Project Bellerophon
783
A.8.0 Risk Analysis
A.8.1 Design Methods
A.8.1.1 Non-Catastrophic Failure
A.8.1.1.1 Avionics
A vast majority of the electronic components included in our launch vehicle design contribute to
the non-catastrophic failure rate. However, the gyro drift rate is the main source of noncatastrophic failure that can be given a specific numerical value from the components selected.
This piece of equipment has the most impact on our ability to meet the mission requirement of
attaining a 300km orbit.
Gyro drift occurs when very small errors in the collected rate data are integrated in an effort to
provide orientation data2.
There are many things that contribute to gyro drift, but their
explanations are complex and are beyond the scope of this section. A gyroscope that is high in
accuracy has a drift rate of less than 0.1 deg/hr2. According to the Survey of Instrumentation and
Measurement, a gyro drift rate of 1 deg/sec is considered acceptable in a warhead seeker or flight
control system, but is only tolerable for a short time in stand-alone navigational applications2.
Electronic components display the gyro drift in terms of the gyro bias. This is the deviation
between the ideal output of the gyro and the actual output1. Because this parameter is one of the
driving forces behind the prices for gyroscopes, we have chosen to accept the use of a gyro with
a larger drift rate. The specified drift rate of our system is 0.5 deg/hr. This is a conservative rate
for a gyroscope being used for our purposes.
One way to prevent this drift rate from becoming a problem is to periodically reset the gyro
output so that it matches the output from other location-sensing devices and thus, eliminate any
accumulated error in measurement1. The accelerometers have a much smaller error rate and our
Authors: Nicole Bryan and Danielle Yaple
Project Bellerophon
784
position data can also be integrated from that data as well as from the positional data of the
gyroscope3.
Other components contributing to the non-catastrophic failure rate include sensor failure, wiring
shorts, malfunctions in the range safety equipment, and failure of the telecommunications
equipment. These, however, did not impact our ability to meet the mission requirements due to
the fact that our sensors are not connected to any system that makes changes to the performance
of the launch vehicle.
References
1
Paniit, Sudhakar M., Zhang, Weibang Modeling Random Gyro Drift Data by Dependent Systems. [online]
Michigan Technological University,
URL:http://ieeexplore.ieee.org/iel5/7/4104218/04104235.pdf?isnumber=4104218&prod=JNL&arnumber=4104235
&arSt=455&ared=460&arAuthor=Paniit%2C+S.M.%3B+Wwibang+Zhang [cited 26 March 2008].
2
Dyer, Stephen A. Survey of Instrumentation and Measurement, Wiley-IEEE, 2001.
3
Wertz, James R., and Wiley J. Larson, eds. Space Mission Analysis and Design, 3rd ed., Torrance, Kluwer
Academic and Microcosm Press, 1999.
Authors: Nicole Bryan and Danielle Yaple
Project Bellerophon
785
A.8.1.1.2 Aerothermal
One parameter that is needed when developing a risk analysis is the standard deviation for drag.
We calculate this value using the following method. First, a randomizer is created that produces
100 random values of angle of attack between zero and ten degrees. Once the list of randomly
generated angle of attack values is obtained, it is used as input to the drag calculation function,
solve_cd.m.
We then ran the drag calculation function for a constant Mach number of 2.5 to produce 100
values for drag at that Mach number. Then, we loaded the values of drag into Excel and used the
standard deviation function to calculate the standard deviation for the drag. The standard
deviation of drag was calculated to be 13.29%. This value was compared to that of Vanguard to
make sure it was reasonable. We found that the value we calculated was reasonable because the
standard deviation of drag for the Vanguard rocket was about 10%.1
The 3.29% difference in standard deviation can be attributed to the difference in launch vehicle
sizes. Our launch vehicles are much shorter than Vanguard’s 23 meter total length. Our tallest
launch vehicle is less than half as tall as Vanguard at 10.27 meters. This shorter total length
results in higher drag on the vehicle. This results in a higher standard deviation, which is a more
conservative estimate.
References
1
“The Vanguard Report”, The Martin Company, Engineering Report No. 11022, April 1960
Author: Chris Strauss
Project Bellerophon
786
A.8.1.1.3 Propulsion
We use the Gaussian Probability Method to account for the uncertainties in our launch vehicle.
To execute this task, each category of design provides standard deviations of various parameters
that have some effect on the size of the launch vehicle. We provide standard deviations of
propellant mass and mass flow rates in liquid engines, hybrid and solid motors. Then we use
these standard deviations to vary the mass flow rate and propellant mass at the start of each
simulation to obtain the correct size of our launch vehicle.
In solid motors, there exist many factors which create variations in several parameters such as
propellant weight, burning rate, density, characteristic velocity and throat area. According to
Professor Heister, in missile applications, limiting variations in ballistic parameters can result in
improved motor case and insulation designs which minimize inert mass.1 We want to minimize
the inert mass because it lowers our GLOM.
Industry standard scales and load cells can determine propellant mass so that the industry can
verify the specification weight ranges of ±0.3% or 3σ. Professor Heister’s paper predicts that
propellant mass values can range from 0.08% to 0.12%. However, because of the small variation
of the parameter, it does not have a huge effect on the burning time dispersions.1 The Monte
Carlo simulation uses the recommended standard deviation of 0.12%.
To determine the burning rate, manufacturers burn small strands of propellant in small ballistic
test motors with different throat sizes. The chamber pressure varies in these small ballistic test
motors. The plot of ln(rb) versus ln(pc) is an empirical representation of the burning rate behavior
of a propellant shown as curve (a) in Fig. 8.1.1.3.1. The related equation is of the burning rate
(rb), St. Robert’s Law, shown in Eq. (A.8.1.1.3.1).2
rb  ap cn
(A.8.1.1.3.1)
where a represents the burn rate coefficient and n represents the burn rate exponent. However,
the equation above does not address the complex thermochemical and combustion processes that
occur when a propellant actually burns.2 Unlike propellant mass, the variation of the burning rate
Authors: Dana Lattibeaudiere, Stephanie Morris
Project Bellerophon
787
is difficult to determine because there exists little research that assesses the variations present
within a batch of propellant tested under constant pressure conditions. According to Humble,
most composite propellants behave as in curves (a) or (d) in Fig. 8.1.1.3.1.
Fig. 8.1.1.3.1 Sample of observed burning rate behavior of solid propellants.
(R. W. Humble, G. N. Henry, W. J. Larson)2
Curves (b) and (c) do not apply because we do not use a double-base propellant in our launch
vehicle. Additionally, temperature sensitivity and throat erosion make it difficult to obtain the
burning rate from full-scale firings.
The parameters  p [%/K], measures temperature sensitivity of burn rate as shown in Eq.
(A.8.1.1.3.2) below.2
p 
 ln( rb )
T
(A.8.1.1.3.2)
pcconst
where T represents the temperature of the propellant grain precombustion.
According to Humble, at higher propellant temperatures, the increased internal energy within the
propellant leads to small increases in burning rate as compared to normal temperature conditions.
Note that in most situations, the small ranges in temperature make this effect small, but not
negligible. Eq. (A.8.1.1.3.3) accounts for this small effect.2
rb  e
( p T )
apcn
Authors: Dana Lattibeaudiere, Stephanie Morris
(A.8.1.1.3.3)
Project Bellerophon
788
where ∆T represents the difference in temperature from the assumed “standard” condition of
15˚C. Typical values range from 0.001 to 0.009 per degree Kelvin.2
Erosion can occur in either of two ways. Erosion can occur because of mass flux shown below in
Eq. (A.8.1.1.3.4), the Lenoir-Robillard model.2
rb  ap 
n
c
G 0.8
0.2
L
  p rb
e
G
(A.8.1.1.3.4)
where α and β represent experimentally determined constants, L represents the length of the
grain and G represents the bore mass flux (kg/m2s). Compressibility can also cause erosion to
occur where the Mach number (M) influences the burning rate as shown in Eq. (A.8.1.1.3.5).2
rb  apcn (1  kM )
(A.8.1.1.3.5)
where k represents the empirical constant that addresses the erosive effects. Fig. 8.1.1.3.2 shows
that erosive burning enhances the burning rate.
Fig. 8.1.1.3.2 Pressure-time curve with and without erosive burning.
(George P. Sutton, Oscar Biblarz)3
Despite the above factors which affect the burning rate, manufacturers use cured strands of
propellant fired at constant pressure to standardize the burning rate of production batches
although ambiguities arise such as bore centerline offset and mandrel misalignment. Using this
technique, manufacturers suggest a burning rate standard deviation of 1% (1σ) which the Monte
Carlo simulation uses.3
Authors: Dana Lattibeaudiere, Stephanie Morris
Project Bellerophon
789
In liquid propellant engines, there are many variables in each launch and usage of the engine.
The main variables that can be calculated and are useful for the Monte Carlo simulation are the
mass of the propellant and the mass flow rate of the propellant. Mass flow rate is directly related
to the thrust of the engine is a standard deviation of thrust. The Vanguard satellite launching
vehicle is a very similar vehicle to the launch system that we have created4. The Vanguard report
publishes their scientifically found standard deviations and average values for each of their
stages. Using these numbers, we calculate our percent standard deviation for propellant mass and
mass flow rate with the Eq.( A.8.1.1.3.6).
% Deviation = (Standard Deviation / Average Value)∗100%
(A.8.1.1.3.6)
The standard deviation for our vehicle is the product of the nominal value and the percent
deviation. The results for calculating the percent deviations can be seen in Table A.8.1.1.3.1.
Table A.8.1.1.3.1 Liquid Propellant Standard Deviations
Propellant Mass
Mass Flow Rate
Average Value4
16,351 kg
110.7 kg/s
Standard Deviation4 Percent Deviation
120
0.7340 %
0.545
0.4923 %
For hybrid motors, we cannot find any historical standard deviations. In order to still be
conservative with the hybrid propellant standard deviations the two standard deviations for solid
and liquid propellants are combined together to obtain a conservative hybrid standard deviation.
Table A.8.1.1.3.2 shows the data for each propellant and the resulting hybrid percent deviation.
Table A.8.1.1.3.2 Propellant Percent Deviations
Mass of Propellant
Mass flow rate
Solid Propellant1,3
0.12
1.0
Liquid Propellant1
0.734
0.4923
Hybrid Propellant
0.854
1.4923
The standard deviation values are included in the Monte Carlo simulation and used to calculate
the deviation of thrust for the launch vehicle.
Authors: Dana Lattibeaudiere, Stephanie Morris
Project Bellerophon
790
References
Heister, S., D., Davis, R., J., “Predicting Burning Time Variations in Solid Rocket Motors,” Journal of Propulsion
and Power, Vol. 8, No. 3, 1992, pp. 564-565.
1
Humble, R. W., Henry, G. N., Larson, W. J., “Solid Rocket Motors,” Space Propulsion Analysis and Design, 1st
ed., Vol. 1, McGraw-Hill, New York, NY, 1995, pp. 327-331.
2
Sutton, G., P., Biblarz, O., “Solid Propellant Rocket Fundamentals,” Rocket Propulsion Elements, 7th ed., Vol. 1,
Wiley, New York, NY, 2001, pp. 434.
3
4
Martin Company, The Vanguard Satellite Launching Vehicle, Engineering Report No.11022. April 1960, pp. 26,
208.
Authors: Dana Lattibeaudiere, Stephanie Morris
Project Bellerophon
791
A.8.1.1.4 Structures
We integrate a percent standard deviation of 2.255% of the nominal inert mass into the noncatastrophic risk analysis. The nominal value chosen for each of our launch vehicles is the total
inert mass that the final design math models produced. The percent standard deviations reflect
the manufacturing tolerances for specific materials after they have undergone certain processes
(rolling, spin forming, casting, etc.).
Our value for percent standard deviation doesn’t come without justification.
Our value
represents a little over 5 times the largest percentage found for the Vanguard (a similar launch
vehicle proportionally).1
Figure A.8.1.1.4.1 is a table of mean values and standard deviations
for the Vanguard found in the Vanguard report.1
Fig. A.8.1.1.4.1: Vanguard Standard Deviation Data
(The Martin Company)
From the mean values and standard deviations in Fig. A.8.1.1.4.1 we calculate percent standard
deviations for each stage. We want to be as conservative as possible with our value so we choose
a percent standard deviation of five times the first stage Vanguard value. Unfortunately, a
miscommunication resulted in our final value increasing to 2.255% which is 5.435 times the first
Author: Brandon White
Project Bellerophon
792
stage Vanguard value. By incorporating a value larger than the Vanguard’s we account for
larger manufacturing tolerances for our structure. Larger tolerances result in less inspection that
a company has to perform when forming certain components. Less inspection equates to lower
cost, satisfying an objective of the project.
Table A.8.1.1.4.1 Vanguard Percent Standard
Deviations for Each Stage
Stage
First
Second
% Standard Deviation
0.415
0.207
We did not want to leave this percent standard deviation only as a function of a reference launch
vehicle. To further justify our value, we found standard manufacturing tolerances for certain
materials. These tolerances, along with dimensional specifications from final design, allow us to
calculate a minimum and maximum inert mass. From those mass values we determine our own
percent standard deviations.
Since the launch vehicle is primarily composed of aluminum, thickness tolerances are found for
cold-rolled aluminum sheets.2 Figure A.8.1.1.4.2 shows an excerpt from Metric Standards for
World Wide Manufacturing.
Thickness (mm)
Tolerance (mm)
Fig. A.8.1.1.4.2: Thickness Tolerances for Cold-Rolled Non-space Grade Aluminum
(The American Society of Mechanical Engineers)
Author: Brandon White
Project Bellerophon
793
The tolerances provided in Fig. A.8.1.1.4.2 represents German national standards. To the best of
our knowledge, we aren’t outsourcing any of our manufacturing to Germany. We made the
assumption that ANSI standards are very similar, if not exact, to the tolerances provided. An
issue that comes into play with this set of tolerances is interpolation for an intermediate sheet
thickness. Figure A.8.1.1.4.2 only provides tolerances when the sheet thickness is an exact
number. Our design methods incorporate very small deviations in thickness and many times the
sheet thickness falls in between two exact values in the table. In order to approximate an
appropriate tolerance to employ in our calculations, we interpolated between two known
tolerances to find a reasonable tolerance.
For manufacturing, we also consider cutting tolerances of the sheet metal. In the case of nonspace grade aluminum, tooling can cut the metal with an area tolerance precision of +/- 0.019 m2.
This precision value represents a 10 ft. by 10 ft. sheet of aluminum with tooling tolerances of +/0.125” in both the length and width dimensions. When working with non-space grade materials
on the basis of minimizing cost, we must be prepared to deal with loose tolerances. When
dealing with companies that work with space grade materials, this area tolerance decreases by a
large margin.
Typically, companies that cut space-grade aluminum can do so at an area
precision of +/- 0.0015 m2. An important aspect to note is that the thickness tolerances are for
non-space grade aluminum. Our design consists of all space-grade aluminum, and we price the
launch vehicle as such. In order to keep consistency, we have to find space-grade tolerances for
cold-rolled aluminum. Table A.8.1.1.4.2 displays our findings.3 The values in the table are based
on standards developed by both ANSI and the Aluminum Association.
Author: Brandon White
Project Bellerophon
794
Table A.8.1.1.4.2 Thickness Tolerances for Space-Grade Aluminum
Greater Than
(mm)
0.00014986
0.000254
0.0004064
0.000635
0.0008128
0.0009906
0.0011938
0.0016002
0.0020066
0.0024892
0.0032004
0.0040132
0.0050038
0.0062992
0.008001
0.0100076
0.016002
0.0249936
0.040005
0.0599948
0.08001
0.0999998
Less Than
(mm)
0.000254
0.0004064
0.000635
0.0008128
0.0009906
0.0011938
0.0016002
0.0020066
0.0024892
0.0032004
0.0040132
0.0050038
0.0062992
0.008001
0.0100076
0.016002
0.0249936
0.040005
0.0599948
0.08001
0.0999998
0.1599946
Tolerance (+/-) (mm)
0.0000254
0.0000381
0.0000381
0.0000381
0.0000381
0.0000508
0.0000508
0.0000508
0.0000635
0.0000889
0.0001016
0.0001524
0.0002286
0.0003048
0.0004318
0.0005842
0.0007874
0.0009906
0.001397
0.001905
0.00254
0.003302
These tolerance values are more exact than the non-space grade values, which results in no
interpolation from our design thicknesses. From this research we calculated standard deviations
for our launch vehicle. We are able to find these standard deviations for both non-space grade
and space-grade materials. The purpose of finding both values is that the results can show
opportunity for further cost minimization in the future. If we can achieve similar standard
deviations with an inexpensive material, it would be in the interest of the project specifications to
consider using non-space grade materials.
Table A.8.1.1.4.3 summarizes percent standard
deviations between non-space grade tolerances, space-grade tolerances, and the actual value
input into Monte Carlo simulations.
Author: Brandon White
Project Bellerophon
795
Table A.8.1.1.4.3 Percent Standard Deviation Comparison
Monte Carlo
Non-Space Grade
Space-Grade
200g Payload
2.255%
2.28%
3.13%
1kg Payload
2.255%
2.32%
2.88%
5kg Payload
2.255%
8.94%
4.42%
Results from this table show that the standard deviation employed into the Monte Carlo
simulations is both reasonable and justified. Also, the standard deviations of the 200 gram and 1
kilogram payloads are smaller when non-space grade tolerances are incorporated. We conclude
that this can be attributed to insufficient accuracy for the thickness tolerances due to interpolating
values. Although, the calculated standard deviations are higher than the ones integrated into the
simulations, we are confident that the manufacturers that we hire can meet the 2.255% deviation
requirements. With the high cost quotes that we received from these manufacturers, we believe
that we are essentially paying for the tight tolerances we require.
The inert mass percent standard deviation of 2.255% is justified through our calculations, but we
would like to see simulations performed in the future with our calculated values for non-space
grade material.
References
1
Klemans, B., “The Vanguard Satellite Launching Vehicle,” The Martin Company, Engineering Report No.11022,
April 1960.
2
Kverneland, K. O., Metric Standards for World Wide Manufacturing, The American Society of Mechanical
Engineers, 1996, N.Y.,N.Y.
3
The Luminum Corporation. http://www.luminum.org/data/dtolaero.html [cited 19 March 2008].
Author: Brandon White
Project Bellerophon
796
A.8.1.1.5 Results
A.8.1.1.5.1: 200g Payload
Table A.8.1.1.5.1.1 200g Monte Carlo simulation comparisons
Variable
Inert Mass Stage 1
Inert Mass Stage 2
Inert Mass Stage 3
Mass Flow Rate Stage 1
Mass Flow Rate Stage 2
Mass Flow Rate Stage 3
Propellant Mass Stage 1
Propellant Mass Stage 2
Propellant Mass Stage 3
Specific Impulse Stage 1
Specific Impulse Stage 2
Specific Impulse Stage 3
Nominal Input
Values
349.4777 kg
153.4559 kg
14.8011 kg
10.689 kg/s
2.728
kg/s
0.194
kg/s
1462.00 kg
566.64 kg
37.26
kg
----
Standard
Deviation
2.255%
2.255%
2.255%
1.4923%
1%
1%
0.854%
0.12%
0.12%
----
Calculated
Means
349.4318 kg
153.4889 kg
14.8081 kg
---1462.26 kg
566.62 kg
37.26
kg
339.4907 s
345.4158 s
346.6426 s
Standard
Deviation
3.213%
3.205%
3.185%
---1.208%
0.169%
0.168%
2.1302%
1.4194%
1.4180%
We now present the 200g payload launch vehicle. For this vehicle, the non-catastrophic failure
rate was 0.01%, which means that it meets the less than 0.15% failure rate condition. The output
histograms of each recorded value for the 200 gram payload case with their calculated final
standard deviations and means are in Figs. A.8.1.1.5.1.1 and A.8.1.1.5.1.2. The output periapsis
histogram has an approximately Gaussian distribution with several outliers toward the lower
periapsis values. The eccentricity histogram is approximately Gaussian.
Author: Alfred Lynam
Project Bellerophon
797
200
number of cases
150
100
50
0
250
300
350
400
450
Periapsis altitude(km)
500
550
Figure A.8.1.1.5.1.1: 200g Periapsis altitude histogram with a 21.803 km
standard deviation and a 437.44 km average.
(Alfred Lynam)
1000
number of cases
800
600
400
200
0
0.25
0.3
0.35
0.4
0.45
0.5
Eccentricity
0.55
0.6
Figure A.8.1.1.5.1.2: 200g Eccentricity histogram with a .0467
standard deviation and a .438 average.
(Alfred Lynam)
Author: Alfred Lynam
0.65
Project Bellerophon
798
A.8.1.1.5.2: 1 kg Payload
Table A.8.1.1.5.2.1 1kg Monte Carlo simulation comparisons
Variable
Inert Mass Stage 1
Inert Mass Stage 2
Inert Mass Stage 3
Mass Flow Rate Stage 1
Mass Flow Rate Stage 2
Mass Flow Rate Stage 3
Propellant Mass Stage 1
Propellant Mass Stage 2
Propellant Mass Stage 3
Specific Impulse Stage 1
Specific Impulse Stage 2
Specific Impulse Stage 3
Nominal Input
Values
281.35 kg
116.07 kg
16.8907 kg
6.73
kg/s
1.88
kg/s
0.23
kg/s
947.90 kg
336.92 kg
45.09
kg
----
Standard
Deviation
2.26%
2.26%
2.26%
1.49%
1%
1%
0.85%
0.12%
0.12%
----
Calculated
Means
281.08 kg
116.03 kg
16.88 kg
---947.86 kg
336.91 kg
45.09 kg
339.41 s
345.67 s
343.49 s
Standard
Deviation
3.21%
3.16%
3.23%
---1.21%
0.17%
0.17%
2.12%
1.42%
1.43%
We now present the 1 kilogram payload case. For this vehicle, the non-catastrophic failure rate
was 0.01%, which means that it meets the less than 0.15% failure rate conditions. The output
histograms of each recorded value for the 1 kilogram payload case with their calculated final
standard deviations and means are in Figs. A.8.1.1.5.2.1 and A.8.1.1.5.2.2. The output periapsis
histogram has an approximately Gaussian distribution which is slightly skewed toward the lower
periapsis values. The eccentricity histogram is approximately Gaussian.
400
number of cases
300
200
100
0
280
300
320
340
360
Periapsis altitude(km)
Author: Alfred Lynam
380
400
420
Project Bellerophon
799
Figure A.8.1.1.5.2.1: 1kg Periapsis altitude histogram with a 15.774 km
standard deviation and a 367.727 km average.
(Alfred Lynam)
1000
number of cases
800
600
400
200
0
0
0.05
0.1
0.15
0.2
Eccentricity
0.25
0.3
Figure A.8.1.1.5.2.2: 1kg eccentricity histogram with a .041094
standard deviation and a .173289 average.
(Alfred Lynam)
Author: Alfred Lynam
0.35
Project Bellerophon
800
A.8.1.1.5.3: 5 kg Payload
Table A.8.1.1.5.3.1 5kg Monte Carlo simulation comparisons
Variable
Inert Mass Stage 1
Inert Mass Stage 2
Inert Mass Stage 3
Mass Flow Rate Stage 1
Mass Flow Rate Stage 2
Mass Flow Rate Stage 3
Propellant Mass Stage 1
Propellant Mass Stage 2
Propellant Mass Stage 3
Specific Impulse Stage 1
Specific Impulse Stage 2
Specific Impulse Stage 3
Nominal Input
Values
842.9585 kg
261.1954 kg
15.0986 kg
23.571 kg/s
4.739
kg/s
0.215
kg/s
4122.85 kg
1009.33 kg
38.37
kg
----
Standard
Deviation
2.255%
2.255%
2.255%
1.4923%
1%
1%
0.854%
0.12%
0.12%
----
Calculated
Means
843.2027 kg
261.2956 kg
15.0978 kg
---4123.581 kg
1009.32 kg
38.37
kg
339.5586 s
345.5977 s
344.7218 s
Standard
Deviation
3.173%
3.193%
3.245%
---1.212%
0.172%
0.170%
2.1224%
1.4209%
1.4287%
We now present the 5 kilogram payload launch vehicle. For this vehicle, the non-catastrophic
failure rate was 0.01%, which means that it meets the less than 0.15% failure rate conditions. The
output histograms of each recorded value for the 5 kilogram payload case with their calculated
final standard deviations and means are in Figs. A.8.1.1.5.3.1 and A.8.1.1.5.3.2. The output
periapsis and eccentricity histograms have approximately Gaussian distributions.
250
number of cases
200
150
100
50
0
450
500
550
Periapsis altitude(km)
Author: Alfred Lynam
600
Project Bellerophon
801
Figure A.8.1.1.5.3.1: 5kg periapsis histogram with a 20.215 km
standard deviation and a 516.546 km average.
(Alfred Lynam)
800
700
number of cases
600
500
400
300
200
100
0
0.25
0.3
0.35
0.4
0.45
0.5
Eccentricity
0.55
0.6
Figure A.8.1.1.5.3.2: 5kg eccentricity histogram with a .050
standard deviation and a .445 average.
(Alfred Lynam)
Author: Alfred Lynam
0.65
Project Bellerophon
802
A.8.1.2 Catastrophic Failure
We define catastrophic failure as a failure in any subsystem that eliminates the possibility of
mission success. Our analysis shows that our vehicles are designed so that non-catastrophic
failure is negligible, and that light variations in launch conditions and vehicle performance will
not jeopardize the mission. Catastrophic failure is usually the most visible and dynamic and
usually the most unexpected. Predicting a vehicle’s catastrophic failure rate without actually
building and flying a test article is an inexact science. Therefore, we look to past launch
successes (and failures) in order to see what a reasonable estimate might be.
Catastrophic risk for a vehicle can be thought of as a sum of the risk for each major component.
This is our first means for analyzing catastrophic risk. The component analysis depends on two
independent studies into launch vehicle failure.1,2 Both studies investigate lifetime failure rates
for launch vehicles from a number of different nations. Possible failure rates are catalogued by
subsystem. When the option existed, we select rates for U.S. systems as opposed to a worldwide
average. Also, when identical systems had different values for success between the two studies,
the more conservative one was chosen.
Table A.8.1.2.1 shows the results of this analysis and our predicted overall success rate of
93.84%. Each of our vehicles has an identical architecture, so this analysis is equally valid
across our three payloads. This success rate meets and exceeds the required 90.00% included in
the design requirements. It is important to remember that the majority of the data collected in
this study applies to mature launch vehicle systems. Due to this, we do not believe that our
vehicles will exceed the required 90.00% success rate initially.
Author: Alan Schwing
Project Bellerophon
803
Table A.8.1.2.1 Component Analysis for Vehicle Catastrophic Risk Percent
Item
Failure Rate
Success Rate
Stage 1 Propellants
2.08 %
97.92 % 1
Stage 2 Propellants
1.44 %
98.56 % 1
Stage 3 Propellants
1.44 %
98.56 % 1
Stage or Payload Separation
Fairing Separation
Electrical
Avionics
0.23 %
0.28 %
0.43 %
0.43 %
99.77 % 2
99.72 % 2
99.57 % 2
99.57 % 2
Total
6.16 %
93.84 % 0
In order to better understand our catastrophic failure rate during preliminary flights, historical
launch behavior is an important tool. Three existing launch vehicles are presented in this
analysis: the Ariane IV, Ariane V, and Pegasus systems. Data regarding their flight history
includes each flight and its status, success or failure.3 This data was examined in order to
understand the trends in reliability as these vehicles developed.
Figure A.8.1.2.1 shows the success rate for these vehicles for initial launches. Contained in the
figure is data for the first 35 Ariane IV and Pegasus launches and the first 17 Ariane V launches.
The percentages on the plot are average success rates based on the total number of launches up to
that point. For these three launch systems, there are a significant number of failed launch
attempts initially resulting in very low success rates. After ten to fifteen launches there appear to
be much more reliable results with very few failures across the board.
Author: Alan Schwing
Project Bellerophon
804
Fig. A.8.1.2.1: Launch success rate for Ariane IV,
Ariane V, and Pegasus systems as a function of launches.
(Alan Schwing)
We assume that our vehicle will show behavior very similar to that of the Pegasus.
In
architecture, our vehicles are much more similar to the Pegasus than the Ariane family because
we are an air-launched system. Also, the Pegasus shows the lowest success rates, so in order to
err on the side of conservatism these rates are more appealing.
Table A.8.1.2.2 Catastrophic Risk Details for Pegasus Launch Vehicle
Number of Launches Cumulative Success Rate Success of Previous 10 Launches
10
60.00 %
60.00 %
20
70.00 %
80.00 %
30
80.00 %
100.00 % s
Table A.8.1.2.2 shows a detailed analysis of the Pegasus launch vehicle history. The cumulative
success rate is identical to that shown in Fig. A.8.1.2.1. To delve a little deeper, success rates
over a smaller number of launches are also shown in the third column. That column better
Author: Alan Schwing
Project Bellerophon
805
illustrates the increase in reliability on a per-launch basis. After twenty flights, Pegasus had no
failures, their success rate from that point onward was 100.00%. This shows the maturity of the
system and the reliability that those early failures bought them.
Our interpretation of the design requirements assumes twelve launches per year. Therefore,
using the above estimates from the Pegasus’ historic success rate, we predict that for the first
year (twelve launches), we will have a success rate of 60.00%. After this first year, we believe
that our vehicle will have a per-launch success rate of 80.00%. Finally, after our second year
(twenty-four launches), we assume that our system is mature and well understood. At this point
the vehicle has a success rate of 93.84% as shown from the component analysis. Historical
launch vehicles have a success rate close to 100.00% near maturity, so it is likely that ours will
be much greater than 93.84%.
The first twelve to twenty-four launches might not be offered to customers and instead be
launched as test flights designed to shake-down the system.
This approach would drive
development costs up by roughly 40M$ to 80$M (the cost for these launches). Another approach
is to offer the very first flights to customers with the understanding that the success rate is low
for these initial flights and that in order to ensure a 90% chance of success, multiple vehicles
might have to be purchased.
References
Chang, I-Shih., Tomei, Edmardo Joe., “Solid Rocket Failures in World Space Launches.” AIAA Paper 2005-3793,
Joint Propulsion Conference and Exhibit, 41st, Tuscon, Az, July, 10-13, 2005.
1
Futron Corporation, Bethesda, MD. “Design Reliability Comparison for SpaceX Falcon Vehicles.” November
2004
2
3
Isakowitz, Steven J., Hopkins, Josshua B., Hopkins, Joseph P., Jr., International Reference Guide to Space Launch
Systems, Fourth Edition, AIAA, New York, 2004.
Author: Alan Schwing