A.3.2.8: Monte Carlo Method and Results
A.3.2.8: Monte Carlo Method and Results
A.3.2.8.1: Monte Carlo Method
We tested the non-catastrophic failure rate of the launch vehicle by using a Monte Carlo simulation. The Monte Carlo simulator’s purpose is to run all three stages of each launch vehicle (200g, 1kg, and 5kg payloads) through the “Main Dynamics and Control” simulator ten thousand times. We define the “failure rate” as the number of times that an orbit is below the periapsis requirement divided by the total number of runs. Conversely, we define the “success rate” as the number of times that an orbit is above the periapsis requirement divided by the total number of runs.
The periapsis requirement for Project Bellorophon is that the periapsis of the final Keplerian orbit must be 300 kilometers above the Earth’s surface. The required success rate is 99.86%, and the disqualifying failure rate is 0.15%. All three launch vehicles (200g, 1kg, and 5kg payloads) must meet the required success rates to establish the overall success of the project. We vary the mass and specific impulse (Isp) inputs for the Monte Carlo simulation from the nominal case inputs so that the Monte Carlo results are non-trivial. The variance of the inputs is pursuant to a bimodal distribution with a maximum of three standard deviations from the mean inputs.
Figure A.3.2.8.1.1: Monte Carlo simulation flowchart.
(Mike Walker)
As illustrated in Fig. A.3.2.8.1, the Monte Carlo algorithm begins by running the “Stage 1 Config” MATLAB file which initializes the initial conditions and other inputs for the simulator. We run the “Main Dynamics and Controls” simulator so that the first stage’s dynamics are completely simulated. The final conditions of the first stage are extracted and defined as the initial conditions of the second stage in the “Stage 2 Config” file. We simulate the second stage’s dynamics using the “Main Dynamics and Controls” simulator. The final conditions of the second stage are extracted and defined as the initial conditions of the third stage in the “Stage 3 Config” file.
We simulate the third stage’s dynamics using the “Main Dynamics and Controls” simulator and extract the final velocity and position. We determine the periapsis and eccentricity of the final orbit from the simulated final velocity and position using Keplerian orbital mechanics. We record these parameters in a text file along with the inputted specific impulse (Isp), inert masses, and propellant masses for each run. If the total number of Monte Carlo runs has not been completed, then the algorithm will loop back to the “Stage 1 Config” file and run again. Otherwise, the program will terminate.
We modify the stages’ configuration files slightly from the nominal case version to include standard deviations of input. The standard deviations are based on percentages of the nominal input mean values. Standard deviation percentages are consistent across all launch vehicles. However, the first stage propellant deviation percentages are higher than the second and third stage percentages because the first stage uses hybrid propulsion and the other stages use solid propulsion. We save the data at the end of each stage.
Standard deviation percentages for the input nominal values are detailed in Tables A.3.2.8.2.1, A.3.2.8.3.1, and A.3.2.8.4.1. These tables also compare the calculated values of the mean and standard deviation percentage for each of the three cases. We call a random distribution function each time a configuration file is called to ensure that each run has unique inputs and therefore unique results. The tables show that the random distribution function actually causes the calculated standard deviation values to be increased by about 40-50%. Thus, the entire Monte Carlo simulation is very conservative.
We ran 10,000 or more simulations for each launch vehicle. The results are organized into several histograms to show the frequency of certain values for periapsis, eccentricity, specific impulse, inert mass, and propellant mass. Each launch vehicle’s results are summarized and graphed separately.
A.3.2.8.2: Monte Carlo 200g results
Table A.3.2.8.2.1 200g Monte Carlo simulation comparisonsVariable / Nominal Input
Values / Standard Deviation / Calculated
Means / Standard
Deviation
Inert Mass Stage 1 / 349.4777 kg / 2.255% / 349.4318 kg / 3.213%
Inert Mass Stage 2 / 153.4559 kg / 2.255% / 153.4889 kg / 3.205%
Inert Mass Stage 3 / 14.8011 kg / 2.255% / 14.8081 kg / 3.185%
Mass Flow Rate Stage 1 / 10.689 kg/s / 1.4923% / -- / --
Mass Flow Rate Stage 2 / 2.728 kg/s / 1% / -- / --
Mass Flow Rate Stage 3 / 0.194 kg/s / 1% / -- / --
Propellant Mass Stage 1 / 1462.00 kg / 0.854% / 1462.26 kg / 1.208%
Propellant Mass Stage 2 / 566.64 kg / 0.12% / 566.62 kg / 0.169%
Propellant Mass Stage 3 / 37.26 kg / 0.12% / 37.26 kg / 0.168%
Specific Impulse Stage 1 / -- / -- / 339.4907 s / 2.1302%
Specific Impulse Stage 2 / -- / -- / 345.4158 s / 1.4194%
Specific Impulse Stage 3 / -- / -- / 346.6426 s / 1.4180%
We now present the 200 gram payload launch vehicle. For this vehicle, the periapsis success rate was 9999 successes per 10000 runs, which means that it met the 99.86% success rate. The output histograms of each recorded value for the 1 kilogram payload case with their calculated final standard deviations and means are in figures A.3.2.8.2.1 and A.3.2.8.2.2. The output periapsis histogram has an approximately Gaussian distribution with several outliers toward the lower periapsis values. The eccentricity histogram is approximately Gaussian.
Figure A.3.2.8.2.1: 200g Periapsis altitude histogram with a 21.803 km
standard deviation and a 437.44 km average.
(Alfred Lynam)
Figure A.3.2.8.2.2: 200g Eccentricity histogram with a .0467
standard deviation and a .438 average.
(Alfred Lynam)
The input histograms for the 200 g payload case represent bimodal distributions. They are listed along with their standard deviations and averages in figures A.3.2.8.5.1 -A.3.2.8.5.9. The input variables that are varied and graphed as histograms are specific impulse (Isp), inert mass, and propellant mass for each stage.
A.3.2.8.3: Monte Carlo 1 kg results
Table A.3.2.8.3.1 1kg Monte Carlo simulation comparisonsVariable / Nominal Input
Values / Standard Deviation / Calculated
Means / Standard
Deviation
Inert Mass Stage 1 / 281.3513 kg / 2.255% / 281.0813 kg / 3.209%
Inert Mass Stage 2 / 116.0688 kg / 2.255% / 116.0305 kg / 3.158%
Inert Mass Stage 3 / 16.8907 kg / 2.255% / 16.8781 kg / 3.225%
Mass Flow Rate Stage 1 / 6.730 kg/s / 1.4923% / -- / --
Mass Flow Rate Stage 2 / 1.880 kg/s / 1% / -- / --
Mass Flow Rate Stage 3 / 0.231 kg/s / 1% / -- / --
Propellant Mass Stage 1 / 947.90 kg / 0.854% / 947.86 kg / 1.208%
Propellant Mass Stage 2 / 336.92 kg / 0.12% / 336.91 kg / 0.170%
Propellant Mass Stage 3 / 45.09 kg / 0.12% / 45.09 kg / 0.169%
Specific Impulse Stage 1 / -- / -- / 339.4066 s / 2.1224%
Specific Impulse Stage 2 / -- / -- / 345.6716 s / 1.4209%
Specific Impulse Stage 3 / -- / -- / 343.4905 s / 1.4287%
We now present the 1 kilogram payload launch vehicle. The periapsis success rate was 9999 successes per 10000 runs, which means that it met the 99.86% success rate. The output histograms of each recorded value for the 1 kilogram payload case with their calculated final standard deviations and means are in figures A.3.2.8.3.1 and A.3.2.8.3.2. The output periapsis histogram has an approximately Gaussian distribution which is slightly skewed toward the lower periapsis values. The eccentricity histogram is approximately Gaussian.
Figure A.3.2.8.3.1: 1kg Periapsis altitude histogram with a 15.774 km
standard deviation and a 367.727 km average.
(Alfred Lynam)
Figure A.3.2.8.3.2: 1kg eccentricity histogram with a .041094
standard deviation and a .173289 average.
(Alfred Lynam)
The input histograms for the 1 kg payload case represent bimodal distributions. They are listed along with their standard deviations and averages in figures A.3.2.8.10 - A.3.2.8.18. The input variables that are varied and graphed as histograms are specific impulse (Isp), inert mass, and propellant mass for each stage.
A.3.2.8.4: Monte Carlo 5 kg results
Table A.3.2.8.4.1 5kg Monte Carlo simulation comparisonsVariable / Nominal Input
Values / Standard Deviation / Calculated
Means / Standard
Deviation
Inert Mass Stage 1 / 842.9585 kg / 2.255% / 843.2027 kg / 3.173%
Inert Mass Stage 2 / 261.1954 kg / 2.255% / 261.2956 kg / 3.193%
Inert Mass Stage 3 / 15.0986 kg / 2.255% / 15.0978 kg / 3.245%
Mass Flow Rate Stage 1 / 23.571 kg/s / 1.4923% / -- / --
Mass Flow Rate Stage 2 / 4.739 kg/s / 1% / -- / --
Mass Flow Rate Stage 3 / 0.215 kg/s / 1% / -- / --
Propellant Mass Stage 1 / 4122.85 kg / 0.854% / 4123.581 kg / 1.212%
Propellant Mass Stage 2 / 1009.33 kg / 0.12% / 1009.32 kg / 0.172%
Propellant Mass Stage 3 / 38.37 kg / 0.12% / 38.37 kg / 0.170%
Specific Impulse Stage 1 / -- / -- / 339.5586 s / 2.1224%
Specific Impulse Stage 2 / -- / -- / 345.5977 s / 1.4209%
Specific Impulse Stage 3 / -- / -- / 344.7218 s / 1.4287%
We now present the 5 kilogram payload launch vehicle. For this vehicle, the periapsis success rate was 10000 successes per 10000 runs, which means that it met the 99.86% success rate. The output histograms of each recorded value for the 1 kilogram payload case with their calculated final standard deviations and means are in figures A.3.2.8.4.1 and A.3.2.8.4.2. The output periapsis and eccentricity histograms have approximately Gaussian distributions.
Figure A.3.2.8.4.1: 5kg periapsis histogram with a 20.215 km
standard deviation and a 516.546 km average.
(Alfred Lynam)
Figure A.3.2.8.4.2: 5kg eccentricity histogram with a .050
standard deviation and a .445 average.
(Alfred Lynam)
The input histograms for the 5 kg payload case represent bimodal distributions. They are listed along with their standard deviations and averages in figures A.3.2.8.5.19 – A.3.2.8.5.27. The input variables that are varied and graphed as histograms are specific impulse (Isp), inert mass, and propellant mass for each stage.
A.3.2.8.5: Input Histograms
The input histograms for the 200g, 1 kg, and 5 kg cases are below:
Figure A.3.2.8.5.1: 200g First stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.2: 200g Second stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.3: 200g Third stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.4: 200g First stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.5: 200g Second stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.6: 200g Third stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.7: 200g First stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.8: 200g Second stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.9: 200g Third stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.10: 1kg first stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.11: 1kg second stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.12: 1kg third stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.13: 1kg first stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.14: 1kg second stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.15: 1kg third stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.16: 1kg first stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.17: 1kg second stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.18: 1kg third stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.19: 5kg first stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.20: 5kg second stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.21: 5kg third stage Isp histogram
(Alfred Lynam)
Figure A.3.2.8.5.22: 5kg first stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.23: 5kg second stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.24: 5kg third stage inert mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.25: 5kg first stage propellant mass histogram
(Alfred Lynam)
Figure A.3.2.8.5.26: 5kg second stage propellant mass histogram.
(Alfred Lynam)
Figure A.3.2.8.5.27: 5kg third stage propellant mass histogram
(Alfred Lynam)
Author: Alfred Lynam