PROJECT MANAGEMENT & DESIGN II
MAE 435
Rocket Project
Final Report
April 23rd, 2015
Group Members:
Nathan Akers
Class Supervisors:
Dr. Sebastian Bawab
Chad Bagley
Dr. Colin Britcher
Paul Campbell
Mr. Michael Polanco
Charles Juenger
Brian Mahan
Chris Skiba
Michael Weber
Michael Wermer
Project Advisor:
Dr. Thomas Alberts
i
Contents
Table of Figures ............................................................................................................................. iii
Nomenclature .................................................................................................................................. v
Abstract ......................................................................................................................................... vii
1.
Introduction ............................................................................................................................. 1
2.
Literature Review .................................................................................................................... 1
3.
Methods ................................................................................................................................... 4
3.1
Structural Design .............................................................................................................. 4
3.2
Fuel and Motor Design ..................................................................................................... 7
3.3
Nozzle Design ................................................................................................................ 10
3.4
Avionics Bay Design ...................................................................................................... 13
3.5
CFD Analysis ................................................................................................................. 14
3.6
Finite Element Analysis ................................................................................................. 15
3.7
Vibration Analysis.......................................................................................................... 16
3.8
Snatch Force Analysis .................................................................................................... 17
3.9
Ejection Charge Analysis ............................................................................................... 18
3.10 Rocket Manufacture ....................................................................................................... 18
3.11 Fuel Manufacture ........................................................................................................... 19
3.12 Motor Burn Testing ........................................................................................................ 21
ii
3.13 Wind Tunnel Testing...................................................................................................... 23
3.14 Launch Testing ............................................................................................................... 25
4.
Results ................................................................................................................................... 25
5.
Discussion .............................................................................................................................. 32
6.
Appendix ............................................................................................................................... 36
6.1 Rocket Drawings ................................................................................................................. 36
6.2 Pictures ................................................................................................................................ 44
6.3 Tables and Plots .................................................................................................................. 47
6.4 Fuel and Combustion Chamber Calculations ...................................................................... 52
6.5 Gantt Chart .......................................................................................................................... 57
6.6 Budget ................................................................................................................................. 58
7.
References ............................................................................................................................. 59
iii
Table of Figures
Figure 1: Rocket Components…………………....………………………...………...…………...5
Figure 2: Rocket Assembly in Inches………………………………………………...…………...6
Figure 3: Avionics Bay Components…………………………………………………………….36
Figure 4: Rocket Upper Body Components…………………………...………………...……….37
Figure 5: Rocket Lower Body Components.………..…………………………………...………38
Figure 6: Rocket Motor Components.…………………………………………………...………39
Figure 7: Propellant Published Values.………………...…………...…………….……...………47
Figure 8: Combustion Chamber Interior.………………………………………………...………..8
Figure 9: Kn vs Web Regression.…………………...…………………………….……...……..…9
Figure 10: Chamber Pressure vs Time……….…….……………………...………...…......…….10
Figure 11: Combustion Chamber with Converging-Diverging Nozzle……….…….…...………10
Figure 12: Velocity Contour Plot.……………………………………………..…….…...………12
Figure 13: Temperature Contour Plot.………………………………………………………...…13
Figure 14: Dual Deployment Wiring Diagram.…………………………...…..………...……….40
Figure 15: Avionics Bay Concept Sketch.………………………..…..………..………...………41
Figure 16: 3D Model of Avionics Bay Interior.……………...………………..………...………42
Figure 17: Coefficient of Drag vs Mach.………………………….…………..………...……….15
Figure 18: Vibration Analysis Input Parameters.…….………………………..………...………17
Figure 19: Amplitude vs Frequency Input Parameters.………………………..………...………17
Figure 20: Rocket Test Stand.……………………..…………………………..…….…...………23
Figure 21: Rocket Test Stand Components.……………………….…………..………...………43
iv
Figure 22: Subsonic Wind Tunnel Diagram.………,………...……...………..…….…...………24
Figure 23: Wind Tunnel Mounted Model.………………...…………………..…….…...………24
Figure 24: Thrust vs Time.…...………………………………………………..…….…...………26
Figure 25: Predicted Rocket Motor Performance.………………………..…...…….…...………47
Figure 26: Predicted Rocket Flight Performance.……………………………..………....………48
Figure 27: Rocket Assembly Predicted Displacement ……………………………...…...………27
Figure 28: Avionics Bay Predicted Displacement……………………………..………...………28
Figure 29: Vibration Analysis Results.………………………………………...………...………49
Figure 30: Avionics Bay Deformation at a Frequency of 926.775 cycles/s…..………...….……29
Figure 31: Frequency vs Standard Deviation Maximum Displacement…………..…….…...…..29
Figure 32: Rocket Failure…….………………………………………………..………...………44
Figure 33: Thrust Force vs Time.………………………….…………………..………...………30
Figure 34: Recovered Combustion Chamber Pieces.……………………...…..………...………44
Figure 35: Design and Burst Pressures for Rocket Motor Casing..………..…..………...………50
Figure 36: Fin Vortices.……...………………………………………………..………...……….31
Figure 37: Rocket Wake……...………………………………………………..………...………45
Figure 38: Rocket Nose Cone Boundary Layer…………………...…………..….……...………46
Figure 39: Test Flight Data..…………………………….……………………..………...………51
v
Nomenclature
r
Propellant Burn Rate
P0
Combustion Chamber Pressure
Kn
Propellant Burn Area to Nozzle Throat Area Ratio
p
Propellant Density
Pexit
Nozzle Exit Pressure
Patm
Atmospheri c Pressure
M throat Nozzle Throat Mach Number
Me
Nozzle Exit Mach Number
T  F Thrust
V exit

m
Mass Flow Rate
A*
Nozzle Throat Area
*
Nozzle Exit Velocity
D
De
Nozzle Throat Diameter
Nozzle Throat Diameter
R
Gas Constant
T0
Combustion Chamber Temperatur e
  k Ratio of Specific Heats
m  m noz Mass Flow Rate
X*
Throat Position
osi
ci
Outer Surface Inhibited (0  inhibited, 1  exposed)
Core Surface Inhibited (0  inhibited, 1  exposed)
ei
dio
Ends Surface Inhibited (0  inhibited, 1  exposed)
Initial Core Diameter
do
lg o
Initial Grain Diameter
Initial Grain Length
n
Ab
Number of Grains
Propellant Burning Area
At
Propellant Throat Area
two
dto
Initial Grain Web Thickness
Propellant Initial Throat Diameter
a
Burn Rate Coefficien t at P0
G
Kv
Erosive Burning Factor
Propellant Erosive Burning Velocity Coefficien t
dc
Combustion Chamber Diameter
vi
Aduct
Flow Area
G*
Erosive Burning Threshold (G *  6)
i
Initial Time Step
j
v grain
Subsequent Time Step
Grain Volume
v free
Free Volume in Combustion Chamber
m grain Grain Mass
m gen
Mass Generation of Combustion Products
m sto
Mass Storage Rate of Products in Combustion Chamber
mass sto Mass of Combustion Products Remaining in Chamber
 sto
Density of Combustion Production in Combustion Chamber
Fty
Yield Strength (ksi)
Ftu
Ult imate Tensile Strength (ksi)
E
Modulus of Elasticity (ksi)

Poisson Ratio
D0
Combustion Chamber Outside Diamter
t
Combustion Chamber Wa ll Thickness
SD
Combustion Chamber Design Safety Factor
 
Fty / Ftu
B
Combustion Chamber Burst Factor
PD
Combustion Chamber Design Pressure (ksi)
PU
Combustion Chamber Burst Pressure (ksi)
AE
Nozzle Exit Area
Cf
Thrust Coefficien t
 nox
Isentropic Efficiency of the Nozzle
t
Time Interval
It
Average Impulse Over Given Time Interval
vii
Abstract
The goal of the project was to design a high-powered rocket capable of deploying a
cosmic ray detector at an apogee of 10,000 feet. Detailed designs of the airframe, propulsion
system, aerodynamic fins, avionics, and recovery system was developed using computer aided
drafting software (CAD). Structural stability was determined through finite element analysis
(FEA). The propulsion system is powered by a Potassium-Nitrate (KNO3) - Dextrose
(C6H12O6) solid propellant. The expected thrust for the fuel was calculated and an optimized
nozzle was designed to accommodate the combustion chamber conditions. Aerodynamic analysis
was performed through modeled computational fluid dynamics (CFD) to determine coefficient of
drag versus Mach number. Scaled rocket subsonic wind tunnel testing was performed to
determine the accuracy of the CFD analysis. The avionics system autonomously records in-flight
data and controls the deployment of the payload, the drogue and main parachutes, ensuring safe
recovery after flight. Theoretical vibration analysis was performed to minimize determine if
there were any harmful effects to the electronics. The analysese and testing performed allowed
the team to extract the necessary information to validate the design and ensure a safe and reliable
flight.
1
1. Introduction
Rockets are a rapid means of payload delivery. Rocket engines operate at high thrusts,
mass flow rates, temperatures, and pressures, with added complexity from multiple staging. The
fuel is often very volatile which can lead to explosions if the wrong part, operating close to the
design limit, fails. This is why rockets are designed based upon the payload.
The requirements of this project are to design a high-powered rocket capable of
deploying a five pound cosmic ray detector at 10,000 feet. These design requirements are the
same as the SEDS (Students for the Exploration and Development of Space) 2014 Rocketry
Competition Rules. However, the 2015 competition goals changed dramatically from the
previous year’s goal, by not requiring a payload. The decision was made to not participate in the
2015 competition, and the rocket was built using the 2014 goal because the primary customer,
the local SEDS chapter, is building the cosmic ray detector to be deployed. The rocket design
was divided into four key areas necessary to satisfy the goal of the project. These included:
structures, propulsions, aerodynamics, and avionics.
2. Literature Review
A high-powered rocket is any rocket weighing more than 53 oz. and using a motor with
more than 160 Newton-seconds total impulse and no more than 40,960 Newton-seconds or a
having average thrust of 80 Newtons. CAD software is used to create an accurate visual
representation of the rocket and its components. The representation is used to perform analysis
that determines design feasibility prior to construction [1]. Finite Element Analysis (FEA) is a
method for determining stress and strain properties on complex solid bodies due to applied forces
[2-4]. FEA is used on the rocket assembly to design for structural integrity and to verify material
2
selection [5]. The analysis is needed to determine if the rocket design is capable of successful
flight. Every new rocket design needs to be tested for structural integrity.
Rocket motors are thrust engines which operate by generating a pressure and momentum
thrust. This is achieved by combusting fuel inside a pressure vessel and expanding the
combustion products through a nozzle. This is typically accomplished in small-scale rockets by
using solid rocket fuels. Commercially available “high-power” hobby rocket motors are typically
made of Ammonium Perchlorate, HTPB and Aluminum, which is similar to the formulations
used in rockets designed for space flight as well as missiles. These formulations, commonly
referred to as APCP (Ammonium Perchlorate Composite Propellant), provide a high specific
impulse relative to other solid formulations. Specific Impulse is a key concept in the design and
of a rocket motor because it is used to describe the efficiency of a fuel based on its weight.
Specific Impulse is the ratio of the amount of thrust a fuel is able to produce to the weight flow
of the propellants [6]. One drawback to using an APCP is the cost to produce; the material cost is
high and the manufacturing process can be difficult. An alternative to APCP which is commonly
used in homemade rockets is Potassium Nitrate-Sugar propellant, sometimes referred to as “RCandy”. This formula is easy to manufacture using readily available materials. The largest
drawback to using these “R-Candy” formulations is the relatively low specific impulse they
produce; meaning to reach a given altitude, more fuel is required. These formulations can be
modified using different sugars such as sucrose, sorbitol, or dextrose, which can impact the
material properties of the casted propellant and influence the ease of manufacturing and the
reliability of the motor itself.
To achieve maximum propulsive efficiency, a rocket motor must sustain a high flow of
heavy particles and elevated combustion temperatures while maximizing the speed of gases
3
through the nozzle exhaust such that the exit pressure is equal to the atmospheric pressure [7, 8].
Exponentially increasing temperatures in the combustion chamber can cause rapid deformation
of the materials of the nozzle and chamber [7]. Combustion gas expansion induces high internal
pressures within the combustion chamber, which further exacerbates component deformation.
Gases that travel through the nozzle carry solid fuel particles, which cause ablation to the throat
section of the nozzle and agglomeration in the nozzle cone [9-11]. This results in decreased
efficiency by altering the nozzle inner dimensions. Additionally, heat transfer in the chamber
must be assessed to determine the effects on deformation of the combustion chamber walls and
to select components such as O-rings that can withstand the heat[12]. The inability to
accommodate these operating conditions results in performance inefficiencies as well as
potential component failure. Determination of a nozzle design, which accommodates maximum
operating conditions, will increase performance and longevity while reducing cost.
Aerodynamics is examined to reduce drag and shock effects. Shapes of nose cones and
fins have been examined in an effort to reduce drag and shock effects [13-18]. Nose cone designs
showed that increased ratios of nose cone length to body length decreased drag, and reduction of
the angle of the nose cone resulted in a smaller shock transition between the nose and the rocket
body during supersonic flight [14, 15]. Fin designs were primarily focused on the flight stability,
but as the size and bluntness of the leading edge increased, the drag and shock effects increased
[13, 16-18]. While the basic aerodynamics of fins and nose cones are known, the final fin
designs and nose cone designs are modified for every rocket based upon mass, and length of the
rocket in order to move the center of pressure aft of the center of mass. This ensures flight
stability by creating a restoring moment if the flight angle of attack is small. If the angle of attack
becomes too large the rocket will become unstable and tumble, but the angle of attack will only
4
become too large if there is a motor failure or there are sudden large gusts of wind. Highpowered rockets can only be launched in wind less than 20 miles per hour.
The avionics bay of a model rocket is typically positioned in the center section of the
rocket, between the forward nose cone and rear motor section. The avionics bay must be
designed to protect the rocket’s sensitive electronics, both from in-flight forces and impact from
landing. The National Aeronautics and Space Administration (NASA) identifies four in-flight
forces: weight, thrust, lift and drag [19]. The avionics bay must be designed to properly house
and support the sensitive electronics, and mitigate the damaging effects from these in-flight
forces. A rapid shock is an additional force the rocket could experience upon landing. The
purpose of the parachute recovery system of the rocket is to lessen this shock load. The
electronics mounting and structural design of the avionics bay must account for both the in-flight
forces and the shock load upon landing. The maximum in-flight forces and shock load are unique
to every rocket because every rocket has a different thrust which affects the in-flight forces and a
different mass which affects the shock load.
3. Methods
3.1 Structural Design
A 3D model of the rocket was created using Autodesk Inventor (Autodesk Inc., San
Rafael, CA) and was developed by designing part assemblies of the different rocket components.
These components (Figure 1) were then joined together to produce the main rocket assembly
(Figure 2).
5
Figure 1: Rocket Components
The first part assembly designed was the avionics bay (Figure 3), which houses the
electrical equipment and connects the top and lower portions of the rocket. The ends are fitted
with two circular end caps and are secured to the tubular housing by two threaded rods with
accompanying fastener hardware. Eyebolts are fitted to the center of the two end caps by
fasteners. These bolts serve as an anchor point for the shock chords after in-flight rocket
separation has occurred. The nose cone (Figure 4) was designed with a hollow cavity
surrounded by the nose cone walls. A circular end cap with a U-bolt and the accompanying
fasteners are fitted to the end of nose cone. The U-bolt serves as another anchor point for the top
shock chord. Connecting the nose cone and the avionics bay is the upper rocket body (Figure 4).
6
The lower rocket body is a tube fitted with six fins (Figure 5). The engine assembly is made up
of a nozzle, outer casing, resin liner and end cap (Figure 6). This is attached to the lower rocket
body using six circular motor mounts attached to inner walls of the bottom rocket body.
Connecting the avionics bay to the nose cone are Kevlar® shock chords. Connecting the nose
cone to the upper body and the avionic bay to the lower body are three shear pins for each
connection. The avionics bay is connected to the upper rocket body using three riveted joints.
Figure 2: Rocket Assembly in Inches
7
3.2 Fuel and Motor Design
The rocket is powered by an “experimental” fuel, meaning that the motor utilizes a
homemade propellant formulation in lieu of a commercially available motor. The formulation of
the fuel was taken from Richard Nakka’s Experimental Rocketry Website [20]. The need to
select a pre-investigated formulation was necessitated by the compressed budget and timeline of
this project as there were not enough resources available to develop a unique formulation of
APCP which could function safely and reliably. The propellant selected for this project was a
mixture of 65% by mass Potassium Nitrate (KNO3) and 35% by mass Dextrose (C6H12O6) [21].
C6H12O6 + 3.31 KNO3  2.116 CO2 + 2.300 CO + 4.512 H2O + 1.424 H2 + 1.655 N2 + 1.585
K2CO3 + 0.133 KOH (Equation 1)
(theoretical combustion reaction at 68atm [21])
This formulation has been tested extensively and is commonly used by high-power
rocketry hobbyists. This relatively inexpensive formula has been proven to be safe and reliable
while being relatively easily manufactured, and produces results within a reasonable margin of
predicted results calculated by using the published values (Figure 7). The preliminary motor
design has been modified using the Microsoft Excel (Microsoft Corporation, Redmond, WA)
spreadsheet (“SRM_2014”) published by Richard Nakka [22]. This spreadsheet functions in
three phases – calculating Kn and throat area, calculating chamber pressure and burn time and
finally calculating thrust and impulse. Each of these functions occurs in an iterative process
using a linear regression of the fuel from the inner core to the outer surface to calculate the next
parameter. The process then continues using the data obtained from the previous iteration until
8
the regression of fuel is complete. This process is detailed step by step in the Appendix 6.5, in
order of the operations as they occur.
A target maximum operating pressure within the combustion chamber (Figure 8) of 1000
pounds per square inch (psia) was based on the final as built nozzle diameter of 0.625”. The
motor casing was constructed from 3.0” outside diameter, 6061-T6 Aluminum tubing with a wall
thickness of 0.125”, giving an inside diameter of 2.75”. The predicted maximum operating
pressure of 1027psia gave a conservative factor of safety of 3.001. This conservative value was
selected because of the need to re-use the motor casing multiple times and the relatively lowimpact of the casing weight on the rocket performance and design.
Figure 8: Combustion Chamber Interior
9
The size of the motor was scaled to deliver the rocket as close to an apogee of 10,000
feet as possible, without exceeding that limit. The operating pressure is achieved by modifying
the Kn value, which is the ratio of burning surface area of the propellant to the area at the throat
of the nozzle [23]. This was done by selecting a commercially available propellant casting mold
and modifying the nozzle diameter to meet the required Kn value. The KNDX propellant burns
progressively in from the center core, causing the exposed surface area to increase (Figure 9), to
a maximum Kn value of 342, resulting in a max pressure in the chamber of 1027psia (Figure
10). Before ignition, the “Kn” value is 276; this is favorable because a higher Kn value allows
for more reliable ignition. Ignition tests using black powder charge igniter have shown that the
motor should ignite completely and reliably.
SRM 2014.XLS
Kn vs Web Regression
400
25
350
250
Kn
15
200
10
150
100
5
50
0
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26
Web Regression (mm)
Kn
Kn max
Kn min
Kn avg
web thickness
342
276
320
Figure 9: Kn vs Web Regression
web thickness
20
300
10
Pmax =
t burn =
t thrust =
1027
1.774
1.849
psi
s.
s.
Figure 10: Chamber Pressure vs Time
3.3 Nozzle Design
The nozzle (Figure 11) was assumed to be isentropic, and compressible flow equations
were used to determine the thermodynamic properties at the subsonic, sonic, and supersonic
regions.
Figure 11: Combustion Chamber with Converging-Diverging Nozzle
11
The properties at each station and the constraint of three inches on the converging section
were used to determine the dimensions of the nozzle. To reach maximum propulsive efficiency,
the exit gases must be expanded through the diverging section of the nozzle so that Pexit equals
Patm. The mass flow rate in the nozzle was determined by
(Equation 2) where
g -1
2g RTc é Pe ù g
Ve =
ê1- ú
g -1 ë Pc û
(Equation 3). Using the mass flow rate and exit velocity, the throat diameter
(Equation 4) and 𝑋 ∗ is a function of the ratio
was determined using the equation
g +1
æ 2 ö 2(g -1)
of specific heats X * (g ) = g ç
(Equation 5). Next, the nozzle exit diameter was found
÷
è g +1 ø
g +1
é
ù
2 êæ Pc ö 2(g -1) ú
by first finding the Mach number at the exit by M e =
-1ú (Equation 6).
ç ÷
g -1 êè Pe ø
êë
úû
Substituting the Mach number into the equation for area ratios provided the diameter for
g +1 ö
æ
2(g -1)
é
ù
æ
ö
æ
ö
4ç 1
2
g -1 2
÷
the exit De = A* ç
M e ÷ú
êç
֍1+
÷ (Equation 7). A simulation of the nozzle
øû
p ç M e ëè g +1 øè
2
÷
è
ø
was run at the determined area and pressure ratios using the CD Nozzle Simulator (engApplets,
Blacksburg, VA).
Using Simulation CFD, (Autodesk Inc., San Rafael, CA) a CFD analysis was performed
on the nozzle with the inlet conditions found in the fuel analysis. The assumptions were a fluid of
air, compressible flow, sea level standard air conditions, steady flow, and no heat transfer. CFD
analysis was run on the nozzle to supply velocity (Figure 12) and temperature (Figure 13)
contour plots of operating conditions. This was used to reveal shockwaves that may occur in the
12
diverging section of the nozzle, with the understanding that this does not provide conclusive
proof. Basic CFD packages such as this run blending algorithms that may smooth out smaller
shocks. The effects of stress on the nozzle were modeled using FEA with a special attention was
paid to the stress concentrations around the screws attaching the nozzle to the combustion
chamber. Also, having an approximation for the heat through the nozzle walls helps select the
proper materials for the two sets of O-rings at the front and rear of the motor. Hand calculations
of the stress concentration also provided additional validations for the models. It was important
to build a motor system that would handle the predicted operating conditions for several launches
and still be light enough to allow for maximum payload weight.
Figure 12: Velocity Contour Plot
13
Figure 13: Temperature Contour Plot
3.4 Avionics Bay Design
The avionics system consists of a redundant, dual-deployment altimeter system capable
of ejecting the drogue and main parachutes. The wiring diagram for the dual-deployment
altimeters is seen in the appendix (Figure 14). The dual-deployment altimeters are programmed
to deploy two separate parachute ejection charges that will allow failure of the rocket airframe
shear pins. The ejection charges and airframe shear pins have been sized using theoretical
analysis, common rocketry calculators [24] and experimental bench testing. The first ejection
charge causes shear pin failure of the rocket’s forward airframe, releasing the drogue parachute
and cosmic ray detector payload. The drogue parachute is deployed at the rocket’s apogee and it
is used to decelerate and stabilize the descent of the rocket. The second ejection charge causes
shear pin failure of the rocket’s rear airframe and deploys the main parachute at approximately
800 feet above the ground.
14
To verify the structural and electrical stability of the avionics bay, vibration analysis and
snatch force analysis were conducted. The vibration analysis was conducted to verify the
displacement due to structural oscillations would not produce damaging forces in the avionics
bay causing failure of the electronics and associated mechanisms. Snatch force analysis was
performed to ensure the avionics bay would not fail during the main parachute deployment.
During main parachute deployment, the parachute ejects from the rocket, but remains connected
by a shock chord attached to an eye bolt mounted to the avionics bay end cap. When the
parachute canopy inflates, the shock chord is jerked under the force from the opening parachute.
At this stage, the avionics bay is acting as a key structural element and is exposed to this
“snatching force.” electronics, electrical mounting structures, parachutes, shock cords, and rocket
airframe shear pins have been ordered. A conceptual sketch has been developed for the
arrangement of the avionics bay section of the rocket and is in the appendix (Figure 15). A 3dimensional model of the avionics bay is in appendix (Figure 16).
3.5 CFD Analysis
Simulation CFD (Autodesk Inc., San Rafael, CA). was chosen to perform aerodynamics
analyses on the rocket. The assumptions for the flow to be used in the CFD analyses were a fluid
of air, compressible flow, sea level standard air conditions, steady flow, and no heat transfer. The
inputs were Mach numbers from 0.1 to 0.8 in step sizes of 0.1. The output criteria to judge the
different designs were coefficient of drag and center of pressure. In order for the design to be
stable the center of pressure had to be below the center of gravity. The Coefficient of Drag vs
Mach number are plotted below (Figure 17).
15
Cd vs Mach
0.4
0.35
0.3
Cd
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
Mach
Figure 17: Coefficient of Drag vs Mach
3.6 Finite Element Analysis
The FEA was carried out through the application of NASTRAN In-Cad (Autodesk Inc.,
San Rafael, CA). Finite element analysis was completed on the rocket. The rocket was analyzed
with the max thrust of 475 pounds applied to the rocket motor. It was opposed by the force of
gravity and a drag force of 88.84 pounds. A standard mesh was developed and then refined to
ensure the success of the analysis. FEA analysis revealed that the rocket had a safety factor of
1.8 at peak thrust and peak drag.
Finite element analysis was also completed on the avionics bay. The opening of the main
parachute exerts a high force on the avionics bay. A maximum force of 129 pounds was
calculated by the avionics team. FEA analysis revealed that during the deploying of the
parachute, there was a safety factor of 1.33.
16
3.7 Vibration Analysis
The completed avionics bay was modeled using Inventor (Autodesk Inc., San Rafael,
CA). This model was imported into Autodesk Simulation Mechanical (Autodesk Inc., San
Rafael, CA) where random vibration analysis was conducted.
Modal analysis was first employed by placing a 5 pound weight at the contact point in the
Inventor model where the forward rocket mass would be resting on the avionics bay. Using the
default 5 frequency inputs (Figure 18), the results generated a cumulative mass excited in all
directions far below the required minimum of 80% mass usage [25]. The number of frequency
inputs was increased to 25 before the analysis examined more than 80% of the cumulative mass.
These 25 modal analysis frequency results were then loaded into the random vibration
analysis tool in Autodesk Simulation Mechanical. Inputs (Figure 19) were required for
frequency and power spectral density, and the inputs were chosen based on the typical power
spectral density input for a model rocket [26]. The theoretical results obtained from the random
vibration analysis produce a displacement result based on the root mean square response. The
root mean square response is the square root of the mean of the squares of a sample. The results
can demonstrate an expected displacement and deformation resulting from the inputs of the 5
pound mass load, 25 frequency points, and power spectral density inputs. These key values are
combined through the modal and random vibration simulation to provide this deterministic
output.
17
Frequency
(Hz)
10
100
1000
2000
1
2
3
4
.
Amplitude
(g2/Hz)
0.002
0.04
0.04
0.02
Figure 18: Vibration Analysis Input Parameters
Amplitude (g2/Hz)
Vibration Testing Input Parameters
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
2500
Frequency (Hz)
Figure 19: Amplitude vs Frequency Input Parameters
3.8 Snatch Force Analysis
In order to perform accurate FEA on the avionics bay, snatch force calculations were
performed. At parachute inflation, an upward force is generated which “jerks” on the shock
cord. This force is directed to the eye bolt mounted to the end cap of the avionics bay. The
maximum force (Fmax) generated is the parachute suspension-line tension. The governing
equation uses units of pounds-force and is outlined below [27].
2𝑊𝑉𝑖
𝐹𝑚𝑎𝑥 = (
𝑔𝑇
[1 −
𝑉𝑓𝑖𝑛𝑎𝑙
𝑉𝑖
]) + 2𝑊 (Equation 8)
W=weight of the rocket, 𝑉𝑖 =fall rate at the beginning of inflation, g=gravitational constant,
T=duration of the inflation sequence, and 𝑉𝑓𝑖𝑛𝑎𝑙 =fall rate at the end of inflation.
A conservative weight of 30 pounds was used for the rocket. The gravitational
constant is in English units and equates to 32.17 ft/sec2. An initial velocity of 90 ft/sec was
18
used which is the maximum velocity that is sustained with the drogue parachute open. The
final velocity will not be slower than 15 ft/sec. Finally, the inflation time will not exceed 2
seconds, if the parachute is properly folded and inserted into the rocket airframe.
3.9 Ejection Charge Analysis
Ejection charge and shear pin sizing was performed, first using theoretical analysis and
commonly available calculators used in model rocketry [23]. Data obtained from these
calculators was used as a baseline to size the ejection charges and shear pins for experimental
testing. Bench tests were performed on the assembled rocket testing the rocket’s separating
force. Proper ejection charge sizing ensures failure of the rocket airframe shear pins during
flight, allowing safe separation of the airframe and deployment of parachutes.
Theoretical ejection charge sizing analysis was conducted to determine the predicted
amount of black powder (BP) needed for proper rocket separation.
𝑉
𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑏𝑙𝑎𝑐𝑘𝑝𝑜𝑤𝑒𝑟 (𝑔𝑟𝑎𝑚𝑠) = 𝑑𝑃 𝑇 (Equation 9).
𝑅
dP=pressure inside rocket during separation is assumed to be 15 psi, V=the volume of the section
under consideration, R=gas constant, and T=absolute temperature during ejection. After the
ejection charges were calculated bench testing was performed to verify the theoretical ejection
charge mass was sized appropriately.
3.10
Rocket Manufacture
The rocket components were fabricated from G12 filament wound glass and arrived from
the manufacturer in accordance with designated specifications. The manufacturer also cut the fin
slits in the lower body tube in accordance with their specified positions. West System epoxy
was used as the fastening agent for assembling the different rocket components. This epoxy
system was employed by mixing three parts of the West System Epoxy resin with one part of the
19
West System Hardener. The motor mount was the first piece of the bottom portion to be
assembled and was epoxied into the bottom rocket body. Next the fins were inserted into their
corresponding holes and glued flush against the motor mounting tube. A thickening agent was
mixed in with the epoxy and fillets were created between the fins and the outside of the rocket
body. The motor retention ring was the final piece to be installed on to the bottom of the lower
rocket body via bolts. The nose cone was assembled by joining the cone to a bulkhead via a bolt
with accompanying nuts. An eyebolt was attached to this bulkhead by a corresponding nut. The
avionics bay was assembled by gluing a centering ring in the middle of the avionics body tube.
The last step was to drill holes in the top body tube, the lower body tube, and the avionics tube
for the necessary connecting hardware. The combustion chamber and nozzle designs were
submitted to the ODU machine shop for manufacture. The steel for the nozzle was provided by
the machine shop.
3.11
Fuel Manufacture
Several manufacturing investigations took place before actual fuel manufacturing began.
This consisted of cooking an inert fuel where the Potassium Nitrate oxidizer was replaced with
Sodium Chloride in the same mass ratios as the fuel mixture. These tests revealed that in order to
make a homogenous mixture which could be molded easily, it is necessary to heat the mixture at
a higher temperature than what is minimally necessary to cause melting of the Dextrose. The
sample batch which was made at a lower temperature did not melt fully and was difficult to work
with when melting occurred. During these tests, the mixture was exposed to an open flame which
showed that the dextrose fuel was resistant to overheating. The dextrose tended to decompose,
rather than burn in the presence of excessive heat. Tests involving a dry mixture of the Potassium
Nitrate and Dextrose confirmed this result, even in the presence of the oxidizer. When exposed
20
to an open flame, prolonged expose to the flame source was necessary to obtain ignition of the
mixture. This is desirable from a safety standpoint while manufacturing fuel to know that there is
a large margin between melting and ignition of the fuel. It was even shown that when the fuel
mixture was put directly onto the heating element, ignition did not occur, rather, the Dextrose
caramelized and decomposed.
The process of manufacturing the fuel required the two components to be thoroughly
mixed together and then heated until the dextrose melts, homogenously binding the two products
together. The particles were mixed together while over heat and stirred constantly throughout
heating to ensure a thorough and homogenous mixture. Initial fuel batches consisted of
Potassium Nitrate and Dextrose particles which were finely milled in a blender before heating;
these batches were highly susceptible to caramelization and burned significantly more slowly
and less energetically than batches which were made with dry ingredients which were not milled
before heating. Although this is counterintuitive to the common design practice to make the dry
particle grain sizes as small as possible to ensure a more efficient burn, the higher rate of
dextrose caramelization had a larger negative impact on burning rate. Later grains which used
the dry ingredients in their native state, which was already fairly fine, showed almost no signs of
caramelization during heating, were much easier to melt homogenously, and the final product
was much lighter in color than earlier batches. A comparison test of samples from each
manufacturing group showed that the unmodified grains burned much faster than the grains
which were milled before heating. To manufacture the fuel, the Potassium Nitrate-Dextrose dry
mixture was heated between 145-150°C. The Dextrose begins to melt at 145°C and caramelize at
157°C so it was important to precisely control the temperature; caramelization of the sugar
negatively impacts the performance of the propellant. The fuel mixture was heated in a heavy
21
bottom metal pot over a hot plate and stirred throughout heating with a silicon spatula. The
constant stirring was necessary to prevent hot spots and premature caramelization of the
Dextrose which would negatively impact fuel performance. After the mixture was fully melted
and homogeneous, the fuel was packed into a 2.493” inside-diameter, cylindrical casting tube,
placed around a cylindrical core with a diameter of 0.6875” and then secured in an air-tight bag
and left to cool and solidify. To form the motor, four of these castings were made at a length of
5.0”; each of these castings is known as a grain. The completed motor is comprised of four
grains, totaling 20” long with an outside diameter of 2.493”, and a core diameter of 0.6875”, for
a total volume of 97.6in3 and theoretical fuel mass of 5.817 lb. A key manufacturing concern
was eliminating as many air gaps in the fuel mixture as possible; any air gaps in the fuel can
cause a rapid increase in burning area of the fuel and therefore cause a spike in pressure. Before
motor fires, the grains were weighed to ensure they were within 95% of their theoretical density.
Tests of the fuel showed that the fuel mixture gave off tremendous amounts of smoke but
burned rather slowly but data for this formulation shows that the mixture burns faster at higher
pressures. Ignition with open flame was inefficient and could not get the entire fuel sample to
ignite rapidly which was necessary for use in a motor. A comparison test showed that a similar
size fuel sample will burn more completely and rapidly when ignited with black powder. Black
powder is the ideal igniter for our motor because achieving target operating pressure at ignition
improves the efficiency of the motor.
3.12
Motor Burn Testing
A test stand was constructed to measure motor thrust (Figure 20, Figure 21). The purpose
of the test stand was to capture the motor thrust and direct it to a single point where it could be
measured accurately with a load cell. It was built with a 2 foot W10 x 22 I beam as the base. 3
22
square 3/16 inch thick steel plates held the motor in place. A 3.125 inch hole was bored in each
of the plates to accommodate the motor. The load cell was bolted to a ½ steel plate at the end of
the stand. The 3 plates aligned the motor so that the front of the motor would be directed into the
load cell, allowing freedom of movement towards the load cell, while ensuring that the motor
stayed secured during a test. The test stand then had 4 holes drilled into it and it was securely
bolted to a pickup truck bed. The truck bed was used since it provided a stable, heavy, mobile
platform. The steel bed also provided some shielding for the testing personnel in the case of a
catastrophic failure. The data was then transmitted into the cab of the truck where it was read by
a computer.
The load cell used was an Interface 1010AF 1000 lb load cell which was fixed to the test
stand as a means of measuring the thrust. A cable was connected to the load cell which attached
it to a National Instruments data acquisition system and a power amplifier. The amplifier
provided the load cell with 15 volts of electricity. The data acquisition system was connected to
a computer through a USB cable. LabView was installed on the computer along with the
appropriate drivers to recognize the data acquisition system. A program was set up on LabView
to record the voltage change across the load cell and convert it to pound force. This program
was also set up to record a point of a data per millisecond.
23
Figure 20: Rocket Test Stand
3.13
Wind Tunnel Testing
Wind tunnel testing was conducted in the subsonic wind tunnel (Figure 22) to obtain a
visualization of the air moving along the rocket body. In order to use the subsonic wind tunnel,
an attachment was machined to connect to the stand which held the model in the tunnel. The
attachment was made from a block of aluminum. Three holes were drilled 1 inch apart into the
top of block. These holes were made so the attachment could be screwed into the testing stand.
A 0.5625 inch hole was drilled through the aluminum for the holding rod to slide into. This rod
was held tightly in place by 2 more screws drilled into the side of the block. Glued onto the rod
was a long wooden shaft which slid inside the back of the model (Figure 23).
24
Figure 22: Subsonic Wind Tunnel Diagram
The model, connected to the test stand, was then placed inside the subsonic wind tunnel.
Using a laser wall and a smoke rod, flow visualization was done to show the different vortices
created by the fins as well as the flow around the body and nosecone.
Figure 23: Wind Tunnel Mounted Model
25
3.14
Launch Testing
Launching the constructed rocket required a TRIPOLI level 1 & 2 rocketry certification.
The level 1 certification was attained by building another rocket for launching with a motor
having an impulse of less than 640 N-s. Level 2 entailed taking a written test and launching the
same rocket with a motor with an impulse of between 640 and 5120 N-s. This certification was
completed in one day moving the project closer to completion and a successful launch. The
original plan was to launch the team rocket at a SEVRA research motor event the next day.
However, the event was cancelled and the next one would not be until after graduation. It was
decided to take the opportunity to test launch the team rocket using a commercial motor. The
motor used was manufactured by Aerotech and had a total impulse of 1762 N-s and an average
thrust of 805 N. The test launch took place on after the certification launches making three
successful launches in one afternoon.
4. Results
A rocket design was completed based upon commercially available rocket designs. These
include the nose cone, avionics bay, rocket outer body, motor mount, fins and engine (Figure 1).
Based upon the design the rocket was manufactured.
The motor was designed to produce a thrust for 1.849 seconds (Figure 10) with an
average thrust of 475 lbf (Figure 24). This configuration will deliver a total impulse of 770.8lbf –
seconds (Figure 25).
26
SRM 2014.XLS
Thrust vs time
500
450
400
Thrust (lbf)
350
300
250
200
150
100
50
0
0.0
0.5
1.0
1.5
2.0
Time (sec.)
F max =
F avg =
t thrust =
475
417
1.849
lbf
lbf
sec.
Figure 24: Thrust vs Time
With an estimated empty-weight for the rocket assembly of 17 pounds and estimated
peak subsonic drag coefficient of 0.321 (Figure 17), this motor configuration will take the rocket
to a predicted altitude of 10757 feet in 26.0 seconds, with a max velocity of 1128 feet/sec (Figure
26).
From the CFD analysis the center of pressure for the rocket was found to be 87.473
inches from the nose cone, and the center of gravity was approximately 57.3935 inches from the
nose cone. This makes the rocket stable during flight because the center of pressure is below the
center of gravity.
Finite element analysis was completed on the rocket assembly (Figure 27.) The rocket
was analyzed with the max thrust of 475 pound applied to the rocket motor. The force of gravity
and a drag force of 88.84 pounds opposed it. A standard mesh was developed and then refined to
27
ensure the success of the analysis. FEA analysis revealed that the rocket had a safety factor of
1.8 at peak thrust and peak drag.
Figure 27: Rocket Assembly Predicted Displacement
Finite element analysis was also completed on the avionics bay (Figure 28). The opening
of the main parachute exerts a high force on the avionics bay. The avionics team calculated a
maximum force of 129 pounds. FEA revealed that during the deploying of the parachute, there
was a safety factor of 1.33.
28
Figure 28: Avionics Bay Predicted Displacement
The vibration analysis displacement results showed the deformation in the normal
direction to the axially applied load, which is the direction the avionics bay bends during
frequency oscillations. These results (Figure 29) presented that the maximum displacement of
the part under loading would not exceed 0.01738060 inches displacement during the frequency
values being considered. An example of the graphical output at one frequency is shown in figure
30. The results also demonstrated the average displacement value to be 0.00635975 inches. From
the truly random nature of the results, some were inclusive as to their effect on the avionics bay
during a simulated rocket flight ascent. At the very low default frequency input values, the
displacement was generated to be an excessively high value. Looking at the expected natural
frequency range of approximately 100-1000 Hz, the results demonstrate a very low expected
displacement based on the location of the load and the input frequency values. These values
range from 0.00064418 inches to 0.01738060 inches of total displacement.
29
Figure 30: Avionics Bay Deformation at a Frequency of 926.775 cycles/s
Figure 31 pictures the truly non-linear nature of random vibration analysis. The theory
behind random vibration analysis is that there is no linear relationship between frequency and the
oscillations produced from the vibrations.
Frequency vs Std Dev Max Displacement
Standard Dev Max
Displacement
0.02000000
0.01500000
0.01000000
0.00500000
0.00000000
0.00E+00 2.00E+02 4.00E+02 6.00E+02 8.00E+02 1.00E+03
-0.00500000
Frequency (Hz)
Figure 31: Frequency vs Standard Deviation Maximum Displacement
Using the parachute force equation and analysis values, the value of force during the
main parachute deployment was determined to be a maximum of 129.94 pounds force (578
30
Newtons) applied directly to the eye bolt on the avionics bay end cap. This did not exceed the
yield stress of the eyebolts.
Bench testing revealed the theoretical black powder amounts were insufficient for
proper rocket separation. After several ejection tests it was determined that 2 grams (30.8
grains) of BP was required for the forward section of the rocket.
The burn test resulted in a burst combustion chamber. After ignition, there was a slight
delay before exhaust was observed, after which thrust occurred for approximately 0.4 seconds.
At the 0.4 second mark, an abrupt pressure change occurred, which caused catastrophic failure of
the motor (Figure 32). During failure the force exerted on the load cell measuring the thrust of
the rocket exceeded the maximum rating of the load cell (Figure 33). The explosion destroyed
the load cell. Remnants of the motor assembly were found at the testing site and have been
photographed (Figure 34) and measured in an attempt to ascertain the failure method. A cursory
investigation indicated a burst failure of the aluminum chamber. The predicted burst pressure for
this assembly was 3936 psi (Figure 35).
Figure 33: Thrust Force vs Time
31
The wind tunnel testing showed the fluid flow along the rocket. The drag was shown
directly by seeing the vortices created by the fins (Figure 36). The wake caused by these vortices
was also seen behind the rocket (Figure 37). When viewing the flow over the nosecone, the
airflow was shown to ride right along the boundary layer on the surface of the rocket (Figure 38).
Overall, the wind tunnel proved to be a useful tool to visualize something that couldn’t otherwise
have been seen.
Figure 36: Fin Vortices
The test launch using a commercial motor was partially successful. After analyzing the
flight data extracted from the altimeters, it appears the back-up altimeter did not operate as
intended. The back-up altimeter never calibrated itself properly prior to launch. That agrees with
why 2 of the 4 charges were expelled (drogue and main, from the primary altimeter only). From
the plot in the appendix (Figure 39), the descent velocity curve is constant which indicates it was
the main parachute that deployed at apogee. However, the plot also indicates that there was a
"main" event at the preset 800 ft. There were two events that took place, agreeing with the
32
expelled ejection charges (2 of 4). The primary altimeter expelled 2 charges, one at apogee and
the other at 800 ft (as programmed). The main charge was expelled at apogee and the drogue
event fired at 800 ft. The drogue chute should have deployed at apogee and the main chute
should have deployed at 800 ft. The 800 ft charge failed to shear the pins in the forward cone.
Even though the drogue parachute did not deploy, the rocket was recovered safely because the
main parachute deployed. The maximum altitude reached was 4407 ft, the maximum velocity
reached was 583 ft/s and the total flight time was 3.75 minutes. The ascent was 16 seconds and
the descent was 3.5 minutes with an average velocity of 22 ft/s.
5. Discussion
The purpose of this project was to design high-powered rocket capable of carrying and
deploying a cosmic ray detector at 10,000 feet. This was done by designing the structure,
propulsions, aerodynamics, and avionics sections of the rocket.
The results of the analyses showed that the rocket design was structurally sound and
aerodynamically stable. FEA predicted the rocket would withstand the expected body forces.
CFD analysis showed the rocket was aerodynamically stable. The vibration analysis
displacement results showed the avionics bay design is acceptable. It was not expected to
experience the damaging vibrations leading to failure. The snatch force analysis predicted a force
that did not exceed the structural limitations of the bolts when a FEA was conducted using the
expected force. The nozzle and combustion chamber were designed to handle the predicted
pressure, thrust, and heat expected from analysis of the fuel. The lowest factor of safety was the
aluminum combustion chamber with a burst safety factor of 3.
Additional calculations predicted the flight performance of the rocket. These calculations
used the average thrust, weight of the rocket, weight of the propellant, mass flow rate, and
33
maximum coefficient of drag to predict a maximum altitude of 10757 feet, which would have
met the purpose of the project. At 10000 feet, the ejection charge analysis calculated the
necessary force to fail the shear pins and safely eject both the parachute and the cosmic ray
detector. However, there were limitations in the analyses and flight predictions.
The CFD analysis was run assuming sea level standard conditions which will not be the
case as the rocket’s altitude changes. The sea level standard conditions resulted in higher
predicted coefficients of drag than occurred during flight. Bench testing of the ejection charges
proved the calculations undersized the charge. For the FEA, the drag was assumed to be a point
force acting in the middle of the body, while in reality, drag affects most of the outer surface of
the rocket. FEA was also conducted at maximum thrust and maximum drag. In reality, maximum
thrust and drag do not occur at the same time so this was a conservative estimate.
Unfortunately, testing only confirmed some of the results of the analyses. Launch testing
was done with a commercial motor to show that the rocket was capable of flight and returning
safely. The rocket succeeded in doing this but not in the way expected. The backup altimeter
never ran through its prelaunch setup and was still in idle mode. This was not noticed so the
flight was conducted without the backup altimeter. Additionally, at apogee the nose cone failed
to separate as designed despite the ejection charge igniting. Instead the shear pins connecting the
avionics bay to the lower rocket body failed which resulted in the main parachute deploying. A
possible reason for this is that the forward nose cone stuck when the charge went off creating a
tension force which sheared the pins connecting the avionics bay to the lower rocket body. This
would be easy to fix in any subsequent launches by using graphite chalk at all connections to
ensure no binding of parts and by verifying that both altimeters have initialized.
34
The test burn of the experimental motor to measure the thrust of the rocket motor resulted
in a catastrophic failure from a sudden over pressurization which exceeded the burst pressure of
the aluminum combustion chamber. Such pressure jumps are typically caused by a rapid increase
of the burning area of the propellant. Possible causes of this are air gaps trapped in the propellant
during manufacturing or an over-powered igniter causing more of the fuel to ignite than just the
inner core surface. Additionally, the black powder ignition charge could have fractured the fuel,
so that a fragment broke off of the grains and plugged the nozzle. This could have also resulted
in the sudden increase in pressure. Contact was made with Richard Nakka, an expert on model
rocketry, to get a second opinion, and his theories agreed with ours.
Based on a review of the report you kindly provided, your design looks
sound. Rocketry is a very unforgiveable beast and sometimes the
smallest of non-perfect details are pounced upon and used to punish
us...this happens to the best of us, even professional motors often
CATO or behave badly on the first try. I've had my share of ruptured
motors taunt me over the years.
A CATO such as yours could only (IMHO) be a result of either:
1) severe propellant fracturing due to an overpowered igniter
2) significant disbonding of the propellant from the casting tube
My bet is on the latter. I have had exactly the same thing happen.
KNDX is kinda tricky to work with in this regard, as it is cast hot,
35
then cools and shrinks, and will often debond while cooling. Sorbitol
propellant is superior as it does not shrink this way...it retains its
flexible nature well past cooling. With KNDX, I use clamping pressure
to prevent the propellant from shrinking radially (this is done with
springs) [28].
Although catastrophic failures are not uncommon in rocketry, it is highly disappointing to
not achieve the intended results. Because of the motor failure, limited time, and budget, it is
impossible to complete a full launch with the experimental motor. Recommendations to future
designers would be to start will smaller scale motors and work up both in size and power.
Arrangements had been made to work with a SEVRA (Sout Eastern Virginia Rocketry
Association) member who is experienced in the design and manufacturing of APCP motors.
However, an experienced user of sugar-powered motors could not be found to assist with this
project. Although there are extensive online resources for this topic, it is no substitute for in
person assistance. It would be recommended that in the future, arrangements be made to work
with a person who has practical experience in the design, manufacturing and testing of rocket
motors.
36
6. Appendix
6.1 Rocket Drawings
Figure 3: Avionics Bay Components
37
Figure 4: Rocket Upper Body Components
38
Figure 5: Rocket Lower Body Components
39
Figure 6: Rocket Motor Components
40
Figure 14: Dual Deployment Wiring Diagram
41
Figure 15: Avionics Bay Concept Sketch
42
Figure 16: 3D Model of Avionics Bay Interior
43
Figure 21: Rocket Test Stand Components
44
6.2 Pictures
Figure 32: Rocket Failure
Figure 34: Recovered Combustion Chamber Pieces
45
Figure 37: Rocket Wake
46
Figure 38: Rocket Nose Cone Boundary Layer
47
6.3 Tables and Plots
Isp
Isp
Parameter
Specific Impulse, ideal
Specific Impulse, measured
137
Units
sec.
sec.
164
C*
Characteristic exhaust velocity,
theoretical
2993 (912)
ft/s (m/s)
C*
Characteristic exhaust velocity,
measured
2922 (891)
ft/s (m/s)
To
Combustion temperature, theoretical
@1000 psia
1437 (1710)
deg Celsius (K)
Density, ideal
Density, as cast
Mass fraction of condensed-phase
products
1.879
1.859
gram/cu.cm.
gram/cu.cm.
-
Ratio of specific heats
Effective molecular wt. of exhaust
products
1.043
plateau
ro
r
Burn rate behavior
Burn rate @ 1 atm.
Burn rate @ 1000 psia
0.509
in/sec
in/sec
Tcr
Auto-ignition temperature
> 300
deg. C.
X
k
M
0.425
g/mole
42.39
0.084
Figure 7: Propellant Published Values
ODU Rocket Team Motor
utilizing KNDX propellant.
Grain mass
Total impulse
Average thrust
Thrust time
Specific Impulse
Motor Classification
2.639
5.817
3428.6
770.8
1854.0
416.8
1.849
132.5
L
kg.
lb.
N-sec.
lb-sec.
N.
lb.
sec.
sec.
1854
Figure 25: Predicted Rocket Motor Performance
48
FLIGHT PERFORMANCE ESTIMATOR FOR HOBBY ROCKETS
(valid for subsonic rockets only)
EzAlt.xls (MS Excel 97)
Version: 1.2
Date: Feb. 2007
Instructions: Enter data in blue text (boxed cells).
Title
Motor average thrust
Motor total impulse
Motor propellant weight
Rocket dead weight
Rocket diameter (max)
Rocket drag coefficient
Motor thrust time
Motor classification
Rocket avg. flight weight
F=
It =
mp =
mr =
D=
Cd =
t=
mra =
Acceleration (average)
a=
or a =
Peak altitude (zero drag)
Time to peak altitude (zero drag)
Max velocity (zero drag)
Burnout altitude (zero drag)
z2 =
t2 =
v1 =
z1 =
Drag Influence number
Peak altitude reduction factor
Time to peak reduction factor
Max velocity reduction factor
Burnout altitude reduction factor
N=
f1 =
f2 =
f3 =
f4 =
example rocket
416.8 lb.
770.8 lb-sec.
5.817 lb.
17.000 lb.
4 inch
0.351
Input data
1.849 sec.
K
19.909 lb.
674 feet/sec 2
20.9 g's
22967
38.7
1186
1097
feet
sec.
feet/sec.
feet
Ideal (no drag
resistance)
614 (valid range between 0 and 900)
0.468
0.621
Drag reduction factors
0.951
0.973
Peak altitude
Time to peak altitude
Max velocity
Z peak =
10757 feet
t peak =
24.0 sec.
V m ax =
1128 feet/sec.
or V m ax =
769 MPH
Burnout altitude
Z bo =
1067 feet
Warning: Rocket is supersonic, results may be invalid
Figure 26: Predicted Rocket Flight Performance
Predicted (with
drag)
49
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Frequency
(Hertz)
1.36E-04
1.98E-04
5.53E-04
7.51E-04
7.72E-04
7.76E-04
2.08E-03
1.95E+02
2.10E+02
3.04E+02
3.39E+02
3.52E+02
3.80E+02
4.16E+02
4.30E+02
4.92E+02
5.05E+02
5.07E+02
5.29E+02
5.85E+02
6.71E+02
7.26E+02
7.77E+02
7.96E+02
9.27E+02
Period
(Seconds)
7.34E+03
5.05E+03
1.81E+03
1.33E+03
1.30E+03
1.29E+03
4.80E+02
5.13E-03
4.77E-03
3.29E-03
2.95E-03
2.84E-03
2.63E-03
2.41E-03
2.33E-03
2.03E-03
1.98E-03
1.97E-03
1.89E-03
1.71E-03
1.49E-03
1.38E-03
1.29E-03
1.26E-03
1.08E-03
Standard Dev Max
Displacement (in)
1079000.00000000
1248900.00000000
41394.60000000
230371.00000000
496343.00000000
344531.00000000
54998.20000000
0.01226130
0.01433770
0.00061118
0.00480574
0.00123981
0.00102288
0.01021890
0.00121483
0.00196697
0.01101720
0.00847359
0.00254671
0.00030500
0.01738060
0.01475140
0.00964377
0.00203377
0.00064418
Figure 29: Vibration Analysis Results
50
Design and Burst Pressures for Rocket Motor Casing
[ Input data in blue text, English or (SI) units]
Casing Dimensions and Design Factors
Do =
t=
SD =
3 in. (mm)
0.125 in. (mm)
3.001
Diameter, outside
wall thickness
Design Safety factor
Material Properties
Fty =
Ftu =
E=
=

B=
37 ksi (MPa)
42 ksi (MPa)
9.9 Msi (MPa)
0.33
0.881
1.277
Yield Strength
Ultimate Strength
Modulus of Elasticity
Poisson Ratio
Fty/Ftu
Burst factor
Design and Burst Pressures
PD =
PU =
SU =
1027 psi (kPa)
3936 psi (kPa)
3.83
Design pressure
Burst pressure
Burst Safety Factor
Elastic Deformation under Pressure *
DD =
Dc =
0.00312 in. (m.)
0.00980 in. (m.)
Change in casing diameter, at P D
Change in casing circumference, at P D
Figure 35: Design and Burst Pressures for Rocket Motor Casing
51
Figure 39: Test Flight Data
52
6.4 Fuel and Combustion Chamber Calculations
Given data for specified fuel mixture KNDX
𝑇0 = 1710 𝐾
Adiabatic flame temperature obtained from GUIPEP/PROPEP software
𝑇 = 1625 𝐾 𝑓𝑜𝑟 𝜂𝑐 = 0.95 (fine grain propellant)
𝑀 = 42.39
𝑘𝑔
𝑘𝑚𝑜𝑙
𝑘 = 1.1308
𝑅 = 196.1
𝑐∗ = √
𝐽
𝑘𝑔 ∗ 𝐾
𝑅 𝑇0
𝑚
= 889
𝑘+1
2 𝑘−1
𝑠
𝑘(
)
𝑘+1
Kn Calculations - Empirically Determined Data by R. Nakka
a
b
c
d
e
KNDX
43.500
0.24168
50.484
-9.9115
163.80
22.224
-0.2240
5095.0
-2460.4
453.48
-35.532
1.0175
calc Kn
-1070
313
363
Range incr
150-400 psi
450-850 psi
900-1300 psi
𝐾𝑛 = 𝑎 + 𝑏 ∗ 𝑃 + 𝑐 ∗ 𝑃2 + 𝑑 ∗ 𝑃3 + 𝑒 ∗ 𝑃4
Where P is Pressure in MPa
Therefore, for target pressure of 1050 psi (7.24 MPa), Kn = 363
Burn Rate Calculations - Empirically Determined Data by R. Nakka
KNDX
a
n
Pressure, psia
psi, in/sec
Pressure, Mpa
14.7
to 113
0.0160
0.619
0.100
to 0.779
113
to 373
0.3105
-0.009
0.779
to 2.572
373
to 860
0.0049
0.688
2.572
to 5.930
860
to 1233
1.4155
-0.148
5.930
to 8.502
1233
to 1625
0.0209
0.442
8.502
to 11.20
a
n
Mpa, mm/sec
8.875
0.619
7.553
-0.009
3.841
0.688
17.20
-0.148
4.775
0.442
53
Calculation of Kn and Throat Area – Assume linear web regression
𝑥𝑖𝑛𝑐 = 𝑥 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 (mm)
𝑑 = 𝑑𝑖𝑜 + 𝑐𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm)
𝐷 = 𝑑𝑜 − 𝑜𝑠𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm)
𝐿 = 𝑙𝑔𝑜 − 𝑒𝑖 ∗ (2 ∗ 𝑛 ∗ 𝑥𝑖𝑛𝑐 ) (mm)
𝑡𝑤𝑒𝑏 = 𝐷 − 𝑑 (mm)
𝜋
𝐴𝑏𝑒 = 𝑒𝑖 ∗ 2 ∗ 𝑛 ∗ 4 ∗ (𝐷2 − 𝑑 2 ) (mm2)
𝐴𝑏𝑐 = 𝑐𝑖 ∗ 𝜋 ∗ 𝑑 ∗ 𝐿 (mm2)
𝐴𝑏𝑠 = 𝑜𝑠𝑖 ∗ 𝜋 ∗ 𝐷 ∗ 𝐿 (mm2)
𝐴𝑏 = 𝐴𝑏𝑐 + 𝐴𝑏𝑒 + 𝐴𝑏𝑠 (mm2)
𝑑𝑡𝑜 = 𝑚𝑎𝑥(𝐴𝑏 )/(𝐾𝑛 @ 𝑃 = 𝑃𝑚𝑎𝑥 )
𝜋
𝐴𝑡 = 4 ∗ (𝑑𝑡𝑜 ∗
𝐾𝑛 =
𝐴𝑏
𝐴𝑡
(𝑡𝑤𝑜−𝑡𝑤𝑒𝑏 ) 2
𝑡𝑤𝑜
) (mm2)
(dimensionless)
54
Calculation of Pressure and Burn Time
In the following set of calculations only n=pressure exponent at Po
𝑥𝑖𝑛𝑐𝑝 = 𝑥 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 𝑓𝑜𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (mm)
𝑑 = 𝑑𝑖𝑜 + 𝑐𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm)
𝐷 = 𝑑𝑜 − 𝑜𝑠𝑖 ∗ (2𝑥𝑖𝑛𝑐 ) (mm)
𝐿 = 𝑙𝑔𝑜 − 𝑒𝑖 ∗ (2 ∗ 𝑛 ∗ 𝑥𝑖𝑛𝑐 ) (mm)
𝑡𝑤𝑒𝑏 = 𝐷 − 𝑑 (mm)
𝜋
𝐴𝑡 = 4 ∗ (𝑑𝑡𝑜 ∗
𝐴∗ =
(𝑡𝑤𝑜−𝑡𝑤𝑒𝑏 ) 2
) (mm2)
𝑡𝑤𝑜
𝐴𝑡
(m2)
10002
𝜋
𝜋
𝐴𝐷𝑈𝐶𝑇 = 4 ∗ 𝑑𝑐 2 − 4 ∗ (𝐷2 − 𝑑 2 ) (mm2)
𝐺 = 𝐺 ∗ − 𝐴𝐷𝑈𝐶𝑇
𝑟 = (1 + 𝑘𝑣 ∗ 𝐺) ∗ 𝑎 ∗ 𝑃0𝑛 (mm/s)
𝑡=
𝑥𝑖𝑛𝑐𝑝
(s)
𝑟+𝑡𝑖
𝜋
𝑉𝑔𝑟𝑎𝑖𝑛 = 4 ∗ (𝐷2 − 𝑑 2 ) ∗ 𝐿 (mm3)
𝑉𝑔𝑟𝑎𝑖𝑛 =
𝑉𝑔𝑟𝑎𝑖𝑛
10003
(m3)
𝑣𝑐
𝑉𝑓𝑟𝑒𝑒 = 10003 −𝑉
𝑔𝑟𝑎𝑖𝑛
(m3)
𝑚𝑔𝑟𝑎𝑖𝑛 = 𝜌𝑔𝑟𝑎𝑖𝑛 ∗ 𝑉𝑔𝑟𝑎𝑖𝑛 /10002 (kg)
𝑚̇𝑔𝑒𝑛 =
𝑚̇𝑛𝑜𝑧 =
𝑚𝑔𝑟𝑎𝑖𝑛,𝑖 −𝑚𝑔𝑟𝑎𝑖𝑛,𝑗
𝑡𝑗 −𝑡𝑖
(𝑃0 −𝑃𝑎𝑡𝑚 )∗1,000,000∗𝐴∗
√𝑅𝑇
(kg/s)
2
∗ √𝑘 ∗ (𝑘+1)
𝑘+1
2
𝑘−1
(kg/s)
𝑚̇𝑠𝑡𝑜 = 𝑚̇𝑔𝑒𝑛 − 𝑚̇𝑛𝑜𝑧 (kg/s)
𝑚𝑎𝑠𝑠𝑠𝑡𝑜,𝑗 = 𝑚̇𝑠𝑡𝑜,𝑗 ∗ (𝑡𝑗 − 𝑡𝑖 ) + 𝑚𝑎𝑠𝑠𝑠𝑡𝑜,𝑖 (kg)
ρprod =
𝑚𝑎𝑠𝑠𝑠𝑡𝑜
𝑉𝑓𝑟𝑒𝑒
(kg/m3)
𝑃0 = ρprod ∗ 𝑅 ∗ 𝑇 + 𝑃𝑎𝑡𝑚 ∗ 1,000,000 (Pa absolute)
𝑃
0
𝑃0 = 1,000,000
(MPa absolute)
𝑃0 = 𝑃0 − 𝑃𝑎𝑡𝑚 (MPa gage)
55
Calculation of Thrust and Impulse
𝑃0 = ρprod ∗ 𝑅 ∗ 𝑇0 + 𝑃𝑎𝑡𝑚 ∗ 1,000,000 (Pa absolute)
𝐴𝑡
𝐴∗ = 10002 (m2)
𝜋
𝐴𝑡 = 4 ∗ (𝑑𝑡𝑜 ∗
(𝑡𝑤𝑜−𝑡𝑤𝑒𝑏 ) 2
𝑡𝑤𝑜
) (mm2)
𝐴𝑒
⁄𝐴 = expansion ratio
𝑡
𝑃𝑒 = [
𝑃0
𝑘
(𝑘−1) 𝑘−1
(1+
)
2∗𝑀2
𝑒
] 𝑤ℎ𝑒𝑛 𝑃𝑒 ≥ 𝑃𝑎𝑡𝑚 , 𝑒𝑙𝑠𝑒 𝑃𝑒 = 𝑃𝑎𝑡𝑚 (Pa absolute)
−1
1
𝑘+1
𝐴𝑒
𝑘+1
, 𝑜𝑝𝑡 = (
)
𝐴𝑡
2
[
1
𝑃𝑒 𝑘
𝑘−1
𝑃𝑒 𝑘
𝑘+1
( ) √(
) (1 − ( )
𝑃0
𝑘−1
𝑃0
]
𝑘+1
𝑘−1
𝑘
2𝑘 2
2 𝑘−1
𝑃𝑒
𝐶𝐹 = 𝜂𝑛𝑜𝑧 √
∗(
)
∗ (1 − ( )
𝑘−1 𝑘+1
𝑃0
)+
(𝑃𝑒 − 𝑃𝑎𝑡𝑚 ∗ 1,000,000)𝐴𝑒
𝑃0 𝐴∗
𝐹 = 𝐶𝐹 ∗ 𝐴∗ ∗ 𝑃0 (N)
𝑡=
𝐼𝑡 =
)
𝐹𝑗 −𝐹𝑖
2
𝑥𝑖𝑛𝑐𝑝
𝑟+𝑡𝑖
(sec)
∗ (𝑡𝑗 − 𝑡𝑖 ) (N-sec)
56
Design of Motor Chamber – Design and Burst Pressure
Fty
Ftu
β=
𝐵 = 𝐴 ∗ β4 + B ∗ β3 + C ∗ β2 + D ∗ β + E
𝑃𝐷 = 2
𝑃𝑈 = 2
Burst Factor Polynomial Coefficients:
A = 9.5833
B=
-33.528
C=
44.929
D = -28.479
E=
8.6475
𝑡 𝐹𝑡𝑦
𝐷𝑜 𝑆𝐷
𝑡 𝐹𝑡𝑦
𝐵
𝐷𝑜
57
6.5 Gantt Chart
58
6.6 Budget
59
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