13 45.8040 5.2584 0.5250 14 45.8040 5.2584 -0.5250 15 45.8040 5.7834 1.0500 16 45.8040 5.7834 0.0000 17 45.8040 5.7834 -1.0500 18 45.8040 6.3084 0.5250 19 45.8040 6.3084 -0.5250 20 45.8040 6.8334 1.0500 21 45.8040 6.8334 0.0000 22 45.8040 6.8334 -1.0500 23 45.8040 7.3584 0.5250 24 45.8040 7.3584 -0.5250 Panel Connect Points for 43 Panels Panel Inboard Outboard Number Point Point 1 1 2 2 1 3 3 2 4 4 3 4 5 3 5 6 3 6 65 7 4 6 8 4 7 9 5 8 10 5 10 11 6 8 12 6 9 13 7 9 14 7 12 15 8 10 16 8 11 17 9 11 18 9 12 19 10 13 20 10 15 21 11 13 22 11 14 23 12 14 24 12 17 25 13 15 26 13 16 27 14 16 28 14 17 29 15 18 30 15 20 31 16 18 32 16 19 33 17 19 34 17 22 35 18 20 36 18 21 37 19 21 38 19 22 39 20 23 40 21 23 41 21 24 42 22 24 43 23 24 Chord Length 0.3840 66 Appendix D Best Fitness vs. Number of Function Calls Output File Plot Best Fitness vs. Number of Function Calls 1 12781.1409493058 22 2158.93799077375 37 1846.62774054338 45 1296.37822732603 121 776.518078551530 157 486.499008026406 241 427.540782168719 276 274.335273449429 344 166.334585000184 379 84.9423373596193 414 60.7299095486573 519 57.4228834720672 594 24.2771689080268 728 6.19279599233190 67 Appendix E Best Fit Member Output File 0.418049484491 ; - 1 rnose/rbody 1.821997761726 ; - 2 lnose/dbody 0.000000000000 ; - 3 fuel type 0.000000000000 ; - 4 star outer R rpvar=(rp+f)/rbody 0.000000000000 ; - 5 star inner ratio=ri/rp 0.000000000000 ; - 6 number of star pts 0.000000000000 ; - 7 llet radius ratio=f/rp 0.000000000000 ; - 8 eps (star PI*eps/N) width 0.000000000000 ; - 9 star point angle deg 0.000000000000 ; - 10 fractional noz len f/ro 0.000000000000 ; - 11 Dia throat/Dbody=Dstar/Dbody 5.500650882721 ; - 12 Fineness ratio Lbody/Dbody 1.067551493645 ; - 13 dia of stage1 meters 0.000572043238 ; - 14 wing semispan/dbody 0.000541782822 ; - 15 wing root chord = crw/dbody 0.852307617664 ; - 16 taper ratio = ctw/crw 40.037799835205 ; - 17 wing LE sweep angle deg 0.406034529209 ; - 18 xLE xLEw/lbody 1.289271235466 ; - 19 tail semispan/dbody 1.016451358795 ; - 20 tail root chord = crt/dbody 0.595359325409 ; - 21 tail taper ratio = ctt/crt 0.914866983891 ; - 22 LE sweep angle deg 0.990155518055 ; - 23 xTEt xTEt/lbody 1.484631061554 ; - 24 auto pilot delay time sec 0.169458851218 ; - 25 initial launch angle deg 2.652890920639 ; - 26 gainp1 - pitch multiplier gain 3.816827058792 ; - 27 gainy1 - yaw multiplier gain 0.000000000000 ; - 28 noz exit dia/dbody -1.100889801979 ; - 29 initial pitch cmd angle (deg) 3.564826965332 ; - 30 gainp2 - angle dif gain in pitch 2160.249023437500 ; - 31 warmas - warhead mass 0.928660809994 ; - 32 time step to actuate nozzle (sec) 0.106755934656 ; - 33 gainy2 - angle dif gain in yaw 0.471196562052 ; - 34 initial launch direction (deg) 0.000751175161 ; - 35 initial pitch cmd angle (deg) 0.633140861988 ; - 36 body CL to base of GF/Dbody 68 2.755671262741 ; - 37 body CL to GF tip/Dbody 0.413074821234 ; - 38 height of GF support base/Dbody 0.456009268761 ; - 39 total height of GF/Dbody 0.155281305313 ; - 40 chord length of GF/Dbody 0.001375171472 ; - 41 AVG GF element thickness/Dbody 2.746897697449 ; - 42 number of cells in GF base corner 1.935090422630 ; - 43 number of cells in GF tip corner 7.035898685455 ; - 44 num. cells in spanwise dir of GF 9.773223876953 ; - 45 num. cells in vertical dir of GF 69 Appendix F Target Fitness Output File Target X Location Target Y Location Miss Distance 15000.0000000000 -5000.00000000000 8.39709064924577 17500.0000000000 -5000.00000000000 17.6984989344853 20000.0000000000 -5000.00000000000 0.55546873193927 22500.0000000000 -5000.00000000000 0.46364226301213 25000.0000000000 -5000.00000000000 0.48115083424448 15000.0000000000 -2500.00000000000 2.81273243773808 17500.0000000000 -2500.00000000000 1.94617933155096 20000.0000000000 -2500.00000000000 10.6224313763366 22500.0000000000 -2500.00000000000 0.71977713125060 25000.0000000000 -2500.00000000000 0.29650714325997 15000.0000000000 0.00000000000000 0.54816644488302 17500.0000000000 0.00000000000000 0.28955690189483 20000.0000000000 0.00000000000000 0.09350200932901 22500.0000000000 0.00000000000000 0.04924306392080 25000.0000000000 0.00000000000000 0.19718661372204 15000.0000000000 2500.00000000000 1.88601935151651 17500.0000000000 2500.00000000000 1.62695897281247 20000.0000000000 2500.00000000000 5.78243053170833 22500.0000000000 2500.00000000000 0.55620057990438 25000.0000000000 2500.00000000000 0.19905981613441 15000.0000000000 5000.00000000000 5.89321404691875 17500.0000000000 5000.00000000000 8.68805009978391 20000.0000000000 5000.00000000000 0.63275712621600 22500.0000000000 5000.00000000000 0.51010608066574 25000.0000000000 5000.00000000000 0.41572819670044 5 5 70