A Phased Trade Study for the Design of a Solid Rocket Motor
by
Andrew Joseph Gandia
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez - Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
April, 2012
(For Graduation August, 2012)
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ........................................................................................................ vi
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
2. Theory and Methodology ............................................................................................ 3
Fundamentals of Rocket Design .................................................................................. 3
Solid Rocket Motor...................................................................................................... 3
Propellants ................................................................................................................... 5
Thrust Calculations ...................................................................................................... 5
Chamber Pressure ........................................................................................................ 6
Range Equation ............................................................................................................ 6
Design Approach ......................................................................................................... 7
Preliminary Design............................................................................................. 7
Trade Studies ...................................................................................................... 8
Final Design ....................................................................................................... 9
3. Results & Discussion ................................................................................................... 10
3.1 Preliminary Design .............................................................................................. 10
Preliminary Design Motor Dimensions ........................................................... 10
Propellant Characteristics................................................................................. 11
Weight Build-Up of Missile ............................................................................. 11
Length Build-Up .............................................................................................. 12
Thrust versus Time ........................................................................................... 13
Nozzle Design .................................................................................................. 14
Chamber Pressure and Hoop Stress ................................................................. 14
ii
Range and Velocity .......................................................................................... 15
Overall Compliance ......................................................................................... 17
3.2 Material Trade ..................................................................................................... 18
3.3 Propellant Trade................................................................................................... 20
3.4 Dimensional Trade............................................................................................... 21
3.5 Final Design ......................................................................................................... 23
Final Design Motor Dimensions ...................................................................... 23
Propellant Characteristics................................................................................. 24
Thrust versus Time ........................................................................................... 26
Nozzle Design .................................................................................................. 27
Chamber Pressure and Hoop Stress ................................................................. 27
Range and Velocity .......................................................................................... 28
Overall Compliance ......................................................................................... 30
4. Conclusion ................................................................................................................... 31
5. References.................................................................................................................... 33
Appendices ...................................................................................................................... 34
Appendix A – Validation of Tools ............................................................................ 34
Appendix B – Summary of Tools .............................................................................. 36
iii
LIST OF TABLES
Table 1: Preliminary Design Motor Inputs ...................................................................... 10
Table 2: Preliminary Design Propellant Inputs ............................................................... 11
Table 3: Preliminary Design Weight Inputs .................................................................... 12
Table 4: Preliminary Design Dimensional Inputs ........................................................... 13
Table 5: Preliminary Design Nozzle Inputs ..................................................................... 14
Table 6: Preliminary Design Compliance Table ............................................................. 17
Table 7: Motor Case Materials [11]................................................................................. 18
Table 8: Chamber Temperature and Material Melting Temperatures ............................. 19
Table 9: Propellant Properties ......................................................................................... 20
Table 10: Propellant Trade Results.................................................................................. 21
Table 11: Final Design Motor Inputs............................................................................... 24
Table 12: Final Design Propellant Inputs ........................................................................ 25
Table 13: Final Design Weight Inputs ............................................................................. 25
Table 14: Final Design Dimensional Inputs .................................................................... 26
Table 15: Final Design Nozzle Inputs ............................................................................. 27
Table 16: Final Design Compliance Table ...................................................................... 30
iv
LIST OF FIGURES
Figure 1: Air to Air Missile [1].......................................................................................... 1
Figure 2: AIM-120A Missile [2] ....................................................................................... 1
Figure 3: Solid Rocket Schematic [6] ................................................................................ 3
Figure 4: Grain Cross Sections [6] .................................................................................... 4
Figure 5: Surface Recession of a Solid Propellant [4] ....................................................... 4
Figure 6: Trade Study Approach for Motor Design .......................................................... 7
Figure 7: Thrust versus Time of Preliminary Design ...................................................... 13
Figure 8: Chamber Pressure versus Time of Preliminary Design ................................... 15
Figure 9: Hoop Stress versus Time of Preliminary Design ............................................. 15
Figure 10: Range versus Time of Preliminary Design .................................................... 16
Figure 11: Velocity versus Time of Preliminary Design ................................................. 16
Figure 12: BurnSim Output for Thrust versus Time of Preliminary Design ................... 17
Figure 12: Stress versus Time of Preliminary Design and Materials .............................. 19
Figure 13: Thrust versus Time of Different Propellant Configuration ............................ 21
Figure 14: Thrust versus Time for different charge lengths ............................................ 22
Figure 15: Thrust versus Time for different conduit diameters ....................................... 23
Figure 17: Thrust versus Time of Final Design ............................................................... 27
Figure 18: Chamber Pressure versus Time of Final Design ............................................ 28
Figure 19: Hoop Stress versus Time of Final Design ...................................................... 28
Figure 20: Range versus Time of Final Design ............................................................... 29
Figure 21: Velocity versus Time of Final Design ........................................................... 29
Figure 22: BurnSim Output for Thrust versus Time of Final Design .............................. 30
Figure 23: Thrust vs. Time for Analytical Approach ...................................................... 34
Figure 24: Thrust vs. Time for BurnSim ......................................................................... 34
Figure 25: Thrust vs. Time Comparison .......................................................................... 35
v
LIST OF SYMBOLS
γ: ratio of specific heats
ε: expansion ratio
ρpropellant: density of propellant (
kg
)
m3
σ: hoop stress (Pa)
Aburn: burn area (m2)
Aexit: nozzle exit area (m2)
A*: throat area (m2)
dcharge: charge diameter (m)
dconduit: conduit diameter (m)
dexit: Exit Diameter (m)
dmotor: motor diameter (m)
FN: thrust (N)
CF,Actual: Actual Thrust Coefficient
k: burning rate constant (
m
)
Pa n
lcharge: charge length (m)
lguidance system: guidance system length (m)
lmissile: total length (m)
lnozzle: nozzle length (m)
ltip: tip length (m)
lwarhead: warhead length (m)
Mw: molecular weight (
g
)
mol
mcasing: casing mass (kg)
mguidance system: guidance system mass (kg)
mmissile: missile mass (kg)
mmiscellaneous comp: miscellaneous components mass (kg)
mpropellant: propellant mass (kg)
mt: mass of missile at current time step (kg)
vi
mwarhead: warhead mass (kg)
n: burning rate exponent
Pchamber: chamber pressure (Pa)
R: gas constant for propellant exhaust gases (
J
)
kg * K
rb: burn rate (m/s)
Δt: time step (s)
tburn: burn time
tcasing: casing thickness (m)
Tchamber: Chamber Temperature (K)
tinsulation: insulation thickness (m)
vburnout: Velocity at burn-out (m/s)
Vt: Velocity of missile at current time step (m/s)
Vt+dt: Velocity of missile at next time step (m/s)
x: displacement (m)
X*: X-Function at the throat
xt+dt: displacement at new time step (m)
xt: displacement at current time step (m)
vii
ACKNOWLEDGMENT
I would first like to thank my Heavenly Father, who has always given me strength and
guidance throughout my life. I would like to recognize my family, who have supported
me since the beginning and have always encouraged me to follow my dreams. I would
also like to recognize Professor Ernesto, who was very supportive in helping me with
this project and throughout my graduate career at Rensselaer.
viii
ABSTRACT
This project utilized a phased trade study approach to design a solid rocket motor for an
air to air missile. The focus of the study was to provide insight into the sensitivity of
changing attributes of a missile motor and noting the impacts that resulted using that
approach. Included in this effort was the utilization of analysis with propulsion
principles and BurnSim Software. To meet the requirements of the motor and missile,
the project was split into three different phases: Development of a Preliminary Design,
trade study sensitivity analyses, and Final Design development. The Preliminary Design
consisted of creating a baseline for the missile and motor that the different sensitivity
studies could be performed on. Although non-compliant to the mission requirements, the
Preliminary Design provided key propulsion outputs that would later be refined and
assessed in the Final Design. The trade studies provided an observation into the
sensitivities of changing different attributes of the motor in the areas of materials,
propellants, and dimensions (charge length and conduit diameter). These trades provided
sensitivity results that were applicable in creating the Final Design.
The Final Design
utilized the Preliminary Design and lessons learned from the trade studies to size the
motor and the missile. The Final Design resulted in a compliant design to the
requirements and validated the application of a phased trade study approach in
understanding the sensitivities in motor and missile design.
ix
1. Introduction
Design for missile solid rocket motors is critical for modern warfare. Missiles are used
amongst militaries all over the world, and provide the capability to precisely strike
specified targets. Missions for different missiles are dependent on the specified type and
required capability. Different applications of missiles include ballistic missiles, anti-ship
missiles, and air to air missiles (Figure 1).
Figure 1: Air to Air Missile [1]
An example of an air to air missile is the AMRAAM AM-120 (Figure 2). This medium
range missile was developed as a result of a Joint Service Operational Requirement in
the post-1985 timeframe. It is powered by a solid rocket motor and can achieve a speed
of Mach 4 in a range in excess of 30 miles. In long range engagements, the AMRAAM
utilizes inertial guidance and receives updated target information from the launch
aircraft. Aircraft utilizing this weapon system today include the F-15, and F-16, and F-18
fighters. [1]
Figure 2: AIM-120A Missile [2]
1
With the understanding of mission requirements and the configuration of the missile, the
design of a motor was investigated to validate compliance to specific requirements.
Richard Nakka [3], R. Clay Hainline [4], and NASA [5] have also addressed numerous
attributes of design of rockets.
This study focused on the design of a solid rocket motor for an air to air missile under
ideal conditions. The ideal conditions included steady state conditions (mass flow in the
chamber equaling mass flow rate at the throat and exit at each time step), constant values
for the ratio of specific heat and molecular weight, and all exhaust gases exiting the
nozzle were assumed to be in the axial direction.
For the requirements, the missile designed needed to have a total mass no greater than
100 kg. The warhead mass provided was 10 kg and the guidance system allocation was
20 kg. The overall length could not exceed 4 meters. The missile needed to also operate
at an altitude of 3,000 m and the motor needed to burn for at least 2 s.
The design approach included a phased approach:
a) A Preliminary Design
b) Trade Studies
c) Final Design
This process provides the framework to develop a compliant design.
2
2. Theory and Methodology
Fundamentals of Rocket Design
Rockets are the oldest type of aerospace propulsion system, dating back 2000 years to
the Han Dynasty. Rockets are propulsion systems that produce thrust by accelerating
mass through a nozzle. Chemical reactions are relied on by many rockets as an energy
source. The propellant utilized in rockets consists of fuel and oxidizer components. The
oxidizer is necessary since rockets are non air-breathing engines. Chemical rockets are
categorized as: Liquid Propellant Rockets, Solid Propellant Rockets, or hybrid propellant
rockets. [6]
Solid Rocket Motor
The solid rocket motor contains the propellant to be burned within a casing. The
propellant takes the solid form called a grain, and once ignited, burns on the surfaces that
are not inhibited by the case [7] [Figure 3].
Figure 3: Solid Rocket Schematic [6]
The grain cross section is a factor in impacting the thrust versus time and performance of
the rocket. For example, as seen from Figure 4, a propellant with a tubular grain design
will have a thrust that will increase over time, while a rod and tube design will have a
3
thrust that is relatively stable over time. The selection of grain design is dependent on
the application.
Figure 4: Grain Cross Sections [6]
In addition, there are two types of burning for a solid rocket motor. The first, end
burning charges, are designed so that burning begins at one end of the chamber until
reaching the other end. The second, and more prominently utilized, is radial burning,
where the propellant cross sectional area changes with time. Radial burning provides
high values of thrust for shorter durations of burn time. Figure 5 shows the surface
recession of a solid propellant with a cross pattern.
Figure 5: Surface Recession of a Solid Propellant [4]
4
Propellants
Solid Rocket Motor Propellants are categorized as homogeneous, heterogeneous, or
composite modified double base. The selection of the propellant, as well as the
determination of the different input parameters of the motor, are fundamental to the
design of the rocket motor. A key parameter for understanding the impact of a propellant
is the burning rate. The law that correlates the burning rate of a propellant to the critical
pressure and propellant characteristics is Vielle’s Law [6]:
rb 
k (10 5 Pc ) n
 k ' Pcn
1000
k ' Pcn  k10 5n 3
(1)
(2)
Where k represents the burning rate constant and n represents the burning rate exponent
for a given propellant.
Thrust Calculations
To determine the thrust, three parameters were determined depending on the design of
the rocket and selection of propellant. These parameters were the thrust coefficient Cf ,
the chamber pressure Pc , and the throat area A*. Utilizing these parameters, the thrust of
the rocket was found utilizing the following expression [6]:
FN  CF , Actual PChamber A*
The thrust over time is dependent on the change in chamber pressure. The chamber
pressure varies with the burn area over time and will be discussed.
5
(3)
Chamber Pressure
The chamber pressure was calculated utilizing input parameters of the rocket from the
expression [6]:
1
Pchamber
  propellantAburnk (10 5 ) RT chamber  1n


1000 A* X *


(4)
Chamber pressure plays a critical role in determining the hoop stress experienced by the
casing of the solid propellant from the expression below [4].

Pchamberd ch arg e ( FactorofSafety)
2 * t ca sin g
(5)
This equation drove the decision of an appropriate material to be utilized.
Range Equation
A fundamental element to meeting missile specifications is understanding the range and
velocity of a rocket. For this project, the following expressions were utilized to calculate
these parameters at different time steps during the motor burn time.
Velocity Equation at each time step:
Vt dt  Vt 
mt
FN
* t
mt
(6)
dv
 FN
dt
(7)
dx
dt
(8)
v
6
Range Equation for each time step:
xt dt  xt  Vt dt t
(9)
Design Approach
To assess the mission objectives for the air to air missile mission, a trade study approach
was developed. The goal of this trade study approach was to first develop a Preliminary
Design that was utilized as a Baseline to perform trade studies. Once sensitivity trade
studies were analyzed, the Preliminary Design and the lessons learned from the trade
studies were utilized to create a Final Design. Figure 6 displays the trade study approach.
Preliminary Design
Trade Studies
Final Design
Figure 6: Trade Study Approach for Motor Design
Preliminary Design
Analytical Approach
To begin the effort, an analytical Preliminary Design was prepared. The overall goal of
this Preliminary Design was to develop a motor and missile to serve as a baseline for
later trades.
7
The necessary input parameters necessary for the motor and missile included propellant
properties, motor diameter, various lengths (charge length, nozzle length), various
thicknesses (motor casing and insulation), nozzle diameter, and material.
Utilizing these inputs, solid rocket propulsion theory equations were used to size the
motor rocket and missile [6]. This approach included analytical calculations based upon
the provided inputs to size the motor. Included in the dependant variables were nozzle
exit area, throat area, chamber pressure, burn time, total mass, thrust, and range.
The final product included tables and plots documenting the results and assumptions for
the Preliminary Design.
BurnSim Comparison
The results of the Preliminary Design through the analytical approach were compared to
the simulation results from the BurnSim Software. BurnSim Software is a tool that
utilizes different input parameters such as charge length, conduit diameter, charge
diameter, propellant properties, and nozzle properties to generate a thrust versus time
plot. This was used as an independent assessment to validate the analytical results.
Trade Studies
Once all parameters were calculated and a Preliminary Design was prepared, the second
phase of the trade study process was to conduct sensitivity analyses for different
attributes of the motor. The following trades were performed for this project:
Material Trade: Utilization of different materials for the motor casing.
Propellant Trade: Selection of different propellants with different parameters as that of
the Preliminary Design.
Dimensional Trade: Changing the charge length and changing the conduit diameter of
the rocket to understand the impact to output parameters. The conduit diameter is
defined as the diameter of the opening within the solid propellant.
8
Final Design
Utilizing the Preliminary Design and the trade study results, a Final Design of the motor
and missile was assessed to meet the mission requirements. A similar analytical
approach to the Preliminary Design to perform iterations of the motor design utilizing
the lessons learned from the trade study was utilized. In addition, the BurnSim Software
was used at this stage to validate the Final Design based upon on the Preliminary Design
and trades. The result of the Final Design included documentation of the output
parameters, a comparison chart of the parameters utilized in the Preliminary Design
versus the Final Design, and a comparison plot from BurnSim.
9
3. Results & Discussion
3.1 Preliminary Design
The Preliminary Design was developed as a starting point to perform the trades to
develop the final motor and missile design. This design is not fully compliant, and the
later trades will investigate different attributes of the missile to meet compliance. This
section reviews the inputs decided upon for the Preliminary Design based on research of
similar missiles and results in a comparison to the requirements.
Preliminary Design Motor Dimensions
The motor diameter was sized based on dimensions similar to those of the AIM-9 family
of Sidewinder missiles [8]. Table 1 reviews the key inputs.
Symbol
Parameter
Dimension
Unit
dmotor
Motor Diameter
0.124
meters
dcharge
Charge Diameter
0.117
meters
lcharge
Charge Length
1.77
meters
tcasing
Casing Thickness
0.0025
meters
tinsulation
Insulation Thickness
0.001
meters
dconduit
Conduit Diameter
0.0784
meters
Table 1: Preliminary Design Motor Inputs
The charge diameter was determined based on the thickness of the casing and insulation
thickness. The casing thickness was determined based upon a motor case sized in [6].
The insulation thickness was determined based on an initial estimate. Further
investigation was conducted for the Final Design for a more accurate insulation
thickness. The charge length was estimated based on a drawing of the AIM-9M [8]. To
determine the size and shape of the conduit diameter, an initial estimate was determined
based upon the sizing of the motor. [6] Further research was performed on the conduit
10
diameter size for the Final Design. The conduit shape was chosen as tubular due to the
need to increase the thrust as the missile reaches its designated target.
Propellant Characteristics
The propellant selected for the Preliminary Design utilized a propellant mixture of
Ammonium Perchlorate (Oxidizer) and Carboxyl-Terminated Polybutadiene (Fuel
Binder). The selection of this propellant was based upon an initial assumption of a need
for a high burn rate constant, which has a large impact on the thrust. This propellant was
eventually compared against other solid motor propellants to determine an appropriate
propellant for the Final Design. Specific propellant characteristics are given in Table 2
[6]
Symbol
Parameter
Dimension
Unit
Tchamber
Chamber Temperature
3370
K
n
Burning Rate Exponent
0.40
--
k
Burning Rate Constant
4.0
--
ρpropellant
Density (kg/m3)
1772
kg/m3
Mw
Molecular Weight
29.3
(g/mol)
R
Gas Constant
284
(J/Kg*K)
γ
Ratio of Specific Heats
1.17
--
C*
Characteristic Velocity
1575
m/s
Table 2: Preliminary Design Propellant Inputs
Weight Build-Up of Missile
The weight of the missile was determined utilizing sizing techniques and estimates based
upon similar sized missiles. Specific values for the mass build-up are given in Table 3.
11
Symbol
Parameter
Dimension
Unit
mwarhead
Warhead Mass
10
kg
mguidance system
Guidance System Mass
20
kg
mpropellant
Propellant Mass
18.6
kg
mcasing
Casing Mass
7.92
kg
mmiscellaneous comp
Miscellaneous Components Mass
14.13
kg
mmissile
Total
70.65
kg
Table 3: Preliminary Design Weight Inputs
The warhead mass was estimated based upon the AIM-9E warhead mass of
approximately 10 kg. [9] The guidance system mass was estimated comparing a
theoretical guidance system allocation and length and utilizing ratio techniques that
value versus the AIM-9M guidance system length [8]. For the Preliminary Design, a
casing thickness was utilized for the entire missile. The material selected for this was
Aluminum 2090-T86, driven by the preliminary selection for a lightweight solution. The
casing mass was determined utilizing an approach mentioned in [10] of utilizing a
hollow cylinder as a representation for the mass of the casing. The miscellaneous
components mass was 25% of the sum of all other components and is representative of
items such as miscellaneous structure, insulation, and fin mass. The factor was based on
a conservative approach to ensure that all other miscellaneous components were
accounted for in the mass build-up. 25% seems reasonable since the total mass of a
missile, such as the AIM-9B, is 70 kg [9]. Note that the casing and propellant will
change as the trades are conducted, but the 25% provides a strong justification for
miscellaneous mass throughout the project.
Length Build-Up
The length of the missile components was based off similar dimensions of that of an
AIM-9M missile drawing [8]. This is critical to understanding the length restrictions as
the charge length is changed for different trade studies. Specific values are given in
Table 4.
12
Symbol
Parameter
Dimension
Unit
lnozzle
Nozzle Length
0.05
m
lcharge
Charge Length
1.77
m
lwarhead
Warhead Length
0.47
m
lguidance system
Guidance System Length
0.79
m
ltip
Tip Length
0.18
m
lmissile
Total Length
3.26
m
Table 4: Preliminary Design Dimensional Inputs
Thrust versus Time
The computed thrust versus time plot is shown for the Preliminary Design. The burn
time for the rocket motor was approximately 0.8 seconds, which is smaller than the
provided requirement. The thrust profile did correlate to the expected burn profile of a
tubular shaped design, where thrust increased with time. The thrust vs time plot is shown
in Figure 7.
Thrust (N)
Thrust vs Time
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
0.1
0.2
0.3
0.4
0.5
Seconds
0.6
0.7
Figure 7: Thrust versus Time of Preliminary Design
13
0.8
Nozzle Design
The nozzle design is essential for producing the optimal thrust necessary to perform the
specified mission. Assumptions for the throat were that the flow was choked at the throat
(M*=1) and that the exit pressure/chamber pressure did not vary with time. Table 5
shows the key parameters for the nozzle.
Symbol
Parameter
Dimension
Unit
Aexit
Exit Area
0.015
m^2
ϵ
Epsilon
4
--
A*
Throat Area
0.0038
m^2
dexit
Exit Diameter
0.1397
m
Table 5: Preliminary Design Nozzle Inputs
The exit area was sized based on the exit diameter provided for the AIM-9M missile [8].
Epsilon was selected based on a theoretical rocket provided in [6]. The throat area was
sized based upon the ratio of exit area to throat area
Chamber Pressure and Hoop Stress
Like thrust, the chamber pressure also varies with time. Utilizing equation 3, the
chamber pressure was found for each time step and calculated to determine the hoop
stress. Figure 8 displays this variation over time.
14
Chamber Pressure vs Time
Chamber Pressure (Pa)
14,000,000
12,000,000
10,000,000
8,000,000
6,000,000
4,000,000
2,000,000
0
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
Figure 8: Chamber Pressure versus Time of Preliminary Design
The hoop stress, utilizing equation 4, will drive the selection of the material for the
pressure chamber of the Final Design.
Hoop Stress vs Time
Hoop Stress (MPa)
350
300
250
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
Seconds
0.6
0.7
0.8
Figure 9: Hoop Stress versus Time of Preliminary Design
Range and Velocity
The range of the Preliminary Design was calculated assuming aerodynamic impacts
were minimal (Lift and Drag effects) and the missile was flying at an altitude of 3000 m
15
throughout its burn time. The range for this missile also analyzed utilizing the range
equation for a given time step, yielding a range of 446 meters during the burn time. The
range versus time is shown in Figure 10.
Range vs Time
500
Range (m)
400
300
200
100
0
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
Figure 10: Range versus Time of Preliminary Design
The velocity of the missile was calculated for the different time steps. The velocity
changes as the thrust changes and is represented at each time step by equation 6 and the
plot is displayed in Figure 11.
Velocity vs Time
1000
Velocity (m/s)
800
600
400
200
0
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
Figure 11: Velocity versus Time of Preliminary Design
16
0.8
BurnSim Validation
The results of the Preliminary Design were also compared against the BurnSim Software
Output. Figure 12 displays the thrust versus time utilizing BurnSim. The burn time
calculated through BurnSim was 0.87 s (versus the 0.8 seconds calculated through the
analytical process. In addition, the max thrust calculated through BurnSim was 14813
lbs (65,892 N). Through the analytical approach, the max thrust was approximately
67,364 N.
Thrust vs Time
70,000
60,000
Thrust (N)
50,000
40,000
30,000
20,000
10,000
0
0
0.1
0.2
0.3
0.4
Time (s)
0.5
0.6
0.7
0.8
Figure 12: BurnSim Output for Thrust versus Time of Preliminary Design
Overall Compliance
The Preliminary Design did not meet the requirements of the project. The key
requirements and deltas are shown in Table 6.
Attribute
Requirement
Preliminary
Percentage
Design
Delta
Compliance
Mass (kg)
100
70.65
29%
Compliant
Length (m)
4
3.26
19%
Compliant
Burn Time (s)
2
0.8
60%
Non-Compliant
Table 6: Preliminary Design Compliance Table
17
With the Preliminary Design defined, the trade studies were utilized to determine which
aspects of the motor and missile design could be improved to meet the requirements.
3.2 Material Trade
The first trade performed for the air to air missile development was an investigation of
materials for the motor case. The material selection and recommendations were based
upon guidance from [4], [5] and [10]. The goal of this trade is to select a material for the
casing that can withstand the hoop stress at all time steps, but also was sensitive to mass
increase. The mass increase was critical to impacting the range equation.
The baseline (Preliminary Design) and three alternative materials were selected. The
selection of these materials were based upon usage on current missiles and
recommended materials. Table 7 displays the materials and properties.
Tracker
Material
Ultimate
Yield
Melting
Strength
Strength
Temperature
(MPa)
(Mpa)
(K)
Density
(kg/m^3)
E Gpa
Baseline
Aluminum 2090-T86
550
520
833.15
2620
76
Alternative 1
AISI 4130 Steel
670
435
1705.35
7850
205
Alternative 2
Pyrolytic Graphite
80
--
3923.15
2220
20
Alternative 3
Ti-6A1-4V
1170
1100
1877.15
4430
113.8
Table 7: Motor Case Materials [11]
Each of the material’s yield stresses were compared against the Preliminary Design
Hoop Stress, shown in Figure 13.
18
Stress vs Time
1200
Stress (MPa)
1000
800
Hoop Stress (Pa)
600
Aluminum 2090-T86
400
AISI 4130 Steel
Pyrolytic Graphite
200
Ti-6A1-4V
0
0
0.1
0.2
0.3
0.4 0.5
time (s)
0.6
0.7
0.8
Figure 13: Stress versus Time of Preliminary Design and Materials
This chart shows an important observation in that for the motor casing thickness of
the motor of 0.0025 m, Titanium, Aluminum, and Steel were sufficient to accommodate
the hoop stress of missile throughout the burn time. Based upon these results, it is
recommended to use Ti-6A1-4V as the material for the Final Design. This result is
driven by the desire to have the strongest material utilized. In addition, there was not a
large mass increase by utilizing the Titanium material.
A key fall-out of this trade was also the selection of an insulation material to
accommodate the high chamber temperature. A comparison of the melting temperatures
for each of the materials is shown below in Table 8:
Chamber
Aluminum 2090-
Temperature (K)
T86 (K)
AISI 4130 Steel (K)
Pyrolytic Graphite (K)
Ti-6A1-4V (K)
3370
833.15
1705.35
3923.15
1877.15
Table 8: Chamber Temperature and Material Melting Temperatures
This figure shows that although Titanium is the strongest material for the casing, the
material with the highest melting temperature is the Pyrolytic Graphite. Therefore, the
Final Design will utilize an insulation material composed of Pyrolytic Graphite.
19
3.3 Propellant Trade
The second trade investigated for the air to air missile and motor design was the
propellant selection. The goal of this trade was to determine the sensitivity of burn time
and thrust utilizing different propellants. The propellant impacts parameters including
burn time, thrust, and the sizing of the missile. Based upon Appendix D of [6] and
provided propellant parameters, 2 alternative propellants were selected to trade against
the baseline selection as shown in Table 9:
Symbol
Tchamber
Parameter
Chamber
Baseline
Alt 1
Alt 2
Unit
(CTPB/AP/AL)
(HTPB/AP/AL)
(PBAN/AP/AL)
3370
3410
3500
K
0.40
0.45
0.33
--
4.0
4.5
3.7
--
Temperature
n
Burning Rate
Exponent
k
Burning Rate
Constant
ρpropellant
Density (kg/m3)
1772
1855
1854.6
kg/m3
Mw
Molecular
29.3
26.1
29.3
(g/mol)
1.17
1.18
1.17
--
1575
1590
1577
m/s
Weight
γ
Ratio of
Specific Heats
C*
Characteristic
Velocity
Table 9: Propellant Properties
Key in the selection of these propellants was the relatively higher and consistent values
of k versus other propellants. [6]
To analyze the impact of these propellants, the thrust versus time for each of these is
plotted in Figure 14 and key results are shown and resulting attributes are shown in
Table 10.
20
Thrust (N)
Thrust vs Time
180,000
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
Baseline
Alternative 1
Alternative 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Time (s)
Figure 14: Thrust versus Time of Different Propellant Configuration
Symbol
Parameter
Baseline
Alt 1
Alt 2
(CTPB/AP/AL)
(HTPB/AP/AL)
(PBAN/AP/AL)
Unit
tburn
Burn Time
0.8
0.4
1.4
Seconds
vburnout
velocity
870
910
902
m/s
x
range
446
247
779
m
FN
Thrust at
67,364
159,844
38,689
N
burnout
Table 10: Propellant Trade Results
Key to note for the results of these alternatives were the deltas amongst some of the key
attributes. As expected, the propellant with the highest thrust will be the fastest to burn
out propellant. This is because the mass flow rate exiting the missile is dependent on the
thrust. Also, the propellant with the lowest burn rate had also the slowest burn time.
Based upon the need to maintain a high thrust and burn time equivalent to roughly 2
seconds, a variation of the baseline propellant will be utilized in accordance to adjusting
the conduit diameter and length. This is further discussed in the Final Design section.
3.4 Dimensional Trade
This trade investigates the impact of changing the charge length and the conduit
diameter of the missile. To exemplify the impact of impacting the length, different
charge length variations were selected to accommodate the need to minimize impacts to
21
the missile overall length, but also to analyze the overall thrust increase at these different
lengths. Figure 15 displays the thrust versus time for these different lengths:
Charge Length vs Thrust
100,000
Thrust (N)
80,000
Charge Length = 1.67 m
60,000
Charge Length = 1.77 m
40,000
Charge Length = 1.87 m
Charge Length = 1.97 m
20,000
Charge Length = 2.07 m
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (s)
Charge Length = 2.17 m
Figure 15: Thrust versus Time for different charge lengths
As expected, as the charge length of the solid rocket motor increased, the thrust of the
motor increased. Interesting to note also was that with a higher charge length, there was
a smaller burn time. It was not a large variation, but a noticeable one of about one tenth
of a second. This was driven by the fact that the increase of the mass flow rate of the
propellant is related to the increase in thrust. Therefore, since more mass is being
burned, the time to burnout is faster.
A comparison of the conduit diameters of the propellant was also performed for this
missile. The main focus of the conduit diameters trade was to understand the impact to
burn time for different conduit diameters.
To assess the conduit diameter impacts, a plot of the thrust versus conduit diameter was
generated and is shown in Figure 16.
22
Conduit Diameter vs Thrust
80,000
70,000
Thrust (N)
60,000
50,000
Conduit Diameter = 0.0784 m
40,000
Conduit Diameter = 0.08 m
30,000
Conduit Diameter = 0.09 m
20,000
Conduit Diameter = 0.1 m
10,000
0
0
0.1
0.2
0.3
0.4
Time (s)
0.5
0.6
0.7
0.8
Figure 16: Thrust versus Time for different conduit diameters
The results showed that as the conduit diameter increased in size, the thrust at burn out
remained approximately constant. There was however, a smaller burn time as the
diameter of the conduit increased. Both these results are expected because the burn area,
calculated for each time step as shown below, largely impacts the burn time and not the
change in force at burn-out:
Aburn    d conduit  l ch arg e
(10)
The main constraint on this trade is the throat area versus the conduit area. If the conduit
area is smaller than the throat area, erosive burning is more likely to occur. Therefore, to
attempt to avoid this, the throat area was kept higher than the burn area. To get higher
performance, the conduit area needed to be at least two times greater than the throat area,
which was the ratio utilized for the Final Design.
3.5 Final Design
To arrive at the Final Design, the results of the trades and Preliminary Design were
utilized to ensure that the requirements were met. This design was compliant to the
design requirements and utilizing lessons learned from the Preliminary Design, the
trades, and further research, the Final Design was formulated.
Final Design Motor Dimensions
For the Final Design, the Preliminary Design was revisited to ensure that the
requirements were met for the project. Table 11 lists the key inputs.
23
Symbol
Parameter
Dimension
Unit
dmotor
Motor Diameter
0.1461
meters
dcharge
Charge Diameter
0.1405
meters
lcharge
Charge Length
1.77
meters
tcasing
Casing Thickness
0.0025
meters
tinsulation
Insulation Thickness
0.0003
meters
dconduit
Conduit Diameter
0.1016
meters
Table 11: Final Design Motor Inputs
The overall motor diameter was changed based on the need to change the charge
diameter and the conduit diameter. Charge diameter was changed based on the need to
change the conduit diameter. The charge length was kept the same because the
dimensional trade resulted in minimal changes to burn time. The casing thickness was
maintained due to the Titanium material providing sufficient capability to react the
pressures seen by the chamber. The insulation material thickness was reduced to reflect
researched thicknesses on the order of 0.012 inches [9]. The conduit diameter was
increased to reflect a throat to conduit area ratio that was greater than 2. The Preliminary
Design had a very small throat to conduit ratio, which leads to erosive burning.
Therefore, the conduit diameter needed to be changed for the Final Design. This drove
changes in the design, specifically the propellant characteristics and the need to
investigate the material selection.
Propellant Characteristics
The propellant selected for the Final Design was a variant of the propellant utilized in
the Preliminary Design, a propellant mixture of Ammonium Perchlorate (Oxidizer) and
Carboxyl-Terminated Polybutadiene (Fuel Binder). This was driven by after further
research [6], the propellant rate of burn needed to be changed to accommodate the higher
chamber pressures and the need for a higher burn time. The characteristics of the Final
Design Propellant are displayed in Table 12.
24
Symbol
Parameter
Dimension
Unit
Tchamber
Chamber Temperature
3371
K
n
Burning Rate Exponent
0.35
--
k
Burning Rate Constant
2.2
--
ρpropellant
Density (kg/m3)
1771.5
kg/m3
Mw
Molecular Weight
29.3
(g/mol)
R
Gas Constant
284
(J/Kg*K)
γ
Ratio of Specific Heats
1.17
--
C*
Characteristic Velocity
1577
m/s
Table 12: Final Design Propellant Inputs
Weight Build-Up
The weight of the missile was maintained relatively similar to the Preliminary Design.
Specific mass inputs are given in Table 13.
Symbol
Parameter
Dimension
Unit
mwarhead
Warhead Mass
10
kg
mguidance system
Guidance System Mass
20
kg
mpropellant
Propellant Mass
23.19
kg
mcasing
Casing Mass
9.52
kg
mmiscellaneous comp
Miscellaneous Components Mass
15.68
kg
mmissile
Total
78.39
kg
Table 13: Final Design Weight Inputs
The design of the warhead and guidance systems utilized the same rationale as the
Preliminary Design. The propellant mass increased for the Final Design because the
conduit and charge diameters became larger. This increase with different propellant
characteristics drove a larger burn time than that of the Preliminary Design. For the Final
Design, a casing thickness was also utilized for the entire missile. However, the material
selected for this design was Titanium (Ti-6A1-4V), driven by the trade study results that
showed Titanium as the most suitable material for the casing and handling the chamber
25
pressure. The factor of 25% for the miscellaneous components assumption was also
utilized for the Final Design.
Length Build-Up
To meet the need of being compatible and integrated onto the aircraft, the length
dimensions and rationale of the missile were kept relatively the same as the Preliminary
Design, with the only exception being a slight variation in the nozzle length due to a
minor configuration change from the Preliminary Design. Specific length inputs are
given in Table 14: Final Design Dimensional Inputs.
Symbol
Parameter
Dimension
Unit
lnozzle
Nozzle Length
0.13
m
lcharge
Charge Length
1.77
m
lwarhead
Warhead Length
0.47
m
lguidance system
Guidance System Length
0.79
m
ltip
Tip Length
0.18
m
lmissile
Total Length
3.34
m
Table 14: Final Design Dimensional Inputs
Thrust versus Time
The thrust versus time plot is shown for the Final Design. The burn time for the rocket
motor was approximately 2.5 seconds, which is higher than that of the Preliminary
Design and the requirement. Like the Preliminary Design, the thrust profile did correlate
to the expected burn profile of a tubular shaped design, where thrust increased with time.
The improvement of burn time was due to the change in propellant characteristics, as
well as the increase in the overall charge diameter of the rocket. The thrust of the motor
was lower than that of the Preliminary Design. The thrust versus time is shown in Figure
17.
26
Thrust vs Time
30,000
Thrust (N)
25,000
20,000
15,000
10,000
5,000
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Time (s)
2
2.1 2.2 2.3 2.4 2.5
Figure 17: Thrust versus Time of Final Design
Nozzle Design
The nozzle design for the Final Design was kept the same as the Preliminary Design.
This was driven by the need to keep the nozzle the same to analyze the impact of other
attributes of the missile and motor. Specific nozzle inputs are given in Table 15.
Symbol
Parameter
Dimension
Unit
Aexit
Exit Area
0.015
m^2
ϵ
Epsilon
4
--
A*
Throat Area
0.0038
m^2
dexit
Exit Diameter
0.1397
m
Table 15: Final Design Nozzle Inputs
Chamber Pressure and Hoop Stress
The chamber pressure and hoop stress were key drivers into the material selection for the
missile. As mentioned, Titanium was chosen as the material for the Final Design because
of its high capability to withstand the chamber pressure through the trades. The Chamber
Pressure versus Time is shown in Figure 18: Chamber Pressure versus Time of Final
Design
27
Chamber Pressure vs Time
Chamber Pressure (Pa)
5,000,000
4,000,000
3,000,000
2,000,000
1,000,000
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Time (s)
2.1 2.2 2.3 2.4 2.5
Figure 18: Chamber Pressure versus Time of Final Design
Figure 19 displays the hoop stress versus time of the casing and the yield strength of
Titanium are compared.
Hoop Stress vs Time
Hoop Stress (MPa)
1200
1000
800
600
Hoop Stress (Mpa)
400
Titanium
200
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Time (s)
2.1 2.2 2.3 2.4 2.5
Figure 19: Hoop Stress versus Time of Final Design
Range and Velocity
Like the Preliminary Design, the range during burn time was calculated assuming
aerodynamic impacts (Lift and Drag Effects) were minimal and the missile was flying at
a steady altitude of 3000 m. The range of the missile was calculated to be 1443 meters
and the range versus time is shown in Figure 20.
28
Range (m)
Range vs Time
1600
1400
1200
1000
800
600
400
200
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Time (s)
2
2.1 2.2 2.3 2.4 2.5
Figure 20: Range versus Time of Final Design
The velocity of the missile was calculated utilizing the same methodology as the
Preliminary Design as shown in Figure 21.
Velocity vs Time
1200
Velocity (m/s)
1000
800
600
400
200
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5
Time (s)
Figure 21: Velocity versus Time of Final Design
BurnSim Validation
Similar to the Preliminary Design, the results of the Final Design were also compared
against the BurnSim Software Output. For the Final Design, the burn time calculated
through BurnSim was 2.46 seconds. The max thrust was calculated to be 6334lbs (28178
N). These results were very similar to the analytical results of approximately 2.5 seconds
and 24235 N. The BurnSim output is shown in Figure 22.
29
Thrust vs. Time
30,000
Thrust (N)
25,000
20,000
15,000
10,000
5,000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2 1.3
Time (s)
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
Figure 22: BurnSim Output for Thrust versus Time of Final Design
Overall Compliance
The Final Design met all project requirements. Compliance was attributed to the
research of performing the trades and integrating those results into the Final Design.
Specific values of compliance are shown in Table 16: Final Design Compliance Table
.
Attribute
Requirement
Preliminary
Final Design
Design
Final Design
Compliance
Mass (kg)
100
70.65
74.48
Compliant
Length
4
3.26
3.34
Compliant
2
0.8
2.5
Compliant
(m)
Burn
Time (s)
Table 16: Final Design Compliance Table
30
4. Conclusion
The development of the motor design for the air to air missile led to many important
conclusions. For motor design, there are many different attributes that can be impacted.
For this project, it was shown that even going from a Preliminary Design to a Final
Design can lead to different results based upon lessons learned throughout the trade
study process.
The Preliminary Design, although creating higher thrusts, had some challenges with the
design, most notably the conduit to throat area ratio. The Preliminary Design also
contained a quick burn time of approximately 0.8 seconds. This concluded in the need to
change the conduit to throat area to be above 2 for the Final Design and the need to
increase the burn time for the Final Design to meet the requirement.
The trade studies yielded important results that helped drive key decision making for the
Final Design. The materials trade presented evidence for the selection of Titanium as the
best suitable material due to its capability to withstand the hoop stress. This was
displayed through the comparison of the hoop stress versus the yield strength of
Titanium for the entire burn time. The propellant trade showed very important results of
the effect of burn rate on burn time. Although the exact propellants from the trade
studies were not utilized for the Final Design, it showed the key sensitivity of burn rate
versus burn time, where the propellants with lower burn rates had a longer burn time.
The dimensions trade displayed that charge length and conduit diameters had impact on
specific parameters (Charge Length impacts predominantly thrust and conduit diameter
predominantly impacts burn time.) A key for this trade was to utilize reasonable conduit
diameters of the propellant of at least two times the throat area.
The Final Design utilized these trades and Preliminary Design to develop a compliant
solution. Three of the major changes in the Final Design included the need to change the
conduit diameter and the change in the propellant. The conduit diameter was altered due
to the need to have a higher diameter at the propellant conduit versus the throat diameter.
This had impacts to the design in which the propellants were analyzed further. The
propellant selected was a variation of the Preliminary Design propellant and was
selected due to its ability to provide longer burn times and still provide significant thrust.
31
The change in propellant with a different conduit diameter drove hoop stress and a check
was done to validate the usage of Titanium as the selected material. With these changes,
the Final Design was compliant to the specified mission parameters with a burn time of
approximately 2.5 seconds. This drove the conclusion that the motor properties analyzed
and traded had influence on sizing the Final Design to meet these specified requirements
and it amplified the importance of analyzing attributes from a system level since many
of the missile attributes impact each other.
Overall, missile and motor design have multiple sensitivities in creating a design. This
project resulted in notable and important results for these specific attributes and drove a
compliant Final Design. However, this study not only utilized analytical results, but also
proved the importance of the trade study process and the importance of utilizing a
phased approach.
32
5. References
[1] AIM 120 AMRAAM Slammer, Pike, John, Copyright 2000 – 2012
http://www.globalsecurity.org/military/systems/munitions/aim-120.htm
[2] Directory of U.S. Military Rockets and Missile AIM-120, Parsch, Andreas,
Copyright 2002 – 2007 http://www.designation-systems.net/dusrm/m-120.html
[3] Nakka, Richard Allan. Solid Propellant Rocket Motor Design and Testing, April
1984, University of Manitoba
[4] R Clay Hainline. Design Optimization of Solid Rocket Motor Grains for Internal
Ballistic Performance. 1998, Southwest Missouri State University
[5] Solid Propellant Grain Design and Internal Ballistics. NASA/SP-8076, March 1972
[6] Ward, Thomas. Aerospace Propulsion Systems. 1st ed Singapore: John Wiley & Sons
(Asia) Pte Ltd., 2010.
[7] Mattingly, Jack D. Elements of Propulsion. 1st ed Virginia: American Institute of
Aeronautics and Astronautics, Inc, 2006
[8] Ball, Jim Sidewinder AIM-9 Missile, Copyright 2008
http://www.rocketryonline.com/jimball/jimball/sidewinder/sidewinder.htm
[9] Raytheon (Philco/General Electric) AAM-N-7/GAR-8/AIM-9 Sidewinder, Parsch,
Andreas, Copyright 2002-2008 http://www.designation-systems.net/dusrm/m-9.html
[10] Fleeman Eugene L. Tactical Missile Design, American Institute of Aeronautics and
Astronautics, 2001.
[11] MATWEB, Matweb, LLC, Copyright 1996-2012, http://www.matweb.com/
33
Appendices
Appendix A – Validation of Tools
To verify the analytical methodology, a validation was performed to compare both the
analytical and BurnSim Software. The analysis was performed for the Final Design (A
radial burning solid rocket motor utilizing a tubular grain cross section.)
For the analytical approach, Microsoft Excel was utilized. Since mass flow rate and
conduit area are dependent on time, time steps of 0.1 seconds were taken. Utilizing this
approach, the burn time of the propellant was approximately 2.5 seconds and the
resulting plot is shown below in Figure 23.
Thrust vs Time
30,000
Thrust (N)
25,000
20,000
15,000
10,000
5,000
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Time (s)
2
2.1 2.2 2.3 2.4 2.5
Figure 23: Thrust vs. Time for Analytical Approach
The BurnSim software, utilizing the same inputs, resulted in the following plot below.
time was calculated to be 2.46 seconds.
Thrust vs. Time
30,000
Thrust (N)
25,000
20,000
15,000
10,000
5,000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2 1.3
Time (s)
1.4
1.5
1.6
1.7
1.8
Figure 24: Thrust vs. Time for BurnSim
34
1.9
2
2.1
2.2
2.3
2.4
2.5
Figure 25 compares the results of both the analytical approach versus BurnSim. The
results outputted below show a percent error of 18% at time equals 0.01 seconds and a
percent error of 13% at time equal to 2.5 s. The rationale behind this percent difference
could be due to rounding estimations and the difference in methodology of calculating
the results. Although there are slight deltas amongst the tools, the BurnSim tool can be
utilized to validate the analytical approach for this effort. In addition, the burn time
comparison is very similar (2.46 seconds for BurnSim, 2.5 seconds for analytical)
Thrust vs. Time
30,000
Thrust (N)
25,000
20,000
15,000
BurnSim
10,000
Analytical
5,000
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Time (s)
2
2.1 2.2 2.3 2.4 2.5
Figure 25: Thrust vs. Time Comparison
35
Appendix B – Summary of Tools
This design project focused on utilizing numerous analytical tools to determine the
required parameters for analysis. This appendix provides a short overview of each of the
tools for each phase of the trade study:
Preliminary Design
Analytical Model: To begin analysis to determine attributes such as the thrust versus
time, chamber pressure, hoop stress, range, and velocity, a model was developed based
off methodology utilized for motor design. [6]. The model was designed for a radial
burning, tubular shaped conduit. Inputs into the tool included all the nozzle inputs
mentioned throughout the project (A*, Aexit), motor attributes (chamber diameter,
conduit diameter, charge length, overall diameter), propellant characteristics, ambient
pressure, and mass inputs. From these, utilizing propulsion principles and equations, the
resulting outputs of the project were determined. A key note to include is for the thrust
versus time plots, there is a very short time period (on the order of 0.01 s), where the
thrust goes from 0 N to the first data points on the plots. This is neglected throughout the
project due to minimal impact on the results and is not shown. In addition, for the
velocity vs time plots, there is an initial velocity of for the missile since the aircraft
carrying the missile is assumed to be in flight.
BurnSim: As mentioned in the project main body, BurnSim was software that utilized
propellant characteristics, motor characteristics (same as the analytical model), and
nozzle characteristics to develop a thrust versus time curve. This provided a validation
for the analytical tool.
Trade Studies
Materials Trade: Once preliminary research was performed on the materials, a
comparison to the hoop stress was done utilizing the analytical model.
36
Propellant Trade: Similar to the materials trade, once preliminary research was
performed on the propellants, the propellants selected were inserted into the analytical
model and the impacts on thrust versus time were analyzed.
Dimensions Trade: Upon completion of preliminary selection of diameters and lengths
to be analyzed, the analytical model was utilized to analyze the impacts of the different
selected lengths and diameters on thrust versus time.
Final Design
Analytical Model: The analytical model utilized for the Final Design was similar to that
of the Preliminary Design. The main delta was the utilization of the new dimensions and
propellant characteristics of the Final Design.
BurnSim: Like the Preliminary Design, a BurnSim validation was performed on the
motor performance.
37