DESIGN OPTIMIZATION OF SPACE LAUNCH VEHICLES USING A GENETIC
ALGORITHM
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
_____________________________
Douglas James Bayley
Certificate of Approval:
_______________________ ____________________
John E. Cochran Roy J. Hartfield, Chair
Professor Associate Professor
Aerospace Engineering Aerospace Engineering
_______________________ ____________________
John E. Burkhalter Christopher J. Roy
Professor Emeritus Assistant Professor
Aerospace Engineering Aerospace Engineering
_____________________
Joe F. Pittman
Interim Dean
Graduate School
DESIGN OPTIMIZATION OF SPACE LAUNCH VEHICLES USING A GENETIC
ALGORITHM
Douglas James Bayley
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 4, 2007
iii
DESIGN OPTIMIZATION OF SPACE LAUNCH VEHICLES USING A GENETIC
ALGORITHM
Douglas James Bayley
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at their expense. The author
reserves all publication rights.
________________________
Signature of Author
________________________
Date of Graduation
iv
VITA
Douglas James Bayley, son of Howard J. Bayley Jr. and Marie (Caione) Bayley,
was born on September 9, 1969 in New Britain, Connecticut. Douglas graduated from
Xavier High School in Middletown, Connecticut in June 1987. He attended Florida
Institute of Technology in the Fall of 1987. Douglas graduated magna cum laude with a
Bachelor of Science Degree in Aerospace Engineering in June 1992. Douglas began
graduate studies in the Fall of 1992 in the Department of Aerospace Engineering, Auburn
University. He completed a Master of Science Degree in Aerospace Engineering in the
Fall of 1994. In January 1995, Douglas began officer training and in May 1995, was
commissioned a Second Lieutenant in the United States Air Force. After assignments in
Montana, Colorado and California, Douglas returned to the Department of Aerospace
Engineering, Auburn University in 2005 to complete his Doctor of Philosophy Degree in
Aerospace Engineering. In 1998, Douglas married his beautiful bride Kelly (Draughon)
Bayley. Douglas and Kelly have been blessed with four wonderful children: Thomas (8),
Sarah (6), Anna (4) and Mary (1).
v
DISSERTATION ABSTRACT
DESIGN OPTIMIZATION OF SPACE LAUNCH VEHICLES USING A GENETIC
ALGORITHM
Douglas James Bayley
Doctor of Philosophy, August 4, 2007
(M.S., Auburn University, 1994)
(B.S., Florida Institute of Technology, 1992)
196 Typed Pages
Directed by Roy J. Hartfield
Disclaimer: The views expressed in this dissertation are those of the author and do not
reflect the official policy or position of the United States Air Force, Department of
Defense, or the U.S. Government.
The United States Air Force (USAF) continues to have a need for assured access
to space. In addition to flexible and responsive spacelift, a reduction in the cost per
launch of space launch vehicles is also desirable. For this purpose, an investigation of the
design optimization of space launch vehicles has been conducted.
Using a suite of custom codes, the performance aspects of an entire space launch
vehicle were analyzed. A genetic algorithm (GA) was employed to optimize the design
of the space launch vehicle. A cost model was incorporated into the optimization process
with the goal of minimizing the overall vehicle cost. The other goals of the design
vi
optimization included obtaining the proper altitude and velocity to achieve a lowEarth
orbit. Specific mission parameters that are particular to USAF space endeavors were
specified at the start of the design optimization process. Solid propellant motors, liquid
fueled rockets, and airlaunched systems in various configurations provided the
propulsion systems for two, three and fourstage launch vehicles. Mass properties
models, an aerodynamics model, and a sixdegreeoffreedom (6DOF) flight dynamics
simulator were all used to model the system.
The results show the feasibility of this method in designing launch vehicles that
meet mission requirements. Comparisons to existing real world systems provide the
validation for the physical system models. However, the ability to obtain a truly
minimized cost was elusive. The cost model uses an industry standard approach,
however, validation of this portion of the model was challenging due to the proprietary
nature of cost figures and due to the dependence of many existing systems on surplus
hardware.
vii
ACKNOWLEDGEMENTS
I would like to thank Dr. Roy J. Hartfield for his guidance and patience during
this long and arduous process. This degree would not have been completed if not for Dr.
Hartfield?s steadfast support. He has been a true role model and mentor. I would also
like to acknowledge the inspiration of Fr. Michael J. McGivney. Most importantly, I
have to thank Kelly, Thomas, Sarah, Anna, and Mary for their love, support and
sacrifices during this endeavor.
viii
Style or journal used:
The American Institute of Aeronautics and Astronautics Journal
Computer software used:
Microsoft Word 2003, IMPROVE
?
3.1 Genetic Algorithm, Compaq Visual FORTRAN
6.6, General Purpose 6DOF Simulation, Tecplot 10, Microsoft Excel 2003
ix
TABLE OF CONTENTS
LIST OF FIGURES .......................................................................................................... xii
LIST OF TABLES............................................................................................................ xv
NOMENCLATURE ......................................................................................................xviii
1.0 INTRODUCTION ....................................................................................................... 1
2.0 CHRONOLOGY OF OPTIMIZATION TECHNIQUES............................................. 3
2.1 Introduction............................................................................................................... 3
2.2 Early Concepts.......................................................................................................... 3
2.2.1 Newton and CalculusBased Methods............................................................... 4
2.2.2 Gradient Methods............................................................................................... 4
2.3 Linear Programming ................................................................................................. 5
2.4 Pattern Search Optimization ..................................................................................... 5
2.5 Design of Experiments.............................................................................................. 7
2.6 Additional Methods .................................................................................................. 8
2.7 Genetic Algorithms................................................................................................... 9
2.8 Recent Launch Vehicle Optimization Work........................................................... 11
3.0 SYSTEM MODELING .............................................................................................. 16
3.1 Introduction............................................................................................................. 16
3.2 Objective Function Link to the Genetic Algorithm (GA)....................................... 16
3.3 Genetic Algorithm (GA)......................................................................................... 19
3.4 Propulsion System Models ..................................................................................... 24
3.4.1 Solid Propellant Rocket Motors....................................................................... 26
3.4.2 Liquid Propellant Rocket Engines ................................................................... 27
3.5 Mass Properties Models.......................................................................................... 29
3.5.1 Mass Properties of Individual Components..................................................... 32
3.5.1.1 Point Mass Example: Electronics ............................................................. 34
3.5.1.2 Cylinder Example: Motor Cases............................................................... 34
3.5.1.3 Sphere Example: Compressed Gas Tank.................................................. 35
x
3.5.1.4 Mass Table................................................................................................ 36
3.5.2 Mass Properties of Entire Launch Vehicle ...................................................... 37
3.5.2.1 Entire Launch Vehicle Mass Properties Example: Phase I....................... 37
3.6 Aerodynamics Model.............................................................................................. 38
3.7 SixDegreeofFreedom (6DOF) Flight Dynamics Simulator................................ 41
3.8 Cost Model.............................................................................................................. 42
3.8.1 Development Cost Submodel .......................................................................... 44
3.8.2 Recurring Cost Submodel................................................................................ 46
3.8.3 Ground and Flight Operations Cost Submodel................................................ 47
3.8.4 Insurance Cost Submodel ................................................................................ 48
3.8.5 Example Calculation........................................................................................ 49
4.0 VALIDATION EFFORTS.......................................................................................... 51
4.1 Introduction............................................................................................................. 51
4.2 Validation Method .................................................................................................. 53
4.2.1 General Description ......................................................................................... 53
4.2.2 Specific Validation Process and Setup ............................................................ 54
4.2.3 Inert and Propellant Mass Fraction Calculations............................................. 56
4.3 ThreeStage Solid Propellant Vehicle vs. Minuteman III ICBM ........................... 57
4.4 FourStage Solid Propellant Vehicle vs. Minotaur I SLV ...................................... 64
4.5 TwoStage Liquid Propellant Vehicle vs. Titan II SLV ......................................... 69
4.6 AirLaunched, TwoStage Liquid Propellant Vehicle vs. QuickReach
TM
.............. 73
4.7 Mass Fractions for ThreeStage Solid/Liquid/Liquid Propellant Vehicle .............. 76
4.8 Summary of Validation Efforts............................................................................... 79
4.8.1 ThreeStage Solid Propellant Launch Vehicle................................................. 80
4.8.2 FourStage Solid Propellant Launch Vehicle .................................................. 80
4.8.3 TwoStage Liquid Propellant Launch Vehicle ................................................ 81
4.8.4 AirLaunched, TwoStage Liquid Propellant Launch Vehicle........................ 81
4.8.5 ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle ......................... 82
5.0 OPTIMIZATION RESULTS...................................................................................... 83
5.1 Introduction............................................................................................................. 83
5.2 Initial Launch Vehicles........................................................................................... 84
5.2.1 Case 1: ThreeStage Solid Propellant Launch Vehicle with Two Goals......... 85
5.2.2 Case 2: ThreeStage Solid Propellant Launch Vehicle with Three Goals....... 91
5.2.3 Conclusions: Initial Launch Vehicle Design Optimizations............................ 94
5.3 Solid Propellant Vehicles........................................................................................ 95
5.3.1 Case 3: ThreeStage Solid Propellant Launch Vehicle (VAFB) ..................... 98
5.3.2 Case 4: ThreeStage Solid Propellant Launch Vehicle (CCAFS) ................. 108
5.3.3 Case 5: FourStage Solid Propellant Launch Vehicle (VAFB) ..................... 113
xi
5.3.4 Case 6: FourStage Solid Propellant Launch Vehicle (CCAFS) ................... 118
5.3.5 Conclusions: Solid Propellant Launch Vehicle Design Optimizations ......... 122
5.4 Liquid Propellant Vehicles ................................................................................... 122
5.4.1 Case 7: ThreeStage Liquid Propellant Launch Vehicle (VAFB) ................. 124
5.4.2 Case 8: ThreeStage Liquid Propellant Launch Vehicle (CCAFS) ............... 127
5.4.3 Case 9: TwoStage Liquid Propellant Launch Vehicle (VAFB) ................... 129
5.4.4 Case 10: TwoStage Liquid Propellant Launch Vehicle (CCAFS) ............... 133
5.4.5 Conclusions: Liquid Propellant Launch Vehicle Design Optimizations ....... 135
5.5 AirLaunched Vehicles ......................................................................................... 136
5.5.1 Case 11: AirLaunched, TwoStage Liquid Propellant Launch Vehicle
(CCAFS) ................................................................................................................. 137
5.5.2 Conclusions: AirLaunched Vehicle Design Optimization ........................... 143
5.6 Mixed Propellant Vehicles.................................................................................... 144
5.6.1 Case 12: ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle (VAFB)
................................................................................................................................. 145
5.6.2 Case 13: ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle
(CCAFS) ................................................................................................................. 149
5.6.3 Conclusions: Mixed Propellant Vehicle Design Optimizations .................... 152
5.7 Launch Vehicle Comparisons............................................................................... 153
5.7.1 Launch Vehicles Comparison of 1,000 lbm Payload Cases .......................... 153
5.7.2 Launch Vehicles Comparison of Propellant Mass Fractions......................... 157
5.7.3 Launch Vehicles Comparison of Inert Mass Fractions.................................. 158
5.7.4 Launch Vehicles Comparison of Cost per Launch Values ............................ 158
6.0 CONCLUSIONS....................................................................................................... 160
6.1 Introduction........................................................................................................... 160
6.2 System Modeling and Validation.......................................................................... 161
6.3 Design Optimizations............................................................................................ 162
7.0 RECOMMENDED IMPROVEMENTS................................................................... 165
7.1 Types of Solid and Liquid Propellants.................................................................. 165
7.2 Aerodynamics Model............................................................................................ 166
7.3 SixDegreeof Freedom (6DOF) Flight Dynamics Simulator .............................. 166
7.4 Cost Model............................................................................................................ 167
7.5 Payload Masses and Orbits ................................................................................... 167
REFERENCES ............................................................................................................... 169
APPENDIX: Mass Table Example for ThreeStage Solid Propellant Launch Vehicle . 176
xii
LIST OF FIGURES
Figure 31. Objective Function Link to the GA................................................................17
Figure 32. Tournament Selection.....................................................................................21
Figure 33. Solid Rocket Motor Schematic.......................................................................26
Figure 34. Definition of Lengths......................................................................................33
Figure 41. ThreeStage Solid Propellant Model vs. Minuteman III ICBM Schematic....60
Figure 42. Minuteman III ICBM Ballistic Flight Profile.................................................63
Figure 43. Validation Model Ballistic Flight Profile........................................................64
Figure 44. FourStage Solid Propellant Model vs. Minotaur I SLV Schematic...............66
Figure 45. TwoStage Liquid Propellant Model vs. Titan II SLV Schematic..................71
Figure 46. Air Launched, TwoStage Liquid Propellant Model vs. QuickReach
TM
Launch Vehicle Schematic..............................................................................74
Figure 47. ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicles Schematic.....79
Figure 51. Progress of Best Performer to Meet Goal #1..................................................87
Figure 52. Progress of Best Performer to Meet Goal #2..................................................87
Figure 53. ThreeStage Solid Propellant Launch Vehicle Schematic..............................89
Figure 54. Altitude vs. Downrange Distance for Best Performer....................................89
Figure 55. Thrust vs. Time for Best Performer................................................................90
Figure 56. Vehicle Mass vs. Time for Best Performer.....................................................90
Figure 57. Velocity vs. Time for Best Performer.............................................................91
xiii
Figure 58. ThreeStage Solid Propellant Launch Vehicle Schematic (Case 2/Run 1).....94
Figure 59. ThreeStage Solid Propellant Launch Vehicle Schematic (Case 2/Run 2).....94
Figure 510. ThreeStage Solid Propellant Launch Vehicle Schematic (VAFB)..............99
Figure 511. Velocity vs. Generation # for ThreeStage Solid Propellant Launch
Vehicle........................................................................................................103
Figure 512. Altitude vs. Generation # for ThreeStage Solid Propellant Launch
Vehicle.........................................................................................................104
Figure 513. Total Vehicle Mass vs. Generation # for ThreeStage Solid Propellant
Launch Vehicle...........................................................................................105
Figure 514. Final Velocity for Best, Worst, and Average Members of Generation
#221.............................................................................................................106
Figure 515. Final Altitude for Best, Worst, and Average Members of Generation
#221.............................................................................................................107
Figure 516. Total Vehicle Mass for Best, Worst, and Average Members of Generation
#221.............................................................................................................107
Figure 517. ThreeStage Solid Propellant Launch Vehicle Schematic (CCAFS)..........109
Figure 518. Mass Improvements for ThreeStage Solid Propellant Launch Vehicles...112
Figure 519. FourStage Solid Propellant Launch Vehicles Schematic (VAFB)............114
Figure 520. FourStage Solid Propellant Launch Vehicle Schematic (CCAFS)............119
Figure 521. ThreeStage Liquid Propellant Launch Vehicle Schematic (VAFB)..........125
Figure 522. ThreeStage Liquid Propellant Launch Vehicle Schematic (CCAFS)........128
Figure 523. TwoStage Liquid Propellant Launch Vehicles Schematic (VAFB)..........130
Figure 524. TwoStage Liquid Propellant Launch Vehicles Schematic (CCAFS)........134
Figure 525. AirLaunched, TwoStage Liquid Propellant Launch Vehicles Schematic
(CCAFS)......................................................................................................141
Figure 526. ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle Schematic
(VAFB)........................................................................................................146
xiv
Figure 527. ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle Schematic
(CCAFS)......................................................................................................150
Figure 528. Total Vehicle Mass Comparison.................................................................155
Figure 529. Cost per Launch Comparison......................................................................156
Figure 530. Propellant Mass Fraction (f
prop
) Comparison..............................................157
Figure 531. Inert Mass Fraction (f
inert
) Comparison.......................................................158
Figure 532. Cost per Launch Comparison for All Launch Vehicles..............................159
xv
LIST OF TABLES
Table 21: Design Variables for a ThreeStage Solid Propellant Launch Vehicle............15
Table 31: Example Design Parameters and Chromosome...............................................19
Table 32: Liquid Propellant Fuels and Oxidizers.............................................................29
Table 33: ThreeStage Solid and Liquid Vehicle Components........................................32
Table 34: Mass Properties of Electronics.........................................................................34
Table 35: Mass Properties of Stage 1 Motor Case...........................................................35
Table 36: Mass Properties of Stage 1 Compressed Gas Tank..........................................36
Table 37: Inputs for Vehicle Cost Example Calculation..................................................50
Table 38: Outputs for Vehicle Cost Example Calculation...............................................50
Table 41: Typical Values of Real World Launch Vehicles..............................................55
Table 42: Example Solid Rocket Motor Mass Fractions..................................................56
Table 43: ThreeStage Solid Propellant Model vs. Minuteman III ICBM Comparison..61
Table 44: FourStage Solid Propellant Model vs. Minotaur I SLV Comparison.............66
Table 45: FourStage Solid Propellant Model Individual Stage Comparison..................67
Table 46: TwoStage Liquid Propellant Model vs. Titan II SLV Comparison................72
Table 47: Air Launched, TwoStage Liquid Propellant Model vs. QuickReach
TM
Launch Vehicle Comparison............................................................................75
Table 48: ThreeStage Solid/Liquid/Liquid Propellant Vehicle Mass Fractions.............77
Table 51: Space Launch Vehicle Design Optimization Cases.........................................84
Table 52: Initial Launch Vehicles Mission Statistics.......................................................85
xvi
Table 53: Case 2 Runs Comparison/ThreeStage Solid Propellant Launch Vehicles......93
Table 54: Solid Propellant Launch Vehicles Mission Statistics.......................................97
Table 55: Summary of ThreeStage Solid Propellant Launch Vehicle Characteristics
(VAFB)...........................................................................................................102
Table 56: Summary of ThreeStage Solid Propellant Launch Vehicle Characteristics
(CCAFS).........................................................................................................111
Table 57: FourStage Solid Propellant Launch Vehicles Comparison...........................115
Table 58: Summary of FourStage Solid Propellant Launch Vehicle Characteristics
(VAFB)...........................................................................................................117
Table 59: Summary of FourStage Solid Propellant Launch Vehicle Characteristics
(CCAFS).........................................................................................................121
Table 510: Liquid Propellant Launch Vehicles Mission Statistics.................................124
Table 511: Summary of ThreeStage Liquid Propellant Launch Vehicle Characteristics
(VAFB).........................................................................................................127
Table 512: Summary of ThreeStage Liquid Propellant Launch Vehicle Characteristics
(CCAFS).......................................................................................................129
Table 513: Summary of TwoStage Liquid Propellant Launch Vehicle Runs
(VAFB).........................................................................................................130
Table 514: Summary of TwoStage Liquid Propellant Launch Vehicle Characteristics
(VAFBRun #2)............................................................................................133
Table 515: Summary of TwoStage Liquid Propellant Launch Vehicle Runs
(CCAFS).......................................................................................................134
Table 516: Summary of TwoStage Liquid Propellant Launch Vehicle Characteristics
(CCAFSRun #2)..........................................................................................135
Table 517: AirLaunched Vehicle Mission Statistics.....................................................137
Table 518: C141 Transport Aircraft Characteristics.....................................................140
Table 519: Summary of AirLaunched, TwoStage Liquid Propellant Launch Vehicle
Characteristics (CCAFS)..............................................................................142
xvii
Table 520: AirLaunched, TwoStage Liquid Propellant Launch Vehicles
Comparison..................................................................................................143
Table 521: Mixed Propellant Launch Vehicles Mission Statistics.................................145
Table 522: Summary of ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle
Characteristics (VAFB)................................................................................148
Table 523: Summary of ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle
Characteristics (CCAFS)..............................................................................152
Table 524: Optimized Cases for 1,000 lbm Payload Mass.............................................154
xviii
NOMENCLATURE
a SystemSpecific Constant Value
a* Speed of Sound at Nozzle Throat
A* Nozzle Throat Area
A
b
Solid Propellant Grain Burn Area
A
e
Nozzle Exit Area
A
exposed
Exposed Area
A
p
Solid Propellant Motor Port Area
altorb Desired Orbital Altitude
alt1 Final Altitude
A
ref
Reference Area
c* Characteristic Exhaust Velocity
CCAFS Cape Canaveral Air Force Station
C
Dflatplate
Coefficient of Drag for Flat Plate
C
development
Development Cost per Launch
C
devtotal
Total Vehicle Development Cost
C
flight ops
Flight Operations Cost per Launch
C
insurance
Insurance Cost per Launch
C
launch
Total Cost per Launch
C
rectotal
Total Vehicle Recurring Cost
C
T
Thrust Coefficient
C
vehicle
Recurring Cost per Launch
CER Cost Estimating Relationship
CER
solid
Solid Rocket Motor Cost Estimating Relationship
CLV Crew Launch Vehicle
?* Gas Density at Nozzle Throat
?
b
Solid Propellant Grain Density
?C
Dcorr
Coefficient of Drag Correction Factor
?v Required Velocity Change (Deltav)
DB DoubleBase Solid Propellant
DARPA Defense Advanced Research Projects Agency
DCX DeltaClipper X
D
throat
Nozzle Throat Diameter
EELV Evolved Expendable Launch Vehicle
? Angular Fraction
f
0d
System Engineering and Integration Factor
f
0p
System Management, Vehicle Integration and Checkout Factor
f
1
Development Standard Factor
f
2
Technical Quality Factor
xix
f
3
Team Experience Factor
f
4
Cost Reduction Factor for Series Production
f
6
,f
7
,f
8
Programmatic Cost Impact Factors
FALCON Force Application and Launch from CONUS
F
ES
Recurring Cost of Solid Rocket Motor
f
inert
Inert Mass Fraction
f
prop
Propellant Mass Fraction
fn Fractional Nozzle Length
fvar Fillet Radius
? Ratio of Specific Heats
g
o
Local Acceleration Due to Gravity
GA Genetic Algorithm
GEM Graphite Epoxy Solid Motor
h
e
Nozzle Exit Enthalpy
h
o
Combustion Chamber Total Enthalpy
H2O295% Hydrogen Peroxide95%
H
ES
Development Cost of Solid Rocket Motor
HMX Cyclotetramethylene Tetranitramine
HTPB HydroxylTerminated Polybutadiene
ICBM Intercontinental Ballistic Missile
inf Inflation Rate
int Interest Rate
IRFNA Inhibited Red Fuming Nitric Acid
I
sp
Specific Impulse
ixx Xaxis Moment of Inertia
iyy Yaxis Moment of Inertia
izz Zaxis Moment of Inertia
l String Bits Length
lgrain Solid Propellant Grain Length
L
o
Launch Site Latitude
LEO Low Earth Orbit
LF2 Liquid Fluorine
LH2 Liquid Hydrogen
LOX Liquid Oxygen
lrate Launch Rate
m& Mass Flow Rate
m
inert
Inert Mass
m
prop
Propellant Mass
max Maximum Design Parameter Value
min Minimum Design Parameter Value
MMH Monomethyl Hydrazine
MYr Man Year
N2O4 Nitrogen Tetroxide
n Population Size
NAFCOM NASA and Air Force Cost Model
xx
NASA National Aeronautics and Space Administration
NASP National Aerospace Plane
nbits Number of Bits in Chromosome String
npay Number of Payments
nsp Number of Star Points
nstg Number of Stages
nunits Number of Units
ORS Operationally Responsive Space
p Learning Factor
P
a
Atmospheric Pressure
P
c
Combustion Pressure
P
e
Exit Pressure
P
avg
Average Payment Value
P
constant
Constant Payment Value
PBAA PolybutadieneAcrylic Acid
PBAN PolybutadieneAcrylic AcidAcrylonitrile Terpolymer
PS Polysulfide
r Solid Propellant Grain Burn Rate
R Gas Constant
R
annual
Annual Reduction Factor
R
bi
Grain Outer Radius
R
i
Inner Star Radius
R
p
Outer Star Radius
RP1 Rocket Propellant1
R
stg1
Radius of Stage 1
R
stg2
Radius of Stage 2
6DOF SixDegreeofFreedom
SLV Space Launch Vehicle
SRB Solid Rocket Booster
T Thrust
T
c
Combustion Temperature
T
o
Combustion Chamber Total Temperature
UDMH Unsymmetrical Dimethylhydrazine
USAF United States Air Force
VAFB Vandenberg Air Force Base
V* Gas Velocity at Nozzle Throat
V
e
Exit Velocity
x SystemSpecific CosttoMass Sensitivity Factor
xcg Center of Gravity Location
y
burn
Solid Propellant Burn Direction
1
1.0 INTRODUCTION
From the early space launch attempts almost 50 years ago up until today, private
companies, government agencies and entire countries have invested large amounts of
capital attempting to lower the price of access to space. Concepts such as the National
Aerospace Plane (NASP), the singlestagetoorbit X33 and the Delta ClipperX (DCX)
have all been valiant attempts at achieving low cost, easy access to space.
Assured access to space and responsive spacelift are two very high priority topics
in the United States Air Force (USAF) space community. As General Kevin P. Chilton,
Commander, Air Force Space Command has put it: ?The rockets we launch into space
carry with them the communication, weather, surveillance, navigation, and other national
assets which are integral to our national security as well as our economy.?
1
Thus,
significant work will continue in order to guarantee that the United States has access to
space and, if necessary, the capability to deny access to an adversary.
As a result, the USAF seeks assured and affordable access to space. The current
USAF vision for achieving this capability is called Operationally Responsive Space
(ORS). One broad outcome of ORS is to produce a launch vehicle with the following
goals: launch a 1,000 lbm payload into lowEarth orbit at a cost of under $5 million and
launch the vehicle within 24 hrs of tasking.
In order to support ORS, the Defense Advanced Research Projects Agency
(DARPA) and the USAF are ?jointly sponsoring the Force Application and Launch from
2
CONUS (FALCON) program to develop technologies and demonstrate capabilities that
will enable transformational changes in global, time critical strike missions.?
2
The goal of
this program is to design a launch vehicle with a prompt global strike capability. The
technologies needed for a prompt global strike capability are essentially the same as those
needed to design a responsive and reliable space launch vehicle.
This dissertation describes an effort to optimize the design of an entire space
launch vehicle that will carry a payload into lowEarth (circular) orbit. The launch
vehicle consists of multiple stages and the design optimization uses a genetic algorithm
(GA) with the goal of minimizing total vehicle weight and ultimately vehicle cost for a
given mission from a given launch site. The entire launch vehicle system is analyzed
using various multistage configurations to reach the desired lowEarth orbit. Three
different types of conventional propulsion systems are considered: solid propellant
motors, liquidfueled rockets, and airlaunched systems using an airborne platform as the
firststage. The vehicle performance modeling required that analysis from four separate
disciplines be integrated into the design optimization process. Those disciplines are the
propulsion characteristics, the mass properties, the aerodynamic characteristics and the
sixdegreeoffreedom (6DOF) flight dynamics characteristics.
3
2.0 CHRONOLOGY OF OPTIMIZATION TECHNIQUES
2.1 Introduction
The goal of design optimization is to find the optimum (from the Latin word
optimus, meaning best) solution to the design problem. The theory of optimization has
an enormous variety of real world applications that can benefit from an optimum
solution. From traffic flow problems to space launch vehicle design, finding an optimum
solution early in the problem solving process can pay huge dividends. Thus, it can be
readily stated that the theory of optimization involves the use of mathematics to facilitate
problem solving. Modern computers, with their incredibly fast computational
capabilities, have turned optimization theory into a rapidly growing branch of applied
mathematics. Numerous different optimization techniques exist from classical methods
to modern evolutionary algorithms.
2.2 Early Concepts
According to Foulds,
3
one of the first recorded uses of optimization theory dates
back to ancient times. In 200 B.C., Archimedes correctly conjectured that the semicircle
was the optimal geometric curve of given length, together with a straight line, that
enclosed the largest possible area. More advanced techniques would not come about
until the 17
th
century with Newton?s development of calculus. Gauss developed the first
formal optimization technique known as steepest descent.
4
2.2.1 Newton and CalculusBased Methods
Newton formulated a straight forward method for determining the local maxima
and minima of a function by using the first derivative of an equation. Setting the first
derivative equal to zero and solving the equation provides the condition for a maximum
or minimum. The sign (positive or negative) of the second derivative can be used as a
test. However, according to Siddal,
4
when multiple functions are used to describe the
behavior of a system, this method can produce numerous nonlinear algebraic equations
that must be solved simultaneously. Determining the actual maximum or minimum value
from these equations can be difficult.
2.2.2 Gradient Methods
Gradient methods such as steepest ascent or steepest descent were developed in
the 19
th
century. These methods attempt to ?march? toward a local maximum (ascent) or
local minimum (descent) by taking steps proportional to the gradient of the function at
the current point. The marching can ?stop? at the point where successive changes in the
function become negligible indicating a maximum or minimum has been reached. This
method can run into problems when there are numerous local maxima or minima. To
avoid getting ?stuck? in these local optima, gradient methods need a reasonable starting
solution to begin the process. As a result, this restricts the possibility of finding a truly
global maximum or minimum value of the function.
In addition, gradient methods are also dependent upon the nature of the function
being analyzed. Since these methods operate on the first derivative of this ?objective?
function, the function must be differentiable in every independent variable. Otherwise, a
5
singularity will result and the ability to ?march? toward a local maximum or local
minimum value will be compromised.
2.3 Linear Programming
Gabasov and Kirillova
5
state that ?Linear programming problems were first
formulated and studied by the Soviet mathematician L.V. Kantorovich in the 1930s. In
the 1940s, the American mathematician G.B. Dantzig developed the simplex method of
solution.? The general linear programming problem involves an objective function and
constraints that are all linear. Additionally, the goal of the optimization is to simply
maximize or minimize the objective function. The simplex algorithm was developed to
analyze feasible solutions to the objective function until no improvement in the objective
function could be made. The search space is modeled in a geometric form such as a
polyhedron. The simplex algorithm simply marches along the outskirts of this shape in
order to find a single optimal point. At this point, either the maximum or minimum
value of the objective function has been found that satisfies the given constraints.
The advantages of linear programming are that this method is very efficient in
practice and it is guaranteed to find a global optimum. The disadvantage is that this
method cannot handle complex problems such as a multivariable optimization because
the objective function and constraints are required to be linear.
2.4 Pattern Search Optimization
When the problem is to maximize a realvalued function with no constraints,
numerous methods exist to find a solution. Many methods look at the behavior of the
function being analyzed and use this information to proceed to the solution. Direct
methods and gradient methods fall into this category. The strategy often involves
6
selecting a point within the domain that is thought to be the most likely place where a
maximum or minimum exists. If no information is available for choosing this point then
one is chosen at random. From there, a particular method is used to generate more points
that move closer and closer to the desired optimum solution.
As described by Foulds,
3
pattern search optimization is a direct search method.
Pattern search, the first direct search method to be examined, was developed by Hooke
and Jeeves in 1961. Direct search methods differ from gradient methods in one important
way. Given a function to be optimized, say f(x), a direct search method requires that f(x)
be evaluated at each point in the optimization process. Gradient methods require the
evaluation of the first derivatives of f(x) at those same points.
The pattern search method is fairly straight forward. The function, f(x), is
evaluated at a chosen point. Then, exploration about this point is done in order to find
the direction of improvement. Slight perturbations of each variable are performed and
f(x) is again evaluated at the chosen point. If an increase is observed (indicating
improvement for a maximizing problem) then, the next variable is evaluated using the
same perturbation. The process continues until all variables have been analyzed and the
final best point has been established. Using the final best point and the original starting
point, a step size is calculated as twice the Euclidean distance between these two points.
Using this step size and the original starting point, the new evaluation point is determined
and the process repeated. Thus, there is a general trend or improvement which Hooke
and Jeeves called a pattern.
The method works well as long as each successful iteration produces a value
closer to the maximum value of the function. The method does have a few short
7
comings. It requires an initial ?guess? to get the process started. The nature of the
function being investigated can also cause problems. If the function has any tightly
curved ridges or sharpcornered contours, the method may be unable to produce any
improvements while still far from a local maximum. Also, the method only finds local
maxima around the point initially chosen (i.e. the initial ?guess?). Thus, it can not
determine a global maximum independently. If numerous local maxima have been
determined then comparison of these local maxima could yield a global maximum.
2.5 Design of Experiments
The concept known as the design of experiments was formulated in the 1920s
when Sir Ronald Fisher wrote his book ?Statistical Methods for Research Workers.? At
the time, experimenters did not have any proven methods for interpreting the vast amount
of data being generated in laboratories. Fisher went about inventing many of the
techniques used today for conducting and analyzing experiments. Through statistical
procedures, he wanted to remove criticism of the results of experiments where the
interpretation of the results and the execution of the experiment were questioned.
Fisher?s
6
influential book ?The Design of Experiments,? written in 1935, discussed the
method known as the analysis of variance. This simple arithmetical procedure
summarizes the experimental results and details the structure of the experiment. This
allows for the proper testing (i.e. interpretation) of the experimental results.
According to Weber and Skillings,
7
?a designed experiment is an experiment in
which the experimenter plans the structure of the experiment.? The authors have
developed specific steps that need to be followed when conducting an experiment. These
steps are as follows: determining the goal of the experiment, defining the variables,
8
establishing the levels or ranges of the variables, designing the experiment, running the
experiment and analyzing the results. A linear statistical model is then used to describe
the structure of the data resulting from the experiment.
In addition to improving physical experiments themselves, the design of
experiments method can be used as an optimization tool. In this application, analytical
models can be used to predict the possible outcomes within a particular experimental
region. The resulting optimal experimental point can then be determined. This process
?designs? the experiment so that the optimum investigation can be performed.
2.6 Additional Methods
The response surface method is an optimization method that is computationally
attractive and straight forward in application. Rodriquez
8
explains that this method takes
a complex and highly nonlinear function and replaces it with a simplified, multi
dimensional surface fit. This response surface fit results in a simple mathematical
representation of the complex function. Both the values of the function and the gradient
at other points can be determined with the response surface. Using a gradientbased
optimizer, the optimal point in the response surface can be determined as well. The
advantage of this method is that it is very efficient and effective at analyzing a complex
function. The disadvantage of the response surface method is that the curve fit must be
an accurate representation of the complex function or an optimal point will not be found.
Monte Carlo design optimization is another popular technique for solving
complex physical and mathematical problems that possess many variables. The Monte
Carlo method is considered to be stochastic in that it generates random numbers in order
to evaluate the function. These values of the function are then averaged in order to
9
estimate the function?s true value. In terms of optimization, the method will randomly
?walk? throughout the design space and tend to move in a direction towards either a
maximum or a minimum. A gradient method can be incorporated to help facilitate the
determination of the optimum value. Monte Carlo methods have the advantage of
searching a very large design space but at the same time are computationally expensive
due to the large amount of random numbers required.
Finally, a method known as particle swarm optimization has been used
successfully in the optimization of physical structures and artificial neural networks. The
method is a populationbased method similar to other evolutionary techniques. The
members of the population follow (or swarm) towards the best performing member and
each member knows its position and velocity compared to the optimum member. In
subsequent generations, the position and velocity of each member are updated in order to
get closer to the characteristics of the optimum member. The advantage of this method is
that it can result in a computationally faster and cheaper method of finding on optimum
solution.
2.7 Genetic Algorithms
Mitchell
9
states that ?Genetic algorithms (GAs) were invented by John Holland in
the 1960s and were developed by Holland and his students and colleagues at the
University of Michigan in the 1960s and 1970s.? Holland wanted to take the
evolutionary processes that are hypothesized to occur in nature (adaptation, survivalof
thefittest, etc.) and incorporate them into a computer system. Holland?s classic 1975
book ?Adaptation in Natural and Artificial Systems? formulates evolutionary/population
based algorithms that can be used to optimize a variety of real world systems.
10
According to Coley,
10
GAs are numerical optimization algorithms built around
natural selection and natural genetics. Sometimes GAs are referred to as evolutionary
optimizers because they mimic some of the processes proposed in Darwinian evolution
theory. Darwinian evolution theorizes that totally new species can be produced via
mutation and random chance. However, unlike evolution, GAs operate more on the
principal of improvement of an initial population of solutions to a design problem rather
than pure optimization. Coley
10
describes the makeup of a typical GA in a number of
ways. First, a GA can be described as a number, or population, of guesses of the solution
to the problem. Second, the GA employs a method of calculating how good or how bad
the individual solutions within the population are. This method is called determining the
fitness of the solution. Additionally, a method for mixing fragments of the better
solutions to form new, on average even better solutions can be used. This method is
called crossover. Finally, the GA is able to use a mutation operator to avoid permanent
loss of diversity within the solutions.
One of the main benefits of using a GA is the fact that the algorithm can start
without a single point, or guess, to get the optimization running. The previously
described direct search and gradient methods all, except design of experiments, need an
initial guess to the problem solution in order to ?march? toward the desired optimized
result; either a maximum or minimum value. The GA uses a population of guesses that
are random and spread throughout the search space. Powerful operators such as
selection, crossover and mutation help direct members of each population toward the
desired goal(s) of the problem. A binary encoding system allows for a host of variables
to be manipulated by the GA and then used in a suite of performance codes. These codes
11
analyze the performance of each member of the population and the GA ranks each one
according to how well that member meets the desired goal(s). In GA terminology, the
objective function is the function that determines the performance of a particular
chromosome (i.e. member of the population).
16
Thus, in this study, the suite of
performance codes, grouped together as a whole, represent the objective function.
At the same time, the GA does have some disadvantages. In using the GA there is
a greater likelihood that a global optimum solution will be found. However, finding this
global optimum is not guaranteed. Even if the GA is in the neighborhood of the global
optimum, there is a possibility through crossover and mutation that the global optimum
may not be selected. Also, the GA does not address the robustness of the individual
design solutions it creates. The GA simply attempts to meet the desired goals and will
adjust the design parameters accordingly. Thus, it is up to the user to ensure the proper
operation of the GA and to verify the results it generates. Finally, the satisfactory
operation of the GA relies on the accuracy of the system models that make up the
objective function. The GA used for this dissertation will be discussed in greater detail in
Chapter 3.
2.8 Recent Launch Vehicle Optimization Work
In recent years, significant work has been done to advance the design and analysis
of entire launch vehicle systems. Extensive research has gone into improving the design
of solid rocket motors. In 1968, Billheimer
11
made one of the first attempts to perform an
automated design of a solid rocket motor. The significance of this study emphasized the
importance of automating the design process. Using a pattern search technique, in 1977,
Woltosz
12
determined five critical design dimensions in order to find an optimum grain
12
geometry for a solid rocket motor. Foster and Sforzini
13
then used the same pattern
search technique to minimize the differences between desired and computed solid rocket
motor ignition characteristics. Finally, in 1980, Sforzini
14
performed an automated
approach to analyzing the internal characteristics of a solid rocket motor and used the
Space Shuttle solid rocket booster for comparison. Using the pattern search optimization
technique, Sforzini
14
was able to generate a head end pressure versus time profile that
closely matched the Space Shuttle solid rocket motor.
In order to analyze the performance of an entire launch vehicle, a suite of
analytical models is required. The aerodynamic characteristics of the vehicle must be
determined over a widerange of flight conditions for the portion of the flight during
which the vehicle is in the atmosphere. It is fortunate that space launch vehicles spend
only a short time in the Earth?s atmosphere. Nonetheless, the aerodynamic characteristics
play an important role and must be analyzed. In the past, gradientbased optimization
techniques have been applied to optimize the aerodynamic characteristics of missiles.
However, as stated previously, these methods have limited capability in finding globally
optimized solutions. In 1990, Washington
15
developed an aerodynamic prediction
package, called AeroDesign, that can be used to determine the aerodynamic constants of
different missiles. Additionally, based on the equations of motion described in Etkin?s
17
book, Anderson
16
developed a sixdegreeoffreedom (6DOF) flight dynamics simulator.
This 6DOF flight dynamics simulator was used to fly the vehicle over a ballistic
trajectory given an initial launch angle.
In 1998, Anderson
16
assembled these performance codes and created an objective
function that could analyze the performance of an entire, singlestage solid propellant
13
rocket vehicle. In addition, Anderson
16,43
wrote a GA that was used to optimize the
performance of these solid rockets given specific goals and using the suite of
performance codes as the objective function.
In 2003 and 2006, Burkhalter et al.
18
and Hartfield et al.
19
took the design
optimization of missile systems further. First, an additional model was created for the
objective function in order to analyze the performance of liquid rocket engines. Second,
the first attempt into multistage configurations was begun with the analysis of both a
twostage, solid propellant tactical missile and a solid propellant boosted ramjet system.
The foundation work necessary to use these system models, in the form of
performance codes, and the GA to pursue the design optimization of space launch
vehicles has been completed. In addition to analyzing the overall performance of each
launch vehicle, a cost model has been developed in order to bring an economic factor into
the optimization process.
The key to the current study is the GA and its ability to find a global optimum
solution to a challenging design problem. Populationbased, evolutionary algorithms,
like the GA, are much more useful than pattern search methods or gradient methods when
investigating a complex, multivariable problem with nondifferentiable objective
functions. Typically, a number of discrete variables will be used by the GA and the
objective function. Also, the functions describing the model can be complex, nonlinear,
and not easily differentiable. This makes pattern search and gradient methods difficult, if
not impossible, to use in solving the problem. In addition, because of the nonlinearity of
these functions, the number of local optima can be significant. Thus, since the pattern
search method uses an initial ?guess? to the solution, the odds of actually hitting the
14
global optimum with this guess are not especially likely. Finally, with the binary
encoding system used by the GA, a large number of variables can be analyzed. Table 21
shows an example of the number and types of variables used to analyze a threestage
solid propellant launch vehicle. A total of 36 different design parameters were used to
optimize the performance of this particular type of space launch vehicle.
Encoding these variables into a single string of bits of length (l) allows the GA to
perform various operations on this single string. Then, the string of variables is decoded
for analysis in the objective function. For the current study, typical string lengths are on
the order of 200 bits. Using the length (l) for the string length, the number of possible
solutions can be expressed as 2
l
. By comparison, a problem that has a string length of 50
bits means that there are 2
50
possible solutions to the problem (1.125 trillion). The GA is
uniquely capable of efficiently analyzing such a large solution space in search of an
optimum.
To summarize, evolutionary techniques have been used to solve a myriad of
design optimization problems.
2024
Significant research has been performed in rocket
based vehicle design optimization using various evolutionary techniques.
2536
A recent
study
37
attempted the design of a satellite launch vehicle using an evolutionary algorithm
to minimize the gross liftoff weight of the vehicle. The vehicle model was based simply
on deltaV requirements for the launch system. The deltaV for the model was
determined by analyzing the performance capability of the Ariane 44L launch vehicle.
Cost has also been considered in some additional studies;
32, 3840
however, this dissertation
represents the first effort of its kind to minimize launch vehicle cost for Earthtoorbit
missions at the preliminary design level using a GA.
15
Table 21: Design Variables for a ThreeStage Solid Propellant Launch Vehicle
Definition (units) Maximum
Value
Minimum
Value
Stage 1
fuel type 9.0 1.0
propellant outer radius ratio 0.80 0.40
propellant inner radius ratio 0.99 0.01
number of star points 13.0 3.0
fillet radius ratio 0.20 0.01
epsilon star width 0.90 0.10
star point angle 40.0 1.0
fractional nozzle length ratio 0.99 0.60
throat diameter (inches) 35.0 5.0
total stage length (inches) 500.0 300.0
stage body diameter (inches) 90.0 70.0
Stage 2
fuel type 9.0 1.0
propellant outer radius ratio 0.80 0.40
propellant inner radius ratio 0.99 0.01
number of star points 13.0 3.0
fillet radius ratio 0.20 0.01
epsilon star width 0.90 0.50
star point angle 40.0 1.0
fractional nozzle length ratio 0.99 0.60
throat diameter (inches) 35.0 5.0
total stage length (inches) 300.0 200.0
stage body diameter (inches) 70.0 60.0
Stage 3
fuel type 9.0 1.0
propellant outer radius ratio 0.80 0.40
propellant inner radius ratio 0.99 0.01
number of star points 13.0 3.0
fillet radius ratio 0.20 0.01
epsilon star width 0.90 0.50
star point angle 40.0 1.0
fractional nozzle length ratio 0.99 0.60
throat diameter (inches) 35.0 5.0
total stage length (inches) 200.0 100.0
stage body diameter (inches) 60.0 50.0
Miscellaneous
nose radius ratio 0.75 0.50
nose length (inches) 100.0 75.0
initial launch angle 89.99 70.0
16
3.0 SYSTEM MODELING
3.1 Introduction
The overall approach taken for modeling the launch vehicle systems considered in
the design optimization process follows the method employed by Anderson,
16
Burkhalter
et al.,
18
Hartfield et al.,
19
and Metts.
20
Modeling of the launch vehicle consists of
employing a suite of performance codes which are based on physical models for the
propulsion system, the mass properties, the aerodynamics, and the vehicle flight
dynamics. All critical vehicle performance parameters are calculated using these
individual system models. The results are used to determine how well the particular
launch vehicle meets the desired goals of the design optimization.
In addition to vehicle performance, a cost model, based on the work of Koelle
41
and Wertz,
38,42
has been incorporated into the objective function. The TRANSCost 7.1
cost model created by Koelle
41
is a massbased model that provides cost estimates for a
variety of launch vehicle types. As a result, the information generated in the mass
properties models is utilized in the cost model.
3.2 Objective Function Link to the Genetic Algorithm (GA)
In general, the design optimization process can be broken into two distinct
operations that are linked together. The objective function determines the performance of
individual members of the population while the GA provides a continuous set of
parameters to be analyzed based on probabilistic selection.
17
Figure 31 shows the program flow with the objective function, how it is linked to the
GA, and the different GA operators.
Program Flow
Objective
Function
Definition
Parameter
Coding
Initial
Population
Initial Population
Statistics
Selection based on
Statistics
Crossover and
Mutation Based on
Probabilities
Stop
Population
Statistics
Max Number
of Generations
Reached?
Yes
No
Initial SetUp
Figure 31. Objective Function Link to the GA
The objective function contains all of the performance codes required to determine the
performance of a particular launch vehicle. In order to analyze different launch vehicles,
some design parameters (e.g. propellant types, nozzle geometry, etc.), whose values can
be altered, are chosen for analysis. Given the design parameters, the objective function
can assemble and model the performance for any of a wide variety of launch vehicles.
In addition to determining the performance of each launch vehicle, the objective
function uses performance criteria to analyze how well a particular launch vehicle meets
the desired goals of the optimization. A quantitative measure is established by the user so
that the objective function can determine which launch vehicles perform ?better? than
18
others. For example, prior to starting the design optimization process, a desired orbital
altitude (altorb) is chosen by the user. Next, the design optimization process is started
and the GA creates a population of candidate launch vehicles. Each candidate launch
vehicle is run through the objective function and attains a final altitude (alt1). Since one
of the goals is to reach the desired orbit, the objective function must compare the two
altitudes. This comparison is done in the following equation:
altorb
altaltorb
answer
1?
= (3.1)
In order to reach the desired orbit, the goal must be to minimize the answer to Equation
(3.1). The launch vehicle with the smallest value of Equation (3.1) is considered to be
the best performer for that generation. It is the highest ranked member of the population
and its characteristics are carried on to the next generation where the process repeats
itself.
Typically, a design optimization problem has one or more desired goals. If the
goal is to find the launch vehicle that maximizes the thrust of a solid rocket motor, then
the objective function must be run numerous times to find the rocket with the desired
characteristics. This can be a tedious and inefficient process which will require trial and
error in order to find the optimum launch vehicle that meets the desired goal. As
Anderson
16
wrote: ?The goal of this research is to remove the human designer from the
tedious task of searching for the optimal parameter set.? The GA can be used in place of
the human designer for the function of evaluating the objective function and deciding on
proposed areas of the design space to explore.
19
3.3 Genetic Algorithm (GA)
The GA for this study was developed by Anderson
43
and uses the biological
concept of generational adaptation to solve a design optimization problem that may
contain numerous local optima. The GA is considered to be an adaptive optimizer. The
process is one in which the GA encodes potential solutions to the design problem into a
numerical string (usually called a chromosome). Some corresponding design parameters
and an example chromosome that might make up a potential solid rocket design solution
are shown in Table 31. This chromosome string illustrates the binary encoding method
used by the GA and described below.
Table 31: Example Design Parameters and Chromosome
Parameter Real Value Binary Form
Number of Star Points 10 1010
Total Stage Length 120 inches 1111000
Stage Diameter 50 inches 110010
Chromosome String 1010111000110010
The chromosome string can then be manipulated via different genetictype operations
such as reproduction, crossover, and mutation. In addition, a selection process is
employed which allows various solutions to compete against one another. The ?better?
or more fit solutions are passed on to subsequent generations while the characteristics of
?lesser? solutions ?die off.? The goal is to create increasingly better solutions as time
successive generations are developed.
Rather than define the first generation of designs from an initial ?guess? provided
by the user, the user specifies a range (maximum, minimum, and resolution) for each
design parameter, and the GA randomly generates a population of candidate solutions
20
from parameters within the prescribed design space. After each candidate is analyzed by
the performance codes that make up the objective function, the GA ranks the candidates
(members) in order of fitness, or how closely they match the objective function.
The process by which possible solutions are converted to a form that can be
manipulated by the GA is known as parameter encoding. The most popular type of
encoding for a GA is binary encoding. Here the design parameters are converted into 1s
and 0s to form the chromosome string that represents a possible solution. Using the
maximum, minimum and resolution values, each design parameter can be converted into
a number of bits using the equation:
()
1
2ln
minmax
ln
+
?
?
?
?
?
? ?
=
resolution
nbits (3.2)
Each design parameter is converted in this way and then all the parameters are strung
together to form one member among a population of possible solutions. The GA can
manipulate each string (or chromosome) and thus produce all the members that make up
one generation. Each individual string can be decoded into real numbers prior to its use
in the objective function.
For this study, a tournamentbased GA is used to control the design process.
Specifically, the IMPROVE
?
3.1 GA, developed by Anderson,
43
has been implemented.
The tournament selection process involves an essentially three step process shown in
Figure 32. First, two members of the current population are chosen at random and
compete against each other. The member that performs better survives to an intermediate
population. The ?losing? member returns to the current population. Next, two more
members from the current population are chosen at random (the ?losing? member could
21
actually be chosen again) and compete. Again, the member that performs better survives
to the intermediate population. Finally, crossover and mutation operations are performed
on the two ?winning? members in order to create two new members that replace the
members lost in the tournament process. This process continues until the new population
has been filled with the required number of new members.
Tournament Selection
Two members of
current population
chosen at random
Dominant
performer placed
in intermediate
population
Crossover and
mutation form
new population
Crossover and
mutation form
new population
Population Filled?
Continue?
Figure 32. Tournament Selection
The tournament selection method is thus used to create the next generation of
members, which will possess characteristics of the previous population but in different
combinations which may result in better overall fitness. When properly configured, the
GA will find solution types that increasingly approach the target fitness over the course
of many generations.
The IMPROVE
?
3.1 GA employs two powerful tools for ensuring that diverse
populations of potential solutions are maintained throughout the design optimization.
First, crossover is the process where two ?parent? solutions exchange portions of their
22
genetic code thus producing two ?offspring.? The goal here is to take advantage of good
genes by promulgating and mixing them with future generations. This helps the
IMPROVE
?
3.1 GA to improve beyond the initial population or any local optima. The
second tool, mutation, allows for the random altering of individual bits/genes that make
up a chromosome. This provides robustness as 1s are randomly switched to 0s and vice
versa. Like crossover, mutation gives the IMPROVE
?
3.1 GA the ability to jump beyond
any local optima that may have been encountered.
The performance codes that make up the objective function analyze each member
generated by the IMPROVE
?
3.1 GA to determine the overall performance of the
vehicle. This allows the IMPROVE
?
3.1 GA to find solution types that not only meet the
target fitness but which are also realistic (i.e. valid). Does one design/solution standout
among all the others? In reality, one design/solution might produce highly desirable
results that meet the goals exactly. However, other designs might not meet the goals
exactly but still produce results that are ?close? to the desired system. In this study, the
answer to finding the optimum design is the IMPROVE
?
3.1 GA. But the GA by itself
does not indicate that a model is valid. In fact, the GA makes no statement at all as to the
validity of the underlying system models. The IMPROVE
?
3.1 GA simply chooses the
optimum design that meets the desired system goals. It is up to the user to take the
information generated by the IMPROVE
?
3.1 GA and make the important decisions as to
which design/solution is the best.
The number of members present in a single population, known as population
sizing, is typically determined prior to the start of the optimization process. The
determination of the number of members required for success of the IMPROVE
?
3.1 GA
23
is not easy. As Anderson
16
wrote: ?Proper population sizing is a seriously debated issue
when using genetic algorithms.? In general, for a complicated design problem with a
large search space, a large population size is required. The population size (n) can be
calculated by knowing the total number of genes required for each member of the
population. In the binary encoding process, the number of genes is represented by the
number of bits making up the chromosome string. Using Equation (3.2), the number of
bits (nbits) is calculated and then used in Equation (3.3).
nbitsn )0.3(= (3.3)
This issue of determining the ideal number of generations for a particular design
problem is also not explicitly defined. If the number of generations is high, the
optimization is more likely to produce an optimum solution. However, a large number of
generations combined with a large number of members in each generation results in
significant computer run times. As the computing speed of modern computers increases,
the ability to run a large amount of generations will improve. At the same time, using a
large number of generations does not necessarily ensure the optimum design will be
found. Other factors such as crossover and mutation also affect genetic diversity and
robustness of the optimization. It is possible that a smaller number of generations could
still produce the optimum solution. Currently, the ideal number of generations to use is a
matter to be decided by the user.
Finally, the IMPROVE
?
3.1 GA uses an additional tool in the design optimization
process. Elitism is the process of preserving the ?elite? member of each population and
to allow that member to survive intact into the next generation. There are both benefits
and drawbacks to elitism. The benefit is that elitism can keep an unfortunate crossover or
24
mutation from wreaking havoc in a particular generation. It allows at least one ?good?
member to be preserved, passed on to the next generation and hopefully improved upon.
The drawback is that elitism can focus the optimization on the current best performer
while at the same time ignoring an even better performer. The use of elitism is again a
decision to be made by the user.
3.4 Propulsion System Models
After the GA has generated values for the set of design parameters, the propulsion
system models are the first performance codes to be analyzed in the objective function.
These models analyze the basic thrust characteristics of solid and liquid propulsion
stages. Fuel and oxidizer properties are preloaded and used to calculate thrust, burn
time, fuel/oxidizer mass, combustion pressure, etc. For example, the grain geometry of a
solid propellant motor can be specified by the GA using the design parameters. From
these values, the entire sealevel thrust profile of the motor is determined in the solid
rocket propulsion model. Similar analyses can be done for the liquid propellant rocket
engines.
For a multistage vehicle, the number of times each particular propulsion system
model is evaluated corresponds to the number of stages in the vehicle. The propulsion
characteristics of each stage are determined separately and in sequence.
It is useful to discuss some of the basic equations associated with rocket
propulsion that are used in the propulsion system models. First, a steady flow assumption
through a choked nozzle is assumed. This allows the mass flow discharged through the
nozzle to be calculated as:
*** VAm
disch
?=& (3.4)
25
Assuming isentropic flow in the nozzle and knowing that V*=a*, the following equation
can be written:
()12
1
1
2
*
?
+
?
?
?
?
?
?
?
?
+
=
?
?
?
?
c
c
disch
RT
PAm& (3.5)
The characteristic velocity of the rocket, c*, can be written as:
()12
1
2
1
*
?
+
?
?
?
?
?
? +
=
?
?
?
?
c
RT
c (3.6)
Substituting Equation (3.6) into Equation (3.5) results in another equation for the rate of
mass discharged through the nozzle:
*
*
c
AP
m
c
disch
=& (3.7)
Additional equations are used to calculate the thrust of the rocket, the thrust coefficient,
and the exit velocity. From the uniform, steady, onedimensional momentum equation,
the thrust, T, is determined to be:
( )
eaee
APPVmT ?+= & (3.8)
The thrust coefficient, C
T
, is defined as:
*AP
T
C
c
T
= (3.9)
Substituting the thrust equation (Equation (3.8)) and the equation for the mass flow rate
discharged through the nozzle (Equation (3.7)) into Equation (3.9) yields:
** A
A
P
P
P
P
c
V
C
e
c
a
c
ee
T
?
?
?
?
?
?
?
?
?+= (3.10)
The uniform, steady, onedimensional energy equation can be used to determine the exit
velocity of the nozzle assuming an adiabatic flow.
26
0
2
2
h
V
h
e
e
=+ (3.11)
Assuming an isentropic expansion through the nozzle and using some thermodynamic
substitutions, the exit velocity can be written as:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
1
0
1
1
2
c
e
e
P
PRT
V (3.12)
3.4.1 Solid Propellant Rocket Motors
The solid propellant rocket propulsion model used in this analysis was developed
by Burkhalter,
44
Sforzini
45
and, with modifications, Hartfield et al.
19
Figure 33 provides
a schematic showing the basic geometry of a solid rocket motor.
Figure 33. Solid Rocket Motor Schematic
Anderson,
16
who used the model extensively, explained that some fundamental
assumptions were made in the formulation of the software used to analyze solid rocket
motors. One assumption is that the combustion of the solid propellant is stable (i.e.
steady). Thus, the rate of mass produced by the burn must equal the rate of the mass
Throat
A*
Nozzle Exit
A
e
disch
m&
V
1
,p
1
V
2
,p
2
y
burn
A
p
lgrain
27
discharged through the throat of the nozzle. If the propellant grain burn area is A
b
and
the burn rate is r, the mass flow rate generated can be written as:
rAm
bbgen
?=& (3.13)
Setting Equation (3.13) equal to Equation (3.7), the mass flow rate generated can
be written as:
disch
c
bbgen
m
c
AP
rAm && ===
*
*
? (3.14)
Equation (3.14) is the basic performance relationship used to determine chamber pressure
and, in turn, sealevel thrust for the solid rocket motor. In this model, the chamber
conditions are considered to be uniform and are ?lumped? into a single variable. For that
reason, this approach to modeling solid rocket motor performance is known as ?lumped
parameter? modeling.
3.4.2 Liquid Propellant Rocket Engines
The liquid propellant rocket propulsion model used in this analysis was created by
Hartfield et al.
19
with additional modifications by Riddle.
21
The oxidizer and fuel are
stored in separate tanks. An inert compressed gas is used to provide pressurization for
both the fuel and oxidizer tanks. The pressure in the fuel and oxidizer tanks is kept
relatively low; around 100 psia. Like the solid propellant motors, the temperature and
pressure in the combustion chamber are assumed to be constant. Also, the thrust goes to
zero at burnout for each stage and there is no delay between burnout of one stage and
ignition of the next stage. The model is also based on the assumption that the
turbopumps are used to pressurize the fuel and oxidizer prior to injection into the
combustion chamber.
28
With some minor changes, Equations (3.4) thru (3.12) are used to determine the
thrust, the thrust coefficient, the mass flow rate, and the exit velocity of the nozzle. The
value of specific impulse, I
sp
, for the propellants is used in the calculation of the thrust
coefficient.
e
e
sp
g
V
I = (3.15)
Solving for V
e
and substituting into Equation (3.8), allows the thrust coefficient to be
written as:
** A
A
P
P
P
P
c
gI
C
e
c
a
c
e
esp
T
?
?
?
?
?
?
?
?
?+= (3.16)
From the coefficient of thrust, the thrust, T, of the rocket can be determined from
Equation (3.8):
Tc
CAPT *=
The mass flow rate can then be determined using the thrust:
*c
Tg
m
e
=& (3.17)
A wide variety of liquid propellant combinations are available for analysis in this study.
The choice of propellants is a design variable for the GA. Riddle
21
writes that propellant
properties such as stoichiometric mixture ratio, combustor total temperature, molecular
weight, characteristic exhaust velocity, sealevel specific impulse, and the ratio of
specific heats are included in the list with their corresponding fuel/oxidizer combination.
A few examples of different liquid propellant fuels and oxidizers available for use in the
design optimization process are shown in Table 32.
29
Table 32: Liquid Propellant Fuels and Oxidizers
Oxidizer Fuel
IRFNA RP1
H2O295% Hydrazine
N2O4 MMH
ClF3 UDMH
LOX LH2
LF2 Ammonia
Riddle
21
has also incorporated a method for determining a variable specific
impulse in order to further enhance the liquid propellant rocket propulsion model. Rather
than use a constant I
sp
, the model uses an equivalence ratio, (?), chosen by the GA to
determine the I
sp
for that particular propellant combination. Using the thermochemical
data of a propellant combination and the equivalence ratio, the I
sp
values can be
approximated using a 5
th
order curve fit. For the current study, this method was not
employed. However, future work should implement this procedure for use in the
optimization of liquid propellant space launch vehicles.
3.5 Mass Properties Models
Upon completion of the propulsion system models, the objective function next
determines the mass properties of the vehicle using the mass properties models. The
mass properties of the vehicle are critical to accurate modeling of the vehicle during its
flight to orbit. The mass of the vehicle is not constant as it burns and ejects propellant
during flight. Thus, the mass properties change with time and the mass properties models
used in this analysis reflect this change.
The physical model of the launch vehicle involves a basic setup where almost all
of the individual components are stacked vertically adjacent to each other. Starting at the
nosecone and working toward the aft end of the vehicle, the mass properties of each
30
component are analyzed individually and in sequence. There is no space between each
component and the lengths of each component are added together to determine the total
stage length. The total vehicle length is determined by summing the stage lengths, the
electronics length and the nosecone length. Also, all components are oriented about the
centerline of the vehicle thus making the entire vehicle axisymmetric.
Some components are not considered to be stacked components. For example, the
motor casing is modeled as a hollowedout cylinder that contains a few of the internal
solid rocket components. The length of the motor case is equal to the length of the
propellant grain. Like the other components, the motor case cylinder is also oriented
along the vehicle center line in order to ensure mass symmetry.
The process of determining the mass properties of the vehicle involves two
sequential steps. First, the mass properties model determines the mass properties of the
individual components of the launch vehicle. These components include the nosecone,
the payload, an electronics section, rocket body casing, propellant tanks and feed lines,
tail fins, and nozzles. Next, using the mass properties of the individual components, the
overall mass properties of the entire vehicle are determined. These values are then
summarized to create a mass properties data file that is used in the sixdegreeoffreedom
(6DOF) flight dynamics simulator model.
The calculations involving the solid and liquid propellants are unique because of
their timedependent nature. Except for the propellants, the mass properties of all the
components of the launch vehicle are constant during the entire flight. Since the
propellants are being burned and ejected thru the nozzle, their mass properties are
changing which, in turn, changes the mass properties of the entire vehicle. These
31
changes must be accounted for. The previously mentioned mass properties data file is the
tool used to keep track of the changing mass properties. At the start of the analysis, the
mass properties of all the propellants are determined and are incorporated into the total
vehicle mass properties. This constitutes the total mass of the vehicle at liftoff. The
table of mass properties is developed starting with the vehicle fully fueled and ready for
launch. Prior to the mass properties model, the burn time and propellant mass history
were determined in the propulsion system models. The timestep and change in
propellant mass from these models are now used in the mass properties model. The total
mass of the vehicle is decreased based on the amount of propellant that has been ejected
by the first stage of the vehicle. After this change, the mass properties of the entire
vehicle are then recalculated. This process continues until all the propellant for the first
stage has been expelled. The same analysis is performed for subsequent stages until the
final burnout time has been reached. At this point in the flight, the thrust is zero, the
propellant mass is zero and the remaining components fly a ballistic trajectory to orbital
altitude.
There are five properties that the mass properties models calculate. Those five
properties are:
1. Mass of the individual component and the entire launch vehicle system
2. Center of gravity relative to the nose of the rocket (xcg) of the individual component
and the entire launch vehicle system
3. Xaxis moment of inertia (ixx) of the individual component and the entire launch
vehicle system
32
4. Yaxis moment of inertia (iyy) of the individual component and the entire launch
vehicle system
5. Zaxis moment of inertia (izz) of the individual component and the entire launch
vehicle system
It should be noted that products of inertia (ixy, iyz, izx) reduce to zero because of the
physical nature of the vehicle. The axisymmetric configuration of all vehicle
components causes the products of inertia to be zero.
3.5.1 Mass Properties of Individual Components
Table 33 lists the different components that make up a threestage solid
propellant launch vehicle and a threestage liquid propellant launch vehicle. All five
mass properties (mass, ixx, iyy, izz, and xcg) are calculated for each of the individual
components listed in the table below.
Table 33: ThreeStage Solid and Liquid Vehicle Components
ThreeStage Solid ThreeStage Liquid
Blunt Nose Blunt Nose
Ogive Section Ogive Section
Payload Payload
Electronics Electronics
Stages 13 Stages 13
Curved Bulkhead Compressed Gas
Ignitor Compressed Gas Tank
Pressure Vessel/Motor Case Fuel
Liner Fuel Tank
Insulation Oxidizer
Nozzle Oxidizer Tank
Propellant Grain Cylindrical Case
Insulation
Engine Assembly
Nozzle
33
The equations for center of gravity and the moments of inertia use a local
coordinate system for the individual component being calculated. However, each stage
does not use the same local coordinate system (Stage 1 is farther from the nose than Stage
3). Thus, the lengths of each stage need to be reflected in the calculations so that the
mass properties are relative to a consistent starting point. The consistent starting point
used in this analysis is the nose of the vehicle.
Since there are numerous components of each stage, the ?shape? of each
component varies and the corresponding equations are used to describe their mass
properties. For the most part, in a typical space launch vehicle, the more common shapes
used are modeled as point masses (payload and electronics), cylinders (motor cases) or
spheres (propellant tanks). Figure 34 defines the lengths that describe the orientation of
a generic threestage vehicle.
Figure 34. Definition of Lengths
54 32
1
1. Length of Stage 1 (length
stg1
)
2. Length of Stage 2 (length
stg2
)
3. Length of Stage 3 (length
stg3
)
4. Length of Electronics (length
elec
)
5. Length of Nosecone (length
nose
)
34
3.5.1.1 Point Mass Example: Electronics
The mass properties of a point mass are described in this section. The electronics
package is modeled as a point mass located directly behind the payload at the front end of
the vehicle. The mass of the electronics (mass
elec
) is a predefined variable determined by
the user. The center of gravity (xcg
elec
) and the moments of inertia (ixx
elec
, iyy
elec
, and
izz
elec
) are calculated using standard dynamics equations for a point mass. In addition,
Figure 34 defines the lengths used in the calculations. Table 34 summarizes the
equations that determine the mass properties of the electronics.
Table 34: Mass Properties of Electronics
Property Calculation
Mass mass
elec
Center of Gravity xcg
elec
= length
nose
+length
elec
/2
Xaxis Moment of Inertia ixx
elec
= 0.0
Yaxis Moment of Inertia iyy
elec
= (mass
elec
)(radius
elec
2
)
Zaxis Moment of Inertia izz
elec
= iyy
elec
3.5.1.2 Cylinder Example: Motor Cases
The dominant shape of the bodies of nearly all launch vehicles is that of a
cylinder. Cylinders are preferred for atmospheric flight because of their low coefficient
of drag characteristics. In space, spheres are preferred because atmospheric drag is no
longer a concern. A cylinder also provides excellent volumetric storage space. For a
solid propellant vehicle, the propellant grain is housed in the center of a cylindrical motor
case. Thus, the motor case can be described as a cylinder with the center core hollowed
out. The distance to the local coordinate system is determined from the lengths defined
in Figure 34. In this case, the distance to the beginning of the 1
st
stage is called
35
length
ph2
, because it encompasses the components that correspond to the second phase of
flight, and is written as:
length
ph2
= length
nose
+ length
elec
+ length
stg3
+ length
stg2
(3.18)
When calculating the mass properties of the 2
nd
stage components, the distance is
called length
ph3
and uses the following lengths defined in Figure 34:
length
ph3
= length
nose
+ length
elec
+ length
stg3
(3.19)
For the 3
rd
stage components, the length
nose
and length
elec
terms are taken into
account in the appropriate equations.
Additionally, the volume of the motor case is needed for the mass calculation.
Knowing the length of the case (length
case
), the radius of the cylinder (radius
case
) and the
thickness of the case wall (thick
case
), the volume of the case can be written as:
V
case
= 2.0 ? (radius
case
)(length
case
)(thick
case
) (3.20)
Using these values for length
ph2
and volume, Table 35 summarizes the equations
involved for determining the mass properties of the 1
st
stage motor case. The density of
the tank material (?
case
) is a predetermined value chosen by the user.
Table 35: Mass Properties of Stage 1 Motor Case
Property Calculation
Mass mass
case
= (?
case
)(V
case
)
Center of Gravity xcg
case
= length
ph2
+ length
case
/2
Xaxis Moment of Inertia ixx
case
= (mass
case
)(radius
case
2
)
Yaxis Moment of Inertia
?
?
?
?
?
?
+=
22
case
3
1
2
1
)mass( Lriyy
case
Zaxis Moment of Inertia izz
case
= iyy
case
3.5.1.3 Sphere Example: Compressed Gas Tank
Propellant tanks used in liquid propellant rockets are usually either spherical or
cylindrical in shape. In order to keep both the fuel and oxidizer tanks properly
36
pressurized, an inert compressed gas, like nitrogen or helium, is fed into the propellant
tanks. Knowing the radius of the gas tank (radius
tank
), the volume of the sphere for the
compressed gas tank can be determined by:
()
3
tantan
3
4
kk
radiusV
?
?
?
?
?
?
=
?
(3.21)
Also, using Eqn. (3.15) and knowing that the length of the tank is length
tank
, Table 36
summarizes the equations involved for calculating the mass properties of the 1
st
stage
compressed gas tank. The value of the density of the tank material (?
tank
) is a
predetermined value established at the start of the modeling process.
Table 36: Mass Properties of Stage 1 Compressed Gas Tank
Property Calculation
Mass mass
tank
= (?
tank
)(V
tank
)
Center of Gravity xcg
tank
= length
ph2
+ length
tank
/2
Xaxis Moment of Inertia
?
?
?
?
?
?
=
3
tantantan
3
2
)(
kkk
radiusmassixx
Yaxis Moment of Inertia
?
?
?
?
?
?
=
2
tantantan
3
2
)(
kkk
xcgmassiyy
Zaxis Moment of Inertia izz
tank
= iyy
tank
3.5.1.4 Mass Table
A mass table is created that summarizes the five mass properties of all the
individual components of the launch vehicle. This table provides a quick visual ?sanity
check? to ensure the calculated values are reasonable and realistic. Errors in the
calculations can easily be determined by closely examining the results listed in the mass
table. An example of a mass table for a threestage solid propellant launch vehicle is
shown in the Appendix.
37
3.5.2 Mass Properties of Entire Launch Vehicle
After the mass properties of the individual components have been determined, the
next step is to determine the mass properties of the entire launch vehicle. This involves
more than just summing the mass properties of all the individual components together.
As previously described, during powered flight, the mass of the vehicle changes with
time and as a result the mass properties of the vehicle must change accordingly.
In order to properly capture the timedependent mass properties, the calculations
for the entire launch vehicle are done systematically in four different phases for a three
stage launch vehicle. As each stage operates and loses mass as propellant burns, the mass
properties are recalculated to take into account the slight changes in center of gravity and
moments of inertia. Each phase corresponds to the time when an individual stage is
firing and producing thrust. Thus, an entire profile from initial launch to 3
rd
stage
burnout is generated. A brief description of each phase is presented here:
1. Phase I: time = 0.0s, Stage 1 firing
2. Phase II: Stage 1 burnout, Stage 1 drops off, Stage 2 firing
3. Phase III: Stage 2 burnout, Stage 2 drops off, Stage 3 firing
4. Phase IV: Stage 3 burnout, ballistic flight, only payload and electronics remain
3.5.2.1 Entire Launch Vehicle Mass Properties Example: Phase I
The calculations for the mass properties of the entire launch vehicle are broken
into two main steps with a few smaller substeps. Each of the two main steps is
performed in each phase of flight with the corresponding components for that stage being
considered in the five mass properties. An example for Phase I of a threestage solid
propellant vehicle is described.
38
The first main step involves two calculations that establish the initial mass
properties of the entire launch vehicle. The five mass properties of all the individual
components, except the propellant grain, are added together. Since the mass of the
propellant grain will be changing, the mass properties of the propellant grain are added in
separately to create the timedependent variables. These timedependent variables of the
mass properties will be continuously updated in the second main step of this process.
Here again, the initial center of gravity of the propellant grain uses the variable, length
ph2
,
to account for the distance from the nose to the 1
st
stage propellant grain. At the
conclusion of this step, the five mass properties of the entire launch vehicle sitting on the
pad ready for launch (time = 0.0s) have been determined. These properties are written to
a data file and saved for future use in the 6DOF flight dynamics simulator.
The second main step is to calculate the timedependent mass properties as the
vehicle?s mass changes during flight. In the solid propellant propulsion model, the thrust
profile of the 1
st
stage is determined. In addition to the thrust calculations, the changes in
the mass of the propellant are calculated. This information is used in a continuous loop
that recalculates the mass properties during each time step during which a small amount
of mass leaves the vehicle. The mass flow is assumed to be constant and steady. Again,
this timedependent information is saved in the data file for future use.
3.6 Aerodynamics Model
In recent years, evolutionary techniques have been applied to aerodynamic design.
Gage and Kroo
46
focused their research on the topological design of nonplanar wings
and used a genetic algorithm to minimize induced drag given a fixed lift. Others have
also attempted design optimizations of various aerodynamic shapes using a genetic
39
algorithm. In 1990, Washington
15
developed an aerodynamic prediction package, called
AeroDesign, which is capable of determining the aerodynamic constants of axisymmetric
missiles of circular crosssection with cruciform wings or fins. AeroDesign has been
chosen for use in the current study.
The AeroDesign model is equipped to handle a wide variety of flow field
conditions. Initial conditions and physical parameters of the vehicle are inputted to the
model. Next, empirical curve fits of wind tunnel data are performed and the aerodynamic
constants are determined across a range of flow field conditions. Vehicle geometry and
other necessary parameters are used to generate an aerodynamic database. AeroDesign
assumes that there are no boundary layers and that no separation occurs.
AeroDesign can analyze either a cone or ogive shape for the nosecone and
assumes a cylindrical body shape. The Mach number regime of the code is from
subsonic to high supersonic/low hypersonic (? Mach 4.0) flow. For this study, the
general shape that has been used is the ogive nosecone and cylindrical body. Four fins in
a ?+? shape configuration are located at the aft end of the 1
st
stage for some of the launch
vehicles analyzed. No other stages have any fins or wings. The body diameter is
constant for all vehicle stages with a few exceptions. AeroDesign has been modified for
the three and fourstage solid propellant vehicle cases along with the airlaunched two
stage case. For these cases, the body diameter is allowed to vary for each stage. The
diameter of the 1
st
stage is equal to or greater than the diameter of the 2
nd
stage. In
AeroDesign, a modification to the calculation of the drag coefficient has been made to
account for differences in the 1
st
and 2
nd
stage diameters as well as the 2
nd
and 3
rd
stage
40
diameters. The drag coefficient for a flat plate (C
Dflatplate
) is used as the baseline
coefficient and is normal to the flow. The correction factor is determined as follows:
ref
osed
DflatplateDcorr
A
A
CC
exp
*=? (3.22)
where
( )
2
2
2
1exp stgstgosed
RRA ?=? (3.23)
and
2
1stgref
RA ?= (3.24)
A similar correction factor has been performed for the difference in the 2
nd
and 3
rd
stage
diameters where the 2
nd
stage diameter is equal to or greater than the 3
rd
stage diameter.
Additional modifications to AeroDesign would be required for a launch vehicle where the
2
nd
stage diameter is equal to or greater than the 1
st
stage diameter.
Using the AeroDesign package along with the known geometric parameters of the
vehicle, the important aerodynamic coefficients are generated. Essential aerodynamic
coefficients are determined for a range of flight Mach numbers and angles of attack. Like
the mass properties, the aerodynamic data generated is saved and used in the 6DOF flight
dynamics simulator.
The aerodynamics model is organized to determine the aerodynamic properties of
the vehicle at each stage of flight. The model is run initially for the entire vehicle with all
stages stacked together. In the 6DOF flight dynamics simulator, the aerodynamics model
is called again after Stage 1 burnout. The aerodynamic properties are calculated again;
this time with the first stage of the vehicle gone and the corresponding velocity and
41
pressure conditions being used. This procedure continues as subsequent vehicle stages
burn out and the vehicle?s geometry changes.
The 6DOF flight dynamics simulator keeps track of variables such as Mach
number, atmospheric pressure, atmospheric density etc. This information in conjunction
with the aerodynamic characteristics and the mass properties of the vehicle is used in
calculating all of the aerodynamic forces on the vehicle in flight.
3.7 SixDegreeofFreedom (6DOF) Flight Dynamics Simulator
The culmination of the vehicle performance analysis occurs in the sixdegreeof
freedom (6DOF) flight dynamics simulator. This system model is probably the most
important part of the objective function since the information it provides is used to judge
vehicle performance. A 7
th
/8
th
order RungeKutta numerical method is used to integrate
the equations of motion. Position, velocity and orientation of the vehicle are determined
at small time intervals throughout the flight. The vehicle flies a ballistic trajectory with
the goal of reaching a lowEarth orbit.
This model begins by reading in mass properties, propulsion data and all initial
conditions for vehicle flight. The equations of motion are then integrated and values such
as altitude, velocity and orientation are saved for postprocessing. The integration
process continues until the vehicle has reached the apogee of the ballistic flight trajectory.
Ideally, this would correspond to the orbital insertion point for a lowEarth, circular orbit.
Two of the goals of the design optimization process are to attain the desired orbital
altitude and orbital velocity for a predetermined lowEarth orbit.
These final velocity and altitude values along with cost/total vehicle mass values
are then sent to the goal determination algorithm. The desired orbital values are pre
42
selected prior to the start of the optimization process and stored in the goal determination
algorithm. These desired orbital values are compared to the values calculated in the
6DOF flight dynamics simulator. The goals of the optimization process are to minimize
the differences between the values produced by the 6DOF flight dynamics simulator and
the desired orbital values.
Finally, it is possible that a final short burn would be required for orbit insertion.
However, this consideration was not explored for the current study. Future work should
incorporate the use of an orbit insertion burn into the operation of the 6DOF flight
dynamics simulator.
3.8 Cost Model
The cost model used in this analysis was derived by Dr. Dietrich E. Koelle and
published in his book Handbook of Cost Engineering for Space Transportation Systems,
Revision 1, with TRANSCost 7.1.
41
The philosophy behind this cost model is different
from cost models of past decades. In the early days of spaceflight, vehicles were
designed without much consideration for cost (the Saturn V moon rocket for example).
After the Apollo program ended, engineers started designing vehicles to fit a specific cost
budget; called DesigntoCost. The idea here was to maximize performance within a pre
determined budget. For current launch vehicles, Dr. Koelle has proposed a new concept
known as ?Cost Engineering.? ?In the case of ?CostEngineering?, its goal is the
minimum vehicle design concept. This means that costs have to be taken into account as
a criterion for each technical decision.?
41
Another promising cost model currently in use today is the NASA and Air Force
Cost Model (NAFCOM).
63
This model is a parametric based cost model developed in
43
1990 and updated in order to estimate the cost of both satellite and launch vehicle space
systems. The use of the NAFCOM was not incorporated into the current study.
However, future design optimization work should include comparisons between the cost
results generated by the TRANSCost 7.1 cost model and the NAFCOM.
The TRANSCost 7.1 cost model fits in well with the current effort of design
optimization of space launch vehicles. In the current study, the designs being optimized
reflect optimum solutions at the preliminary design stage. The TRANSCost 7.1 cost
model provides a powerful tool to aid in the minimization of vehicle launch costs at this
stage of the design process. An additional economic model developed by Wertz
42
is used
in order to refine the TRANSCost 7.1 cost model so that a cost per launch value can be
calculated.
The TRANSCost 7.1 cost model is broken into three submodels. These submodels
are: the Development Cost Submodel, the Recurring Cost Submodel, and the Ground and
Flight Operations Submodel. For this study, the cost of launch insurance is also included
in the calculations to provide additional realism to the cost per flight determination.
Thus, the cost per launch is determined by using the following equation:
C
launch
= C
development
+ C
vehicle
+ C
flightops
+ C
insurance
(3.25)
The various submodels employ systemlevel Cost Estimation Relationships (CERs) to
predict cost. These CERs are the backbone of the model and provide the cost of a system
in a generic unit called the ?ManYear (MYr).? The reason for using the MYr as the
costing unit is that this unit provides firm cost data which is valid internationally and free
from annual changes due to inflation and other factors. ?For each of the technical
44
systems, a specific CER has been derived which is mostly massrelated with the basic
form of:
x
aMCER = (3.26)
where: CER = cost (MYr)
a = systemspecific constant value
M = mass (kg)
x = systemspecific costtomass sensitivity factor?
41
The values of ?a? and ?x? in Equation (3.26) are determined for specific types of launch
vehicle systems using a data fit of the costtomass relationships of a group of similar
systems.
3.8.1 Development Cost Submodel
Using historical data of previously built and flown solid propellant rocket motors,
the basic CER for the Development Cost Submodel can be written as:
53.0
)2.19( MCER
solid
= (3.27)
Similar CERs have been established for liquid propellant rockets along with turbofan and
turbojet engines.
Additionally, the CER alone does not provide the entire cost picture for a
particular submodel. Various cost factors have been introduced to make the cost
calculation more realistic. Some examples of these cost factors are the Development
Standard Factor (f
1
) and the Team Experience Factor (f
3
). These factors can either
increase or decrease the system cost. For example, a more experienced team that is
designing a launch vehicle should be able to keep costs down since they can take
advantage of prior knowledge. Thus, a more experienced team will have a numerically
lower Team Experience Factor (f
3
).
45
For the solid propellant rocket motor, the complete CER for the Development
Cost Submodel (H
ES
) of a singlestage vehicle is:
31
53.0
))(2.19( ffMH
ES
= (3.28)
Finally, the total development cost (C
devtotal
) of an entire multistage vehicle can
be determined by summing up the CERs for each individual stage and employing
additional cost factors such as the system engineering/integration factor (f
0
) and
programmatic cost impact factors (f
6
, f
7
, f
8
).
8760
)( fffHfC
EStotaldev ?
=
?
(3.29)
Thus, one equation has been derived for the total development cost (Development
Cost Submodel) of a multistage solid propellant launch vehicle. For different types of
propulsion systems, the corresponding CER can be used in Equation (3.29) to reflect the
development cost of the entire vehicle.
The total value for the development cost calculated in Equation (3.29) is typically
not used in the cost per launch determination (C
development
). Wertz
42
presented a method
where this total value is spread out over a predetermined period of time (usually the
number of years of the contract) to produce a yearly development cost. This method
takes into account inflation along with interest rate. Using C
devtotal
from Equation (3.29),
amortization assumes a constant payment (P
constant
) over time in real (thenyear) dollars
of:
()
npay
totaldev
tcons
C
P
?
?
+?
=
int11
int*
tan
(3.30)
46
where ?int? is the interest rate and ?npay? is the number of payments. For a given
constant inflation rate, ?inf?, the annual reduction, R
annual
, in the value of money can be
written as:
()inf1
1
+
=
annual
R (3.31)
Thus, the average payment, P
avg
, will be reduced to:
( )
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
annual
npay
annualannual
tconsavg
R
R
npay
R
PP
1
1
tan
(3.32)
To determine the cost per launch of launch vehicle development, the yearly development
cost, P
avg
, is divided by the launch rate per year, lrate, which produces the value of
C
development
used in Equation (3.24).
lrate
P
C
avg
tdevelopmen
= (3.33)
The launch rate per year, lrate, multiplied by the number of years of the contract, npay,
results in the number of units, nunits, to be built. This value for the number of units will
be used in the Recurring Cost Submodel calculations.
3.8.2 Recurring Cost Submodel
The Recurring Cost Submodel is developed in a similar way to the Development
Cost Submodel. The basic recurring cost CER is used with some different cost factors.
The Learning Factor (p) is introduced and is used to determine the Cost Reduction Factor
for Series Production (f
4
). The Learning Factor takes into account the reduction of effort
required after the initial vehicle rolls off the production line and experience is gained in
producing more and more identical units. Typically, the value of the Learning Factor (p)
47
ranges between 0.70 and 0.95. Knowing this value and the number of units, ?nunits?, to
be produced, the value of f
4
is determined by:
?
=
nunits
p
nunits
nunits
f
1
4
2ln
ln1
(3.34)
Using the solid propellant rocket motor as an example again, the CER for the
Recurring Cost Submodel can be written as:
4
395.0
))()(42.2( fMnunitsF
ES
= (3.35)
where ?nunits? is the number of units being built.
In order to calculate the total recurring cost (C
rectotal
), the system management,
vehicle integration and checkout factor (f
0
) is used to get:
EStotalrec
FfC
0
=
?
(3.36)
The value used in the cost per launch calculation (C
vehicle
) is obtained by taking the result
of Equation (3.36) and dividing it by the number of units being built.
nunits
C
C
totalrec
vehicle
?
= (3.37)
3.8.3 Ground and Flight Operations Cost Submodel
As stated by Koelle,
41
?assessment and modeling of launch vehicles? operations
cost is the most difficult task compared to development cost and recurring cost
modeling.? It is not an easy task to accurately estimate the costs associated with
preparing a launch vehicle for flight.
Based on the work done by Wertz,
42
for expendable launch vehicles, the flight
operations cost is typically $0.5 million to $1.0 million per mission. The amount of $1.0
million (in 2003 dollars) is used for C
flightops
in the Ground and Flight Operations Cost
48
Submodel and in Equation (3.24). This value will be constant throughout the design
optimization process for all launch vehicle types.
3.8.4 Insurance Cost Submodel
The Insurance Cost Submodel (C
insurance
) is modeled as a percentage of the launch
vehicle recurring cost (C
vehicle
). This insurance covers only the launch itself and does not
include the cost of replacing the payload in the event of launch failure. Also, this model
typically represents the upper limit of insurance cost. Most likely, insurance cost will
drop as launch vehicle reliability is established with successful initial flights. Since this
study focuses on the preliminary design, using the upper limit insurance cost is a prudent
choice.
According to Wertz,
42
a typical insurance cost is on the order of 15% of the
launch vehicle recurring cost. Thus, in the cost model being used for this study, C
insurance
for the cost per launch determination is calculated by multiplying C
vehicle
by 15%. Thus,
the cost of production of an individual launch vehicle drives the cost to insure it. This
makes sense when considering that it is much more expensive to insure a luxury car as
opposed to an economy car, for example.
The inclusion of the insurance cost is more for realism since the overall cost
model is massbased. Due to the massbased nature of the CERs, minimizing the mass of
the vehicle should minimize the cost of the vehicle (C
vehicle
) as well. Since the insurance
is simply a percentage of C
vehicle
, the results are not affected by the insurance cost.
However, the goal here is to present as realistic a value as possible for this preliminary
design study.
49
3.8.5 Example Calculation
An example cost calculation for a threestage, solid propellant rocket is presented
here. The vehicle is designed to carry a 1,000 lbm payload into a lowEarth orbit. The
launch takes place from Cape Canaveral AFS, FL. The MYr unit has been converted to a
dollar value representing currency in 2003. The values for the masses of each stage
(mstg1, mstg2, mstg3) are determined in the mass properties models and converted to
kilograms in the cost model.
Assumptions have been made for the input values of the Launch Rate (lrate),
Number of Payments (npay), Development Standard Factor (f
1
), Team Experience Factor
(f
3
) and the Learning Factor (p). A common sense, realistic approach has been used for
these values. Since the USAF is attempting to field a responsive launch vehicle that can
launch quickly and rapidly, an annual launch rate of 15 is a reasonable value. The
number of payments reflects a 15 year contract. The Development Standard Factor (f
1
)
and the Team Experience Factor (f
3
) can have a range from 0.4 to 1.4. Choosing values
of 0.90 for both factors represents a standard project and a company/team with some
related experience. For this study, investigating a stateoftheart design with a brand
new company is not the goal. Thus, the more ?middle of the road? value of 0.90 makes
sense. According to Koelle,
41
the Learning Factor (p) for space systems ranges between
0.80 and 1.0. Here again, choosing a value of 0.85 for the Learning Factor (p) is
reasonable.
Table 37 summarizes the inputs that are used to calculate the cost per launch for
the threestage solid propellant launch vehicle described above. Table 38 describes the
results of the cost determination process. The Total Launch Vehicle Cost per Launch is
50
determined from Equation (3.24). From the results in Table 38, it can be seen that this
particular launch vehicle is moderately priced ($50.41 million) for an expendable, solid
propellant launch vehicle.
Table 37: Inputs for Vehicle Cost Example Calculation
Mass of Stage 1 (mstg1) 22,186 kg
Mass of Stage 2 (mstg2) 13,390 kg
Mass of Stage 3 (mstg3) 5,470 kg
Launch Rate (lrate) 15
Number of Payments (npay) 15
Number of Units (nunits) 225
Development Standard Factor (f
1
) 0.90
Team Experience Factor (f
3
) 0.90
Learning Factor (p) 0.85
Table 38: Outputs for Vehicle Cost Example Calculation
Total Development Cost (C
dev
) $18.60 million
Cost Reduction Factor for Series Production (f
4
) 0.364
Total Recurring Cost (C
veh
) $26.78 million
Total Flight Operations Cost (C
ops
) $1.0 million
Total Insurance Cost (C
ins
) $4.02 million
Total Launch Vehicle Cost per Launch (C
lnch
) $50.41 million
51
4.0 VALIDATION EFFORTS
4.1 Introduction
The employment of system modeling in a preliminary design process can provide
a variety of results that represent different solutions to the stated design problem. Some
results may produce a seemingly more desirable solution than other results. Simply
taking these particular results as the optimum solution may lead to an unrealistic design
that cannot to be reproduced in the real world. Basically, results that have not been
validated could lead to designs that are not physically attainable. Thus, it is very
important to validate the results that have been generated so that the degree of confidence
in the accuracy of the modeling effort can be ascertained. In addition, these validation
efforts are important because the outcome of the design optimization process is highly
dependent on the accuracy of the system modeling. It should be noted that, for
preliminary design purposes, validating the accuracy of the system modeling is always
necessary regardless of whether or not design optimization is being performed. To that
end, Cosner et al.
47
provide the definition of validation:
?The process of determining the degree to which a model is an accurate
representation of the real world from the perspective of the intended uses of the
model.?
Roy
48
adds that validation deals with the physics of the process that the system model is
attempting to simulate.
52
Models form a crucial element of engineering design since they are the link
between the design parameters being employed and the actual system performance. Of
course, models are not perfect representations of real world systems. Uncertainty, both
mathematical and physical, is introduced when computations are performed and
assumptions are made. Additionally, predictive models can often produce a number of
possible designs that meet the specified goals of the system using different combinations
of the design parameters. This can either be a benefit or it can complicate the search for
the desired design solution. Finally, statistical variations in the performance of physical
systems must be taken into consideration when performing model validation. The
importance of the validity of system models takes on greater significance in highly
complex systems, including space launch vehicles, where the models must incorporate
numerous design disciplines such as propulsion, aerodynamics and flight dynamics.
The various models used in the current study have been validated independently
in previous work performed by their respective aerospace engineering researchers. The
following sentences describe the work performed by these researchers. The solid
propellant propulsion model was developed and validated by Burkhalter,
44
Sforzini
45
and,
with modifications, Hartfield et al.
19
This model was then used by Anderson
16
and
Metts
20
in the investigation and reverse engineering of small to mediumsized solid
propellant tactical missiles. Jenkins
18
developed the liquid propellant propulsion model
and then used an existing real world system, the SCUDB short range ballistic missile, to
successfully validate the model. The aerodynamics model developed by Washington,
15
known as AeroDesign, has been an industry standard since 1990. The sixdegreeof
freedom (6DOF) flight dynamics simulator has been used extensively in the previous
53
work. Finally, Koelle
41
has used a large supply of real world data to develop the Cost
Estimate Relationships (CERs) used in his TRANSCost 7.1 cost model. The use of this
historical data has resulted in a successful validation of his cost model.
The goal for this study is to incorporate all these system models into one
comprehensive model that represents an entire multistage space launch vehicle. With
confidence in the validity of the individual system models, validation of the different
stages as well as the entire launch vehicle can be undertaken. However, simply stating
that ?since the individual system models are valid then the entire launch vehicle must be
valid as well? is not sufficient. A comprehensive validation of the entire launch vehicle,
that is as detailed as possible, has been performed as follows.
4.2 Validation Method
4.2.1 General Description
The method used in this study for model validation follows the method used by
Jenkins
18
and Riddle
49
for the validation of a singlestage liquid propellant rocket model.
The present method begins by researching and choosing a launch vehicle that is similar to
the system being modeled by the various physical models included in the objective
function. In the case of the work performed by Jenkins
18
and Riddle,
49
a liquid propellant
rocket was being reverse engineered. As a result, the liquid fueled SCUDB was chosen
as the real world system. Next, as much information on the chosen vehicle is determined
and appropriately hardcoded into the appropriate input locations for the objective
function. This information includes physical size, thrust values and/or propellant types.
The objective function, along with other design parameters, is then manipulated in an
attempt to reproduce the characteristics of the real world example. Also, in order to
54
attempt to match the real world example more closely, a design optimization can be run
using the genetic algorithm (GA). The purpose of this optimization is not to maximize or
minimize any particular vehicle performance characteristics but rather to allow the GA to
?fine tune? the model by choosing the remaining unknown parameters so that the
resulting vehicle matches, as closely as possible, the real world example. If the objective
function can produce a launch vehicle that is strikingly similar to the real world example,
given the real world example?s known and GAdetermined parameters, then the validity
of the model is substantially strengthened.
4.2.2 Specific Validation Process and Setup
For the current study, four specific system model validations have been
performed. The availability of information on real world launch vehicles drove the
selection of the types of system models to validate. The system models that have been
validated are: the three and fourstage solid propellant launch vehicles, the twostage
liquid propellant launch vehicle and the airlaunched, twostage liquid propellant launch
vehicle. An additional comparison involving a threestage solid/liquid/liquid launch
vehicle has also been performed.
The same validation method was used for each of the four system models with
slight variations in the setup depending on the known parameters of the real world
example launch vehicle. Given the known parameters, the system model was
manipulated in an attempt to match the physical properties and the performance
characteristics of the real world example. The known parameters of the real world
example were:
55
 payload mass to orbit
 desired altitude and velocity
 individual stage geometry (diameter and length)
 individual stage propellants
 individual stage burn time (used for liquid propellant vehicles)
In order to more closely model United States Air Force (USAF) space launch
vehicle systems, the objective function was configured to include the latitude and
longitude coordinates of the two primary USAF launch sites: Vandenberg AFB, CA
(VAFB) and Cape Canaveral AFS, FL (CCAFS). Also, typical payload, final altitude
and final velocity values for the four real world examples were chosen as direct inputs to
the objective function. These values are summarized in Table 41.
Table 41: Typical Values of Real World Launch Vehicles
Model Real World
Example
Payload Altitude Velocity Launch
Site
ThreeStage
Solid
Minuteman III
ICBM
2,540 lbm 750,000 ft 22,000 ft/s VAFB
FourStage
Solid
Minotaur I
SLV
738 lbm 2,430,000 ft 25,004 ft/s VAFB
TwoStage
Liquid
Titan II
SLV
7,000 lbm 607,000 ft 25,600 ft/s CCAFS
AirLaunched,
TwoStage
Liquid
QuickReach
TM
Launch
Vehicle
1,000 lbm 700,000 ft 25,532 ft/s CCAFS
One important check on model validity for launch vehicles involves the
calculation of propellant and inert mass fractions. These calculations can be performed
using the resulting mass properties for the launch vehicle generated in the previously
described validation method. These computed mass fractions are then compared to
typical mass fractions for previously built and flown launch vehicles. The historical data
56
for these real world launch vehicles provides a record of successfully designed launch
vehicle systems. For example, from Humble et al.,
50
the propellant mass fraction (f
prop
)
of solid rocket motors typically ranges from 0.80 to 0.95. Table 42 shows some example
propellant mass fractions for existing solid rocket motors.
Table 42: Example Solid Rocket Motor Mass Fractions
50
Motor Designation f
prop
Shuttle Advanced Solid Rocket Motor (ASRM) 0.895
Titan IV Solid Rocket Booster (SRB) 0.815
ORBUS 21 0.936
Star 48B 0.939
Pegasus 1
st
Stage Orion 50 0.898
Pegasus 3
rd
Stage Orion 38 0.859
4.2.3 Inert and Propellant Mass Fraction Calculations
Various mathematical tools can be used to determine the size of a particular
launch vehicle. The inert mass fraction and the propellant mass fraction are two of those
tools. Humble et al.
50
write the equations for inert mass fraction (f
inert
) and propellant
mass fraction (f
prop
) as shown below in Equations 4.1 and 4.2:
inertprop
inert
inert
mm
m
f
+
= (4.1)
inertprop
prop
prop
mm
m
f
+
= (4.2)
where m
prop
is the mass of the propellant and m
inert
is the mass of the vehicle or stage
minus the mass of the propellant and the mass of the payload (i.e. the dry mass). In
addition, manipulating Equations 4.1 and 4.2 yields the relationship between these two
types of mass fractions as written in Equation 4.3.
inertprop
ff ?=1 (4.3)
57
In their discussion of real world launch vehicles, Humble et al.
50
provide some
historical data for both solid and liquid propellant launch vehicles. The data summarizes
the typical values of propellant mass fraction for solid propellant motors as shown in
Table 42. The inert mass fraction is chosen to describe the mass properties of liquid
propellant rockets.
4.3 ThreeStage Solid Propellant Vehicle vs. Minuteman III ICBM
For solid propellant launch vehicles, much of the historical data is in the form of
the propellant mass fraction. The historical data shows that the propellant mass fraction
for solid propellant motors ideally should be around 0.90 based on the typical value of
specific impulse (I
sp
) for solid motors. There are essentially two explanations for why
this value for the propellant mass fraction is used. First, for a given launch vehicle, with
a known inert mass, the portion of the total vehicle mass not used for inert mass
components must be divided between the payload mass and the propellant mass. The
goal would be to minimize the propellant mass (hence minimize the propellant mass
fraction) in order for the vehicle to be able to carry more payload mass to orbit. Equation
4.2 provides one way to determine the propellant mass fraction but the values from this
equation range from 0 to 1. The propellant mass fraction must be somewhere between
these two values. The propellant mass (m
prop
) can be determined using a version of the
ideal rocket equation given by Humble et al.
50
and written in Equation 4.4.
58
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
osp
osp
gI
v
inert
inert
gI
v
pay
prop
ef
fem
m
1
11
(4.4)
where m
pay
is the payload mass, I
sp
is the specific impulse, g
o
is the local acceleration of
gravity, ?v is the required change in velocity and f
inert
is the inert mass fraction.
Knowing that the maximum vacuum I
sp
for a solid propellant rocket motor is about 290s,
using Equation 4.4 and knowing the inert mass of the vehicle, the minimum propellant
mass fraction will be approximately 0.90 for Earthtoorbit missions. In addition, many
actual solid rocket motors use steel as the casing material which adds to the inert mass
and leaves even less mass available for payload and other useful components. The
historical data has shown that the value of 0.90 provides enough propellant to achieve
orbit while leaving a large enough portion of the vehicle?s mass for payload.
For the threestage solid propellant vehicle model, the objective function has been
manipulated and the resulting best performer has been compared to a real world example.
The real world example launch vehicle chosen for the objective function to match is the
Minuteman III intercontinental ballistic missile (ICBM), a threestage solid propellant
strategic weapon. The Minuteman III ICBM is launched from Vandenberg AFB, CA for
testing purposes. Thus, the two vehicles being compared for this validation process are
analyzed using Vandenberg AFB, CA as the launch site. The propellant mass fractions
for the Minuteman III ICBM were determined from published values of the rocket?s
specifications.
59
Currently, there are no landbased, threestage solid propellant launch vehicles
used by any spacefaring nation to put satellites into orbit. The Pegasus launch vehicle,
operated by Orbital Sciences Corporation, is a threestage solid propellant launch vehicle
but it is airlaunched from a modified L1011 aircraft. Even though the Minuteman III
ICBM is not designed to attain orbital velocity, it is comparable to a space launch
vehicle; attaining a burnout altitude of 750,000 feet and a burnout velocity of 22,000
feet per second. According to the United States Air Force LGM30G Minuteman III Fact
Sheet
51
?The Minuteman III is a strategic weapon system using a ballistic missile of
intercontinental range.?
A schematic comparing the launch vehicle generated by the model and the
Minuteman III ICBM is shown in Figure 41. The Minuteman III ICBM has a
significantly more conical nose than the nose generated by the model. Also, the
Minuteman III ICBM uses an interstage skirt between the between the first and second
stages due to the diameter differences. This skirt was not used in the model thus showing
the slightly larger diameter first stage more dramatically. Finally, the firststage of the
Minuteman III uses four nozzles whereas the firststage of the model is designed to use a
single nozzle. Generally, the schematic shows a very good match between the two
launch vehicles.
Table 43 summarizes the physical and performance characteristics generated by
the threestage solid propellant launch vehicle model compared to the Minuteman III
ICBM. The bold* values were direct inputs into the objective function based on the
published characteristics of the Minuteman III ICBM. All other values for the model
were calculated using the various system models that make up the objective function.
60
Figure 41. ThreeStage Solid Propellant Model vs. Minuteman III ICBM Schematic
(Ref. 52: http://www.globalsecurity.org/wmd/systems/images/us_nuke_minuteman301)
61
Table 43: ThreeStage Solid Propellant Model vs. Minuteman III ICBM Comparison
Parameter Model Minuteman III ICBM
Payload* 2,540 lbm 2,540 lbm
Total Vehicle Weight 75,870 lbm 79,432 lbm
Total Vehicle Length 61.33 ft 59.90 ft
Total Vehicle f
prop
0.9102 0.8925
Final Altitude 768,682 ft 750,000 ft
Final Velocity 22,071 ft/s 22,000 ft/s
Stage 1
Stage Length* 295.20 in 295.20 in
Stage Diameter* 66.00 in 66.00 in
Propellants* PBAA/AP/Al PBAA/AP/Al
Total Stage Weight 49,882 lbm 50,486 lbm
m
prop
45,853 lbm 45,371 lbm
m
inert
4,029 lbm 5,115 lbm
f
prop
0.9192 0.8987
Burnout Time 67.72 s 61.00 s
Burnout Altitude 119,963 ft 100,000 ft
Stage 2
Stage Length* 184.00 in 184.00 in
Stage Diameter* 52.00 in 52.00 in
Propellants* PBAA/AP/Al PBAA/AP/Al
Total Stage Weight 15,514 lbm 15,432 lbm
m
prop
13,993 lbm 13,669 lbm
m
inert
1,521 lbm 1,764 lbm
f
prop
0.9020 0.8857
Burnout Time 130.23 s 126.00 s
Burnout Altitude 406,931 ft 300,000 ft
Stage 3
Stage Length* 90.00 in 90.00 in
Stage Diameter* 52.00 in 52.00 in
Propellants* PBAA/AP/Al PBAA/AP/Al
Total Stage Weight 7,666 lbm 9,520 lbm
m
prop
6,897 lbm 7,055 lbm
m
inert
769 lbm 882 lbm
f
prop
0.8997 0.8889
Burnout Time 189.23 s 191.00 s
Burnout Altitude 768,682 ft 750,000 ft
Burnout Velocity 22,071 ft/s 22,000 ft/s
A good match between the model and the Minuteman III ICBM has been
obtained. The primary difference is that the threestage solid propellant vehicle model
62
weighs about 3,500 pounds less than the Minuteman III ICBM. This weight difference
comes about due to lower values of the inert mass in the model. This also causes the
propellant mass fractions of the model to be approximately 2% higher than the propellant
mass fractions of the Minuteman III ICBM. The model does not fully account for all the
inert mass components thus resulting in lower values of the inert mass for each stage.
This limitation on the mass properties model has implications for the actual design
optimizations. Since the model underestimates the inert mass, an adjustment to the
results of the validation model and the optimized vehicles would be required. For this
study, an adjustment to the model was not performed. Future work to address this issue
could be in the form of an updated mass properties model or incorporation of a correction
factor into the existing mass properties model. Since the values of the propellant mass
are quite accurate when compared to the Minuteman III ICBM, lower values of inert
mass will cause higher propellant mass fractions. However, the propellant mass fractions
of the model are still accurate for a solid propellant launch vehicle.
The strength of this model is its ability to reproduce the performance
characteristics of the Minuteman III ICBM. This can be seen in the values generated for
the final altitude (768,682 ft vs. 750,000 ft) and final velocity (22,071 ft/s vs. 22,000 ft/s).
The model accurately predicts the payload position and velocity at burnout similar to the
conditions of the Minuteman III ICBM. The burnout times and burnout altitudes for the
different stages also match the published trajectory of the Minuteman III ICBM. Figure
42 shows a diagram of the Minuteman III ICBM ballistic flight profile. The vehicle is
being modeled up until the point in the diagram that says ?Third Stage Jettison?
(approximately the first 191 seconds of flight). The vehicle flight profile for the
63
validation model is shown in Figure 43. The burnout altitudes and burnout times for
both vehicles match fairly closely. However, the Minuteman III ICBM comes out of its
launch facility at a much steeper launch angle than the validation model. The reason for
this is that the 6DOF flight dynamics simulator used with the validation model is not
configured to perform a programmed pitchover maneuver.
Figure 42. Minuteman III ICBM Ballistic Flight Profile
(Ref. 53: http://www.geocities.com/minuteman_missile/diagrams.htm)
64
Figure 43. Validation Model Ballistic Flight Profile
4.4 FourStage Solid Propellant Vehicle vs. Minotaur I SLV
Like the threestage solid propellant launch vehicle, the objective function for the
fourstage solid propellant launch vehicle model has been manipulated using the known
values of a real world system. The real world comparison vehicle chosen for this case is
the Minotaur I Space Launch Vehicle (SLV). The Minotaur I SLV is a fourstage solid
propellant launch vehicle used to carry small to medium payloads into low Earth orbit.
The Minotaur I SLV has been launched from both the east and west coasts of the United
States. For this validation, the west coast launch site, Vandenberg AFB, CA, has been
chosen.
The Minotaur I SLV is owned and operated by Orbital Sciences Corporation.
According to the Minotaur I SLV User?s Guide,
54
?The Minotaur I launch vehicle was
developed by Orbital for the United States Air Force (USAF) to provide a cost effective,
65
reliable and flexible means of placing small satellites into orbit.? This launch vehicle
utilizes surplus stages of decommissioned Minuteman II ICBMs along with stages from
Orbital Sciences Corporation?s airlaunched Pegasus launch vehicle.
A schematic showing the fourstage solid propellant vehicle generated by the
model and the Minotaur I SLV is shown in Figure 44. Another good match between the
two vehicles has been obtained. However, one of the differences in the two vehicles is
that the skirt between the first and second stages has not been included in the model.
Also, the model uses a single nozzle for the first stage while the Minotaur I SLV has four
nozzles which is the configuration on the first stage of the Minuteman III ICBM. The
reason for this is that the propulsion system model for solid propellant motors was
created for a single nozzle rocket. This is not a significant difference for this preliminary
design study.
The comparison between the characteristics of the fourstage solid propellant
launch vehicle model generated using the objective function and the Minotaur I SLV is
shown in Table 44 and Table 45. The bold* values were direct inputs into the objective
function based on the published characteristics of the Minotaur I SLV.
66
Figure 44. FourStage Solid Propellant Model vs. Minotaur I SLV Schematic
(Ref. 55: http://www.orbital.com/SpaceLaunch/Minotaur/index.html)
Table 44: FourStage Solid Propellant Model vs. Minotaur I SLV Comparison
Parameter Model Minotaur I SLV
Payload* 738 lbm 738 lbm
Total Vehicle Weight 78,090 lbm 79,800 lbm
Total Vehicle Length 64.58 ft 63.02 ft
Total Vehicle f
prop
0.9185 0.8998
Final Altitude 2,425,999 ft 2,430,000 ft
Final Velocity 25,002 ft/s 25,004 ft/s
Cost per Launch $51.95 million $52.05 million
Adjusted Cost per Launch $29.76 million
Advertised Cost per Launch $20.00 million
67
Table 45: FourStage Solid Propellant Model Individual Stage Comparison
Parameter Model Minotaur I SLV
Stage 1
Stage Length* 295.20 in 295.20 in
Stage Diameter* 66.00 in 66.00 in
Propellants* PBAA/AP/Al PBAA/AP/Al
Total Stage Weight 49,882 lbm 50,486 lbm
f
prop
0.9192 0.8987
Burnout Time 68.32 s 61.30 s
Burnout Altitude 123,706 ft 103,968 ft
Burnout Velocity 5,078 ft/s 4,919 ft/s
Stage 2
Stage Length* 162.00 in 162.00 in
Stage Diameter* 52.00 in 52.00 in
Propellants* PBAA/AP/Al PBAA/AP/Al
Total Stage Weight 15,840 lbm 15,432 lbm
f
prop
0.9131 0.8857
Burnout Time 130.73 s 128.10 s
Burnout Altitude 440,142 ft 382,669 ft
Burnout Velocity 10,267 ft/s 9,512 ft/s
Stage 3
Stage Length* 145.20 in 145.20 in
Stage Diameter* 50.00 in 50.00 in
Propellants* HTPB/AP/Al HTPB/AP/Al
Total Stage Weight 9,396 lbm 9,520 lbm
f
prop
0.9136 0.9086
Burnout Time 201.62 s 203.50 s
Burnout Altitude 976,766 ft 801,054 ft
Burnout Velocity 19,694 ft/s 19,208 ft/s
Stage 4
Stage Length* 60.00 in 60.00 in
Stage Diameter* 38.00 in 38.00 in
Propellants* HTPB/AP/Al HTPB/AP/Al
Total Stage Weight 1,957 lbm 1,966 lbm
f
prop
0.8719 0.8642
Burnout Time 554.33 s 763.80 s
Burnout Altitude 2,425,999 ft 2,430,000 ft
Burnout Velocity 25,002 ft/s 25,004 ft/s
A good match has been obtained for the fourstage solid propellant vehicle model
and its corresponding real world example. In this case, the model produces a vehicle
68
very similar to the Minotaur I SLV. The total weight of the vehicle produced by the
model is about 1,700 pounds less than the weight of the Minotaur I SLV. Again, this
difference is most likely due to the bias error in the mass properties model. There is also
a large difference in the Stage 4 burnout time for the model versus the Minotaur I SLV.
During the firing of the first three stages, the model and the Minotaur I SLV match
burnout time, burnout velocity and burnout altitude very well. After the third stage burns
out, the Minotaur I SLV uses a coast phase of about 400 seconds before firing the fourth
stage for orbit insertion. Using the model, the value for this coast phase was determined
to be 200 seconds in order to achieve the required orbital parameters.
As with the threestage solid propellant model, the propellant mass fractions of
the fourstage solid propellant model match the propellant mass fractions of the Minotaur
I SLV within about 2% for each stage. Again, the model produces mass fractions that are
slightly higher than the actual values of the Minotaur I SLV.
While the vehicle model being used in this study provides a reasonably high
fidelity analysis of system performance, it cannot fully reproduce a real world system
down to the specifics of individual components. Sutton
60
describes the eight components
that make up the first stage of the Minuteman ICBM. These components are the
propellant, internal and external insulation, the liner, the igniter, the nozzle, the motor
case, and other miscellaneous components. Assuming that the other three stages of a
fourstage solid propellant launch vehicle have the same components then the total
number of components rises to 32. The total number of components becomes 35 when
the nosecone, the payload and an electronics/avionics package are included. For a liquid
propellant launch vehicle, the total number of vehicle components would be even higher.
69
An important point to remember is that the vehicle models being generated in this study
represent a preliminary design level model. As a result, it would be quite difficult for the
model to produce a very detailed representation of each vehicle component. This type of
component analysis is usually done during later stages of the design engineering process.
Next, the subjective nature of vehicle cost is apparent in the cost comparison.
Using the published mass values for the Minotaur I SLV and applying the cost model
developed for this study, the cost per launch of the Minotaur I SLV is $52.05 million in
2003 dollars. This is roughly about the same as the cost per launch of the vehicle
generated by the model ($51.95 million vs. $52.05 million). In reality, there is no
recurring cost for the first two stages of the Minotaur I SLV since those stages come from
the surplus Minuteman II ICBMs that have already been built. An adjustment to the cost
model to allow for this yields a cost per launch of the Minotaur I SLV to be $29.76
million. Finally, the advertised cost per launch of the Minotaur I SLV is given as $20.00
million.
52
There is currently no data in an open source format that explains how the
$20.00 million value is determined. Overall, the results obtained for the performance
characteristics and the mass properties are sufficient for this preliminary design study.
4.5 TwoStage Liquid Propellant Vehicle vs. Titan II SLV
The historical data have shown that for a liquid propellant vehicle, the inert mass
fraction is used rather than the propellant mass fraction. As stated in Humble et al.,
50
the
inert mass fraction for a multistage, liquid propellant launch vehicle typically falls in the
range of 0.04 to 0.14. Other data from Humble et al.
50
shows that the average value for
f
inert
for liquid propellant rockets is around 0.17. The same principals of mass fraction
and total impulse apply to a liquid propellant vehicle as is done for a solid propellant
70
vehicle. Enough propellant mass is needed to launch the payload into the required orbit.
The ideal rocket equation (Equation 4.4) drives this calculation.
As in the case of the solid propellant launch vehicle, the objective function for a
liquid propellant launch vehicle has been manipulated and the characteristics of the
resulting vehicle have been analyzed. Initial attempts to validate the multistage liquid
propellant vehicle model focused on a threestage liquid propellant launch vehicle.
However, finding a realworld threestage liquid propellant launch vehicle similar to the
one being analyzed in the current study was a difficult task. One real world example, the
Zenit3SL,
62
proved difficult to model due to incomplete information on the vehicle?s
launch trajectory.
Thus, the focus changed to using a twostage liquid propellant launch vehicle for
the model validation. There are numerous current, as well as previously built and flown,
twostage liquid propellant space launch vehicles on the market. One launch vehicle
proved to be the best real world example to use for model validation. The Titan II SLV
was chosen for the availability of information on the vehicle. The Titan II launch vehicle
dates back to the 1960?s when it was first used as an ICBM. It was also used by NASA
to launch manned capsules into orbit as part of the Gemini program. According to the
United States Air Force Titan II Space Launch Vehicle Profile Fact Sheet
56
?The Titan II
Space Launch Vehicle is a modified Titan II ICBM that can lift approximately 4,200
pounds into polar orbit.? The Titan II SLV has been retired in favor of the Delta IV and
Atlas V launch vehicles. The Delta IV and Atlas V were developed under the USAF?s
evolved expendable launch vehicle (EELV) program. The last Titan II SLV was
71
launched in 2004. The launch site for these liquid propellant launch vehicles has been
chosen to be Cape Canaveral AFS, FL.
Figure 45 shows a schematic comparing the launch vehicle generated by the
model and the Titan II SLV. The stage diameter is the same for both the first and second
stages. Also, the nosecone of the Titan II SLV is a more blunt shape than the nosecone
on the model. This is due to the blunted ogive model used by the mass properties model.
Figure 45. TwoStage Liquid Propellant Model vs. Titan II SLV Schematic
(Ref. 57: http://www.globalsecurity.org/space/systems/t2.htm)
The comparison between the characteristics of the twostage liquid propellant
launch vehicle model and the Titan II SLV are shown in Table 46. The bold* values
were direct inputs into the objective function based on the published characteristics of the
Titan II SLV.
72
Table 46: TwoStage Liquid Propellant Model vs. Titan II SLV Comparison
Parameter Model Titan II SLV
Payload* 7,000 lbm 7,000 lbm
Initial Vehicle
Weight
290,499 lbm 339,000 lbm
Total Vehicle Length 101.13 ft 103.00 ft
Total Vehicle f
inert
0.0819 0.0818
Final Altitude 608,431 ft 607,000 ft
Final Velocity 26,248 ft/s 25,600 ft/s
Stage 1
Burn Time* 140.00 s 140.00 s
Stage Diameter* 10.00 ft 10.00 ft
Propellants* N2O4/Hydrazine N2O4/Hydrazine
Total Stage Weight 220,836 lbm 259,850 lbm
m
prop
207,544 lbm 245,000 lbm
m
inert
13,293 lbm 14,850 lbm
f
inert
0.0602 0.0571
Stage Vacuum Thrust 480,556 lbf 488,337 lbf
Stage 2
Burn Time* 180.00 s 180.00 s
Stage Diameter* 10.00 ft 10.00 ft
Propellants* N2O4/Hydrazine N2O4/Hydrazine
Total Stage Weight 57,357 lbm 63,939 lbm
m
prop
52,629 lbm 58,640 lbm
m
inert
4,728 lbm 5,299 lbm
f
inert
0.0824 0.0829
Stage Vacuum Thrust 100,569 lbf 100,000 lbf
The validity of the liquid propellant launch vehicle model is established in the
strong comparison between the twostage liquid propellant model and the Titan II SLV.
The inert mass fractions are typical values for liquid propellant vehicles and the thrust
characteristics of each stage are almost exactly those of the Titan II SLV. As with both
solid propellant vehicle models, there are differences in the total vehicle mass of the two
liquid propellant vehicles that are being compared. The model designs the vehicle to be
about 48,500 pounds lighter than the Titan II SLV. This is seen in the difference in the
total amount of fuel and oxidizer propellant in the first stage of the two launch vehicles
73
(207,544 lbm vs. 245,000 lbm). All the other physical and performance characteristics of
the twostage liquid propellant model closely match those of the Titan II SLV.
4.6 AirLaunched, TwoStage Liquid Propellant Vehicle vs. QuickReach
TM
In order to provide additional validation of the liquid propellant launch vehicle
model, an airlaunched, twostage liquid propellant case has been analyzed. Like the
previous multistage liquid propellant vehicle, finding a realworld example of an air
launched, twostage liquid propellant vehicle was not possible. However, currently, there
is an airlaunched, twostage liquid propellant launch vehicle in development by a
company known as AirLaunch LLC. The QuickReach
TM
launch vehicle
58
is designed to
be a responsive smalllift vehicle used to launch small satellites into Low Earth Orbit
(LEO) within a 24 hour callup for a launch price of $5 million.
The QuickReach
TM
launch vehicle is airlaunched from the cargo bay of a C17
transport aircraft. Some performance characteristics of this vehicle have been published.
However, the specific mass properties of the QuickReach
TM
launch vehicle are not
available so only a limited comparison can be made. Specifically, the values for inert
mass and propellant mass would allow for a more direct comparison to the vehicle
generated using the system model.
A schematic showing a test article of the QuickReach
TM
launch vehicle being
deployed from a C17 and the launch vehicle designed by the system model are shown in
Figure 46. One important difference can be seen in the size of the first stage nozzle.
The first stage nozzle for the QuickReach
TM
launch vehicle is much larger than the one
generated in the model. Additional work on the model?s generation of the first stage
74
nozzle would need to be done once more specific information on the QuickReach
TM
launch vehicle becomes available.
Figure 46. Air Launched, TwoStage Liquid Propellant Model vs. QuickReach
TM
Launch Vehicle Schematic
(Ref. 59: http://www.designationsystems.net/dusrm/app4/quickreachslv.jpg)
Table 47 shows the characteristics of the air launched, twostage liquid
propellant launch vehicle model with the QuickReach
TM
launch vehicle. Because the
QuickReach
TM
launch vehicle is relatively new, there is not a large amount of
information on the specifics of the vehicle. However, enough data exist to make the
comparison a useful one. The validity of the liquid propellant model is also strengthened
through this comparison.
75
Table 47: Air Launched, TwoStage Liquid Propellant Model vs. QuickReach
TM
Launch
Vehicle Comparison
Parameter Model QuickReach
TM
Launch Vehicle
Payload* 1,000 lbm 1,000 lbm
Initial Altitude* 35,000 ft 35,000 ft
Initial Velocity* 760 ft/s 760 ft/s
Total Vehicle Weight 74,633 lbm 72,000 lbm
Total Vehicle Length 62.40 ft 66.00 ft
Total Vehicle f
inert
0.12 unknown
Final Altitude 738,783 ft 700,000 ft
Final Velocity 25,619 ft/s 25,532 ft/s
Stage 1
Stage Diameter* 7.00 ft 7.00 ft
Propellants* LOX/Kerosene LOX/Propane
m
prop
60,796 lbm unknown
m
inert
5,130 lbm unknown
f
inert
0.08 unknown
Stage Vacuum Thrust 171,825 lbf 172,000 lbf
Stage 2
Stage Diameter* 7.00 ft 7.00 ft
Propellants* LOX/Kerosene LOX/Propane
m
prop
3,945 lbm unknown
m
inert
1,516 lbm unknown
f
inert
0.27 unknown
Stage Vacuum Thrust 25,333 lbf 24,000 lbf
As with the previous model validation cases, the objective function has been
manipulated in order to generate a vehicle for the purposes of matching the real world
example. Table 47 shows that a good match was obtained in total vehicle weight and
performance characteristics between the model and the QuickReach
TM
launch vehicle.
The total vehicle mass of the model is slightly higher than the total vehicle mass of the
QuickReach
TM
launch vehicle (74,633 lbm vs. 72,000 lbm). However, the final altitude
and the final velocity of the model are higher but still relatively close to the
QuickReach
TM
launch vehicle. This provides confidence in the ability of the vehicle
generated by the model to reach the desired orbit. Also, the values for vacuum thrust of
76
each stage are very similar. The Stage 1 thrust for the model is slightly lower than the
QuickReach
TM
launch vehicle (171,825 lbf vs. 172,000 lbf) whereas the Stage 2 thrust is
slightly higher (25,333 lbf vs. 24,000 lbf).
The inert mass fractions for this liquid propellant launch vehicle, also shown in
Table 47, fall within the desired range of typical inert mass fraction values published in
Humble et al.
50
4.7 Mass Fractions for ThreeStage Solid/Liquid/Liquid Propellant Vehicle
In addition to traditional launch vehicle configurations, the case of a mixed
propellant system has been analyzed. This case is a threestage vehicle that employs a
solid propellant firststage along with liquid propellant second and third stages. There are
numerous real world examples of mixed propellant systems. Most use solid propellant
motors as thrust augmentation to a firststage liquid propellant rocket engine. The Space
Shuttle solid rocket boosters (SRBs) are used to provide the liftoff thrust for the shuttle
stack. The Delta II SLV uses up to nine graphite epoxy solid motors (GEMs) during
operation of the liquid propellant firststage. However, the case of a purely solid
propellant firststage along with liquid propellant second and third stages is a unique one.
A real world example currently being developed by the National Aeronautics and Space
Administration (NASA) is the Ares I Crew Launch Vehicle (CLV). NASA has proposed
a twostage; first stage solid/second stage liquid combination launch vehicle for the Ares
I CLV. The Ares I CLV would provide a good real world example for any future model
validation attempts.
Even though no real world examples exist for the threestage solid/liquid/liquid
launch vehicle, analysis of the mass fractions will give validity to the model being used.
77
The historical data of Humble et al.
50
will continue to be referenced in order to compare
the results from the model with previously designed and flown vehicles. The solid
propellant first stage will use the propellant mass fraction and the liquid propellant
second and third stages will use the inert mass fraction. The reason for using two types
of mass fractions is that the historical data used by Humble et al.
50
focuses on the
propellant mass fraction for solids and the inert mass fraction for liquids.
For this case, two pure design optimizations using the GA were performed. There
were three goals involved in each optimization. The first two were for the vehicle to
attain the desired lowEarth orbit. The third goal was to minimize the total vehicle mass.
Design optimizations have been performed for launch out of Vandenberg AFB, CA and
Cape Canaveral AFS, FL. Table 48 summarizes the vehicle mass fraction for each of
the three stages of the two vehicle models produced by the GA.
Table 48: ThreeStage Solid/Liquid/Liquid Propellant Vehicle Mass Fractions
3Stage
Solid/Liquid/Liquid
(Vandenberg Launch)
3Stage
Solid/Liquid/Liquid
(Cape Canaveral Launch)
Stage 1 (solid)
f
prop
0.8991 0.8699
Stage 2 (liquid)
f
inert
0.1274 0.0748
Stage 3 (liquid)
f
inert
0.0957 0.1128
Schematics of the resulting launch vehicles are shown in Figure 47. Both
vehicles are very similar in geometry as well as their mass fractions. While not ideal, the
mass fractions listed in Table 48 are consistent with the values published in Humble et
al.
50
The propellant mass fraction for Stage 1 of both vehicles is slightly lower than the
desired value of 0.90 for solid rocket motors. Since this value is lower than 0.90, the
78
inert mass for this stage is probably too high. This indicates some adjustment to the
modeling of the Stage 1 inert mass needs to be done in order to bring the propellant mass
fractions closer to 0.90. The inert mass fractions of the two liquid stages fall within the
expected range (0.04 to 0.14) for liquid propellant rocket engines.
Overall, the model for this threestage solid/liquid/liquid propellant launch vehicle
configuration appears to be accurate. It should be emphasized that these threestage
solid/liquid/liquid propellant launch vehicles have not been compared to any existing
experimental data. The validity of the model is limited to the comparison of the mass
fractions of the individual vehicle stages. Certainly, future real world examples would
help strengthen the current model?s validity. The use of variable diameter stages might
also provide additional insight into the modeling of this type of launch vehicle. As
previously stated, the Ares I CLV is proposed to have a solid propellant first stage and a
liquid propellant second stage along with a second stage diameter that is larger than the
first stage diameter.
79
Vandenberg Launch Cape Canaveral Launch
Figure 47. ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicles Schematic
4.8 Summary of Validation Efforts
Four successful model validations have been performed for four different types of
launch vehicles. The validity of the system models used in the objective functions has
been established through the use of comparisons with real world examples. While the
launch vehicle models do not exactly match their respective real world counterparts, the
results are encouraging. These validation efforts should instill confidence that the launch
vehicle models have produced realistic designs at this preliminary design level.
80
4.8.1 ThreeStage Solid Propellant Launch Vehicle
The threestage solid propellant launch vehicle model has been validated against
the Minuteman III ICBM. A good match between these two vehicles has been made with
the model producing a vehicle with a lower weight than the Minuteman III ICBM. This
difference is likely related to the ability of the model to reproduce individual system
components.
The performance values for the vehicle model and the Minuteman III ICBM
match very well. This can be seen in the values generated for the final altitude of the
model versus the Minuteman III ICBM (768,682 ft vs. 750,000 ft) and the final velocity
(22,071 ft/s vs. 22,000 ft/s).
4.8.2 FourStage Solid Propellant Launch Vehicle
For the validation of the fourstage solid propellant launch vehicle, the Minotaur I
SLV has been used as the real world example. Again, a good match has been obtained
between the vehicle model and the Minotaur I SLV. The model generates a vehicle
within 1,700 pounds of the total vehicle mass of the Minotaur I SLV.
In terms of performance, the first three stages of the vehicle model match the
velocity and altitude of the Minotaur I SLV. However, after Stage 3 burnout, the
Minotaur I SLV goes into a coast phase for approximately 400 seconds. Upon
completion of the coast phase, Stage 4 is fired for orbit insertion. For the vehicle model,
the same type of trajectory was modeled. However, the vehicle model produced a coast
phase of 240 seconds. A recommended improvement of the 6DOF flight dynamics
simulator would be to incorporate an orbit insertion burn for future design optimization
work. This recommendation will be discussed further in Chapter 8.
81
4.8.3 TwoStage Liquid Propellant Launch Vehicle
The Titan II SLV has been chosen to validate the twostage liquid propellant
launch vehicle model. As in the previous two cases, the vehicle model successfully
reproduced the characteristics of the Titan II SLV with some slight differences.
Comparing the performance of the model versus the Titan II SLV, the vehicle model was
very accurate in final altitude (608,431 ft vs. 607,000 ft) and final velocity (26,248 ft/s vs.
25,600 ft/s).
One important difference was the total vehicle mass with the model coming in at
about 40,000 pounds lighter than the Titan II SLV. This is seen in the difference in the
amount of propellant in the first stage (207,543 pounds vs. 245,000 pounds). Additional
work could be done to investigate the parameters (such as mass flow rate and fuelto
oxidizer ratio) used to calculate the propellant mass. These parameters could then be
compared to the same parameters of the Titan II SLV.
4.8.4 AirLaunched, TwoStage Liquid Propellant Launch Vehicle
An additional validation of the liquid propellant model has been performed using
the QuickReach
TM
launch vehicle as the real world example. The QuickReach
TM
launch
vehicle is still in its initial development stages so detailed information about the vehicle is
currently unavailable. However, enough information is available to make a comparison
with the vehicle model. A good match has been obtained between the vehicle model and
the QuickReach
TM
launch vehicle.
Vehicle performance was again the strength of the launch vehicle model versus
the QuickReach
TM
launch vehicle. The system model was able to accurately match the
final altitude (738,783 ft vs. 700,000 ft) and final velocity (25,619 ft/s vs. 25,532 ft/s) of
82
the two launch vehicles. Also, the values of vacuum thrust for each stage were similar to
the published values of the QuickReach
TM
launch vehicle (Stage 1: 171,825 lbf vs.
172,000 lbf and Stage 2: 25,333 lbf vs. 24,000 lbf). The main difference was the size of
the Stage 1 nozzle. The nozzle on the QuickReach
TM
launch vehicle is much larger than
the nozzle generated in the model. This is an area where additional work on the model
can be performed. Also, when more specific data on the QuickReach
TM
launch vehicle is
published, the system model will be improved upon.
4.8.5 ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle
A basic validation of threestage solid/liquid/liquid propellant launch vehicles has
been performed. A real world example launch vehicle does not exist for this type of
launch vehicle configuration. Two pure design optimizations have been performed for
each of the two launch vehicles based on the choice of launch site (Vandenberg AFB, CA
and Cape Canaveral AFS, FL). As a result, a comparison of the propellant and inert mass
fractions has been performed for the resulting best performers from the design
optimization processes. The mass fractions of these two launch vehicles match well the
established values for propellant and inert mass fraction for space launch vehicle systems.
83
5.0 OPTIMIZATION RESULTS
5.1 Introduction
The motivation behind the current study is to support the important national
security effort of assured access to space. The results demonstrate the feasibility of
designing launch vehicles using a genetic algorithm (GA). In addition, one overall goal
has been to investigate the possibility of improving launch vehicle design. The results
also have the added benefit of tying a cost model to the design optimization effort.
The philosophy behind choosing different cases is to provide a widerange of
space launch vehicle types for the purposes of finding an optimum design. The thirteen
cases that were accomplished encompass different types of propulsion systems as well as
launches from the two primary United States Air Force (USAF) space launch sites: Cape
Canaveral AFS, FL (CCAFS) and Vandenberg AFB, CA (VAFB).
The cases are broken down into five different sets of results. The first section
encompasses the initial launch vehicle optimizations that demonstrate the feasibility of
the process. Next, solid propellant rockets have been optimized for both three and four
stage configurations. The third section covers the optimization of two and threestage
liquid propellant rockets. Twostage, airlaunched vehicles were then analyzed using
liquid propellant systems. Finally, threestage vehicles with a solid propellant first stage
and liquid propellant second and third stages have been optimized.
84
The results show the positive and negative aspects of each type of launch vehicle
design. Cost per launch values along with performance and physical characteristics are
summarized for each vehicle that has been optimized. In certain cases, the results are
summarized to show how these vehicles have demonstrated improvement in cost and
performance over their real world example counterpart launch vehicle. Information
describing some of the important characteristics of the design optimization cases is
presented in Table 51.
Table 51: Space Launch Vehicle Design Optimization Cases
Case # 1
st
Stage 2
nd
Stage 3
rd
Stage 4
th
Stage Goal(s) Launch
Site
Initial
Vehicles
5.2.1 1 solid solid solid Get to orbit VAFB
5.2.2 2 solid solid solid Min. mass VAFB
Solids
5.3.1 3 solid solid solid Min. cost VAFB
5.3.2 4 solid solid solid Min. cost CCAFS
5.3.3 5 solid solid solid solid Min. cost VAFB
5.3.4 6 solid solid solid solid Min. cost CCAFS
Liquids
5.4.1 7 liquid liquid liquid Min. cost VAFB
5.4.2 8 liquid liquid liquid Min. cost CCAFS
5.4.3 9 liquid liquid Min. cost VAFB
5.4.4 10 liquid liquid Min. cost CCAFS
AirLaunch
5.5.1 11 airlaunch liquid liquid Min. cost CCAFS
Mixed
5.6.1 12 solid liquid liquid Min. cost VAFB
5.6.2 13 solid liquid liquid Min. cost CCAFS
5.2 Initial Launch Vehicles
The design optimization of space launch vehicles begins with two cases that
represent initial vehicles optimized using the system models along with the genetic
algorithm (GA). These initial vehicles provide insight to the optimization process and
show how well the GA works at meeting the desired goals. These cases are not meant to
be the best solutions to the launch vehicle preliminary design problem. The reason for
85
this is that the system models used for these initial vehicles have not gone through the
model validation process.
For this initial analysis, the threestage solid propellant launch vehicle has been
chosen for the design optimization process. Additionally, a typical lowEarth orbit has
been chosen for the mission with the launch site being Vandenberg AFB, CA. The
launch direction is due North in order to obtain a polar orbit. A nominal 1,000 pound
payload provides the mass to be carried into orbit by the launch vehicle. The two cases
analyzed in this section use slightly different goals for the GA to optimize.
5.2.1 Case 1: ThreeStage Solid Propellant Launch Vehicle with Two Goals
The purpose of this initial design optimization is to determine if a threestage
solid propellant rocket can be designed that will achieve a lowEarth orbit. The desired
mission statistics for this vehicle are listed in Table 52.
Table 52: Initial Launch Vehicles Mission Statistics
Payload Mass 1,000 lbm
Launch Site Vandenberg AFB, CA (34.6? N, 120.6? W)
Launch Direction Due North (0? Azimuth or i=90?/polar orbit)
Desired Orbital Velocity 24,550 ft/s
Desired Orbital Altitude 2,430,000 ft
The two goals for this optimization are:
Goal #1: minimize the difference between the desired orbital velocity and the
actual velocity of the vehicle
Goal #2: minimize the difference between the desired orbital altitude and the
actual altitude of the vehicle
These goals assume that all the vehicle propellant has been consumed upon reaching the
desired orbital parameters. Note that this first case is merely a demonstration of the
86
feasibility of finding a workable solution using the GA. The design optimization for this
threestage solid propellant rocket uses a population size of 400 members. The reason for
the large population size is due to the large number of design variables (34 for the three
stage solid propellant launch vehicle). The number of design variables, along with their
particular range of values, results is a relatively long chromosome string (over 100 bits).
As previously mentioned, Anderson
16
used Equation (3.3) to determine the population
size for his design optimizations. Thus, in order to maintain sufficient genetic diversity
and to avoid falling into a local optimum, the large population size is required. Also, the
value of 400 members was the maximum value available for use in the GA.
The design optimization was intended to run for 150 generations but the optimum
solution that met both goals was actually achieved by generation #43 (see Figures 51 and
52). Figures 51 and 52 show how the best performer improved as each generation
proceeded. In Figure 51, the velocity goal was met by generation #23, while in Figure
52, the altitude goal was met by generation #43.
The behavior of the best performer in Figures 51 and 52 is interesting in that,
during certain generations, there is a large change in performance. For example, there is
significant improvement in the best performer?s ability to meet the altitude goal from
generation #22 to generation #23. This improvement highlights the powerful GA
operators of crossover and mutation. These operators allow the GA to choose values of
the different design variables that facilitate better vehicle performance. In addition, the
complexities of launch vehicle design can also be seen in the improvement of the altitude
and velocity goals. The right change in a particular variable (like nozzle throat area) at
the right place and at the right time can greatly affect vehicle performance.
87
Figure 51. Progress of Best Performer to Meet Goal #1
Figure 52. Progress of Best Performer to Meet Goal #2
The best performing vehicle of the final generation was used to display the actual
performance of the optimum design. This vehicle would thus be the threestage solid
rocket that most closely meets the desired performance goals. The design variables (i.e.
88
the 34 GA variables) that created the best performer were run in a singlerun format to
generate data files which are summarized in the following paragraphs and figures.
Figures 53 thru 57 show the important performance parameters of this vehicle.
Looking at Figure 53, it appears that the GA selected very similar dimensions for the 1
st
and 2
nd
stages of the rocket and a relatively small sized 3
rd
stage in order to meet both
design goals. The ballistic flight trajectory of the vehicle is shown in Figure 54 where
the altitude has reached the orbital altitude at the top of the parabolic trajectory. The next
three figures display the changes in vehicle performance with time. Figure 55 shows the
Thrust vs. Time characteristics. Three distinct ?humps? in the curve show where the
thrust in each stage tails off and then the next stage abruptly starts. There is no delay
between the end of one stage and the start of another. Figure 56 (Vehicle Mass vs.
Time) shows how the mass of the vehicle continuously decreases until burnout of Stage 3
at approximately 107 seconds. Finally, the Velocity vs. Time plot in Figure 57 displays
the increase in velocity during powered flight and then the slight decrease in velocity
during ballistic flight to maximum altitude.
89
Figure 53. ThreeStage Solid Propellant Launch Vehicle Schematic
Figure 54. Altitude vs. Downrange Distance for Best Performer
90
Figure 55. Thrust vs. Time for Best Performer
Figure 56. Vehicle Mass vs. Time for Best Performer
91
Figure 57. Velocity vs. Time for Best Performer
5.2.2 Case 2: ThreeStage Solid Propellant Launch Vehicle with Three Goals
The second design optimization demonstration again uses a threestage solid
propellant launch vehicle with a third goal added. The first two goals are the same as the
goals used in Case 1. However, in addition to getting to orbit, it would be useful to
minimize the total vehicle mass at liftoff. Thus, the third goal is to minimize the system
mass while still attaining the desired orbital parameters. For these initial launch vehicles,
the cost model has not been included in the design optimization process. The goal at this
point is to focus on minimization of the total vehicle mass.
Also, with the addition of the third goal, a slightly different optimization process
has been used. Instead of trying to meet the goals individually, a single, global solution
is found. This global solution attempts to optimize the design to produce a single vehicle
that meets all three design goals. However, not all three goals are weighted equally. The
two orbital goals (velocity and altitude) have a higher priority because if these goals are
not met (i.e. the vehicle does not get to the desired orbit) then the mission is a failure.
92
Finally, in order to make the GA perform better, the goals are normalized by
dividing each of the three differences by their particular desired value. This prevents the
GA from trying to optimize one goal which could have differences of 5,000 ft/s (velocity
goal) and another goal which could have differences of 150,000 ft (altitude goal). The
GA is also configured in a nonpareto format so that one overall fitness function is used
in the optimization process.
Goal #1: minimize the difference between the desired orbital velocity and the
actual velocity of the vehicle divided by the desired orbital velocity
Goal #2: minimize the difference between the desired orbital altitude and the
actual altitude of the vehicle divided by the desired orbital altitude
Goal #3: minimize the difference between a desired minimum mass and the total
vehicle mass divided by the desired minimum mass
Some interesting results have been obtained for this case. Two different design
optimization runs were performed. The first run was configured for the objective
function to throw out any vehicles that had final velocity and final altitude values that
were below the desired orbital values. The second run relaxed this restriction and
allowed for offdesign vehicles to be considered in the optimization process. It was
theorized that the goal minimization in the GA would account for the differences and
ensure that the desired orbital values would be met. The results for these runs are
summarized in Table 53.
93
Table 53: Case 2 Runs Comparison/ThreeStage Solid Propellant Launch Vehicles
Run #1 Run #2
Actual Altitude 2,433,559 ft 2,439,733 ft
Actual Velocity 24,820 ft/s 23,944 ft/s
Total Vehicle Mass 115,068 lbm 104,553 lbm
Total Vehicle Diameter 6.59 ft 5.74 ft
Total Vehicle Length 109.37 ft 110.66 ft
These results show that the optimum vehicle for each run is very similar.
However, in order to meet the velocity requirement, a slightly larger rocket is required.
The differences in total vehicle mass and vehicle diameter show that attaining the desired
orbital velocity is driven by vehicle mass. The difference is an additional 10,000 pounds
which, as a result, will most likely add to the cost per launch of the vehicle.
Also, normalizing the goals forces the GA to choose reasonable values for the
various design parameters. For the members of each generation, the three answers
corresponding to each of the three goals are within a factor of 10 of each other due to this
normalization process. This ensures that the GA attempts to minimize all three goals in a
consistent manner. Figures 58 and 59 show the schematics for the two different
vehicles analyzed in the Case 2 design optimization runs.
94
Figure 58. ThreeStage Solid Propellant Figure 59. ThreeStage Solid Propellant
Launch Vehicle Schematic (Case 2/Run 1) Launch Vehicle Schematic (Case 2/Run2)
5.2.3 Conclusions: Initial Launch Vehicle Design Optimizations
Initial design optimizations for threestage solid propellant launch vehicles have
been performed. These optimizations served two purposes. The first was to demonstrate
that the objective function and the GA together could be used to design a launch vehicle
that achieves the desired lowEarth orbit. This was done successfully with the first case
where the best performer met both design goals in a relatively small number of
generations.
The second purpose was to still meet the desired orbital altitude and orbital
velocity goals but also to add an additional goal of minimizing the total vehicle mass.
95
Two different design optimization runs were accomplished for this second case. The
first run produced an optimized vehicle that met the desired altitude and velocity goals
well and reduced the total vehicle mass from the previous case (115,068 lbm vs. 155,853
lbm). The second run did not quite meet the desired altitude and velocity goals as well as
the first run but lowered the total vehicle mass to 104,553 pounds. Thus, it has been
shown that the trend toward reducing the total vehicle mass and still meeting the
performance requirements can be accomplished.
For comparison purposes, the threestage Minuteman III ICBM has a total vehicle
mass of 79,432 pounds and the fourstage solid propellant Minotaur I SLV has a total
vehicle mass of 79,800 pounds. The threestage solid propellant launch vehicles
analyzed here have significantly higher values of total vehicle mass. The reason for this
is that these launch vehicles are preliminary design optimization runs. The vehicle model
used for these runs has not been through the model validation process.
Thus, the next step will be to implement the previously discussed model
validation work and use the objective function and GA to investigate the improvement of
the real world launch vehicle examples.
5.3 Solid Propellant Vehicles
The results for the design optimization of three and fourstage solid propellant
launch vehicles are presented in this section. The four cases investigated for these solid
propellant launch vehicles are an extension of the analysis performed during model
validation. The models developed during the validation efforts have been used in the
design optimization process with the goal of improving on the real world examples.
96
Table 54 shows the mission statistics for the four different solid propellant
launch vehicles that have been optimized. The payloads (1,000 lbm) for both threestage
solid vehicles are different than the payload used for the model validation (2,540 lbm).
The reason for this is that the real world example for the threestage solid propellant
vehicles is the Minuteman III ICBM. The payload of the Minuteman III ICBM is the
reentry system which weighs 2,540 pounds. However, the Minuteman III ICBM is not an
orbital vehicle. Thus, a more typical payload (1,000 lbm) for the threestage orbital
vehicles has been chosen. The payload for the fourstage solid/VAFB matches the
typical payload of the Minotaur I SLV (738 lbm). In an attempt to expand the
performance of the fourstage solid/CCAFS vehicle, the payload mass was increased to
1,000 pounds.
The lowEarth orbit parameters for all four cases are essentially the same except
for the fourstage solid/VAFB. The desired orbital velocity for this case (25,004 ft/s) is
slightly higher than the desired orbital velocities of the other three cases (24,550 ft/s).
The reason for this is that certain launches out of VAFB are slightly retrograde in order to
attain a sunsynchronous orbit. This orbit requires the vehicle to attain a higher velocity
in order to overcome the eastward rotation of the Earth. Thus, the velocity goal of the
fourstage solid/VAFB is to match the 25,004 feet per second value.
97
Table 54: Solid Propellant Launch Vehicles Mission Statistics
ThreeStage
Solid
ThreeStage
Solid
FourStage
Solid
FourStage
Solid
Minotaur I
SLV
Payload 1,000 lbm 1,000 lbm 738 lbm 1,000 lbm 738 lbm
Launch
Site
VAFB CCAFS VAFB CCAFS VAFB
Launch
Direction
Due North
0? Azimuth
Due East
90? Azimuth
Due North
0? Azimuth
Due East
90?
Azimuth
Slightly
Northwest
Orbit
Type
i=90?
polar orbit
i=28.4?
prograde orbit
i=90?
polar orbit
i=28.4?
prograde
orbit
i=97.5?
sunsync
orbit
Desired
Orbital
Altitude
2,430,000 ft 2,430,000 ft 2,430,000 ft 2,430,000 ft 2,430,000 ft
Desired
Orbital
Velocity
24,550 ft/s 24,550 ft/s 25,004 ft/s 24,550 ft/s 25,004 ft/s
Upon completion of the analysis of the initial launch vehicles, the next step in the
design optimization of solid propellant launch vehicles is to incorporate the cost model
described in Chapter 3, the TRANSCost 7.1 cost model, into the design optimization
process. This is essentially a minor modification to the sequence of operations in the
objective function. The information from the mass properties model is used in the
TRANSCost 7.1 cost model and the cost per launch of the vehicle is determined. Also,
Goal #3 is changed to a straight minimization of the system mass rather than a
minimization of the differences between two values. Since the TRANSCost 7.1 cost
model is massbased, the decision has been made for Goal #3 to minimize the total
vehicle mass rather than the cost per launch value. A minimized total vehicle mass
should produce the minimum cost per launch vehicle. The three goals are listed below.
Goal #1: minimize the difference between the desired orbital velocity and the
actual velocity of the vehicle divided by the desired orbital velocity
98
Goal #2: minimize the difference between the desired orbital altitude and the
actual altitude of the vehicle divided by the desired orbital altitude
Goal #3: minimize the total vehicle system mass divided by a desired total
vehicle system mass
As in the previous design optimizations, the velocity and altitude goals are
weighted higher than the total vehicle mass goal. This is to ensure that, at a minimum,
the vehicle reaches the desired orbital values.
5.3.1 Case 3: ThreeStage Solid Propellant Launch Vehicle (VAFB)
A successful design optimization of a threestage solid propellant launch vehicle
has been performed. Figure 510 shows a schematic of the best performer resulting from
the design optimization of the threestage solid propellant vehicle. The launch vehicle
was designed to carry a 1,000 pound payload and the launch site was chosen to be
Vandenberg AFB, CA. Additional mission statistics for this launch vehicle are described
in Table 54.
99
Figure 510. ThreeStage Solid Propellant Launch Vehicle Schematic (VAFB)
The important characteristics of the optimized threestage solid propellant vehicle
are shown in Table 55. The final altitude of the vehicle is within 10,000 feet of the
desired orbital altitude. The final velocity is slightly above the desired orbital velocity
which ensures the payload reaches orbit. Thus, the optimized vehicle meets the required
orbital parameters thus reaching orbit and ensuring mission success.
The GA chose different propellants for each of the three solid propellant stages.
The first stage solid propellant is polybutadieneacrylic acid (PBAA)/ammonium
perchlorate (AP)/aluminum (Al). Sutton
60
describes this type of propellant as a
composite propellant that forms a heterogeneous grain. The AP is used as the crystalline
100
oxidizer and the PBAA is the fuel. In addition, solid aluminum (Al) powder is added to
enhance combustion. The AP and Al are held together in the matrix with the PBAA as
the binder. The PBAA/AP/Al propellant is one of the more common solid propellants
because of its relatively high I
sp
(260s265s) and moderate density (0.064 lbm/in
3
). The
second stage also uses a composite propellant with the same oxidizer and fuel as the first
stage but with a different binder. The GA chooses the polysulfide/ammonium
perchlorate/aluminum (PS/AP/Al) propellant for this stage. Finally, an energetic
propellant that is based on the solid propellant used in the Star 37 solid rocket motor has
been developed. This motor has a high I
sp
(292.6s) for a solid propellant which makes it
an attractive choice for performance reasons. Knowing that the Star 37 solid propellant is
based on the hydroxylterminated polybutadiene (HTPB)/AP/Al propellant, the necessary
data was determined and loaded into the appropriate solid propellant model. The GA
chose this propellant for the third stage of the optimized vehicle.
From a mass fraction perspective, the entire vehicle as well as the individual
stages, all closely match the desired value of propellant mass fraction (0.90) given by
Humble et al.
50
The propellant mass fraction (f
prop
) for the entire vehicle is 0.9072 which
is as expected. For each of the three individual stages (Stage 1: 0.9139, Stage 2: 0.8882,
Stage 3: 0.9241), reasonable values for the propellant mass fraction have been produced.
The first stage thrust provides the initial high thrust (331,972 lbf) required to lift
the vehicle off the ground with a thrusttoweight ratio of over 3to1. The Stage 2 and
Stage 3 thrust values drop off but still provide the necessary force to get the payload into
the desired lowEarth orbit.
101
Finally, the cost per launch of $49.71 million in 2003 dollars is a moderate price
for an expendable launch vehicle. It is not exceptionally cheap but it also is not
prohibitively expensive. The total vehicle mass of 89,906 pounds is the reason for this
particular price. This value of the total vehicle mass is an improvement over the vehicles
optimized in Cases 1 and 2. However, this value is over 10,000 pounds heavier than the
Minuteman III ICBM (79,432 lbm) and the Minotaur I SLV (79,800 lbm). The
Minuteman III ICBM is not an orbital vehicle. This would suggest that an additional
10,000 pounds is required to enable the launch vehicle to go from suborbital speed
(22,000 ft/s) to orbital velocity (24,550 ft/s). Additionally, the actual vehicle flight
trajectory might also play a factor in attaining the desired orbital velocity. The sequence
of powered flight/coast phase/orbit insertion burn has been used for existing launch
vehicles to provide the final velocity boost into orbit.
102
Table 55: Summary of ThreeStage Solid Propellant Launch Vehicle Characteristics
(VAFB)
Entire Vehicle Stage 1
Final Altitude 2,439,276 ft Stage Length 33.05 ft
Final Velocity 24,595 ft/s Stage Diameter 5.98 ft
Total Vehicle Mass 89,906 lbm Stage Weight 53,550 lbm
Total Vehicle Length 77.45 ft Initial Thrust 331,972 lbf
Total Vehicle f
prop
0.9072 Propellants PBAA/AP/Al
Nosecone Length 6.61 ft m
prop
48,939 lbm
Cost per Launch $49.71 million m
inert
4,611 lbm
f
prop
0.9139
Stage 2
Stage Length 19.28 ft
Stage Diameter 5.32 ft
Stage Weight 21,351 lbm
Initial Thrust 82,093 lbf
Propellants PS/AP/Al
m
prop
18,965 lbm
inert
2,386 lbm
f
prop
0.8882
Stage 3
Stage Length 16.50 ft
Stage Diameter 4.38 ft
Stage Weight 13,801 lbm
Initial Thrust 67,192 lbf
Propellants Energetic Star 37
m
prop
12,754 lbm
inert
1,047 lbm
f
prop
0.9241
The progress of the design optimization of this threestage solid propellant launch
vehicle is shown in Figures 511 thru 513. These three figures show how the GA
progressed from one generation to the next in order to meet each of the three design
optimization goals. The reason for the jagged nature of the plots is that only a small
sampling of the over 200 generations was used to generate the plots. Analyzing each
generation would have been a work intensive effort that would not have yielded
significantly different results. The overall trend of each plot is the important information
103
that should be taken away. Figure 511 shows how the initial generations produced a
launch vehicle with a final velocity of around 22,000 feet per second. As the
optimization progressed, the GA was able to create vehicles that were much closer to the
desired orbital velocity of 24,550 feet per second. Finally, the best performer of
generation #221 produced a launch vehicle with a final velocity of 24,595 feet per
second.
Figure 511. Velocity vs. Generation # for ThreeStage Solid Propellant Launch Vehicle
The trend for the final altitude, shown in Figure 512, was the opposite of the
trend for the final velocity. Initially, the GA designed launch vehicles that had a final
altitude of over 4,000,000 feet. Again, as the optimization progressed, the GA was able
to meet the final velocity goal while, at the same time, reducing the final altitude to the
desired value of 2,430,000 feet. The best performer for this optimization produced a final
altitude of 2,439,276 feet which is a good match for the desired orbital altitude.
104
Figure 512. Altitude vs. Generation # for ThreeStage Solid Propellant Launch Vehicle
For the total vehicle mass, the GA was simply trying to minimize this value while
still meeting the desired orbital parameters. Thus, the trend, as shown in Figure 513,
should be for decreasing values of total vehicle mass. The optimization starts with a total
vehicle mass of over 120,000 pounds and by generation #221, the mass has been reduced
to 89,906 pounds. Overall, the design optimization is a success due to the GA?s ability to
match the desired orbital goals while minimizing the total vehicle mass.
105
Figure 513. Total Vehicle Mass vs. Generation # for ThreeStage Solid Propellant
Launch Vehicle
The design optimization for this threestage solid propellant launch vehicle ran for
221 generations. This was sufficient to produce a vehicle that met the desired orbital
parameters and minimized the total vehicle mass. A closer look at a few of the members
of this generation provides additional insight into the results generated by the GA.
Figures 514 thru 516 show the values of final velocity, final altitude and total vehicle
mass for three members from generation #221. The ?best? member represents the best
performer of the design optimization process. The results for this best performer are
summarized in Table 55. The member that made it through the objective function and
was ranked lowest within the population was chosen as the ?worst? member. Finally, the
average member was a randomly chosen member with results that were between the best
and worst members.
106
The results for the best member have been previously discussed. Both the final
velocity and the final altitude of the average member are fairly close to the desired orbital
parameters. The reason for this is because the goals for the orbital parameters are
weighted more heavily than the total vehicle mass goal. However, the total vehicle mass
of the average member is well above the total vehicle mass of the best member. This
justifies why the average member is not ranked higher within the population. Finally, all
the results of the worst member differ considerably from the results of the best member.
While the design parameters of the worst member produce a workable launch vehicle, the
design does not meet the mission requirements and, as a result, would result in mission
failure.
Final Velocities for Generation #221
19000
20000
21000
22000
23000
24000
25000
Best Average Worst
F
i
n
a
l
V
e
l
o
c
i
ty
(
ft/
s)
Figure 514. Final Velocity for Best, Worst, and Average Members of Generation #221
107
Final Altitudes for Generation #221
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
4000000
4500000
5000000
Best Average Worst
Fi
na
l
A
l
t
i
t
ude
(
f
t
)
Figure 515. Final Altitude for Best, Worst, and Average Members of Generation #221
Total Vehicle Masses for Generation #221
84000
86000
88000
90000
92000
94000
96000
98000
100000
102000
104000
Best Average Worst
T
o
t
al
V
eh
i
cl
e M
ass (
l
b
m
)
Figure 516. Total Vehicle Mass for Best, Worst, and Average Members of Generation
#221
108
5.3.2 Case 4: ThreeStage Solid Propellant Launch Vehicle (CCAFS)
The same approach for the Vandenberg launch design optimization is performed
for the Cape Canaveral launch. The main difference is the direction of launch and the
subsequent benefit received due to the Earth?s eastward rotation. Up until this point in
the design optimizations, the vehicles have been analyzed using Vandenberg AFB, CA as
the launch site. Typical space launches out of Vandenberg follow a polar trajectory
either due North/South or slightly retrograde. For these launches, the eastward rotation
of the Earth provides no initial velocity benefit. In fact, for the slightly retrograde
launches, the Earth?s rotation is a hindrance.
Eastward launches out of Cape Canaveral AFS, FL are popular for the initial
velocity boost imparted to a launch vehicle simply because the Earth is rotating in the
direction of launch. Sellers
61
writes this velocity boost as given by Equation 5.1:
()
oEarth
L
s
km
V cos4651.0
?
?
?
?
?
?
= (5.1)
where L
o
is the launch site latitude. From this equation, it should be noted that the
greatest velocity boost comes from an eastward launch at the equator where L
o
= 0?.
For the design optimization of space launch vehicles being launched out of Cape
Canaveral AFS, FL, Equation 5.1 has been incorporated into the sixdegreeoffreedom
(6DOF) flight dynamics simulator model. In the end, this velocity boost should result in
some improvement in total vehicle mass over a similar launch vehicle performing the
same mission out of Vandenberg AFB, CA.
109
Figure 517 shows a schematic of the best performer of the design optimization of
the threestage solid propellant launch vehicle for a due East launch out of Cape
Canaveral AFS, FL.
Figure 517. ThreeStage Solid Propellant Launch Vehicle Schematic (CCAFS)
The results of the best performer for the design optimization process involving
launch out of Cape Canaveral AFS, FL are shown in Table 56. There is some cost
($48.48 million vs. $49.71 million) and mass (89,884 lbm vs. 89,906 lbm) savings for
this case as compared to the previous case but overall the two vehicles are very similar.
The main difference is that the GA designed this vehicle to be about seven feet shorter
than the Vandenberg vehicle.
110
The GA also chose different propellants for Stage 1 and Stage 2 of the Cape
Canaveral vehicle. For Stage 1, the polybutadieneacrylic acidacrylonitrile terpolymer
(PBAN)/ammonium perchlorate (AP)/aluminum (Al) composite propellant was chosen.
This propellant has very similar characteristics to the PBAA/AP/Al propellant
combination. The hydroxylterminated polybutadiene (HTPB)/AP/Al propellant was
chosen for the second stage and according to Sutton,
60
this propellant has better
performance characteristics than the PBAN binder. The energetic Star 37 propellant was
chosen again for Stage 3. The conclusion here is that the GA is looking for a propellant
with high I
sp
characteristics in order to get into orbit. The higher I
sp
of the energetic Star
37 is the best choice for this application among all the available solid propellants.
Like the Vandenberg vehicle, this optimized vehicle has excellent values of
propellant mass fraction. The overall vehicle propellant mass fraction of 0.9111 makes it
a very realistic system for Earthtoorbit missions. All three individual stages also have
very good values of propellant mass fraction (Stage 1: 0.9190, Stage 2: 0.9027, Stage 3:
0.9056).
The cost per launch and total vehicle mass savings are not as dramatic as one
would have liked. This is likely because the eastward launch out of Cape Canaveral does
not provide as significant a boost that could result in significant mass savings.
111
Table 56: Summary of ThreeStage Solid Propellant Launch Vehicle Characteristics
(CCAFS)
Entire Vehicle Stage 1
Final Altitude 2,448,695 ft Stage Length 30.64 ft
Final Velocity 24,575 ft/s Stage Diameter 6.25 ft
Total Vehicle Mass 89,884 lbm Stage Weight 55,926 lbm
Total Vehicle Length 69.86 ft Initial Thrust 417,563 lbf
Total Vehicle f
prop
0.9111 Propellants PBAN/AP/Al
Nosecone Length 6.36 ft m
prop
51,397 lbm
Cost per Launch $48.48 million m
inert
4,530 lbm
f
prop
0.9190
Stage 2
Stage Length 21.99 ft
Stage Diameter 5.30 ft
Stage Weight 25,090 lbm
Initial Thrust 112,301 lbf
Propellants HTPB/AP/Al
m
prop
22,650 lbm
inert
2,441 lbm
f
prop
0.9027
Stage 3
Stage Length 11.11 ft
Stage Diameter 4.56 ft
Stage Weight 7,658 lbm
Initial Thrust 26,165 lbf
Propellants Energetic Star 37
m
prop
6,935 lbm
inert
723 lbm
f
prop
0.9056
Finally, a comparison has been made for the threestage solid propellant launch
vehicles generated in Cases 1 thru 4 of this study. The improvements in total vehicle
mass for these different cases are shown in Figure 518. This figure shows that when
choosing to launch a 1,000 pound payload into a lowEarth orbit, the preliminary design
of a possible launch vehicle can and should go through a number of iterations to find the
optimum solution. The total vehicle mass improves from Case 1 to Case 4 as a stepby
step process is employed to work towards the best performer. For Case 1, the goal was
112
simply to reach the desired orbit. The two runs for Case 2 added the goal of minimizing
the total vehicle mass while putting two constraints on the GA. The first run ignored
vehicles with velocity and altitude performance that were not above the desired orbital
values, The second run relaxed this restriction and allowed the GA to keep offdesign
members even though they didn?t meet the desired parameters Finally, Cases 3 and 4
used the models from the validation efforts to optimize the threestage solid propellant
launch vehicle. The total vehicle mass of the Minuteman III ICBM is included for the
sake of comparison. With a multivariable design problem like a space launch vehicle,
the odds of finding the single ?best? solution of all the possible solutions are low. Thus,
the focus here has been on improving the launch vehicle designs rather than on finding a
single optimum solution.
Total Vehicle Mass for ThreeStage Solid
Propellant Cases
10000
30000
50000
70000
90000
110000
130000
150000
170000
C
a
s
e 1/
G
e
t
t
o
o
rb
i
t
C
a
se
2
/
Ig
n
o
re
v
e
h
i
cl
e
s
C
a
s
e 2/
A
l
l
o
w
o
f
f
d
e
s
i
g
n
Ca
s
e
3
/V
A
F
B
mo
d
e
l
Ca
s
e
4
/
C
CA
F
S
mo
d
e
l
MM
II
I
I
CB
M
ThreeStage Solid Propellant Cases
T
o
t
al
V
eh
i
cl
e M
ass (
l
b
m
)
Figure 518. Mass Improvements for ThreeStage Solid Propellant Vehicles
113
5.3.3 Case 5: FourStage Solid Propellant Launch Vehicle (VAFB)
The design optimization for the fourstage solid propellant launch vehicles is
quite similar to the threestage solid propellant vehicles (Cases 1 thru 4). The launch
sites and desired orbital velocity and orbital altitude values are described in Table 54.
However, additional changes have been made in an attempt to mirror an existing, real
world launch vehicle. The Minotaur I Space Launch Vehicle (SLV) is a fourstage solid
propellant rocket operated by Orbital Sciences Corporation. The first two stages of the
Minotaur I SLV consist of decommissioned Minuteman II ICBMs. The third and fourth
stages employ the second and third stages of Orbital Sciences Corporation?s Pegasus
launch vehicle.
The fourstage solid propellant launch vehicle design optimization attempts to
either match or improve upon the Minotaur I SLV design. A few modifications to the
design parameters are incorporated to allow for the possibility of a launch vehicle to be
generated by the GA that is similar to the Minotaur I SLV. In no way do these
modifications ensure that the Minotaur I SLV design is chosen. It is left up to the GA
and the objective function to choose the specific design parameters. Also, the fourstage
solid propellant launch vehicle has some characteristics not found in the threestage solid
launch vehicle. These differences are listed below:
 the payload mass is changed to 738 lbm for Vandenberg launch
 use of coast period after Stage 3 burnout
 Stage 4 is located inside the nosecone fairing
Figure 519 shows a schematic of the fourstage solid propellant launch vehicles
analyzed in the current study. The launch site of all three vehicles was chosen to be
114
Vandenberg AFB, CA. The left image is the Minotaur I SLV in flight. The middle
schematic is the vehicle produced in the model validation process. The optimized vehicle
is shown on the right. A comparison summary of these fourstage solid propellant launch
vehicles is shown in Table 57. The results of the design optimization of this vehicle are
summarized in Table 58. These results show significant improvement over the Minotaur
I SLV.
Figure 519. FourStage Solid Propellant Launch Vehicles Schematic (VAFB)
(Ref. 55: http://www.orbital.com/SpaceLaunch/Minotaur/index.html)
115
Table 57: FourStage Solid Propellant Launch Vehicles Comparison
Minotaur I
SLV
Validation
Model
Optimized
Vehicle
Payload 738 lbm 738 lbm 738 lbm
Total Vehicle Weight 79,800 lbm 78,090 lbm 60,690 lbm
Total Vehicle Length 63.02 ft 64.58 ft 69.46 ft
Total Vehicle f
prop
0.8998 0.9185 0.8976
Final Altitude 2,430,000 ft 2,425,999 ft 2,430,505 ft
Final Velocity 25,004 ft/s 25,002 ft/s 25,036 ft/s
Cost per Launch $52.05 million $51.95 million $46.07 million
Advertised Cost per Launch $20.00 million
The best performer from the design optimization of the fourstage solid propellant
launch vehicle weighs 19,000 pounds less than the Minotaur I SLV. An important note
concerning the mass properties model should be mentioned again. As was discussed in
the model validation section, the mass properties model underestimates the inert mass
values. Thus, the reduction in total vehicle mass is most likely not as high as described
here. Future work on the mass properties model will address this issue. With that in
mind, the design optimization of the fourstage solid propellant launch vehicle has still
resulted in a fairly substantial mass savings.
The mass savings comes primarily from the difference in the mass of the Stage 1
propellants. The Stage 1 propellant mass in the optimized vehicle is 30,017 pounds
versus 45,371 pounds for the Minotaur I SLV. This reduction in propellant mass was
probably also aided by the choice of propellants for the individual stages. The GA chose
fairly energetic composite and doublebased propellants that posses a higher I
sp
than
some of the more common solid propellants (260s ? 265s). The Stage 1 and Stage 2
propellants were chosen to be HTPB/AP/Al which has an I
sp
, according to Sutton
60
, of
around 267s. The energetic Star 37 propellant was chosen for Stage 3. This propellant is
116
an HTPB/AP/Al derivative with an I
sp
of 292.6s. Finally, an energetic doublebased
propellant, DB/APHMX/Al, was chosen for Stage 4 of the vehicle. From Sutton,
60
this
propellant uses HMX mixed into the propellant thus reducing the amount of AP. The
HMX is a crystalline nitramine or explosive that provides higher performance for this
type of solid propellant (I
sp
=275s). The conclusion regarding these propellants is that the
higher I
sp
provides higher performance for the vehicle and thus reduces the amount of
propellant required to achieve orbit.
The optimized vehicle also provides about a $6 million savings in cost per launch
over the Minotaur I SLV. Using the TRANSCost 7.1 cost model for the mass values of
the Minotaur I SLV resulted in a cost per launch of $52.05 million. The optimized four
stage solid propellant launch vehicle yielded a cost per launch of $46.07 million. It
should be noted that the advertised cost per launch of the Minotaur I SLV is $20 million.
This is attributed to the use of decommissioned Minuteman II ICBMs for the first two
stages of the Minotaur I SLV.
The performance characteristics for the optimized vehicle also closely match the
desired orbital altitude and orbital velocity parameters. The total vehicle propellant mass
fraction of 0.8976 is right in line for the mass fraction of a solid propellant launch
vehicle. The propellant mass fractions of the individual stages also produce excellent
results (Stage 1: 0.9137, Stage 2: 0.9000, Stage 3: 0.8250, Stage 4: 0.8448).
There is a difference in the coast times for the orbit insertion burn of the two
vehicles. After Stage 3 burnout, the Minotaur I SLV uses a coast time of 400s before
Stage 4 ignites to put the payload into its final orbit. The GA chose a coast time of 55s
between Stage 3 burnout and Stage 4 ignition. The issue of coast time will be addressed
117
in future work associated with the recommended improvements of the 6DOF flight
dynamics simulator.
Table 58: Summary of FourStage Solid Propellant Launch Vehicle Characteristics
(VAFB)
Entire Vehicle Stage 1
Final Altitude 2,430,505 ft Stage Length 24.10 ft
Final Velocity 25,036 ft/s Stage Diameter 5.04 ft
Total Vehicle Mass 60,690 lbm Stage Weight 32,852 lbm
Total Vehicle Length 69.46 ft Initial Thrust 239,820 lbf
Total Vehicle f
prop
0.8976 Propellants HTPB/AP/Al
Nosecone Length 6.54 ft m
prop
30,017 lbm
Cost per Launch $46.07 million m
inert
2,835 lbm
f
prop
0.9137
Stage 2
Stage Length 20.21 ft
Stage Diameter 4.58 ft
Stage Weight 21,609 lbm
Initial Thrust 98,829 lbf
Propellants HTPB/AP/Al
m
prop
19,448 lbm
inert
2,161 lbm
f
prop
0.9000
Stage 3
Stage Length 11.84 ft
Stage Diameter 3.87 ft
Stage Weight 3,743 lbm
Initial Thrust 35,065 lbf
Propellants Energetic Star 37
m
prop
3,088 lbm
inert
655 lbm
f
prop
0.8250
Stage 4
Stage Length 4.76 ft
Stage Diameter 3.21 ft
Stage Weight 1,490 lbm
Initial Thrust 18,459 lbf
Propellants DB/APHMX/Al
m
prop
1,259 lbm
inert
231 lbm
f
prop
0.8448
118
5.3.4 Case 6: FourStage Solid Propellant Launch Vehicle (CCAFS)
The approach for the design optimization of the fourstage solid propellant launch
vehicle with launch out of Cape Canaveral AFS, FL was slightly different than the
Vandenberg launch. Since the eastward launch out of Cape Canaveral provides a slight
velocity boost, the payload for this case has been increased to 1,000 pounds. The vehicle
should have a higher payload carrying capacity due to the Earth?s eastward rotation.
Also, since the target orbit is a typical lowEarth orbit, the desired orbital velocity
(24,550 ft/s) corresponds to the desired orbital altitude (2,430,000 ft). A schematic
showing the optimized vehicle for this case is shown in Figure 520. The results of the
design optimization of this vehicle are summarized in Table 59.
119
Figure 520. FourStage Solid Propellant Launch Vehicle Schematic (CCAFS)
Another successful design optimization has been obtained for the fourstage solid
propellant launch vehicle. The total vehicle mass (72,080 lbm) comes in at 7,700 pounds
less than the Minotaur I SLV (79,800 lbm). The same issue with the mass properties
model is applicable here as well. Thus, even with an increase in payload mass, the design
optimization still improves on the total vehicle mass characteristics of the Minotaur I
SLV. As in the previous case, the GA chose more energetic propellants for the Stage 3
and Stage 4. The DB/APHMX/Al propellant was chosen for Stage 3 and the
HTPB/AP/Al was chosen for Stage 4.
120
The performance characteristics of the optimized vehicle essentially match
exactly the desired orbital altitude and the desired orbital velocity. As in the previous
case, the propellant mass fractions are all appropriate values for a solid propellant launch
vehicle. The propellant mass fraction of the total vehicle is 0.8894 which is only slightly
lower than the ideal 0.90 value.
The cost per launch ($48.98 million) for the optimized vehicle is comparable for a
typical expendable launch vehicle. As with the total vehicle mass, the cost per launch for
the optimized vehicle is lower than the cost per launch of the Minotaur I SLV ($52.05
million). However, the cost per launch of the optimized vehicle does not approach the
advertised cost per launch of the Minotaur I SLV which is $20 million per launch.
The GA chose a very different coast time for this vehicle than the coast time of
the Vandenberg launch vehicle. The coast time for the Cape Canaveral optimized vehicle
is 301s which is much closer to the 400s of the Minotaur I SLV. This would seem to
indicate that the use of Stage 4 for the orbit insertion burn requires additional analysis.
Specifically, the operation of the 6DOF flight dynamics simulator and how it takes the
coast time into account needs to be further investigated. The results of the design
optimization for this vehicle provide an attractive fourstage solid propellant launch
vehicle that meets the mission requirements.
121
Table 59: Summary of FourStage Solid Propellant Launch Vehicle Characteristics
(CCAFS)
Entire Vehicle Stage 1
Final Altitude 2,432,143 ft Stage Length 25.68 ft
Final Velocity 24,592 ft/s Stage Diameter 5.33 ft
Total Vehicle Mass 72,0780 lbm Stage Weight 38,958 lbm
Total Vehicle Length 71.83 ft Initial Thrust 358,962 lbf
Total Vehicle f
prop
0.8894 Propellants PBAA/AP/Al
Nosecone Length 7.27 ft m
prop
34,500 lbm
Cost per Launch $48.48 million m
inert
4,457 lbm
f
prop
0.8856
Stage 2
Stage Length 20.13 ft
Stage Diameter 4.98 ft
Stage Weight 26,422 lbm
Initial Thrust 176,198 lbf
Propellants PBAN/AP/Al
m
prop
24,130 lbm
inert
2,292 lbm
f
prop
0.9133
Stage 3
Stage Length 12.29 ft
Stage Diameter 3.44 ft
Stage Weight 4,174 lbm
Initial Thrust 129,442 lbf
Propellants DB/APHMX/Al
m
prop
3,515 lbm
inert
659 lbm
f
prop
0.8420
Stage 4
Stage Length 4.46 ft
Stage Diameter 2.91 ft
Stage Weight 1,279 lbm
Initial Thrust 28,477 lbf
Propellants HTPB/AP/Al
m
prop
1,077 pbm
inert
202.50 lbm
f
prop
0.8417
122
5.3.5 Conclusions: Solid Propellant Launch Vehicle Design Optimizations
Four successful design optimizations have been performed for three and four
stage solid propellant launch vehicles. All four vehicles meet the mission requirements
and the optimization process generates improvements in total vehicle mass and cost per
launch. In addition, the propellant mass fractions are consistent with the historical data
for solid propellant vehicles. This fact further strengthens the argument that the
preliminary designs are realistic and could be developed into real world systems.
The use of energetic solid propellants seems to be the key to meeting the
performance parameters while at the same time minimizing total vehicle mass.
Throughout the design optimizations, the GA consistently chose the more energetic
propellants rather than propellants with lower density but less desirable I
sp
.
5.4 Liquid Propellant Vehicles
The results for the design optimization of two and threestage liquid propellant
launch vehicles have also been investigated. Like the solid propellant cases, the liquid
propellant launch vehicles have been analyzed for launch out of Vandenberg AFB, CA
and Cape Canaveral AFS, FL. Model validation was performed for the twostage liquid
propellant case, so the design optimization of these vehicles was an attempt to improve
on the real world example. General comments were made on the results for the three
stage liquid propellant cases and how those results compared to real world examples.
However, the model validation applies to both types of liquid propellant launch vehicles.
The mission statistics for the liquid propellant launch vehicles that have been
optimized are shown in Table 510. The payload, desired orbital altitude and desired
orbital velocity for the threestage vehicles were chosen as typical values for responsive
123
spacetype missions. Also, these values allow the threestage vehicles to be compared to
other types of optimized launch vehicles currently being investigated in this study. For
the twostage liquid propellant cases, initially, the values of payload, desired orbital
altitude and desired orbital velocity were chosen to mirror those same values used in the
model validation. The twostage liquid propellant launch vehicle model was validated
using the Titan II SLV. However, during the design optimization process, it became
necessary to adjust some of these values. Three different design optimization runs have
been performed for both of the twostage liquid propellant launch vehicle cases. The
desired orbital altitude and the desired orbital velocity were the same values as those used
to validate the Titan II SLV. However, the issue of attaining the desired orbit forced the
payload mass to be changed from 7,000 pounds to 1,000 pounds.
The three goals for the design optimization of these vehicles were the same as
those for the solid propellant launch vehicle cases. The first two goals were to minimize
the differences between the desired orbital parameters and the actual vehicle performance
parameters. The third goal was a direct minimization of the total vehicle mass. Using the
TRANSCost 7.1 cost model allowed for the minimization of the cost per launch since the
total vehicle mass has been minimized. However, the results for these liquid propellant
cases show that the decision to minimize the total vehicle mass rather than the cost per
launch itself may have been an incorrect one.
124
Table 510: Liquid Propellant Launch Vehicles Mission Statistics
ThreeStage
Liquid
ThreeStage
Liquid
TwoStage
Liquid
TwoStage
Liquid
Payload 1,000 lbm 1,000 lbm 1,000 lbm 1,000 lbm
Launch Site VAFB CCAFS VAFB CCAFS
Launch
Direction
Due North
0? Azimuth
Due East
90? Azimuth
Due North
0? Azimuth
Due East
90? Azimuth
Orbit Type i=90?
polar orbit
i=28.4?
prograde orbit
i=90?
polar orbit
i=28.4?
prograde orbit
Desired
Orbital
Altitude
2,430,000 ft 2,430,000 ft 656,000 ft 656,000 ft
Desired
Orbital
Velocity
24,550 ft/s 24,550 ft/s 25,532 ft/s 25,532 ft/s
5.4.1 Case 7: ThreeStage Liquid Propellant Launch Vehicle (VAFB)
The results for the design optimization of the threestage liquid propellant launch
vehicle are presented here. Figure 521 shows a schematic of the threestage liquid
propellant vehicle with Vandenberg AFB, CA as the launch site. The mission statistics
are described in Table 510. From Figure 521, it can be seen that Stage 1 is the largest
of the three stages while the other two stages are significantly smaller. The fin geometry
has been predetermined for aerodynamic stability purposes.
125
Figure 521. ThreeStage Liquid Propellant Launch Vehicle Schematic (VAFB)
The results of the design optimization for the threestage liquid propellant rocket
are shown in Table 511. In terms of total vehicle mass and cost per launch, this rocket is
much heavier and more expensive than the threestage solid propellant vehicle launched
out of Vandenberg. The additional weight of engine structure, propellant tanks and
separate fuel and oxidizer are just a few of the reasons for this difference. Liquid
propellant launch vehicles are also known to be more expensive to design and build and
this is reflected in the results for the cost per launch. Yet, $100.11 million is a relatively
moderate cost for a liquid propellant vehicle. For the sake of comparison, a current,
126
operational threestage liquid propellant launch vehicle, the Zenit3SL, has an advertised
cost per launch of $90 million.
62
The choices of liquid propellants available for the design optimization have been
limited to accommodate the calculations in the cost model. The only option for the
oxidizer is liquid oxygen (LOX) and the options for fuels are restricted to storable fuel
types (Hydrazine, RP1, etc.). Liquid hydrogen (LH2) is not an option because it requires
special consideration in the cost model. The TRANSCost 7.1 cost model employs
separate development and recurring Cost Estimating Relationships (CERs) for launch
vehicles using LH2 as the fuel. These CERs result in much higher development and
recurring costs for LH2 fueled vehicles. Since one of the goals of this study is to
minimize vehicle cost per launch, the choice has been made not to use LH2 as an option.
Of course, the combination of LOX/LH2 provides close to the highest I
sp
for all liquid
fueled systems. Future design optimizations should look at using only LOX/LH2 as the
launch vehicle?s propellants. Then, a comparison of the resulting total vehicle mass and
cost per launch values for the best performers can be made with the results of other types
of liquid propellants.
From the results in Table 511, the GA chose LOX/Hydrazine as the oxidizer/fuel
combination of all three stages for the optimized vehicle. The choice of LOX/Hydrazine
in this case represents the propellant combination with the highest sealevel I
sp
(313.0s)
and thus makes sense for the design optimization process for this threestage liquid
propellant launch vehicle.
127
In addition, the performance characteristics of the optimized vehicle (final altitude
and final velocity) closely match the desired orbital values. As a result, this launch
vehicle performs well in launching the required payload to its intended lowEarth orbit.
Table 511: Summary of ThreeStage Liquid Propellant Launch Vehicle Characteristics
(VAFB)
Entire Vehicle Stage 1
Final Altitude 2,545,061 ft Stage Length 70.26 ft
Final Velocity 24,732 ft/s Stage Diameter 6.41 ft
Total Vehicle Mass 130,136 lbm Stage Weight 119,267 lbm
Total Vehicle Length 102.49 ft Initial Thrust 387,903 lbf
Total Vehicle f
inert
0.1094 Propellants LOX/Hydrazine
Nosecone Length 7.31 ft Stage 2
Cost per Launch $100.11 million Stage Length 14.63 ft
Stage Diameter 6.41 ft
Stage Weight 6,358 lbm
Initial Thrust 35,207lbf
Propellants LOX/Hydrazine
Stage 3
Stage Length 8.29 ft
Stage Diameter 6.41 ft
Stage Weight 2,310 lbm
Initial Thrust 7,125 lbf
Propellants LOX/Hydrazine
5.4.2 Case 8: ThreeStage Liquid Propellant Launch Vehicle (CCAFS)
Again, similar to the threestage solid propellant cases, a design optimization of a
threestage liquid propellant launch vehicle has been performed with launch out of Cape
Canaveral AFS, FL. The mission statistics for this launch vehicle are described in Table
510. Launching eastward out of Cape Canaveral provides an initial velocity boost that,
in theory, should produce a lower mass vehicle than the Vandenberg launch. Figure 522
shows a schematic of the best performer from the design optimization of this threestage
liquid propellant launch vehicle. The large size of the 1
st
stage is again evident with
smaller 2
nd
and 3
rd
stages.
128
Figure 522. ThreeStage Liquid Propellant Launch Vehicle Schematic (CCAFS)
The results for the design optimization of the threestage liquid propellant launch
vehicle are shown in Table 512. The benefits of launching out of Cape Canaveral AFS,
FL are apparent. The best performer weighs almost 26,000 pounds less than the
Vandenberg vehicle. The cost per launch is also significantly less ($93.68 million vs.
$100.11 million) than the Vandenberg vehicle. This brings the cost per launch of this
vehicle more in line with existing liquid propellant launch vehicles.
Like the Vandenberg vehicle, the GA has chosen LOX/Hydrazine as the
propellants for all three stages. These propellants provide the best performance and thus
are the logical choice to aid in minimizing the mass and subsequently the cost of the
vehicle. Geometrically, this vehicle has essentially the same body diameter as the
129
Vandenberg vehicle. However, the mass savings is a result mainly of a reduction in the
length of the rocket. The Cape Canaveral vehicle is only 88.69 feet long whereas the
Vandenberg vehicle is 102.49 feet long. This difference in length comes about due to the
differences in the Stage 1 length of both vehicles.
Also, like the Vandenberg vehicle, the performance characteristics of this
optimized vehicle match well the desired orbital parameters. This threestage liquid
propellant launch vehicle would be good choice for further development of the design
into a real world system.
Table 512: Summary of ThreeStage Liquid Propellant Launch Vehicle Characteristics
(CCAFS)
Entire Vehicle Stage 1
Final Altitude 2,433,952 ft Stage Length 57.17 ft
Final Velocity 24,604 ft/s Stage Diameter 6.30 ft
Total Vehicle Mass 104,354 lbm Stage Weight 91,702 lbm
Total Vehicle Length 88.69 ft Initial Thrust 321,273 lbf
Total Vehicle f
inert
0.1178 Propellants LOX/Hydrazine
Nosecone Length 6.55 ft Stage 2
Cost per Launch $93.68 million Stage Length 14.91 ft
Stage Diameter 6.30 ft
Stage Weight 8,256 lbm
Initial Thrust 35,537 lbf
Propellants LOX/Hydrazine
Stage 3
Stage Length 8.06 ft
Stage Diameter 6.30 ft
Stage Weight 2,249 lbm
Initial Thrust 8,567 lbf
Propellants LOX/Hydrazine
5.4.3 Case 9: TwoStage Liquid Propellant Launch Vehicle (VAFB)
The results of the design optimization process of a twostage liquid propellant
launch vehicle launched out of Vandenberg AFB, CA are presented here. Two distinct
design optimization runs have been performed with changes in each one due to vehicle
130
performance. A schematic of the resulting optimized launch vehicles from these two runs
is shown in Figure 523. The overall trend seen in these runs was the challenge of
meeting the desired orbital altitude and orbital velocity requirements. A summary of the
two runs is shown in Table 513. The results from Run #2, shown in Table 514,
provided the best performing vehicle and produced the lowest total mass vehicle.
Run #1 Run #2
Figure 523. TwoStage Liquid Propellant Launch Vehicles Schematic (VAFB)
Table 513: Summary of TwoStage Liquid Propellant Launch Vehicle Runs (VAFB)
Run #1 Run #2
Payload 7,000 lbm 1,000 lbm
Desired Altitude 656,000 ft 656,000 ft
Actual Altitude 653,691 ft 652,269 ft
Desired Velocity 25,532 ft/s 25,532 ft/s
Actual Velocity 22,667 ft/s 25,537 ft/s
Total Vehicle Mass 172,989 lbm 159,432 lbm
Cost per Launch $87.41 million $85.96 million
131
An initial design optimization that employed variable first and second stage
diameters for the launch vehicle was performed. This optimization was an attempt to
improve on the Titan II SLV that was used in the validation of the liquid propellant
model. The run was very successful at improving on the vehicle mass of the Titan II
SLV with the resulting vehicle having a mass (187,409 lbm) substantially lower than that
of the Titan II SLV (339,000 lbm). However, while the optimized vehicle was able to
reach the desired orbital altitude, the final velocity (23,145 ft/s) was less than the desired
orbital velocity (25,532 ft/s). This would make it difficult for the payload to remain in its
proper orbit and thus runs the risk of mission failure.
For the next design optimization (Run #1), the decision was made to eliminate the
variable diameter aspect of the vehicle and employ a constant diameter configuration.
This produced a more streamlined vehicle and reduced the total vehicle mass.
Unfortunately, both the final altitude and final velocity for the vehicle were much lower
than the desired orbital values. The constant diameter helped reduce the drag but it
forced the second stage diameter to be the same as the first stage diameter. The use of
more energetic liquid propellants is one possibility that may help solve this particular
problem. The use of LOX/Kerosene in the first stage and LOX/Hydrazine in the second
stage does not provide the highest I
sp
for liquid propellants. Allowing the GA to use
LOX/LH2 in the future would help improve the performance.
Since only storable liquid propellants were used in order to avoid the higher costs
associated with using LH2, the decision was made to reduce the payload mass for the
final design optimization run (Run #2) presented here. The vehicle model has already
improved upon the design of the Titan II SLV even though the performance values were
132
not ideal. Using the value of 1,000 pounds for the payload mass allows the twostage
liquid propellant launch vehicle to be compared to other types of vehicles that have been
optimized in this study. The optimized vehicle for this final run continued the trend of
reducing the total vehicle length while increasing the vehicle diameter. This resulted in
the performance parameters producing a good match with the desired performance
values.
One other interesting note, both vehicles have a cost per launch of around $85
million in 2003 dollars. The reason for this is that the inert mass of each vehicle is
slightly different. The vehicle from Run #2 has the lowest inert mass of the two vehicles.
The TRANSCost 7.1 cost model does not use the propellant mass in the development and
recurring cost models. The cost of the propellants would be considered in the ground and
flight operations part of the cost model which was not considered for this study.
Overall, the design optimization of a twostage liquid propellant launch vehicle
has been successful. The results in Table 514 show that the optimized vehicle meets the
mission requirements while significantly reducing total vehicle mass. The inert mass
fractions of the total vehicle and the individual stages fall within the range of values
expressed by Humble et al.
50
In terms of cost per launch, the optimized vehicle is more
expensive than a solid propellant launch vehicle.
133
Table 514: Summary of TwoStage Liquid Propellant Launch Vehicle Characteristics
(VAFBRun #2)
Entire Vehicle Stage 1
Final Altitude 652,267 ft Stage Length 36.29 ft
Final Velocity 25,537 ft/s Stage Diameter 11.52 ft
Total Vehicle Mass 159,432 lbm Stage Weight 121,405 lbm
Total Vehicle Length 76.37 ft Initial Thrust 245,399 lbf
Total Vehicle f
inert
0.1051 Propellants LOX/Kerosene
Nosecone Length 17.84 ft m
prop
112,993 lbm
Cost per Launch $85.96 million m
inert
8,412 lbm
f
inert
0.0693
Stage 2
Stage Length 20.24 ft
Stage Diameter 11.52 ft
Stage Weight 32,912 lbm
Initial Thrust 54,574 lbf
Propellants LOX/Ammonia
m
prop
28,783 lbm
inert
4,129 lbm
f
inert
0.1255
5.4.4 Case 10: TwoStage Liquid Propellant Launch Vehicle (CCAFS)
The design optimization of twostage liquid propellant launch vehicles launched
out of Cape Canaveral AFS, FL followed the same approach as the previously described
Vandenberg case. Again, two distinct optimization runs have been performed with minor
differences between each one. A schematic of the resulting best performers from these
two runs is shown in Figure 524. With the eastward launch and accompanying velocity
boost, there was some savings in total vehicle mass. However, the overall trend seen in
these two runs was still the challenge of meeting the desired orbital altitude and orbital
velocity requirements. A summary of the two runs is shown in Table 515. The results
from Run #2, shown in Table 516, provided the best performing vehicle and produced
the lowest total mass vehicle.
134
Run #1 Run #2
Figure 524. TwoStage Liquid Propellant Launch Vehicles Schematic (CCAFS)
Table 515: Summary of TwoStage Liquid Propellant Launch Vehicle Runs (CCAFS)
Run #1 Run #2
Payload 7,000 lbm 1,000 lbm
Desired Altitude 656,000 ft 656,000 ft
Actual Altitude 622,405 ft 660,170 ft
Desired Velocity 25,532 ft/s 25,532 ft/s
Actual Velocity 25,294 ft/s 25,531 ft/s
Total Vehicle Mass 141,046 lbm 135,121 lbm
Cost per Launch $79.53 million $79.04 million
As with the previous case, the design conditions have been changed in order to
incorporate a constant diameter launch vehicle. This change for Run #1 resulted in
further reduction in the total vehicle mass but the performance values for the final altitude
135
(622,405 ft) and the final velocity (25,294 ft/s) were reduced. It is interesting that for this
optimized vehicle, the GA chose to increase the overall length of the vehicle.
Finally, the second design optimization (Run #2) produced an optimized vehicle
that matched well the desired altitude and velocity parameters while reducing the total
vehicle mass further. This vehicle was also the least expensive ($79.04 million) of all the
twostage liquid propellant launch vehicles being analyzed. Of course, a decrease in
payload mass from 7,000 pounds to 1,000 pounds should produce much better
performance and the results in Table 515 show this.
Table 516: Summary of TwoStage Liquid Propellant Launch Vehicle Characteristics
(CCAFSRun #2)
Entire Vehicle Stage 1
Final Altitude 660,170 ft Stage Length 33.25 ft
Final Velocity 25,531 ft/s Stage Diameter 11.09 ft
Total Vehicle Mass 135,121 lbm Stage Weight 104,492 lbm
Total Vehicle Length 72.51 ft Initial Thrust 201,370 lbf
Total Vehicle f
inert
0.1124 Propellants LOX/Kerosene
Nosecone Length 18.53 ft m
prop
97,074 lbm
Cost per Launch $79.04 million m
inert
7,419 lbm
f
inert
0.0710
Stage 2
Stage Length 18.72 ft
Stage Diameter 11.09 ft
Stage Weight 25,521 lbm
Initial Thrust 39,513 lbf
Propellants LOX/Ammonia
m
prop
21,965 lbm
inert
3,556 lbm
f
inert
0.1393
5.4.5 Conclusions: Liquid Propellant Launch Vehicle Design Optimizations
A number of successful design optimizations of two and threestage liquid
propellant launch vehicles have been performed. All four vehicles meet the mission
136
requirements for their respective orbital altitude and orbital velocity constraints. Also,
the inert mass fractions meet the required values for liquid propellant systems.
As was the case in the solid propellant vehicles, the use of higher I
sp
propellants
can have an effect on vehicle performance and sizing. Future work in the design
optimization of liquid propellant launch vehicles should look at the use of LOX/LH2 as a
possible propellant combination. Also, the use of a variable diameter design seemed to
complicate the design of the twostage liquid vehicles more than the solid propellant
vehicles. Of course, the GA chose a much more dramatic difference in the stage
diameters of the liquid vehicles versus the solid vehicles. Additional work in the
modeling of the aerodynamics of variable diameter vehicles would prove useful for future
design optimization endeavors.
5.5 AirLaunched Vehicles
A design optimization has been performed for an airlaunched, Earthtoorbit
launch vehicle system. This system employs a twostage liquid propellant launch vehicle
deployed from the cargo bay of a USAF C141 transport aircraft. In addition, the air
launched, twostage liquid propellant launch vehicle has a real world counterpart, the
QuickReach
TM
launch vehicle, which was used in the model validation efforts.
Table 517 summarizes the mission statistics for this case. Payload, desired
orbital altitude and desired orbital velocity are typical for a lowEarth orbit mission.
They are essentially the same as the values used in the design optimizations of the three
and fourstage solid propellant launch vehicles and the two and threestage liquid
propellant launch vehicles.
137
Table 517: AirLaunched Vehicle Mission Statistics
TwoStage Air
Launched Liquid
Payload 1,000 lbm
Launch Site C141 Cargo Bay
Launch Direction Due East
90? Azimuth
Orbit Type i=28.4?
prograde orbit
Desired Orbital Altitude 2,430,000 ft
Desired Orbital Velocity 24,550 ft/s
5.5.1 Case 11: AirLaunched, TwoStage Liquid Propellant Launch Vehicle (CCAFS)
The results of the design optimization of an airlaunched, twostage solid
propellant launch vehicle are presented here. The concept of an airlaunched vehicle is as
follows: one stage of the launch vehicle is replaced with an aircraft platform that would
act as the ?launch site? or, what is commonly called Stage 0, of the rocket. This concept
is not new. Orbital Sciences Corporation deploys its Pegasus space launch vehicle from
the underside of a modified L1011 aircraft. The Pegasus is a threestage solid propellant
launch vehicle that has been quite successful to date. Also, a similar effort is currently
ongoing where the QuickReach
TM
launch vehicle is being developed by AirLaunch
LLC.
57
The QuickReach
TM
launch vehicle is a twostage, liquid propellant vehicle as
opposed to a solid propellant vehicle. Additionally, the QuickReach
TM
launch vehicle is
to be deployed from the cargo bay of an unmodified USAF C17A transport aircraft. In
recent months, a fullscale mockup of this launch vehicle has been successfully drop
tested.
The current study involves a demonstration of the feasibility of finding a
workable solution using the GA. The goals of this design optimization are the same as
138
previous cases: minimize orbital velocity and orbital altitude differences along with
minimizing total vehicle mass. The TRANSCost 7.1 cost model has again been used but
it only estimates the cost per launch of the launch vehicle itself. The cost of the airborne
platform would also need to be considered in order to get a more accurate cost estimate.
When considering the cost of the airborne platform, there are different options
that can be taken. First, an aircraft can be specifically designed and built to
accommodate the launch vehicle to be employed. This option provides the most
flexibility but is probably the most expensive. Second, an existing aircraft can be
modified for use specifically for the launch vehicle. This option is probably not as
expensive but will impose some restrictions on the design of the launch vehicle. The
vehicle must be designed to accommodate specific aircraft physical characteristics.
Finally, an existing, unmodified aircraft can be chosen and the launch vehicle can be
designed and built specifically for launch on this aircraft. This is the approach that
AirLaunch LLC is taking for their QuickReach
TM
launch vehicle.
For a design optimization using the GA, any of these airborne platform options
will work. However, the objective function requires some initial conditions in order to
analyze the flight trajectory of the rocket. Those conditions are the airspeed of the
aircraft and the cruise altitude. An existing aircraft has been used to provide that data for
use in the 6DOF flight dynamics simulator. The USAF C141 transport aircraft is a good
candidate for this application. Since these aircraft have been retired, the possibility exists
that one or more C141s could be obtained by a launch vehicle provider for a reduced
cost and used as the launch platform for this particular launch vehicle (similar to the
Minotaur I SLV?s use of decommissioned Minuteman II ICBMs).
139
In order to properly configure the objective function, a few other assumptions
have been made. Again, the 6DOF flight dynamics simulator needs initial information in
order to analyze the flight of the vehicle. Specifically, the latitude and longitude of the
launch site are required inputs. For an airlaunched case, technically, the aircraft could
fly almost anywhere in the world to launch the rocket (provided the aircraft can be
refueled). For example, the aircraft could fly down to the equator and launch the rocket
in an eastward direction to maximize the benefit of the Earth?s rotation. However, for
the sake of consistency with other design optimizations that have been performed, the
latitude and longitude values have been chosen to be the same as Cape Canaveral AFS,
FL. Also, it is assumed that the C141 moves out of the way essentially instantly in order
for the rocket to fire its first stage motor. Lastly, it is assumed that a system will be
available to orient the vehicle to its initial launch angle instantly after deployment from
the cargo bay of the C141 aircraft. Certain design parameters for the GA are specified in
such a way so that the rocket fits into the cargo bay of the C141. Vehicle length and
diameter are restricted so that the GA cannot choose values that would result in a rocket
larger than the aircraft?s cargo bay. Table 518 summarizes the characteristics of the C
141 transport aircraft used in this study. It is assumed that the aircraft is flying directly
over the Cape Canaveral launch pads at an altitude of 35,000 feet and with a velocity of
733.33 feet per second in the direction of launch.
140
Table 518: C141 Transport Aircraft Characteristics
Cargo Bay Dimensions 93 ft (length) x 10 ft (width) x 9 ft (height)
Cargo Bay Load Capacity 70,000 lbm
Launch Altitude 35,000 ft
Launch Velocity 733.33 ft/s
A successful design optimization has been performed for an airlaunched, two
stage liquid propellant launch vehicle. The mission statistics are described in Table 517
and the C141 transport aircraft is used as the airborne platform. In addition, a direct
comparison to the QuickReach
TM
launch vehicle can be made because it is also a two
stage liquid propellant launch vehicle. Figure 525 shows a schematic of the airlaunched
twostage liquid propellant launch vehicles that have been analyzed. The left image
shows a mockup of the QuickReach
TM
launch vehicle being deployed from a C17A
transport aircraft. The middle schematic is the launch vehicle produced in the validation
process. The best performer from the design optimization process is shown in the
schematic on the right. Table 519 summarizes the important characteristics of the
optimized launch vehicle.
141
Figure 525. Air Launched, TwoStage Liquid Propellant Launch Vehicles Schematic
(CCAFS)
(Ref. 59: http://www.designationsystems.net/dusrm/app4/quickreachslv.jpg)
The launch vehicle resulting from the design optimization process is very similar
to the QuickReach
TM
launch vehicle with some improvements in the physical
characteristics. Both launch vehicles have essentially the same total vehicle mass (72,049
lbm vs. 72,000 lbm). However, the optimized vehicle shows improvement in both total
vehicle length and the individual stage diameters. The QuickReach
TM
launch vehicle has
a total length of 66 feet whereas the optimized vehicle?s total length is 64.11 feet. The
diameter of both stages of the QuickReach
TM
launch vehicle is 7 feet. The optimized
vehicle has a 6.86 feet diameter for Stage 1 and a 5.32 feet diameter for Stage 2. These
improvements are important since the launch vehicle has to fit into the small confines of
a C141?s cargo bay.
As with previous liquid propellant launch vehicles, the GA chose LOX/Hydrazine
as the oxidizer/fuel combination for Stage 1. These propellants have the highest I
sp
of the
142
available choices so this makes sense for the GA to choose the best performing
propellants. However, the GA chose the LOX/Ammonia combination for Stage 2 of the
launch vehicle. This is an interesting choice since the sealevel I
sp
of LOX/Ammonia
(294.0s) is much lower than the sealevel I
sp
of LOX/Hydrazine (313.0s).
In addition, the inert mass fractions for this case match well the typical values for
liquid propellant engines. The total vehicle inert mass fraction is 0.1029 which falls in
the accepted range of values.
Table 519: Summary of AirLaunched, TwoStage Liquid Propellant Launch Vehicle
Characteristics (CCAFS)
Entire Vehicle Stage 1
Final Altitude 699,990 ft Stage Length 33.21 ft
Final Velocity 25,532 ft/s Stage Diameter 6.86 ft
Total Vehicle Mass 72,049 lbm Stage Weight 49,991 lbm
Total Vehicle Length 64.11 ft Initial Thrust 127,081 lbf
Total Vehicle f
inert
0.1029 Propellants LOX/Hydrazine
Nosecone Length 6.25 ft m
prop
45,951 lbm
Cost per Launch $61.99 million m
inert
4,039 lbm
f
inert
0.0808
Stage 2
Stage Length 22.65 ft
Stage Diameter 5.32 ft
Stage Weight 20,118 lbm
Initial Thrust 72,412 lbf
Propellants LOX/Ammonia
m
prop
17,783 lbm
inert
2,335 lbm
f
inert
0.1161
A comparison between the airlaunched, twostage liquid propellant launch
vehicle resulting from the design optimization and the QuickReach
TM
launch vehicle are
shown in Table 520. The optimized vehicle closely matches the physical characteristics
of the QuickReach
TM
launch vehicle. Also, the optimized vehicle is able to attain the
same desired orbital parameters as the QuickReach
TM
launch vehicle. There are some
143
differences in the vacuum thrust of both stages of each vehicle. The GA has designed a
much more powerful Stage 2 thrust for the optimized vehicle (72,412 lbf vs. 24,000 lbf).
Table 520: AirLaunched, TwoStage Liquid Propellant Launch Vehicles Comparison
QuickReach
TM
Launch Vehicle
Validation
Model
Optimized
Vehicle
Payload 1,000 lbm 1,000 lbm 1,000 lbm
Total Vehicle Weight 72,000 lbm 74,633 lbm 72,049 lbm
Total Vehicle Length 66.00 ft 62.40 ft 64.11 ft
Final Altitude 700,000 ft 738,783 ft 699,990 ft
Final Velocity 25,532 ft/s 25,619 ft/s 25,532 ft/s
Stage 1 Vacuum
Thrust
172,000 lbf 171,825 lbf 127,081 lbf
Stage 2 Vacuum
Thrust
24,000 lbf 25,333 lbf 72,412 lbf
5.5.2 Conclusions: AirLaunched Vehicle Design Optimization
The design optimization of an airlaunched, twostage liquid propellant launch
vehicle has been performed. This optimization has shown the feasibility of this system
and the fact that the objective function and GA can meet the mission requirements. The
optimized liquid propellant launch vehicle can fit into the cargo bay of the C141
transport aircraft. However, the total vehicle mass of the optimized vehicle is slightly
higher than the payload capacity of the C141.
The cost per launch of the optimized launch vehicle is quite attractive ($61.99
million). This cost per launch value is lower than all the other liquid propellant launch
vehicles. This cost per launch does not include the cost of the airborne platform. Of
course, using existing transport aircraft like the C141 or C17 should keep this cost at a
minimum.
Additional design optimizations for this system should include using different
combinations of liquid propellants. Consideration should be made for the fact that these
144
launch vehicles are launched from an airborne platform. This would have an affect on
the choice of propellants since they would have special transport requirements. Also,
design optimizations should investigate launching the vehicle from locations other than
Cape Canaveral AFS, FL. Flying the transport aircraft down to the equator and launching
in an eastward direction would provide additional useful information on the preliminary
design of this type of launch vehicle. Finally, the case of an airlaunched, twostage solid
propellant launch vehicle should be investigated.
5.6 Mixed Propellant Vehicles
The last two cases analyzed in the current study are mixed propellant launch
vehicles with larger payloads and different desired orbital parameters. These launch
vehicles employ a unique configuration of a first stage solid propellant motor along with
second and third stage liquid propellant rocket engines. The use of mixed propellant
systems on space launch vehicles is not a new idea. Strapon solid rocket motors are
often used to provide thrust augmentation at liftoff for a variety of launch vehicles (like
the Space Shuttle and Delta II). However, in the current study, the 1
st
stage solid
propellant motor is designed to provide all the liftoff thrust for the vehicle. Then, the
two liquid propellant stages take over and accelerate the payload to orbit.
In order to investigate the characteristics of a medium to heavy lift launch vehicle
capability, the payload mass has been increased to 10,000 pounds. In addition, most of
the previous design optimizations have used a relatively high value for the desired orbital
altitude of 700 km (2,430,000 ft). For these final two cases, the desired orbital altitude
has been lowered to an altitude of 300 km (984,252 ft). As a result of this lower orbit, the
desired orbital velocity requirement has increased to 25,328 feet per second in
145
accordance with the appropriate orbital mechanics calculations. Table 521 summarizes
the mission statistics for the two launch vehicles being optimized.
Table 521: Mixed Propellant Launch Vehicles Mission Statistics
ThreeStage
Solid/Liquid/Liquid
ThreeStage
Solid/Liquid/Liquid
Payload 10,000 lbm 10,000 lbm
Launch Site VAFB CCAFS
Launch Direction Due North
0? Azimuth
Due East
90? Azimuth
Orbit Type i=90?
polar orbit
i=28.4?
prograde orbit
Desired Orbital
Altitude
984,252 ft 984,252 ft
Desired Orbital
Velocity
25,328 ft/s 25,328 ft/s
5.6.1 Case 12: ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle (VAFB)
The results for the design optimization of a threestage solid/liquid/liquid
propellant launch vehicle with launch out of Vandenberg AFB, CA are presented here.
The schematic of the best performer from the optimization process for this vehicle is
shown in Figure 526. The choice of a constant diameter vehicle was chosen prior to the
start of the design optimization process. The fairly large 2
nd
stage can be seen in the
schematic as well as the relatively smaller 1
st
and 3
rd
stages. Also, this vehicle is longer
(155.45 ft) than all the other previously optimized vehicles investigated in this study.
The large launch vehicle size is to be expected due to the increase in the payload massto
orbit requirement.
146
Figure 526. ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle Schematic
(VAFB)
The important characteristics of the optimized threestage solid/liquid/liquid
propellant launch vehicle are shown in Table 522. Two design optimizations have been
run for this case. The first run produced a launch vehicle with some generally good
characteristics. However, the optimized vehicle failed to meet one of the desired goals.
The final velocity of this vehicle (23,595 ft/s) was well below the desired orbital velocity
(25,328 ft/s). As a result, this design optimization was declared an unusable solution. A
second design optimization was prepared with one important change in the optimization
147
process. The previous design optimization run limited the choice of available liquid
propellants for the GA. The reason for this was to facilitate the operation of the
TRANSCost 7.1 cost model by using only storable liquid propellants. These liquid
propellants (like LOX/Kerosene and LOX/Hydrazine) do not have as good performance
characteristics as other propellants (like LOX/LH2). For the second design optimization
run, it was decided to open up the available choices for the GA to include higher I
sp
liquid
propellants (like LOX/LH2 and LF2/LH2). The TRANSCost 7.1 cost model was adjusted
in order to account for the possibility of the GA choosing these types of liquid
propellants.
The success of this decision is seen in the performance characteristics of the
optimized vehicle. The final altitude (984,726 ft) and final velocity (25,330 ft/s) closely
match the desired orbital parameters. The total vehicle mass (180,673 lbm) also has been
reduced significantly from the total vehicle mass of the previous run (226,589 lbm). The
choice of higher sealevel I
sp
propellants definitely has made the difference. The GA
chose LF2/LH2 (I
sp
= 410s) for the Stage 2 propellants and LF2/Hydrazine (I
sp
= 363s) for
the Stage 3 propellants. These propellants are excellent choices to increase the
performance of the optimized launch vehicle. However, there is a downside to using
these propellants. The toxicity of LF2 makes it highly undesirable as a propellant. Also,
the TRANSCost 7.1 cost model was adjusted for the use of these propellants and the result
was a higher cost per launch ($125.08 million) than the previous run ($109.75 million).
148
Overall, this design optimization has produced a successful launch vehicle that
closely matches the desired orbital parameters. The mass fractions for this optimized
vehicle are also desirable and all three values fall within the range for the desired mass
fractions for solid and liquid propellant launch vehicles.
Table 522: Summary of ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle
Characteristics (VAFB)
Entire Vehicle Stage 1 (solid)
Final Altitude 984,726 ft Stage Length 28.26 ft
Final Velocity 25,330 ft/s Stage Diameter 6.54 ft
Total Vehicle Mass 180,673 lbm Stage Weight 64,874 lbm
Total Vehicle Length 155.45 ft Initial Thrust 623,082 lbf
Nosecone Length 13.36 ft Propellants DB/AP/Al
Cost per Launch $125.08 million m
prop
58,330 lbm
inert
6,545 lbm
f
prop
0.8991
Stage 2 (liquid)
Stage Length 87.83 ft
Stage Diameter 6.54 ft
Stage Weight 73,606 lbm
Initial Thrust 221,433 lbf
Propellants LF2/LH2
m
prop
64,226 lbm
inert
9,380 lbm
f
inert
0.1274
Stage 3 (liquid)
Stage Length 24.00 ft
Stage Diameter 6.54 ft
Stage Weight 29,851 lbm
Initial Thrust 101,114 lbf
Propellants LF2/Hydrazine
m
prop
26,994 lbm
inert
2,858 lbm
f
inert
0.0957
149
5.6.2 Case 13: ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle (CCAFS)
The results for the final design optimization case of the current study are
presented here. This threestage solid/liquid/liquid propellant launch vehicle has been
optimized for launch out of Cape Canaveral AFS, FL. In addition, the mission statistics
for this vehicle are described in Table 521. The schematic of the best performer from
the optimization process for this vehicle is shown in Figure 527. This optimized vehicle
is very similar to the Vandenberg launch vehicle. The 2
nd
stage is the largest stage and
the overall vehicle length is large (142.66 ft).
150
Figure 527. ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle Schematic
(CCAFS)
The important characteristics of the optimized threestage solid/liquid/liquid
propellant launch vehicle are shown in Table 523. The benefits of launching in the
eastward direction are clearly seen in the performance results. The final altitude
(985,320 ft) and the final velocity (25,377 ft/s) of this optimized vehicle both closely
match the desired orbital altitude and desired orbital velocity values. Thus, the slight
velocity boost given by the Earth?s rotation provides the difference for getting to orbit.
151
The other performance characteristics of the optimized vehicle yield good results.
The propellant mass fraction of the solid propellant 1
st
stage is a respectable 0.8699. The
inert mass fractions of the two liquid propellant stages (Stage 2: 0.0748 & Stage 3:
0.1128) also fall within the acceptable range for these types of engines. These results
show that this particular launch vehicle model is a good one and the results are realistic.
The only possible draw back to this vehicle is the cost per launch ($117.21
million). While still less expensive than the Vandenberg vehicle, the cost per launch for
this optimized vehicle is higher than the cost per launch values of all the other previously
optimized cases. Of course, the payload mass for the solid/liquid/liquid propellant launch
vehicles (10,000 lbm) is ten times greater than the payload mass (1,000 lbm) of the
previous cases. Because of this larger payload, a larger and more expensive launch
vehicle should be expected.
152
Table 523: Summary of ThreeStage Solid/Liquid/Liquid Propellant Launch Vehicle
Characteristics (CCAFS)
Entire Vehicle Stage 1 (solid)
Final Altitude 985,320 ft Stage Length 37.64 ft
Final Velocity 25,377 ft/s Stage Diameter 6.66 ft
Total Vehicle Mass 242,884 lbm Stage Weight 82,915 lbm
Total Vehicle Length 142.66 ft Initial Thrust 1,781,882 lbm
Nosecone Length 10.17 ft Propellants PBAN/AP/Al
Cost per Launch $117.21 million m
prop
72,128 lbm
inert
10,786 lbm
f
prop
0.8699
Stage 2 (liquid)
Stage Length 68.51 ft
Stage Diameter 6.66 ft
Stage Weight 121,273 lbm
Initial Thrust 283,503 lbf
Propellants LOX/Hydrazine
m
prop
112,203 lbm
inert
9,070 lbm
f
inert
0.0748
Stage 3 (liquid)
Stage Length 24.33 ft
Stage Diameter 6.66 ft
Stage Weight 27,045 lbm
Initial Thrust 109,584 lbf
Propellants LOX/Hydrazine
m
prop
23,993 lbm
inert
3,052 lbm
f
inert
0.1128
5.6.3 Conclusions: Mixed Propellant Vehicle Design Optimizations
Two successful design optimizations of a unique threestage solid/liquid/liquid
launch vehicle have been performed. The results provide some interesting insight into
the challenge of meeting mission requirements (orbital altitude and orbital velocity) while
at the same time reducing total vehicle mass and cost per launch. There are certainly
many parameters to consider and the GA provides an effective tool to aid in the analysis
of space launch vehicles.
153
The optimized vehicle for the Vandenberg launch initially did not match the
desired orbital velocity and this prompted an analysis of vehicle mass and propellant
choice considerations. Allowing for the use of more energetic propellants to increase
performance and reduce weight was proposed as a possible solution. As a result, a
second design optimization was performed that allowed for the choice of other liquid
propellants. This proved to be successful as the new optimized vehicle closely matched
the desired orbital values. The downside was that the cost per launch for this vehicle rose
substantially.
The optimized vehicle for the Cape Canaveral launch met the desired orbital
parameters most likely due to the velocity boost provided by the Earth?s eastward
rotation. The same argument of variable diameter stages and more energetic propellants
applies to the Cape Canaveral launch vehicle in order to reduce total vehicle mass while
maintaining vehicle performance.
5.7 Launch Vehicle Comparisons
5.7.1 Launch Vehicles Comparison of 1,000 lbm Payload Cases
The first comparison of the launch vehicles produced by the various design
optimizations performed in this study is presented here. In order to make a useful
comparison, all the launch vehicles being considered for this comparison should have a
common reference point. In this comparison, the common reference point is the payload
mass launched into orbit. The payload mass is a direct input to the objective function and
an important variable in the design optimization process. Of the thirteen different design
optimizations performed in this study, eight cases used the value of 1,000 lbm for the
payload mass. These eight cases are summarized in Table 524.
154
Table 524: Optimized Cases for 1,000 lbm Payload Mass
Case # Optimized Launch Vehicle
3 ThreeStage Solid (VAFB)
4 ThreeStage Solid (CCAFS)
6 FourStage Solid (CCAFS)
7 ThreeStage Liquid (VAFB)
8 ThreeStage Liquid (CCAFS)
9 TwoStage Liquid (VAFB)
10 TwoStage Liquid (CCAFS)
11 AirLaunched, TwoStage Liquid (CCAFS)
The comparison of the total vehicle mass for these eight cases is shown in Figure
528. This comparison shows that the liquid propellant vehicles tend to have the highest
total vehicle mass of all the launch vehicles being compared; the one exception is the air
launched twostage liquid propellant vehicle (72,049 lbm).
All of the solid propellant vehicles weigh around 80,000 pounds whereas all of
the liquid cases weigh over 100,000 pounds (the airlaunched twostage liquid being the
one exception). Thus, in general, the broad conclusion can be made that in order to
achieve a particular lowEarth orbit with a 1,000 pound payload, the total mass of the
launch vehicle will need to be at least 70,000 pounds.
155
Total Vehicle Mass for 1,000 lbm Payload Cases
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
3stg
solid
(VAFB)
3stg
solid
(CCAFS)
4stg
solid
(CCAFS)
3stg liq
(VAFB)
3stg liq
(CCAFS)
2stg liq
(VAFB)
2stg liq
(CCAFS)
2stg air
(liqd)
Optimized Cases
Tot
a
l
V
e
hi
c
l
e
M
a
s
s
(
l
bm
)
Figure 528. Total Vehicle Mass Comparison
The comparison of the cost per launch values for these same nine cases is shown
in Figure 529. Since the cost estimating relationships (CERs) used in the TRANSCost
7.1 cost model are massbased, the less expensive launch vehicles are the solid propellant
vehicles because they weigh less than the liquid propellant vehicles. What is interesting
is the cost per launch values of the two and threestage liquid propellant vehicles. The
threestage liquid propellant vehicles have a lower total vehicle mass than the twostage
liquid propellant vehicles. However, the threestage liquid propellant vehicles are more
expensive than the twostage liquid propellant vehicles. This implies that, for the liquid
propellant vehicles, minimizing the total vehicle mass did not result in a minimized cost
per launch. Future work should be done where the third design goal is a direct
minimization of the cost per launch.
The reason for this is the use of the system engineering/integration factor (f
0d
) in
the Development Cost Submodel and the system management/vehicle
integration/checkout factor (f
0p
) used in the Production Cost Submodel. These two
156
factors address the cost associated with developing a multistage vehicle along with the
complexities of producing it and integrating the different stages. Thus, both of these
factors are dependent on the number of stages of the vehicle. The equations used for
these two factors are written by Koelle
41
as:
nstg
d
f 04.1
0
= (5.2)
nstg
p
f 025.1
0
= (5.3)
where nstg is the number of stages in the vehicle. Equations 5.2 and 5.3 show that a
threestage launch vehicle could have a higher cost than a twostage vehicle even if the
total mass of the threestage vehicle is lower than the total mass of the twostage vehicle.
Figure 529 also shows that, overall, liquid propellant launch vehicles are more
expensive than solid propellant launch vehicles. Thus, if cost is a driving concern in the
design of a launch vehicle, solid propellant systems probably should be considered early
on in the systems engineering process.
Cost per Launch for 1,000 lbm Payload Cases
0
20
40
60
80
100
120
3stg solid
(VAFB)
3stg solid
(CCAFS)
4stg solid
(CCAFS)
3stg liq
(VAFB)
3stg liq
(CCAFS)
2stg liq
(VAFB)
2stg liq
(CCAFS)
2stg air
(liqd)
Optimized Cases
C
os
t
pe
r
La
unc
h (
m
i
l
l
i
on)
Figure 529. Cost per Launch Comparison
157
5.7.2 Launch Vehicles Comparison of Propellant Mass Fractions
The next comparison involves the propellant mass fractions (f
prop
) that were
determined for all the solid propellant launch vehicles. As previously explained in
Humble et al.,
50
the propellant mass fraction is used to size solid propellant launch
vehicles. The preferred value for propellant mass fraction, as determined from previously
flown and built launch vehicles, is 0.90 for solid propellant vehicles. Thus, one goal of
the design optimizations that have been performed was to ensure that the propellant mass
fractions for the various optimized launch vehicles met the desired propellant mass
fraction criteria.
Figure 530 shows the propellant mass fractions for the various solid propellant
launch vehicle cases optimized in the current study. These values represent the
propellant mass fraction of the entire vehicle stack. The propellant mass fractions for all
the optimized vehicles fall within the range of 0.89 to 0.90. Overall, these values of
propellant mass fraction closely match the desired value.
Propellant Mass Fraction (fprop) for Solid Propellant Vehicles
0.85
0.86
0.87
0.88
0.89
0.9
0.91
0.92
3stg solid
(VAFB)
3stg solid
(CCAFS)
4stg solid
(VAFB)
4stg solid
(CCAFS)
3stg sll
(VAFB)
3stg sll
(CCAFS)
Optimized Cases
P
r
o
p
el
l
an
t M
a
ss
F
r
ac
t
i
o
n
Figure 530. Propellant Mass Fractions (f
prop
) Comparison
158
5.7.3 Launch Vehicles Comparison of Inert Mass Fractions
Figure 531 shows the inert mass fractions for the various liquid propellant launch
vehicles optimized in the current study. All of these results are well within the required
range and produce very good inert mass fractions for the liquid propellant launch
vehicles. This makes their high degree of accuracy similar to the accuracy of the solid
propellant launch vehicles. Thus, the mass fractions for both solid and liquid propellant
launch vehicles compare very well to real world launch vehicles.
Inert Mass Fraction (finert) for Liquid Propellant Vehicles
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
2stg liq
(VAFB)
2stg liq
(CCAFS)
3stg liq
(VAFB)
3stg liq
(CCAFS)
2stg air
(liqd)
3stg sll
(VAFB)
3stg sll
(CCAFS)
Optimized Cases
I
n
er
t
M
a
s
s
F
r
ac
t
i
o
n
Figure 531. Inert Mass Fractions (f
inert
) Comparison
5.7.4 Launch Vehicles Comparison of Cost per Launch Values
Finally, one of the main goals of the current study was to attempt to minimize the
cost per launch of Earthtoorbit launch vehicles. A variety of optimizations were
performed in order to investigate the feasibility of lowering launch costs for space launch
vehicles. While the results presented here are promising, the cost per launch values did
not produce a truly economical launch vehicle. Many factors must be considered when
determining cost and certainly not all were considered in the cost model used. Yet, the
159
TRANSCost 7.1 cost model provides a very powerful tool for analysis at this preliminary
design level. The least expensive cost per launch ($46.07 million) is a fairly good price
for an expendable launch vehicle. However, it is also much higher than the $5 million
cost per launch desired by the USAF Operationally Responsive Space (ORS) program.
The costs per launch values, in millions of dollars, for all the launch vehicles
considered in this study are shown in Figure 532. One item to note is that not all the
launch vehicles considered used the same payload mass and not all attained the same
lowEarth orbit. However, this comparison does show the overall trend that solid
propellant vehicles tend to be less expensive than liquid propellant launch vehicles. Also,
when solids and liquids are combined (as in the threestage solid/liquid/liquid cases) the
cost per launch is even higher than for single propellant types of systems. Thus, if
possible, careful consideration should be taken for launch vehicle designs that employ
different types of propellants since higher costs are likely.
Cost per Launch for All Launch Vehicles
0
20
40
60
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140
3
s
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Optimized Cases
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Figure 532. Cost per Launch Comparison for All Launch Vehicles
160
6.0 CONCLUSIONS
6.1 Introduction
A variety of design optimizations have been performed in an attempt to
demonstrate the possibility for improvement upon current space launch vehicle design.
The analysis from these optimizations has shown encouraging results with regards to the
possibility for improvement of launch vehicle systems.
The design optimization processes undertaken in this study provide insight into
possible launch vehicle design improvement. The focus has been on generating useful
results that can aid the systems engineering process at the preliminary design level. The
goals of the optimization process have been to minimize total launch vehicle mass and
cost per launch while at the same time meeting the desired orbital parameters.
The genetic algorithm (GA), in conjunction with the objective function, provides
a powerful tool for finding the optimum solution for a given set of design goals. For the
highly complicated space launch vehicle system, the GA and objective function have
been successfully implemented and some interesting launch vehicle designs have
emerged. One unique aspect of this analysis is the implementation of a cost model used
to predict the cost per launch of each optimized vehicle.
The optimized launch vehicle results are within the confines of the launch vehicle
design class that has been modeled. The optimized vehicles fall into the class of launch
vehicles categorized as small to medium lift vehicles. These vehicles also employ
161
conventional forms of propulsion based on heritage vehicle technology. It is likely that
additional improvement in the design goals may be gained by the use of additional
technology in the form of propulsion and vehicle shaping. The use of airbreathing
propulsion systems along with lifting body shapes could bring about better performing
and more cost effective space launch vehicles.
6.2 System Modeling and Validation
An entire multistage launch vehicle has been modeled using a suite of system
models in the form of performance codes and then validated against real world systems.
The propulsion characteristics, mass properties, aerodynamic characteristics and vehicle
flight dynamics have been successfully integrated to analyze the launch vehicle
performance from liftoff to orbit insertion. In addition, a massbased cost model has
been successfully added to provide cost per launch data on the optimized launch vehicle.
The important task of model validation has been performed on four specific
launch vehicle types. A threestage solid propellant vehicle, a fourstage solid propellant
vehicle, a twostage liquid propellant vehicle and an airlaunched, twostage liquid
propellant vehicle have been validated against similar real world launch vehicles. The
use of propellant and inert mass fractions also strengthened the validity of system models
being used.
One important limitation has been identified in the model validation process. The
mass properties model tended to underestimate the mass values of certain vehicle
components. Specifically, the inert mass of the solid propellant launch vehicles was less
than the inert mass of the real world examples. Precisely estimating the mass properties
of space launch vehicles is a difficult task. Future work could address this issue by either
162
updating the mass properties model with the most current and detailed information
available or including a correction factor for the inert mass in the current model.
6.3 Design Optimizations
The design optimizations performed in this study have produced informative
results in the preliminary design of space launch vehicles. Thirrteen different cases that
covered a broad range of vehicle types showed the feasibility of minimizing total vehicle
mass and cost per launch while meeting the specified orbital parameters. The resulting
optimized vehicles are not only realistic but are also improvements on their real world
counterparts.
Three specific trends have been noted throughout the results of these
optimizations. First, propellant choice had a large effect on vehicle performance and
meeting the desired goals. Often the GA would attempt to use the more energetic solid
and liquid propellants in order to meet the desired orbital parameter goals. This would
make sense since those goals were weighted higher than the third design goal.
Second, along the same lines as propellant choice, the payload mass for each
launch vehicle played a large role in overall vehicle size and performance. All the
vehicles that used a payload mass of less than 1,000 pounds were able to closely match
the desired orbital parameters. In addition, the total vehicle mass and cost per launch for
these vehicles fell within the typical range for this class of launch vehicles. A few
vehicles were optimized using a payload mass of 7,000 lbm or higher. The challenge of
lifting higher mass payloads into orbit was realized in the resulting higher total vehicle
mass and cost per launch values obtained for these heavier lift vehicles.
163
Lastly, the cost per launch values for the optimized vehicles were good but did
not result in breakthroughs in launch vehicle cost. The Operationally Responsive Space
(ORS) program has the goal of getting launch costs down to $5 million per launch for
1,000 pound payloads. The lowest cost per launch obtained from the results of this study
is $46.07 million for the fourstage solid propellant launch vehicle. A difference of over
$40 million exists between the desired ORS program launch cost and the results obtained
here. From the design optimizations performed in this study, the likelihood of achieving
the $5 million per launch value does not seem feasible. A significant breakthrough in
propulsion or materials technology would be required to bring the cost per launch down
to that price. As a result, additional work still needs to be done in order to reduce the cost
of access to space.
One final note should be made on the issue of cost and the choice of design goals.
Early in the process, the decision was made to minimize the total vehicle mass as the
third design optimization goal. Since the cost model being used was massbased, this
seemed like a logical choice that would result in a minimized cost per launch value. For
the most part, the results of the design optimization supported this choice. However, the
analysis of the liquid propellant launch vehicles resulted in a different conclusion. In the
end, a fourth goal should have been added that directly minimized the cost per launch
value.
To summarize, the purpose of the current effort was to demonstrate the viability
of optimizing launch vehicles using a genetic algorithm (GA) along with an objective
function containing detailed preliminary design level models. The three goals of the
design optimization process were to match the desired orbital altitude and orbital velocity
164
values along with minimizing the total vehicle mass and also vehicle cost. Both model
validation and design optimizations using a GA have been performed for a wide variety
of space launch vehicle configurations. The results show the feasibility of this approach
in the improvement of launch vehicle design. The best example is the fourstage solid
propellant launch vehicle. The validated model closely matched the real world example
launch vehicle. The subsequent design optimization improved on the real world example
resulting in a significant mass savings. Thus, through the validation of the system models
and the resulting design optimizations, the stated purpose of this study has been
accomplished.
165
7.0 RECOMMENDED IMPROVEMENTS
The preliminary design level results generated in this study for the design of space
launch vehicles are very useful in the continuous effort to improve space transportation
systems. For future design optimization work, it is recommended that the following
improvements be implemented.
7.1 Types of Solid and Liquid Propellants
As mentioned in Chapter 6, the choice of propellants available in the optimization
process can greatly affect the results. Expanding the different types of propellants
available for the genetic algorithm (GA) to choose or allowing the GA to design the
propellants directly could improve the performance of the launch vehicles. Specifically,
more energetic propellants would increase the I
sp
and characteristic exhaust velocity (c*)
which should lower the overall weight of the propellants.
For the solid propellants, two additional types of propellants (DB/APHMX/Al
and the energetic Star 37) were added to the existing list of available solid propellant
types. As it turned out, the GA ended up choosing these propellants in the design
optimizations that were run. On the liquid propellant side, the choices were restricted to
only a few types of storable propellant combinations (LOX/Kerosene, LOX/Hydrazine,
etc.) Allowing the GA to choose one of the most common propellant combinations,
LOX/LH2, should greatly enhance the liquid propellant launch vehicle results.
166
7.2 Aerodynamics Model
The aerodynamics model used in this study, AeroDesign, is configured to model
only certain aerodynamic shapes. AeroDesign can only analyze conical and ogive shaped
nosecones and cylindrical shaped center bodies. For the most part, this is not a problem
since many rocket and missile designs employ these shapes. However, for more complex
vehicle designs, AeroDesign needs to be altered in order to generate the aerodynamic
characteristics of different shaped vehicles.
An improvement on the modeling for variable diameter vehicles should be done.
A very basic estimate for the increased drag associated with variable diameter shapes was
used in this study. This estimate was probably on the high side since most launch
vehicles have skirts and interstages that allow for a smooth geometric transition between
stages of differing diameters. Also, AeroDesign needs to be modified in order to analyze
the aerodynamic characteristics of strapon boosters common in many modern launch
vehicle designs.
7.3 SixDegreeof Freedom (6DOF) Flight Dynamics Simulator
The 6DOF flight dynamics simulator used in this study analyzed a basic ballistic
flight trajectory. Future work should investigate using the autopilot and pronav
applications in the 6DOF flight dynamics simulator to allow for greater accuracy in
attaining the desired orbital altitude and orbital velocity. For this preliminary study, the
use of the ballistic trajectory provided the required analysis for the flight of the vehicle.
However, an autopilot and inertial navigation system would greatly enhance the flight
trajectory with the subsequent benefit of optimizing the amount of payload to orbit.
These capabilities are currently available in the 6DOF flight dynamics simulator. They
167
should be implemented in order to investigate another aspect of the design of space
launch vehicles: the optimization of launch vehicle flight trajectory.
For this study, the use of a coast time between stages and an orbit insertion burn
were modeled in the fourstage solid propellant launch vehicle. This type of modeling
needs to be extended to all the launch vehicle types. Like the autopilot and pronav
applications, coast times and an orbit insertion burn would greatly increase the accuracy
of the launch vehicle in attaining the desired orbit.
7.4 Cost Model
The TRANSCost 7.1 cost model used in this study provided excellent insight into
the calculation of cost per launch values for optimized space launch vehicles. However,
the model could be improved to more accurately reflect insurance costs as well as ground
and flight operations costs.
7.5 Payload Masses and Orbits
Finally, performing design optimizations using a wider variety of payload masses
would provide even more insight to launch vehicle design. A majority of the payloads
used in this study weighed 1,000 pounds so that vehicle comparisons could be made.
Also, this payload class is what the USAF is looking for when it comes to responsive
space launch vehicles. However, the USAF, along with civil agencies and commercial
companies, launch much larger payloads into orbit. The launch vehicles that carry these
types of payloads need to be analyzed as well.
Also, all of the launch vehicles in this study carried their payloads to lowEarth,
circular orbits. Future work should look at launches to semisynchronous orbit as well as
to geosynchronous transfer orbit.
168
Finally, the two primary United States launch sites, Vandenberg AFB, CA and
Cape Canaveral AFS, FL were the only two launch sites considered in this study. While
these two sites are the primary launch sites for the USAF, other worldwide launch sites
are used for space missions. Alternative launch sites such as Wallops Island, VA or
Kourou, French Guiana should be considered for future work.
169
REFERENCES
1. Chilton, K.P., ?Assured Access to Space in a Competitive World?, High Frontier: The
Journal for Space and Missile Professionals, Vol. 3, No. 1, 2006, pp. 23.
2. System Description: Force Application and Launch from CONUS (FALCON),
http://www.globalsecurity.org/space/systems/falcon.htm.
3. Foulds, L.R., Optimization Techniques: An Introduction, SpringerVerlag, New York,
NY, 1981.
4. Siddal, J.N., Optimal Engineering Design: Principles and Applications, Mercell
Dekker Inc., New York, NY, 1982.
5. Gabasov, R.F. and Kirillova, F.M., Methods of Optimization, Optimization Software
Inc., New York, NY, 1988.
6. Fisher, R.A., The Design of Experiments: 8
th
Edition, Hafner Publishing Company,
New York, NY, 1971.
7. Weber, D.C. and Skillings, J.H., A First Course in the Design of Experiments: A
Linear Models Approach, CRC Press, Boca Raton, FL, 2000.
8. Rodriguez, D.L., ?Response Surface Based Optimization with a Cartesian CFD
Method?, AIAA Paper 20030465, Presented at the 41
st
AIAA Aerospace Sciences
Meeting, Reno, NV, January 2003.
9. Mitchell, M., An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA,
1998.
170
10. Coley, D.A., An Introduction to Genetic Algorithms for Scientists and Engineers,
World Scientific Publishing Co., Singapore, 1999.
11. Billheimer, J.S., ?Optimization and Design Simulation in Solid Rocket Design?,
AIAA Paper 68488, Presented at the 3
rd
AIAA Solid Propulsion Conference, June 1968.
12. Woltosz, W.S., ?The Application of Numerical Optimization Techniques to Solid
Propellant Rocket Motor Design?, Masters Thesis, Auburn University, March 1977.
13. Foster, W.A., Jr., and Sforzini, R.H., ?Optimization of Solid Rocket Motor Igniter
Performance Requirements?, AIAA Paper 781016, Presented at the 14
th
AIAA/SAE
Joint Propulsion Conference, Las Vegas, NV, July 1978.
14. Sforzini, R.H., ?An Automated Approach to Design of Solid Rockets Utilizing a
Special Internal Ballistics Model?, AIAA Paper 801135, Presented at the 16
th
AIAA/SAE/ASME Joint Propulsion Conference, July 1980.
15. Washington, W.D., ?Missile Aerodynamic Design Program?, 1980, 1990.
16. Anderson, M.B., ?Design of a Missile Interceptor Using a Genetic Algorithm?,
Doctoral Dissertation, Auburn University, 1998.
17. Etkin, Bernard, Dynamics of Atmospheric Flight, John Wiley & Sons Inc., New
York, NY, 1972.
18. Burkhalter, J.E., Jenkins, R.M., and Hartfield, R.J., ?Genetic Algorithms for Missile
Analysis ? Final Report?, Submitted to Missile and Space Intelligence Center, Redstone
Arsenal, AL, 35898, February 2003.
19. Hartfield, R.J., Burkhalter, J.E., Jenkins, R.M., and Metts, J.G., ?Genetic Algorithm
Upgrade ? Final Report?, Submitted to Missile and Space Intelligence Center, Redstone
Arsenal, AL, 35898, June 2005, Reference Contract No. HHM40204P0061.
171
20. Metts, J.G., ?Determination of Key Parameters for Reverse Engineering Solid Rocket
Powered Missiles?, Masters Thesis, Auburn University, 2006.
21. Riddle, D.B., ?Design Tool Development for Liquid Propellant Missile Systems?,
Masters Thesis, Auburn University, 2007.
22. Hartfield, R.J., Jenkins, R.M., and Burkhalter, J.E., ?Ramjet Powered Missile Design
Using a Genetic Algorithm,? AIAA Paper 20040451, Presented at the 42
nd
AIAA
Aerospace Sciences Meeting, Reno, NV, January 2004.
23. Budianto, J., and Olds, J., ?Design and Deployment of a Satellite Constellation Using
Collaborative Optimization,? Journal of Spacecraft and Rockets, Vol. 41, No. 6, 2004,
pp. 956963.
24. Lian, Y., and Liou, M., ?MultiObjective Optimization of Transonic Compressor
Blade Using Evolutionary Algorithm,? Journal of Propulsion and Power, Vol. 21, No. 6,
2005, pp. 979987.
25. Brown, N., and Olds, J., ?Evaluation of Multidisciplinary Optimization (MDO)
Technique Applied to a Reusable Launch Vehicle,? AIAA Paper 2005707, Presented at
the 43
rd
AIAA Aerospace Sciences Meeting, Reno, NV, January 2005.
26. Tsuchiya, T., and Mori, T., ?Orbital Design of TwoStagetoOrbit Space Planes with
Airbreathing Engines,? Journal of Spacecraft and Rockets, Vol. 42, No. 1, 2005, pp. 90
97.
27. Casalino, L., and Pastrone, D., ?Optimal Design and Control of Hybrid Rockets for
Access to Space,? AIAA Paper 20053547, Presented at the 41
st
AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Tucson, AZ, July 2005.
172
28. Anderson, M.B., Burkhalter, J.E., and Jenkins, R.M., "Design of an Air to Air
Interceptor Using Genetic Algorithms", AIAA Paper 994081, Presented at the AIAA
Guidance, Navigation and Control Conference, Portland, OR, August 1999.
29. Anderson, M.B., Burkhalter, J.E., and Jenkins, R.M., ?Intelligent Systems Approach to
Designing an Interceptor to Defeat Highly Maneuverable Targets?, AIAA Paper 2001
1123, Presented at the 39
th
AIAA Aerospace Sciences Meeting, Reno, NV, January 2001.
30. Chernyavsky, B., Stepanov, V., Rasheed, K., Blaize, M., and Knight, D., ?3D
Hypersonic Inlet Optimization Using a Genetic Algorithm?, AIAA Paper 983582,
Presented at the 34
th
AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Cleveland,
OH, July 1998.
31. Shelton, J., Frederick, R., Wilhite, A., ?Launch Vehicle Propulsion Design with
Multiple Selection Criteria,? AIAA Paper 20053851, Presented at the 41
st
AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Tucson, AZ, July 2005.
32. Portz, R., ?Launch Vehicle Design Features for Minimum Cost,? AIAA Paper 2004
3562, Presented at the 40
th
AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Fort
Lauderdale, FL, July 2004.
33. Braun, R.D., Moore, A.A., and Kroo, I.M., ?Collaborative Approach to Launch
Vehicle Design,? Journal of Spacecraft and Rockets, Vol. 34, No. 4, 1997, pp. 478486.
34. Wollam, J., Kramer, S., and Campbell, S., ?Reverse Engineering of Foreign Missiles
via Genetic Algorithm,? AIAA Paper 20000685, Presented at the 38
th
AIAA Aerospace
Sciences Meeting, Reno, NV, January 2000.
173
35. Anderson, M.B., Burkhalter, J.E., and Jenkins, R.M., ?MultiDisciplinary Intelligent
Systems Approach to Solid Rocket Motor Design, Part I: Single and Dual Goal
Optimization,? AIAA Paper 20013493, Presented at the 37
th
AIAA/ASME/SAE/ASEE
Joint Propulsion Conference, Salt Lake City, UT, July 2001.
36. Anderson, M.B., Burkhalter, J.E., and Jenkins, R.M., ?MultiDisciplinary Intelligent
Systems Approach to Solid Rocket Motor Design, Part II: Multiple Goal Optimization,?
AIAA Paper 20013494, Presented at the 37
th
AIAA/ASME/SAE/ASEE Joint Propulsion
Conference, Salt Lake City, UT, July 2001.
37. Briggs, G.P., Ray, T., and Milthorpe, J.F., ?Evolutionary Algorithm use in
Optimization of a Launch Vehicle Stack Model,? AIAA Paper 2007364, Presented at the
45
th
AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2007.
38. Wertz, J.R., ?Responsive Launch Vehicle Cost Model,? AIAA Paper RS220042004,
Presented at the 2
nd
AIAA Responsive Space Conference, Los Angeles, CA, April 2004.
39. Unal, R., Braun, R.D., Moore, A.A., and Lepsch, R.A., ?Design Optimization for
Cost Using Genetic Algorithms,? Proceedings of the 17
th
Annual Conference of the
International Society of Parametric Analysts, May 1995, pp. 183191.
40. Olds, J.R. and Lee, H., ?Application of a New Economic Analysis Tool to a Two
StagetoOrbit RBCC Launch Vehicle,? AIAA Paper 19964092, Presented at the 6
th
AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization,
Bellevue, WA, September 1996.
41. Koelle, D.E., Handbook of Cost Engineering for Space Transportation Systems,
Revision 1, with TRANSCost 7.1, TransCostSystems, Ottobrunn, Germany, 2003.
174
42. Wertz, J.R., ?Economic Model of Reusable vs. Expendable Launch Vehicles?,
Presented at the IAF Congress, Rio de Janeiro, Brazil, October 2000.
43. Anderson, M.B., ?Users Manual for IMPROVE
?
Version 2.8: An Optimization
Software Package Based on Genetic Algorithms?, Sverdrup Technology Inc. / TEAS
Group, Eglin AFB, FL, March 6, 2001.
44. Burkhalter, J.E., ?Rocket Propulsion II Course Notes?, Auburn University, Auburn,
AL, 1985.
45. Sforzini, R.H., ?Notes on SolidPropellant Rocket Motors Including Jet Propulsion
Fundamentals?, Auburn University Engineering Extension Service, 1983.
46. Gage, P., and Kroo, I., ?A Role of Genetic Algorithms in a Preliminary Design
Environment,? AIAA Paper 19933933, Presented at the AIAA Aircraft Design, Systems,
and Operations Meeting, Monterey, CA, August 1993.
47. Cosner, R.R., Oberkampf, W.L., Rumsey, C.L., Rahaim, C.L., Shih, T.P., ?AIAA
Committee on Standards for Computational Fluid Dynamics: Status and Plans?, AIAA
Paper 2006889, Presented at the 44
th
AIAA Aerospace Sciences Meeting and Exhibit,
Reno, NV, January 2006.
48. Roy, C.J., ?Introduction to Computational Fluid Dynamics Course Notes?, Auburn
University, Auburn, AL, 2006.
49. Riddle, D.B., Hartfield, R.J., Burkhalter, J.E., Jenkins, R.J., ?Genetic Algorithm
Optimization of Liquid Propellant Missile Systems,? AIAA Paper 2007362, Presented at
the 45
th
AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2007.
50. Humble, R.W., Henry, G.N., and Larson, W.J., Space Propulsion Analysis and
Design, The McGrawHill Companies, Inc., New York, NY, 1995.
175
51. U.S. Air Force Fact Sheet, LGM30G Minuteman III ICBM, Air Force Space
Command, November 2006.
52. System Description: Weapons of Mass Destruction, LGM30 Minuteman III,
http://www.globalsecurity.org.
53. Minuteman ICBM History Website, Minuteman ICBM Diagrams, Minuteman III ?
Ballistic Flight Profile, http://www.geocities.com/minuteman_missile/index.html.
54. Minotaur I User?s Guide, Release 2.1, Orbital Sciences Corporation, January 2006.
55. System Description: Orbital Sciences Corporation, Space Launch Systems, Minotaur
Space Launch Vehicle, http://www.orbital.com/SpaceLaunch/Minotaur/index.html.
56. U.S. Air Force Fact Sheet, Titan II Space Launch Vehicle, Air Force Space
Command, March 2001.
57. System Description: Space, Titan II SLV, http://www.globalsecurity.org.
58. AirLaunch LLC Fact Sheet, QuickReach
TM
Launch Vehicle, November 2006.
59. System Description: Directory of U.S. Military Rockets and Missiles, AirLaunch
QuickReach, http://designationsystems.net/dusrm.
60. Sutton, G.P., Rocket Propulsion Elements: An Introduction to the Engineering of
Rockets, 6
th
Edition, John Wiley & Sons, Inc., New York, NY, 1992.
61. Sellers, J.J., Astore, W.J., Giffen, R.B., and Larson, W.J., Understanding Space: An
Introduction to Astronautics, 3
rd
Edition, McGrawHill Primis Custom Publishing, New
York, NY, 2003.
62. System Description: Zenit Rocket, http://en.wikipedia.org/wiki/Zenit_rocket.
63. NASA Exploration Systems Architecture Study (ESAS) ? Final Report, November
2005, pp. 669710.
176
APPENDIX: Mass Table Example for ThreeStage Solid Propellant Launch Vehicle
Part Name mass xcg ixx iyy izz
blunt nose 5.16 3.47 421.7 354.3 354.3
ogive section 98.66 47.99 74196.5 170582.7 170582.7
payload 1000.00 39.67 0.0 1573340.4 1573340.4
electronics 100.00 91.33 0.0 834129.9 834129.9
curved bulk 3 31.60 106.86 11200.8 366521.5 366521.5
ignitor 3 15.00 106.86 0.0 171275.5 171275.5
press vessel 3 318.25 174.35 220358.8 10318916.4 10318916.4
liner 3 87.55 174.35 60406.1 2838684.6 2838684.6
insulation 3 461.79 177.87 318607.7 15545957.4 15545957.4
nozzle 3 132.62 275.32 458636.9 9954230.1 9954230.1
grain 3 12754.48 175.22 5411860.2 412149855.0 412149855.0
curved bulk 2 99.04 305.65 51602.5 9278345.8 9278345.8
ignitor 2 26.00 305.64 0.0 2428887.2 2428887.2
press vessel 2 1056.52 392.56 1075261.6 166281361.0 166281361.0
liner 2 136.12 392.56 137662.8 21422199.8 21422199.8
insulation 2 717.93 396.84 726092.7 115412630.9 115412630.9
nozzle 2 350.34 510.18 5307228.4 91819317.4 91819317.4
grain 2 18965.35 393.62 13556946.2 2989514231.7 2989514231.7
curved bulk 1 125.14 537.55 82393.7 36203426.3 36203426.3
ignitor 1 26.00 537.55 0.0 7512900.0 7512900.0
press vessel 1 2233.91 704.27 2872844.1 1131358075.5 1131358075.5
liner 1 287.90 704.27 368185.4 145806550.1 145806550.1
insulation 1 1518.52 709.08 1941968.7 779359905.1 779359905.1
nozzle 1 419.34 905.86 9145618.0 343356082.9 343356082.9
grain 1 48939.05 705.46 44013937.8 24812033772.8 24812033772.8