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
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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