Design of a Rocket Motor Casing - Rensselaer at Hartford

Design of a Rocket Motor Casing

by
Devon K. Cowles
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF MECHANICAL ENGINEERING

Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic Institute
Hartford, Connecticut
May, 2012

i

© Copyright 2012
by

Devon K. Cowles
All Rights Reserved

ii

CONTENTS

LIST OF TABLES............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF EQUATIONS................................................................................................... vii
LIST OF SYMBOLS ........................................................................................................ ix
Glossary ........................................................................................................................... xii
ABSTRACT ................................................................................................................... xiii
1. Introduction\Background ............................................................................................. 1

1.1 Rockets............................................................................................................... 1
1.2 Mission Requirements........................................................................................ 2
1.3 Structural Requirements..................................................................................... 3
2. Methodology................................................................................................................ 4
2.1 Assumptions....................................................................................................... 4
2.2 Flight Performance............................................................................................. 4
2.3 Motor thrust requirements.................................................................................. 6
2.4 Motor Casing Sizing .......................................................................................... 9
2.5 Material ............................................................................................................ 10

2.5.1 Aluminum Alloy .................................................................................. 10
2.5.2 Composite Material.............................................................................. 11
3. Results........................................................................................................................ 15
3.1 Engine Parameters............................................................................................ 15
3.2 Aluminum Alloy Casing Design...................................................................... 15
3.2.1 Aluminum Casing Geometry ............................................................... 15
3.2.2 Finite Element Analysis ....................................................................... 16
3.3 Composite Casing Design ................................................................................ 19
3.3.1 Layup.................................................................................................... 19
3.3.2 Composite Casing Geometry ............................................................... 20

iii

3.3.3 Finite Element Analysis ....................................................................... 22
4. Conclusion ................................................................................................................. 26
References........................................................................................................................ 28
Appendix A...................................................................................................................... 29

iv

LIST OF TABLES

Table 1.1 Mission Requirements ....................................................................................... 2
Table 1.2 Tomahawk Cruise Missile Specifications ......................................................... 2
Table 1.3 Available Composite Propellant........................................................................ 3
Table 2.1 E357 T-6 Casted Aluminum............................................................................ 11
Table 2.2 Hexcel Intermediate Modulus Carbon Fiber/Resin Properties........................ 11
Table 3.1 Engine Parameters ........................................................................................... 15
Table 3.2 Aluminum Engine Weight............................................................................... 16
Table 3.3 Laminate Properties Calculated by CLT ......................................................... 20
Table 3.4 Composite Engine Weight............................................................................... 21
Table 3.5 Laminate Properties in ANSYS....................................................................... 22
Table 3.6 Composite Casing Stress and Margins ............................................................ 24

v

LIST OF FIGURES

Figure 1.1 Typical Rocket Components ............................................................................ 1
Figure 2.1 Rocket Free Body Diagram.............................................................................. 5
Figure 2.2 Aluminum Engine Casing Concept.................................................................. 9
Figure 2.3 Composite Engine Casing Concept.................................................................. 9
Figure 2.4 Finite Element Load and Boundary Conditions............................................. 10
Figure 3.1 Aluminum Alloy Casing Detail...................................................................... 16
Figure 3.2 Maximum Stress Aluminum Engine Casing Upper....................................... 17
Figure 3.3 Maximum Stress Aluminum Engine Casing Lower ...................................... 17
Figure 3.4 Assembly Flight Path ..................................................................................... 18
Figure 3.5 Flight Performance......................................................................................... 18
Figure 3.6 Rocket Angle, Altitude and Velocities........................................................... 19
Figure 3.7 Composite Casing Detail................................................................................ 21
Figure 3.8 FEA Geometry for Composite Casing ........................................................... 22
Figure 3.9 Load and Boundary Conditions Composite Casing ....................................... 23
Figure 3.10 Maximum Total Deformation Composite Casing........................................ 23
Figure 3.11 Top Radius Stress Composite Casing .......................................................... 24
Figure 4.1 Flight Performance Comparison .................................................................... 27

vi

LIST OF EQUATIONS

Equation 2.1 – Flight path angle to ground ....................................................................... 4
Equation 2.2 – Axial acceleration at 1,000 feet................................................................ 5
Equation 2.3 – Axial acceleration below1,000 feet ........................................................... 5
Equation 2.4 – Radial acceleration above 1,000 feet......................................................... 5
Equation 2.5 – Radial acceleration below 1,000 feet ........................................................ 5
Equation 2.6 – Axial velocity ............................................................................................ 5
Equation 2.7 – Radial velocity........................................................................................... 5
Equation 2.8 – Axial dispacement ..................................................................................... 5
Equation 2.9 – Radial dispacement .................................................................................. 5
Equation 2.10 – Horizontal distance and altitude matrix .................................................. 6
Equation 2.11 – Propellant burn Area ............................................................................... 6
Equation 2.12 – X Function............................................................................................... 6
Equation 2.13 – Nozzle exit area....................................................................................... 6
Equation 2.14 – Nozzle throat area.................................................................................... 6
Equation 2.15 – Combustion chamber pressure ................................................................ 7
Equation 2.16 – Propellant burn rate ................................................................................. 7
Equation 2.17 – Expansion ratio........................................................................................ 7
Equation 2.18 – Exit pressure............................................................................................ 7
Equation 2.19 – Ideal thrust coefficient............................................................................. 7
Equation 2.20 – Actual thrust coefficient .......................................................................... 7
Equation 2.21 – Specific impulse ...................................................................................... 7
Equation 2.22 – Propellant core area ................................................................................. 8
Equation 2.23 – Propellant core volume............................................................................ 8
Equation 2.24 – No-core propellant volume...................................................................... 8
Equation 2.25 – Propellant volume ................................................................................... 8
Equation 2.26 – Propellant mass ....................................................................................... 8
Equation 2.27 – Mass flow ................................................................................................ 8
Equation 2.28 – Burn time................................................................................................. 8
Equation 2.29 – Total impulse........................................................................................... 8

vii

Equation 2.30 – Laminae Stress/Strain relationship........................................................ 12
Equation 2.31 – Laminae Reduced Compliance Stress/Strain relationship .................... 12
Equation 2.32 – Laminae Global Reduced Compliance Stress/Strain relationship........ 12
Equation 2.33 – Transformation matrix........................................................................... 12
Equation 2.34 – Laminae Transformed Stress/Strain relationship .................................. 12
Equation 2.35 – Laminae global x stiffness..................................................................... 12
Equation 2.36 – Laminae global y stiffness..................................................................... 13
Equation 2.37 – Laminae global shear modulus.............................................................. 13
Equation 2.38 – Laminae Poisson’s ratio xy ................................................................... 13
Equation 2.39 – Laminae Poisson’s ratio yx ................................................................... 13
Equation 2.40 – Laminae Global Reduced Stiffness Stress/Strain relationship............. 13
Equation 2.41 – Reduced Stiffness Matrix definition ..................................................... 13
Equation 2.42 – Extensional stiffness matrix .................................................................. 13
Equation 2.43 – Couplingstiffness matrix ....................................................................... 13
Equation 2.44 – Bending stiffness matrix........................................................................ 13
Equation 2.45 – Laminate Load/Strain relationship ........................................................ 13
Equation 2.46 – Tsai-Hill failure criteria......................................................................... 14
Equation 3.1 – Yield stress margin of safety................................................................... 16
Equation 3.2 – Ultimate stress margin of safety.............................................................. 17

viii

LIST OF SYMBOLS

Motion
a – Acceleration (ft/s2)
awing – Lift/Mass of Missile Wing. (32.174 ft/s2)
Cd – Coefficient of Drag
Cl – Coefficient of Lift
F – Engine Thrust (lbf)
g – Acceleration of Gravity on Earth (32.174 ft/s2)
l/d ratio – Ratio of Cl to Cd
tn – Time at Increment n (s).
Ψ – Engine Thrust Relative to Horizontal (degrees)
ρair – Density of Air (lb/ft3)
θ – Direction of Flight Relative to Horizontal (degrees)
v – Velocity in Rocket Coordinates (ft/s)
x – Axial Displacement in Rocket Coordinates (ft)
y – Radial Displacement in Rocket Coordinates (ft)

ix

Engine
A* – Nozzle Throat Cross-Sectional Area (in2)
Ab – Propellant Burning Area (in2)
Aconduit – Area of Core in Propellant Charge (in2)
Ae – Cross Sectional Area of Exhaust Cone (in2)
Cf – Thrust Coefficient
dc – Propellant Outer Diameter (in)
de – Diameter of Exhaust Cone (in)
ε – Expansion Ratio
Fn – Engine Thrust (lbf)
γ – Specific Heat Ratio
Isp – Specific Impulse (s)
It – Total Impulse (lbf-s)
k – Propellant Burn Rate Factor
Lc – Length of Propellant (in)
ṁ – Mass Flow Rate (lbm/s)
mc – Mass of Propellant (lbm)
n – Propellant Burn Rate Factor
Pc – Chamber Pressure (psi)
Po – Atmospheric Pressure (psi)
R – Gas Constant (lbf-in/lbm-R)
rb – Propellant Burn Rate (in/s)
tb – Propellant Burn Time (s)
ρp – Density of Propellant (lbm/in3)
Tc – Propellant Burn Temperature (°R)
V0 – Volume of No Core Propellant (in3)
Vc – Propellant Volume (in3)
Vconduit – Conduit Volume (in3)
X*– Non-Dimensional Mass Flow Rate in Nozzle Throat

x

Material
E – Young’s Modulus (psi)
ε – Normal Strain (in/in)
γ – Shear Strain (in/in)
Fcy – Yield Compressive Strength (psi)
Fcu – Ultimate Compressive Strength (psi)
Fty – Yield Tensile Strength (psi)
Ftu – Ultimate Tensile Strength (psi)
Fsu – Ultimate Shear Strength (psi)
G – Shear Modulus (psi)
M.S.yld-comp – Yield Strength Margin of Safety – Compressive
M.S.ult-comp – Ultimate Strength Margin of Safety – Compressive
M.S.yld-tensile – Yield Strength Margin of Safety – Tensile
M.S.ult-tensile – Ultimate Strength Margin of Safety – Tensile
ν – Poisson’s Ratio
σ – Normal Stress (psi)
[Q] – Laminae Reduced Stiffness Matrix (psi)
[Q̅ ] – Laminae Transposed Reduced Stiffness Matrix (psi)
[S] – Laminae Reduced Compliance Matrix (in2/lb)
[S̅ ] – Laminae Transposed Reduced Compliance Matrix (in2/lb)
τ –Shear Stress (psi)

xi

Glossary

Adiabatic – A thermodynamic process in which heat is neither added nor removed from
the system.

ANSYS – Software created by ANSYS Inc. used for finite element analysis.
BurnSim – Software created by Gregory Deputy to simulate the performance of a solid

propellant rocket motor.
BATES – A cylindrical solid propellant configuration with a cylindrical core.
CATIA – Software created by Dassault Systémes to perform 3 dimensional computer

aided design.
CLT – Classical Laminate Theory used to calculate laminate properties from the

properties of the individual layers.
Condi Nozzle – A convergent/divergent nozzle.
Isentropic – A thermodynamic process in which there is no change in entropy of the

system.
Laminae – A single layer of a composite matrix.
Laminate – A stack of laminae.

xii

ABSTRACT

The purpose of this project is to design a ground launched rocket booster to meet
specific mission requirements. The mission constraints include minimum speed,
maximum flight altitude as well as length and weight limits. The mission is to launch a
3,000 lb payload such as a Tomahawk cruise missile to an altitude of 1,000 feet and
accelerate the missile to 550 MPH (807 fps). To meet these mission requirements, the
weight of the rocket body should be as light as possible while maintaining the required
structural integrity and reliability. The motor parameters such as the nozzle size,
expansion ratio, propellant size and shape are determined through an iterative process.
The thrust performance from a preliminary motor design is used to calculate the
resulting flight performance based on the calculated thrust overcoming gravity, inertia
and aerodynamic drag of the booster rocket and cruise missile assembly. The engine
nozzle parameters are then varied to meet the mission requirements and to minimize
excess capability to ensure a weight efficient motor. The initial motor casing design
will be made of a light weight casted aluminum. The aluminum motor design will be
compared to a design made of a fiber and resin composite material. The composition and
layup of the composite material and the thickness of the aluminum material will be
designed to meet industry standard safety margins based on the material’s strength
properties. This paper will present the calculated engine parameters as well as the
engine weight and engine size for both the aluminum casing and the composite casing.

xiii

1. Introduction\Background
1.1 Rockets

Rockets are a type of aircraft used to carry a payload at high speeds over a wide range of
distances depending on the design. Rockets are powered by a reaction type engine
which uses chemical energy to accelerate and expel mass through a nozzle and relies on
the principals of Sir Isaac Newton’s third law of motion [1] to propel the rocket forward.
Rocket engines use either solid or liquid fuel. They carry both the fuel and the oxidizer
required to convert the fuel into thermal energy and gas byproducts. The gas byproducts
under pressure are then passed through a nozzle which converts the high pressure low
velocity gas into a low pressure high velocity gas.
The following figure shows the different components of a typical rocket [2].

Figure 1.1 Typical Rocket Components

1

1.2 Mission Requirements

This rocket is a ground launched booster that is used to launch a payload such as a
Tomahawk cruise missile to a prescribed altitude and to a required velocity. The
mission can be viewed in two phases. In the first phase, the booster is on the ground at
rest and accelerates the payload vertically to 1,000 feet. In the second phase, the booster
accelerates the payload horizontally to 550 MPH (807 fps). The rocket engine must be
sized appropriately to meet the mission requirements as summarized in Table 1.1. The
Tomahawk cruise missile specifications are listed in Table 1.2. The cruise missile in this
mission will use an onboard gas turbine engine to continue flight once the missile has
reached 1,000 ft altitude and 550 MPH (807 fps). In the horizontal portion of the flight,
the cruise missile will deploy the stowed wings to provide lift which will allow the thrust
of the booster to be used solely to accelerate the missile to the appropriate speed. Once
the missile has reached the target altitude and speed and the solid propellant has been
consumed, the booster will be jettisoned from the cruise missile assembly to fall back to
earth. The total assembly is limited to 3,500 lbm and the payload is 2,700 lbm. The
properties of the fuel to be used in this mission are shown in Table 1.3.

Table 1.1 Mission Requirements Value Units
0 - 1,000 ft
Altitude range
Minimum Velocity 550 MPH
3,500 lbm
Maximum Mass 2700 lbm
Payload Mass

Table 1.2 Tomahawk Cruise Missile Specifications

RGM 109D Length (in) Diameter (in) Weight (lb)
219 20.9 2700

2

Table 1.3 Available Composite Propellant AP (70%)
CTPB (12%)
Oxidizer %
Fuel Binder % AL (16%)
Metallic Fuel % Epoxy (2%)
Curative %
Flame Temperature (R) 6,840
Burning Rate Constants
.0341
k 0.4
n
Density (slinch/in3) 1.64E-4
Molecular Weight (kg/kmole) 29.3
Gas Constant (lb-in/slinch-R)
Ratio of Specific Heats 238,662.7
Characteristic Velocity (in/s) 1.17
62008

1.3 Structural Requirements

The rocket engine casing must be able to withstand the internal engine pressure loads
and the force applied to the payload through the attachment point. In some locations, the
casing materials must be able to withstand high pressures and elevated temperatures due
to the combustion of the fuel.
In this project, the casing design will be determined based on the stress analysis using
closed form equations and the finite element method. The nozzle and casing will be
sized using E357-T6 aluminum alloy and then resized using carbon fiber/resin composite
materials. The components will be sized based on the maximum load and pressure the
casing will be subjected to during the mission. This maximum load will be referred to as
the limit load.

3

2. Methodology

2.1 Assumptions

The following assumptions are made for the motor design to simplify the analysis.
1) The booster is an ideal rocket.
2) The specific heat ratio (γ) of the exhaust gases is constant throughout the booster.
3) Flow through the nozzle is adiabatic, isentropic and one dimensional.
4) There is no loss of total pressure during combustion.
5) The flow area in the combustion chamber is large compared to the nozzle area so
the velocity at the nozzle entrance is negligible.
6) All of the exhaust gasses exit the nozzle in the axial direction.
7) The nozzle is a fully expanding Condi nozzle.
8) The coefficient of drag for the payload and booster assembly is 0.75
9) In the rocket combustion chamber, there is a 2mm (0.079 inch) liner is made of a
material of sufficient properties to keep the casing temperatures below 300
degrees Fahrenheit.

2.2 Flight Performance

Thrust is required to accelerate the payload, fuel and motor casing mass to 1,000 feet and
550 MPH (807 fps) overcoming the forces of gravity, mass inertia and aerodynamic
body drag. As the propellant is consumed, the thrust increases and the mass of the
booster assembly decreases. As a result, the axial and the radial acceleration, velocity
and displacement is calculated in a discretized fashion for time steps of 0.01 seconds.
The displacements are then transformed from the rocket reference frame to the ground
reference frame to determine altitude and horizontal velocity.
The flight path is predetermined to transitions from a vertical flight to a horizontal path
based on the function [3]:

ࣂ = ‫ିܛܗ܋‬૚ ቀ ࢇ࢒࢚࢏࢚࢛ࢊࢋ ቁ [2.1]

࢚ࢇ࢘ࢍࢋ࢚ ࢇ࢚࢏࢚࢛ࢊࢋ

θ is the angle of the rocket axis to the ground as shown in Figure 2.1. The rocket at the

beginning of the launch is vertical (θ=90°).

4

Figure 2.1 Rocket Free Body Diagram

The axial acceleration of the body is calculated by the following equation when the
cruise missile wings are deployed at 1,000 feet:

ࢇ࢞ = ࡲ ‫ܛܗ܋‬ሺ࣐ െ ࣂሻ െ ࢇ࢝࢏࢔ࢍ െ ࢉࢊ ࣋ࢇ࢏࢘ ࢜૛࢞ ࡭ࢌ࢘࢕࢔࢚ െ ࢍ ࢙࢏࢔ࣂ [2.2]
࢓ ࢊ࢒ ࢘ࢇ࢚࢏࢕ ૛࢓

The axial acceleration is calculated by the following equation below 1,000 feet:

ࢇ࢞ = ࡲ ‫ܛܗ܋‬ሺ࣐ െ ࣂሻ െ ࢉࢊ ࣋ࢇ࢏࢘ ࢜૛࢞ ࡭ࢌ࢘࢕࢔࢚ െ ࢍ ࢙࢏࢔ࣂ [2.3]
࢓ ૛࢓

The acceleration in the direction perpendicular to the cruise missile wingspan plane is

calculated as follows when the cruise missile wings are deployed at 1,000 ft:

ࢇ࢟ = ࡲ ‫ܖܑܛ‬ሺ࣐ െ ࣂሻ ൅ ࢇ࢝࢏࢔ࢍ െ ࢉࢊ ࣋ࢇ࢏࢘࢜࢟૛࡭࢈࢕ࢊ࢟ െ ࢍ ࢉ࢕࢙ࣂ [2.4]
࢓ ૛࢓

And the equation below 1,000 ft is:

ࢇ࢟ = ࡲ ‫ܖܑܛ‬ሺ࣐ െ ࣂሻ െ ࢉࢊ ࣋ࢇ࢏࢘࢜࢟૛࡭࢈࢕ࢊ࢟ െ ࢍ ࢉ࢕࢙ࣂ [2.5]
࢓ ૛࢓

The axial (x) and radial (y) velocity is calculated using:

࢜࢞ሺ࢚࢔ሻ = ࢜࢞ሺ࢚࢔ି૚ሻ ൅ ࢇ࢞ሺ࢚࢔ି૚ሻ ∗ ሺ࢚࢔ െ ࢚࢔ି૚ሻ [2.6]
࢜࢟ሺ࢚࢔ሻ = ࢜࢞ሺ࢚࢔ି૚ሻ ൅ ࢇ࢟ሺ࢚࢔ି૚ሻ ∗ ሺ࢚࢔ െ ࢚࢔ି૚ሻ [2.7]
And the displacement is similarly calculated:

࢞ሺ࢚࢔ሻ = ࢞ሺ࢚࢔ି૚ሻ ൅ ࢜࢞ሺ࢚࢔ି૚ሻ ∗ ሺ࢚࢔ െ ࢚࢔ି૚ሻ [2.8]
࢟ሺ࢚࢔ሻ = ࢟ሺ࢚࢔ି૚ሻ ൅ ࢜࢟ሺ࢚࢔ି૚ሻ ∗ ሺ࢚࢔ െ ࢚࢔ି૚ሻ [2.9]

5

The displacement values are then transformed into the ground reference frame to
determine the horizontal distance and the altitude.

൤ࢎ࢕࢘࢏ࢠ࢕ࢇ࢔࢒࢚࢚࢏ࢇ࢚࢒࢛ ࢊࢊ࢏ࢋ࢙ሺ࢚࢚ࢇ࢔࢔ሻࢉࢋሺ࢚࢔ሻ൨ = ൤ࢎ࢕࢘࢏ࢠ࢕ࢇ࢔࢒࢚࢚࢏ࢇ࢚࢒࢛ ࢊࢊ࢏ࢋ࢙ሺ࢚࢚ࢇ࢔࢔ିࢉ૚ሻࢋሺ࢚࢔ି૚ሻ൨ ൅ ቂ࢓࢔ െ࢓࢔ቃ ൤࢟࢞ሺሺ࢚࢚࢔࢔ ሻ െ ࢟࢞ሺሺ࢚࢚࢔࢔ିି૚૚ሻሻ൨ [2.10]
ሻ െ
Where m=cos(θ) and n=sin(θ).

2.3 Motor thrust requirements

The equations above are dependent on the thrust (F) of the booster engine. The thrust is
calculated using equations found in the Aerospace Propulsion Systems textbook [4]. For
the preliminary sizing of the rocket motor, these closed form equations are used to
calculate the engine performance. The maximum diameter of the engine is sized to be
similar to that of the cruise missile. The length of the propellant is limited to 26 inches
to minimize the length of the booster motor. Knowing the diameter and the length of the
charge, the burn diameter can be calculated:

‫ܣ‬௕ = ߨ ∙ ݀௖ ∙ ‫ܮ‬௖ [2.11]

The x-function is the non-dimensional mass flow of the motor and is calculated by:

ംశభ

ܺ∗ = √ߛ ቂ ଶ ቃమሺംషభሻ [2.12]

ఊାଵ

The exhaust cone diameter is a variable that can influence thrust and is adjusted as

needed in the design to get the appropriate thrust based on a given expansion ratio. The

chosen diameter for this rocket motor is 13 inches. The exit area is calculated from the

cone diameter.

‫ܣ‬௘ = గ ݀௘ଶ [2.13]
ସ

The nozzle area is calculated from the exit area and the prescribed expansion ration

epsilon. The expansion ratio can be adjusted in the design phase in an iterative nature to

achieve the required thrust.

‫ = ∗ܣ‬஺೐ [2.14]

ఌ

The chamber pressure can now be calculated based on the propellant properties and the

nozzle area.

6

భ [2.15]

ܲ௖ = ൤ఘ೛∙஺஺∗್∙ඥ௑ோ∗ ∙்೎൨భష೙

The burn rate of the propellant is sensitive to the chamber pressure. The burn rate is

calculates as:

‫ݎ‬௕ = ݇ ∙ ‫݌‬௖௡ [2.16]

As can be seen in the previous two equations, the chamber pressure is dependent on

the burn rate and the burn area. Both the burn rate and the burn area are increasing as

the propellant is consumed which provides a progressive burn rate. To minimize this

effect, creative cross sectional areas can be made so that the total area does not increase

with propellant consumption.

In order to calculate the thrust coefficient, the exit velocity or exit Mach number needs

to be calculated. Unfortunately due to the complex nature of the following equations, an

iterative process is used to solve for Me.

ംశభ ംశభ

ߝ = ଵ ቀ1 ൅ ఊିଵ ∙ ‫ܯ‬௘ଶቁమሺംషభሻ ቀ ଶ ቁమሺംషభሻ [2.17]
ெ೐ ଶ [2.18]
ఊାଵ

షം

ܲ௘ = ܲ௖ ቀ1 ൅ ఊିଵ ∙ ‫ܯ‬௘ଶቁംషభ
ଶ

Knowing the exit velocity and the chamber pressure, the thrust coefficient is calculated

as shown.

ംశభ ംషభ
ඨଶఊమ ቂ ଶ ቃംషభ ቈ1 െ ቀ௉೐ቁ
‫ܥ‬ி = ം ቉ ൅ ஺೐ ቂ௉೐ି௉೚ቃ [2.19]
஺∗
ఊିଵ ఊାଵ ௉೎ ௉೎

These calculations are based on an ideal nozzle with full expansion. Due to thermal and
other losses, the actual thrust coefficient will be about 90% of the ideal thrust
coefficient.

‫ܥ‬௙ ௔௖௧௨௔௟ = 90% ∙ ‫ܥ‬ி [2.20]

A measure of the efficiency of the rocket design is the specific impulse. The specific

impulse can provide an idea of what the propellant flow rate is required for the given

thrust. The theoretical specific impulse is calculated by:

‫ܫ‬௦௣ = ஼ ∗ ∙஼ಷೌ೎೟ೠೌ೗ [2.21]
௚

7

The area of the core in the BATES type fuel configuration or a cylindrical configuration
should be four times the area of the nozzle to prevent erosive burning. From this area,
the volume of the core can be calculated using the propellant length. This core volume
is then subtracted from the propellant volume to give the actual propellant volume.
From this volume, the mass of the propellant can be determined.

‫ܣ‬௖௢௡ௗ௨௜௧ = 4 ∙ ‫∗ܣ‬ [2.22]
ܸ௖௢௡ௗ௨௜௧ = ‫ܮ∗ܣ‬௖ [2.23]

ܸ଴ = ‫ܣ‬௕‫ܮ‬௖ [2.24]

ܸ௖ = ܸ଴ െ ܸ௖௢௡ௗ௨௜௧ [2.25]

݉௖ = ߩ௣ ∙ ܸ௖ [2.26]

To calculate the burn time, the mass flow rate is determined and then the burn time is

calculated based on the propellant mass.

݉ሶ = ி೙ [2.27]

ூೞ೛∙௚

‫ݐ‬௕ = ௠೎ [2.28]
௠ሶ

An important characteristic of the motor performance is the total impulse. This is the

average thrust times the burn time.

‫ܫ‬௧ = ‫ܨ‬ே ∙ ‫ݐ‬௕ [2.29]

These calculations are performed in Microsoft Excel. The internal iterative solver in
Excel is used to determine the appropriate nozzle diameter and exhaust cone diameter to
meet the mission requirements. The Excel spreadsheet also calculates additional engine
parameters including chamber pressure which is required to properly size the structural
components of the engine casing. BurnSim software is then used to more accurately
calculate the engine thrust, chamber pressure and the mass flow. These parameters are
then imported into Excel to calculate the flight performance based on the BurnSim
results.

8

2.4 Motor Casing Sizing

Based on the thrust load and the chamber pressure, the stresses in the initial casing
design is analyzed using closed form equations provided in Roark’s Handbook [5].
ANSYS finite element software is then used to determine the stresses in the final casing
design. . The stresses are compared to the yield and ultimate strength of the material.
An aerospace standard factors of safety of 1.5 for ultimate strength and 1.15 for yield
strength. The metal alloy version of the rocket casing is to be made of a casted E357-T6
aluminum. Casting the casing will minimize the number of bolted joints and maximize
the strength of the structure which will minimize the weight. The composite material
version of the rocket motor casing will be designed of a similar shape.

Liner Propellant

Nozzle Casing Integral igniter
housing
Figure 2.2 Aluminum Engine Casing Concept Filler

Liner Propellant

Nozzle Thrust Plate/
Casing Igniter Housing

Figure 2.3 Composite Engine Casing Concept Filler

9

Figure 2.4 shows a typical 2 dimensional axisymmetric finite element model used to
analyze the motor casing. Pressure is applied to the internal surfaces up to the nozzle
where the pressure drops to near ambient values. A thrust load is applied to the top
surface in the axial direction (depicted as “C”). This load application represents where
the thrust is transferred to the payload. The casing is grounded at the end of the nozzle.
The load is typically distributed throughout the nozzle and not concentrated on the end
but the nozzle is structurally sturdy and not an area of concern for this project.

Figure 2.4 Finite Element Load and Boundary Conditions

2.5 Material

2.5.1 Aluminum Alloy
The Table 2.1 shows the material properties for E357T-6 cast aluminum prepared per
AMS 4288 [6]. A casting design is chosen to minimize the need for bolted joints which
will simplify the design and maximize weight efficiency. This aluminum is used since it
has a relatively high strength to weight ratio for a casted alloy.

10

Table 2.1 E357 T-6 Casted Aluminum ν ρ (lb/in3)
AMS 4288 Ftu (ksi) Fty (ksi) Fcy (ksi) Fsu (ksi) E (ksi)

T=72°F 45 36 36 28 10.4E3 0.33 0.097

T=300°F 39 37 - - 10.6E3 - -

2.5.2 Composite Material

A carbon epoxy composite material from Hexcel [8] is chosen for the composite version
of the casing. The properties for unidirectional fibers are shown in Table 2.2. The
maximum casing temperature is 300°F and so the strength is reduced by 10% based on
similar material trends. The strength is further reduced by 50% as an industry standard
ultimate strength safety factor.

Table 2.2 Hexcel Intermediate Modulus Carbon Fiber/Resin Properties

Ftu 1 Fcu 1 Ftu 2 Fcu 2 F12 E1 E2 G12 ν12
(psi) (psi) (psi) (psi) (psi) (psi) (psi) (psi)

room
348,000 232,000 11,000 36,200 13,800

temperature

300°F 313,200 208,800 9,900 32,580 12,420 2,466,000 1,305,000 638,000 0.27

1.5 Safety 208,800 139,200 6,600 21,720 8,280
Factor

This unidirectional material is layered several layers thick into a laminate. Some of the
layers will be at different angles from the others to tailor the material for the mission
loads. This allows the composite material to be optimized to minimize weight without
sacrificing strength. The overall laminate properties will be calculated based on the
material properties in Table 2.2 utilizing Classical Laminate Theory and Kirchoff’s
Hypothesis [7]. The following assumptions are made:

1) Lines normal to the midplane of a layer remain normal and straight and normal
during bending of the layer.

2) All laminates are perfectly bonded together so that there is no dislocation
between layers.

11

3) Properties for a layer are uniform throughout the layer.
The resulting stress strain relationship of the laminate is defined by:

ሼߝሽ = ሾܵሿሼߪሽ
This equation is expanded to:

૚ ିૅ૚૛ ିૅ૚૛ ૙ ૙ ૙‫ې‬
۳૛ ۳૛
‫ ۍ‬۳૚ ‫ۑ‬
‫ૅିێ‬૚૛ ‫ۑ‬
‫ࢿࢿࢿࢽێێۍێ‬૛૚૛૜૜‫ېۑۑۑ‬ ‫ ێ‬۳૛ ૚ ିૅ૛૜ ૙ ૙ ૙ ‫ۑ‬ ࣌૚
‫ࢽێ‬૛૜‫ۑ‬ ‫ૅିێ‬૚૛ ۳૛ ۳૛ ૙ ‫ۑ‬ ‫ۓ‬ ࣌૛ ۗ
‫ࢽۏ‬૚૛‫ے‬ ‫ ێ‬۳૛ ିૅ૛૜ ૚ ૙ ૙ ۖ ۖ
۳૛ ۳૛ ૙
= ૚ ૙ [2.30]
‫ێ‬ ૙ ૙ ૙ ۵૛૜ ૙ ‫ۑ‬ ‫۔‬૙ۘ
‫ێ‬ ૚ ૙ ‫ۑ‬ ۖ૙ۖ
૙ ૙ ૙ ۵૚૛ ‫࣎ە‬૚૛ۙ
‫ێ‬ ૙ ‫ۑ‬
‫ێ‬ ‫ۑ‬

‫ێ‬ ૙ ૙ ૙ ૙૙ ૚‫ۑ‬
‫ۏ‬
۵૚૛‫ے‬

The equation can be reduced to the following:

ࢿ૚ ࡿ૚૚ ࡿ૚૛ ૙ ࣌૚ [2.31]
൝ ࢿ૛ ൡ = ൥ࡿ૛૚ ࡿ૛૛ ૙ ൩ ൝ ࣌૛ ൡ
ࢽ૚૛ ૙ ૙ ࡿ૟૟ ࣎૚૛

Where [S] in Equation 2.12 is the reduced compliance matrix. This matrix is

transformed for each layer to equate the properties into the laminate coordinate system

as follows:

ࢿ࢞ ࡿ૚૚ ࡿ૚૛ ૙ ࣌࢞ [2.32]
ቐ ࢿ࢟ ቑ = ሾࢀሿି૚ ቎ࡿ૛૚ ࡿ૛૛ ૙ ቏ ሾࢀሿ ൝ ࣌࢟ ൡ
૚ ૚ ࣎࢞࢟
૛ ࢽ࢞࢟ ૙ ૙ ૛ ࡿ૟૟

Where

࢓૛ ࢔૛ ૛࢓࢔ [2.33]
ሾࢀሿ = ൥ ࢔૛ ࢓૛ െ૛࢓࢔ ൩

െ࢓࢔ ࢓࢔ ࢓૛ െ ࢔૛

This can be represented by:

ࢿ࢞ ഥࡿ૚૚ ഥࡿ૚૛ ࡿഥ૚૟ ࣌࢞ [2.34]
൝ ࢿ࢟ ൡ = ቎ࡿഥ૚૛ ഥࡿ૛૛ ࡿഥ૛૟቏ ൝ ࣌࢟ ൡ
ࢽ࢞࢟ ഥࡿ૚૟ ࡿഥ૛૟ ࡿഥ૟૟ ࣎࢞࢟
The global properties for the laminate can be calculated as follows:

ࡱ࢞ = ૚ [2.35]
ഥࡿ૚૚

12

ࡱ࢟ = ૚ [2.36]
ഥࡿ૛૛

ࡳ࢞࢟ = ૚ [2.37]
ഥࡿ૟૟

ࣇ࢞࢟ = െ ࡿഥ૚૛ [2.38]
ࡿഥ૚૚

ࣇ࢟࢞ = െ ࡿഥ૚૛ [2.39]
ࡿഥ૛૛

In the laminate coordinate system, the stress to strain relationship for a single layer can

be written as:

ሼ࣌ሽ = ሾࡽഥሿሼࢿሽ [2.40]

where

[Q̅ ] = [S̅ ]-1 [2.41]

To create the overall laminate load to strain relationship, the ABD matrix is created as

follows:

∑ ( )N _ [2.42]
[2.43]
Aij = Qijk zk − zk−1 [2.44]
k =1

Bij = ∑ ( )N _ −
Qijk z 2 z2
k k −1

k =1

Dij = ∑ ( )N _Qijkz 3k − z3
k −1

k =1

Where zk is the z-directional position of the ply number k. In a symmetric layup, z=0 at
the midplane and is positive in the lower layers and negative in the upper layers.

The complete load to strain relationship matrix is:

 NX  AA1121 A12 A16 B11 B12 B16   ε 0 
 NY    X 
  A 22 A 26 B12 B22 B26   
 ε 0 
A 26 A66 B16 B26 B66  Y 
B12 B16 D11 D12 D16 
MN XXY  = AB1116 κγ 0X0XY [2.45]

 κκ
M   B12 B22 B26 D  0 
M Y   B16 B26 B66 D12 D 22 D 26  Y 
XY  D16 D 26 66
0
XY

13

The maximum stress for the laminate is based on Tsai-Hill failure criteria for each
layer. The laminate will be considered to have failed when any layer exceeds the
maximum allowed stress. An Excel spreadsheet is used to calculate the stress in the
layers based on the laminate stress from the finite element model. The layer stresses are
used with the Tsai-Hill equation to determine a static margin. The Tsai-Hill failure
criteria equation is as follows:

ቂ࣌૚ ૛ െ ቂ ࣌૚࣌૛ ቃ ൅ ቂ࣌૛ቃ૛ ൅ ቂ࣎૚૛ቃ૛ < ૚ [2.46]
ࢄ૚ ࢄ૛ࢄ૛ ࢅ ࡿ
ቃ

Where

X1=F1t if σ1>0 and F1c if σ1<0
X2=F1t if σ2>0 and F1c if σ2<0

Y=F2t if σ2>0 and F2c if σ2<0
S=F12

14

3. Results

3.1 Engine Parameters

The predicted engine parameters based on the chosen nozzle diameter, expansion ratio
and fuel size are shown in Table 3.1.

Table 3.1 Engine Parameters Value Units
Parameter 13,481 lb
1,104 psi
Maximum Thrust 120,150
Max Chamber Pressure lbf-s
237 s
Total Impulse 20.87 in
Specific Impulse 6.55 in
Burn Diameter in
Conduit Diameter 26 s
Propellant Length 12.88 in
3.28 in
Burn Time 8.02
Nozzle Diameter
Nozzle Exit Diameter 6.0
Expansion Ratio 2.86
Exit Mach Number
Optimal Thrust 1.61

Coefficient 1.45
Thrust Coefficient Actual

3.2 Aluminum Alloy Casing Design

3.2.1 Aluminum Casing Geometry

Based on the engine parameters shown in Table 3.1, an engine casing is designed and
optimized for weight based on the material strength as shown in Table 2.1. The
maximum casing temperature is 300°F and so the material properties are reduced from
the room temperature properties as shown in the table. Figure 3.1 shows the final

15

dimensions of the engine casing. Table 3.2 shows the final weight of the aluminum
engine casing assembly.

Figure 3.1 Aluminum Alloy Casing Detail

Table 3.2 Aluminum Engine Weight Weight (lb)
Component 172
507
Engine Casing 50
Fuel 7
736
Liner/Filler
Nozzle
Total

3.2.2 Finite Element Analysis

Finite element analysis is performed using ANSYS Workbench V13. The loads and
boundary conditions are applied as shown in Figure 2.4 in section 2. The results are
shown in Figure 3.2 and Figure 3.3. The margins of safety are calculated with a 1.5
safety factor on the ultimate strength and a 1.15 safety factor on the yield strength.

ࡹࡿ࢟࢒ࢊ = ࡲ࢚࢟ െ ૚ [3.1]
૚.૚૞×࣌࢓ࢇ࢞

16

ࡹࡿ࢛࢒࢚ = ࡲ࢚࢛ െ ૚ [3.2]
૚.૞×࣌࢓ࢇ࢞

With a maximum stress of 25,817 psi, the margin of safety for the aluminum casing is

0.24 for yield strength and 0.01 for ultimate strength.

Figure 3.2 Maximum Stress Aluminum Engine Casing Upper

Figure 3.3 Maximum Stress Aluminum Engine Casing Lower
17

The predicted flight performances based on the total assembly weight and predicted
engine thrust is shown in Figure 3.4,Figure 3.5 and Figure 3.6.

Flight Path

1400

1200

Altitude (ft) 1000 altitude (ft)
800

600

400

200

0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Ground Distance (ft)

Figure 3.4 Assembly Flight Path

Flight Performance

14000 1000
12000
10000 800

8000Thrust (lb) 600 Thrust (lb)
6000 Altitude (ft) and Velocity (ft/sec)
4000 altitude (ft)
2000
horizontal velocity
0 400 (ft/sec)
0
200

0
2 4 6 8 10 12

TIme

Figure 3.5 Flight Performance

18

θ (degrees) Flight Performance
Altitude (ft) and Velocity (ft/sec)
90

1000
80

70
800

60

50 600 θ (degrees)
altitude (ft)

40 horizontal velocity
(ft/sec)
vertical velocity

400 (ft/sec)
30

20
200

10

00
0 2 4 6 8 10 12
TIme

Figure 3.6 Rocket Angle, Altitude and Velocities

3.3 Composite Casing Design

The composite version of the engine casing will have the same general shape as the
aluminum version but will be made of wound fibers over a sand mold. The fibers will
be coated in an epoxy resin. The thickness of the material will be tailored to optimize
the weight and strength of the structure. For to ease manufacturing and analysis, the
casing will be of uniform thickness.

3.3.1 Layup

The composite layup is [902/02/45/-45]s. Each layer is 0.006 inches thick. The sub-
laminate has 12 layers and the sub-laminate is layered 7 times. The engine casing is 0.5
inches thick made up of a total of 84 layers. The 0 degree orientation is in-line with the
casing axis but following the contour of the shell from top to bottom and the 90 degree
orientation is in the hoop direction. This layup will give strength in the hoop direction
for the pressure loading with the 90 degree fibers. The 0 and 45 degree fibers give the
laminate strength for bending in the curved geometry at the top and bottom of the casing
to react thrust load.

19

The overall properties of this layup are calculated using classical laminate plate theory.
The resulting properties of the laminate are shown in Table 3.3. The stress allowable for
this laminate would ultimately be determined through physical testing of the laminate.
The safety factors are calculated based on the unidirectional material properties and CLT
with Tsai-Hill failure criteria.
Table 3.3 Laminate Properties Calculated by CLT

Ex Ey Ez Gxy Gxz Gyz
νxy νzx νzy

10^6 psi 10^6 psi 10^6 psi 10^6 psi 10^6 psi 10^6 psi

10.68 10.68 1.73 2.54 0.53 0.53 0.20 0.38 0.38

3.3.2 Composite Casing Geometry
Based on the engine parameters shown in Table 3.1, an engine casing is designed and
optimized for weight based on the material strength as shown in Table 2.2. Figure 3.7
shows the final dimensions of the engine casing. Table 3.4 shows the final weight of the
composite engine casing assembly.

20

Figure 3.7 Composite Casing Detail

Table 3.4 Composite Engine Weight Weight (lb)
Component 97
507
Engine Casing 50
Fuel 7
1
Liner/Filler 662
Nozzle

Thrust Plate
Total

21

3.3.3 Finite Element Analysis

A two dimensional axi-symmetric analysis is performed similar to the aluminum casing.

The two dimensional geometry is split into segments as shown in Figure 3.8 so that the

coordinates of the finite elements in the curved sections can be aligned with the

curvature of the geometry. This allows the material properties to be, as will the fibers,

aligned with the geometric curvature. The material stiffness properties as applied in

ANSYS are shown in Table 3.5. These values are the same as in Table 3.3 but

transposed to align with the coordinate system used in ANSYS. In the ANSYS model,

the hoop direction is the z-coordinate, the axial direction is the y-coordinate and the

radial direction or through thickness is the x-coordinate.

Throat Top

Radius Throat

Bottom Cone Radius

Top Barrel Cone

Top Radius

Bottom Radius

Figure 3.8 FEA Geometry for Composite Casing

Table 3.5 Laminate Properties in ANSYS

Ex Ey Ez Gxy Gxz Gyz νxy νzx νzy
0.20
6 6 6 6 6 6
10 psi 10 psi 10 psi 10 psi 10 psi 10 psi

1.73 10.68 10.68 0.53 0.53 2.54 0.06 0.06

Figure 3.9 shows the load and boundary conditions similar to that of the aluminum
casing shown in Figure 2.4. The additional remote displacement is used on the top
section of the casing to represent the bonded thrust ring shown in Figure 2.3. This

22

constraint prevents the edges of the top hole from expanding or contracting radially but
allows all rotations and axial displacement. Figure 3.10 shows the deformation of the
casing and Figure 3.11 shows the peak stresses in the top curved section. A summary of
the margin of safety is listed in Table 3.6.

Figure 3.9 Load and Boundary Conditions Composite Casing

Figure 3.10 Maximum Total Deformation Composite Casing
23

Figure 3.11 Top Radius Stress Composite Casing

Table 3.6 Composite Casing Stress and Margins

Stress (psi)

Location Axial Axial Hoop Hoop Shear Shear Margin
min max min max min max
794.8 2.081
Cone -20144 -4729.2 -4897.3 -566.72 -312.46 674.68 3.716
183.22 5.415
Cone radius -13465 -6717 -6074.4 -1876.2 -722.42
504.71 0.696
Nozzle -9718.6 -8095.4 -2745.3 295.22 -426.46
1809.9 0.718
Throat Top
-14915 24486 -413 28774 -2014.8 1851.8 1.163

Radius 1116.6 1.198
1158.1 0.793
Bottom -13401 24279 15498 28629 -123.22 5475.1 0.129

Bottom -546.32 17727 3015.8 23340 -1188.7
Radius

Barrel 3701.5 13574 13057 24296 -1144.8

Top Radius -14147 29048 2951.3 19953 -1482.9

Top -21888 34018 8241.7 40858 1248.8

The lowest margin in the composites is similar to that of the aluminum casing in the top
section which is reacting the thrust forces as well as internal pressures. These margins
include the temperature knock downs as well as the 1.5 safety factor. Since composites

24

behave as a brittle material in that they do not significantly plastically deform prior to
failure, only ultimate margins are calculated.

25

4. Conclusion

A rocket motor provides a great deal of power for a short duration of time. In this
project, a solid fuel rocket motor is designed to produce over 13,000 lb of thrust for
almost 13 seconds which is capable of lifting over 3,000 lb of mass to a height of 1,000
feet and accelerate it to over 550 mph. There are many options for size and shape of the
propellant which can have a great influence on the thrust profile. A simple cylindrical
propellant shape was utilized in this project for simplicity but other options can be
explored. The thrust profile is progressive in that the thrust increases with time. The
chamber pressure is a moderate pressure of about 1,000 psi. The pressure makes it
feasible to use metal alloy and composite casings. The advantage of the composite is the
high strength to weight which allows for weight savings. For this design, the weight
savings is only 74 lb in an assembly that weighs more than 3,000 lbs. This weight
savings provides marginal flight performance increase as shown in Figure 4.1. Further
refinements can be done for the composite casing design to decrease the thickness in
high margin locations. Varying the thickness will require ply drop offs or fiver
terminations which requires special stress analysis. Overall, composites can be more
expensive and more technically challenging to manufacture than metal alloys. A further
cost and manufacturing analysis would need to be performed to determine if the use of
composites is justified.

26

1000 Aluminum vs Composite Flight Performance
800
600 900
400
200 800
0
0 700

600

500

400

300

200

100

0
2 4 6 8 10 12

TIme

Figure 4.1 Flight Performance Comparison
Altitude (ft) Aluminum Casing
Horizontal Velocity (ft/sec) Altitude
Composite Casing
Altitude
Aluminum Casing
Velocity

27

References

[1] Newton, Isaac. The Mathematical Principles of Natural Philosophy, pg 19 1729

[2] "Solid Rocket Motor." Wikipedia: The Free Encyclopedia. Wikimedia
Foundation, Inc. 2 February 2012. Web. 19 May. 2008

[3] Sutton, George Paul. Rocket Propulsion Elements. New York: John Wiley &
Sons, 1992

[4] Ward, Thomas A. Aerospace Propulsion Systems. Singapore: John Wiley &
Sons, 2010

[5] Young, Budynas, and Sadegh. Roark’s Formulas for Stress and Strain. New
York: McGraw-Hill, 2011.

[6] Metallic Materials Properties Development and Standardization (MMPDS-05)
U.S. Federal Aviation Administration.

[7] Hyer, M. W., and S. R. White. Stress Analysis of Fiber-reinforced Composite
Materials. Pennsylvania: DEStech Publications, 2009

[8] Hexcel (2005, March) Prepreg Technology. Pg 26 Retrieved February 02,
2012, from http://www.hexcel.com/Resources/DataSheets/Brochure-Data-
Sheets/HexForce_Technical_Fabrics_Handbook

28

Appendix A

29