undenied supply delivery system - career account web service

Undenied Supply Delivery System

Trinity Aerospace

Kyle Duckering

Noah Marquand

Nicholas D. Turo-Shields

Sree Vellanki

Final Project Report: AAE 25100-001 Introduction to Aerospace Design

Fall 2016

1 Trinity Aerospace

Nomenclature

AFB = Air Force Base (only considering AFBs in the contiguous United States) AHP = Analytic Hierarchy Process α = Angle of attack 𝛽 = launch azimuth CAD = computer aided design CL = Coefficient of Lift CL, max = Maximum Coefficient of Lift CD = Coefficient of Drag finert = Inert mass fraction FoS = factor of safety g0 = gravitational acceleration at sea-level, 9.81 m/s2 h0 = initial altitude of launch hA = altitude of orbit above Earth’s surface is = orbit inclination Isp = specific impulse 𝜆 = latitude of rocket launch m0 = mass of rocket stage mpay = mass of rocket payload 𝜇E = Earth’s gravitational constant, 3.986 * 105 km3/s2 ⍵E = rotation rate of Earth, 7.292 * 10-5 rad/s OMS = orbital maneuvering system P A = Power output by engines P R = Power required by plane to be able to operate at a given altitude. rA = altitude at upper edge of the atmosphere + radius of Earth R.C. = rate of climb of the aircraft rE = radius of Earth, 6378 km ΔV = change in velocity ΔVdeorbit = ΔV required to enter Earth’s atmosphere from a circular orbit ΔVDrag = drag loss ΔVE = ΔV provided by earth ΔVeffective = effective ΔV ΔVGravity = gravity loss ΔVideal = ideal ΔV required for launch ΔVSteering = steering loss Vorbit = velocity of a circular orbit VLO = velocity of aircraft at liftoff Vplane = velocity of the aircraft at rocket launch Vstall = stall velocity of aircraft

Abstract

This paper entails the development of a proposal for an “undenied supply delivery system” (USDS) that can deliver a 2000 kg payload from the United States to any populated area in the world. The USDS consists of three components: an aircraft, a rocket, and an orbital maneuvering system (OMS). The aircraft must be able to take off from any air force base (AFB) in the continental United States and fly 500 nautical miles away from any land. It must be capable of flying at 50,000 feet and carrying the entire rocket and OMS. The aircraft must drop the rocket and then return to the nearest AFB. The rocket ignites in air and puts the OMS and payload into a 400 km circular orbit with an inclination that allows its ground track to cover the target area. The OMS is expected to stay in orbit for a maximum of 24 hours. The OMS is responsible for deorbiting at the right time and location such that the payload can land safely, accurately, and timely.

2 Trinity Aerospace

The rocket was designed first. The maximum ΔV required for the mission by developing several flight plans and determining the ΔV of the worst-case scenario. Then, the Analytic Hierarchical Process was employed to determine the relative importances of cost, mass, and reliability for an optimal system. Trinity Aerospace researched existing rocket stages, and using the data acquired on propellant type, specific impulse (Isp), reliability, payload mass, inert mass, and propellant mass, the finert of each stage, including the OMS, was determined. Equipped with this information, stages were scaled down to optimal sizing, taking into account the reduction of performance characteristics from the larger, existing engines. Once this research was completed, rocket design began. Incorporating a factor of safety, an optimization code written in Python ran through each stage of the rocket’s mission and determined the masses, costs, and reliabilities of every possible combination of stages that was researched. The values were weighted according to the Analytic Hierarchical Process, and the optimal system was output. The volume, radius, and length for each stage were calculated, and a CAD model was developed in Autodesk Inventor to display the accurate sizes and so that the rocket and the plane could be virtually interfaced. The aircraft was designed to carry the rocket from any air force base out to sea. The maximum distance the aircraft would have to fly for a mission was determined, as well as the minimum thrust required to take off at the air force bases with the highest altitudes and shortest runways. Military transport aircraft were researched as well as engines that could provide the required thrust. Finally, the engines and airfoil were chosen such that the aircraft could attain the required lift and thrust to complete the mission.

I. Mission Overview With today’s ongoing military operations and humanitarian aid providing services, one of the major problems faced by countries such as the United States is the lack of a strong and consistent supply chain capable of delivering necessary supplies continuously to regions of the world in need. Although delivering the supplies by land or by sea are generally the most effective methods, they may not be feasible due to physical or political obstacles. Isolated locations, either geographically or deep within “enemy” territory, can effectively prevent land and sea routes from accessing the populations that live in that area. The next logical choice is an air-based delivery system, but this also proves difficult. Remote regions sustainably supplied by aircraft only with great difficulty, and hostile countries across the world can deny access to the airspace just as they can deny access to land routes. Space based systems are another alternative, but are highly expensive and are not easily adjusted to different locations and time schedules due to the great difficulties involved in changing an orbit, particularly once the orbital maneuvering system is in its final orbit. Since humanitarian crises are rarely convenient timewise (or otherwise), a space based system alone is not feasible for delivering a time-critical payload. Trinity Aerospace has developed an “Undenied Supply Delivery System” that circumvents the aforementioned challenges. This system is highly versatile, able to be deployed from any air force base (AFB) located in the continental United States, and deliver the payload to the destination in less than 36 hours. This USDS combines aspects of the air-based and space-based systems to deliver any 2000 kilogram payload almost anywhere, with minimal advanced notice. The system is as follows: First, the payload is attached to the orbital maneuvering system. The OMS is loaded into the fairing of two stage rocket which is then loaded onto the carrier aircraft. Given the coordinates of the city to receive the payload, the aircraft flies 500 nautical miles into the Pacific Ocean or the Atlantic Ocean, where the aircraft climbs to 50,000 feet and slows down to the slowest safe speed.The aircraft then aligns itself with the launch azimuth required to launch into the target latitude inclination and releases the rocket from its fixture inside. The rocket can then begin its ascent while the aircraft returns to the nearest United States Air Force base. The first two stages of the rocket deliver the OMS to a 40 km circular orbit. The first stage is dropped over the ocean during launch, and the second stage deposits the OMS into orbit before performing a deorbit burn. Once it is over the final destination, the OMS performs its own deorbit maneuver and delivers the payload into the atmosphere, where its onboard heatshield and landing system take over the descent to the target.

II. Aircraft Design A. Preliminary Research and Considerations The design of the aircraft began with research into military transport aircraft currently in existence. Special attention was paid to the payload capabilities, as this is what our company foresaw as the most important aspect of aircraft design. Furthermore, comparisons were made to other military transport aircraft to determine approximate relationships between aircraft empty mass, fuel mass, payload mass, and maximum takeoff mass. These linear

3 Trinity Aerospace

approximations were used to determine the size requirements of the aircraft based on the mass of the payload, the fuel capacity, and the takeoff mass. As stated in the Request For Proposal, the complete system must be able to be deployed from any Air Force Base located in the contiguous United States. To be able to make the claim that our system can launch from every contiguous U.S. AFB, our company compiled a list of every AFB in the contiguous U.S., the longest runway at each AFB, and the elevation of each AFB (Appendix A). The reasoning for this was that each runway will have different air densities based on its altitude, different values of acceleration due to gravity, and different runway lengths, all of which are critical factors to consider in the design of a new aircraft. Using these data, a MATLAB program was developed to calculate the thrust required to take off from each of these runway scenarios, assuming that all the air force bases have standard atmospheric conditions. When each “thrust required” value were returned, it was stored and compared with the other thrust required values. At the end of the iteration, the highest thrust required value of all of the different AFB scenarios was chosen as the absolute minimum thrust that the aircraft must produce to be able to take off from any AFB. With the requirement of the project stating that the aircraft must release the rocket payload no less than 500 nautical miles from the coast or a populated area, planning was required for the plane’s flight path to ensure that it could launch from any AFB, fly out to 500 nmi from the coast, and then return to the nearest or most convenient AFB. Further complexity and considerations were made necessary by requests from the Space System design team to navigate the aircraft to a 35° latitude to reduce the propellant required for later plane changes while in orbit. These conditions allowed us to develop requirements for range, and further develop the aircraft design and narrow down the choice of engines. The final requirements to be taken into consideration were the minimum altitude that the aircraft must be capable of flying at, and the slow speed clause. The request for proposal specified “10,000 ft above typical commercial aircraft and must deploy the rocket at the lowest possible speed.” Our company took the typical commercial aircraft cruise altitude to be 39,000 ft. A factor of safety (FoS) of 1.1 was decided for the altitude buffer zone to account for adverse atmospheric conditions, and to ensure the rocket would not interfere with nor be interfered with by other aerial bodies. At higher aircraft speeds, the rocket may be released in an uncontrolled manner and be unsuccessful in completing its mission. For all of Trinity’s calculations, we set 50,000 ft as the target altitude for deploying the rocket and subsequent systems. Additionally, the slowest speed that our aircraft will launch the rocket will be VLO = 1.2Vstall. A FoS of 1.2 is standard for takeoff.

B. Assumptions During the design and performance of analysis, our company had to employ several approximations and assumptions. First, all locations were assumed to adhere to standard atmosphere conditions to simplify the model. Next, the carrier aircraft’s engines were modeled as a turbojet to satisfy the Breguet Range Equation (Eq. 1), noting that turbofan engines utilize a turbojet core do drive the vehicle and produce thrust. For said Range calculations, the aircraft was assumed to be flying at the service ceiling of 50,000 ft (15, 240 m) for the duration of the mission, thus removing the issue of takeoff and landing fuel inefficiencies for this model. Then, mass relationships from comparable aircraft were assumed to be viable models used in the carrier aircraft sizing parameters (see figure 1.). Lastly, the thrust provided by a turbofan engine is constant over all flight speeds.

) Rjet = √ 8ρ S∞ * 1

ct* CD

√CL* (√W 0 − √W 1

(1) C. Aerodynamics The payload configuration of having the rocket carried inside the aircraft’s body was chosen over having the rocket carried beneath or above was for aerodynamic reasons. Carrying the rocket under wing was not considered due to the significant mass of the rocket, as well as the fact that the drag would be unbalanced and the aircraft would experience adverse moments in all three directions. Rockets are not designed to be aerodynamic, they are designed to get out of the atmosphere as quickly as possible. As such, we decided to design an airplane large enough to carry the rocket on the interior. For our aircraft, we decided to use a NACA 2412 airfoil. This airfoil was chosen because because it has a relatively high cl, max of 1.6 and a cd, 0 of 0.005. Since these are infinite wing values, our analysis, we estimated the finite wing and aircraft body effects by adjusting these values to be CL, max = 1.5 and CD, 0 of 0.04. For takeoff, we

4 Trinity Aerospace

used the lift coefficient in a takeoff configuration (α = 0) of CL = 0.6. These results were used in all of the following aircraft calculations. D. Performance To calculate the minimum thrust that the airplane needs to have, every air force base in the country was researched and analyzed. The runway lengths and elevations were the primary factors, as thrust required is directly affected by both. Shorter runways and higher altitudes are adverse to a jet engine’s performance during takeoff, so the aircraft was designed to successfully lift off from both the AFB with the shortest runway (Columbus Air Force Base Auxiliary Field, 1920 m)1, and at the highest elevation (Hill Air Force Base, 1459 m)2, as shorter runway lengths and higher altitudes are adverse to a jet engine’s performance during takeoff. The highest amount of thrust required was for Keesler Air Force Base in Biloxi, Mississippi with a value of 816,600 N. To calculate this value, we started off by calculating the take-off velocity needed. This was done by using Eq. (2):

.2 V LO = 1 √ 2Wp SC∞ L,max

(2)

Our plane had to be designed with service ceiling of 50,000 ft. This was accomplished by incrementing the thrust provided by the engined and evaluating the corresponding service ceiling until the aircraft was capable of flying at the service ceiling of 15,240 m(3). This script gave us a value of TR = 1,380,000 N total(4). This ended up being considerably higher than the thrust required to take off from an AFB, so the Thrust required to climb to the service ceiling became the new limiting factor, and Trinity can confidently state that our aircraft is more than capable of taking off from any AFB.

.C. R = WP −PA R

(3) To calculate the minimum range that the airplane needs to have, we calculated the longest flight path possible from an air force base to 500 nautical miles away from any land and used that as the benchmark. The Hudson Bay is not large enough for the plane to safely launch the rocket away from populations, and the orbital inclinations are very limited in the Arctic Circle. Because plane changes are more expensive energetically for orbits than a longitude change of an aircraft, only the two coasts along the continental United States were considered for flight planning. We found that the farthest AFB from both the Atlantic or Pacific coasts was Minneapolis St. Paul Air Reserve Station. This AFB is 1656 km away from the Atlantic coast, and even farther away from the Pacific coast in our benchmarking process. Following the shortest flight path to sea, the aircraft is at a latitude of 35° when it starts to fly over the Atlantic Ocean. From this point, we added the 500 nautical miles due East, which brought the distance flown over sea to 940 km. An additional 940 km must be added for the return flight from the launch location back to the nearest AFB on the East coast. This came out to a total of approximately 3500 km. To further anticipate any unexpected situations, used a factor of safety of 1.5 for the range out aircraft must be capable of travelling. This maximum required range is 1.5 * 3500 = 5,250 km(31). In our range calculations, we decided to plan a worst-case scenario. We considered the worst-case scenario as having to launch from Minneapolis St. Paul Air Reserve Station (our longest distance over land) and then out to 500 nautical miles off the coast. The worst possible case we could consider during our mission execution was if something were to happen just before the the launch of the rocket. If this were the case, Trinity Aerospace would choose not to launch the rocket and would have to return with the rocket and not launch. This extra mass would more require more fuel from the aircraft, and although we do not anticipate needing to make use of this ability, we wanted our aircraft to be capable of functioning on the most extreme mission under the worst possible circumstances.

5 Trinity Aerospace

E. Mass Estimation Information of various masses from military transport aircraft(6) was compiled and a linear relationship was made between different aircraft mass values and the empty masses.

F. Cost Using the relationship shown in the mass comparison, we determined that to carry a payload (the rocket that delivers the payload) of 42,000 kg, our aircraft had to have an empty (inert) mass of 67,500 kg. With the cost of this inert mass of $10,000/kg, the cost of the inert mass of the aircraft comes out to $675,000,000.

The fuel required for our worst-case scenario as described earlier comes out to be 142,000 kg. With the cost of Jet A being given as $1000/kg, the cost of fuel comes out to be $1,420,000.

The combination of the fuel cost and inert mass cost brings the total cost of the aircraft to $676,420,000 G. Engines Extensive research was conducted on aircraft engines, comparing and contrasting values for thrust and weight. Optimizing for the largest thrust to weight ratio, special attention was given to the engines used in most of today’s military transport vehicles. These aircraft included the C-5M Super Galaxy, AN 124, C-17, etc. Looking at the engines of the aircraft along with other engines used by the armed forces, we finalized on one which gave us the best results. To compare the usefulness of the rocket, we looked at the amount of dry thrust the engine outputs and the weight of the engines in question(3). The final engine at the end of the decision process was the GE90-85B Turbofan Engine. This engine had a significant amount of thrust output which is an optimal amount for our airplane-rocket system. This engine weighs 18,260 kg and can produce a thrust of 376,764 N per engine. We decided to use four of these engines on our plane based on the performance requirements from the RFP and the Space System Team, bringing the total amount of thrust of the aircraft to 1507057 N. This amount surpasses the required amount of thrust needed to achieve the necessary altitude of 50,000 ft by approximately 207,000 N. Although this margin is large, and although there might be better suited engines, we chose this one specifically because, it gives significant leeway if the mass of the rocket’s payload or the rocket’s propellant mass increases, then these engines will still be able to produce the necessary amount of thrust needed to successfully complete the mission(4). To further demonstrate the strength of the engine design decision, the values of mass and dry thrust provided by the GE engine were compared to that of another turbofan engine, the GE90-85B. Although the GE90-85B engine was able to provide more thrust than the GE CF6-80C2A, the original decision still stood because the second engine was much more massive than the previously determine “optimal” engine. If four GE90-85B were used, the total mass of the aircraft from the engines alone would be 73,040 kg, which is 24,790 kg more than the total mass of the

6 Trinity Aerospace

aircraft using six of the GE CF6 engines. It was determined that the less massive GE CF6 engines allowed the aircraft to have a higher payload capacity for more fuel or a larger rocket, or a longer range for the rocket so it can be launched at more preferred locations, better optimized for the required orbit and timing of the payload destination(5).

III. Space System Design

A. Introduction The space system consists of the rocket and the orbital maneuvering system. The rocket is carried inside the airplane until it is 500 nautical miles away from land, and then is dropped from the airplane at the lowest possible speed to avoid serious issues with drag and so the rocket can orient itself upwards out of the atmosphere. It must deposit the OMS into a 400 km altitude circular orbit in space. The OMS must be in orbit for 24 hours or less, and the ground track of the orbit must cover the target area so that the payload can be released in the atmosphere and land in the right location. B. Flight Planning Assuming the coordinates of the destination are known, the orbit required of the Orbital Maneuvering System will be designed to always have a ground track to cover that area on the Earth. The orbit of the OMS must have an inclination angle of at least the latitude of the destination in degrees. If the inclination of the orbit is less than the latitude of the target, then the ground track of the orbit will never intersect the target, and the payload will not land where it needs to. The most efficient latitude to launch the rocket carrying the OMS from is the same latitude as the target. At this latitude, the Earth provides the most “free” ΔV to the rocket, because the rocket would be launched in the same direction as the Earth’s rotation, East. However, the aircraft will rarely be able to reach this optimal latitude due to the already taxing ranges required of it. If the inclination is particularly high, and the timing of the launch is off, the orbit may need to have a larger inclination than the latitude of the target. If this is the case, an orbital plane change may help the situation. However, designing a rocket capable of reaching different inclinations is less complex and more energy efficient than designing an OMS to perform significant plane changes. The rocket can be launched at different azimuths to reach the desired location at a certain time within the 24 hour time period that the OMS is in orbit. As such, the OMS was not designed to perform plane changes. The inclination of an orbit launched at the equator is simply the angle relative to the equator that the rocket is launched at. At latitudes other than the equator, different launch azimuths are required to reach the same inclination. This relationship is given by Eq. (1).

𝛽 = sin-1(cos(i) / cos(𝜆)) (4)

To appropriately size the rocket, the worst case scenario was chosen as the flight path. The worst conditions are a 35° latitude at launch, and a 70° orbit inclination. These values were chosen because 35° is the lowest latitude that the airplane can feasibly reach from every AFB in the continental United States, and 70° is the highest orbital inclination because the largest city containing over 100,000 people, Norilsk, Russia, is at a latitude of 69° (7). Regions with populations less than 100,000 people are assumed not to be notable. With the launch altitude and inclination chosen, calculations to determine the total ΔV required for the mission were performed. First, the “free” ΔV provided by the Earth is given by Eq. (2).

ΔVE = ⍵E * rE * cos(𝜆) * sin(𝛽) + Vplane

(5)

When 𝜆 = 35°, i = 70°, and Vplane = 0 m/s, Eq. (4) yields 𝛽 = 52.8°. When those values are plugged into Eq. (5), ΔVE = 0.348 km/s. The final velocity is calculated using Then, the velocity of of an orbit at 400 km above the Earth’s surface is calculated to determine the ideal ΔV required by launch.

7 Trinity Aerospace

Vorbit = (𝜇E / (rE + hA))1/2 (6)

When hA = 400 km, Vorbit = 7.821 km/s. Subtracting Eq. (5) from Eq. (6), ΔVideal is calculated. ΔVideal = 7.510 km/s. Drag, gravity, and steering losses, as well as a factor of safety must be incorporated to ensure the rocket can complete its mission of depositing the OMS in the proper orbit. Using linear approximations, these losses can be estimated based on the altitude of launch, and the effective ΔV is determined.

ΔVDrag = 150 - 0.0075 * h0 (m/s) (7)

ΔVGravity = 150 - 0.075 * h0 (m/s) (8)

ΔVSteering = 200 (m/s) (9)

ΔVeffective = FoS * (ΔVideal + ΔVDrag + ΔVGravity + ΔVSteering) (10)

Converting m/s to km/s, Eq. (4) yields ΔVDrag = 0.150 km/s, Eq. (5) yields ΔVGravity = 0.150 km/s, and ΔVSteering = 0.20 km/s. Plugging into Eq. (7) with FoS = 1.5, ΔVeffective = 8.811 km/s.

Figure 2. ΔV as a function of target latitude ΔV required for a rocket launched at a 35 degree latitude to an orbital inclination equal to target latitude Because the rocket was designed to put the OMS in any orbit required for the delivery of the payload to the

given location, the OMS only has to house and deorbit the payload. The ΔV required to deorbit was found by constructing a Hohmann transfer from the initial orbit to the edge of the atmosphere, without the recircularization burn so that the OMS is not orbiting the Earth at the edge of the atmosphere.

ΔVdeorbit = Vorbit - (2 * (-𝜇E / (rE + rA) + 𝜇E / rA))1/2

(11)

Eq. (8) yields ΔVdeorbit = 0.09 km/s. Once the OMS reaches the edge of the atmosphere, atmospheric drag will further reduce its velocity, and when it ejects the payload, the parachutes and landing systems stored inside the payload capsule will ensure the package reaches its target. C. Scoring In order to optimize the rocket, three factors were considered: cost, mass, and reliability. The Analytic Hierarchy Process (AHP) was employed to weight these values against each other (Appendix B). First, three “consistency” matrices were created to make pairwise comparisons between cost, mass, and reliability for the most cost efficient system, the least massive system, and the most reliable system. Optimizing for cost alone, it was determined that

8 Trinity Aerospace

cost is very strongly prefered over mass (ranking of 7). This ranking was chosen because although the cost is the most important factor, mass is still important to the aircraft design, and mass must still be considered. Reliability is strongly prefered over mass (ranking of 5), and cost is mildly prefered over reliability. Cost and reliability are close in priority because it is assumed that a failed rocket will require a relaunch, greatly increasing the cost. Optimizing for mass alone, it was determined that mass is very strongly prefered over cost (ranking of 7). This ranking was chosen because although mass is the most important factor, the whole system must still be competitive with previously existing systems that can accomplish the same mission. Mass is strongly prefered over reliability (ranking of 5) because having to relaunch the rocket affects the cost much more than it affects the mass. There is no preference between cost and reliability while optimizing for mass because they are so closely related. Optimizing for reliability alone, it was determined that cost is strongly prefered over mass (ranking of 5) because reliability is more closely related to cost. A less reliable system would cost more because there is a higher likelihood of needing a relaunch, but also a more reliable system would use better parts, which could cost more. This strong correlation is why mass is the least important factor in this matrix. Reliability is most strongly prefered over cost (ranking of 9) because even though reliability and cost are closely tied, this matrix is optimizing reliability. The next step of AHP is to normalize the data in the three consistency matrices. The numbers in the matrix are divided by the sum of all the numbers in their respective columns so that the numbers in each column of the “normalized matrices” add up to 1. A fourth column is added to each matrix, containing the average of the three values in their respective rows. Finally, a scoring matrix is created with the three criteria in the in the first column, the second column containing weightings, and the last three columns with the three criteria again. The weights for mass, cost, and reliability are 0.6, 0.25, and 0.15, respectively. Mass was weighted the highest because it has it is not only related exponentially to the space system’s mission ΔV, but the aircraft design was also heavily dependent on the rocket’s mass. The cost was weighted 0.25 because the overall system must be competitive in the market. Finally, reliability was weighted 0.15 because almost all the rockets considered during the research phase had success rates of over 92.8%, and the design team is currently unequipped to compare rockets with varying success rates and missions (i.e. the European Space Agency’s Vega has a 100% success rate with 8 launches, and Russia’s Soyuz-U has a 97.3% success rate with 785 launches). The column containing the averages from the normalized matrix for cost is input into cost row of the final scoring matrix, the column containing the averages from the normalized matrix for mass is input into the mass row of the final scoring matrix, and the column containing the averages from the normalized matrix for reliability. The rows of the scoring matrix are multiplied by the weights, and the columns are added to produce the final weighted rankings. Mass is ranked first at 0.475, reliability is ranked second at 0.267, and cost is ranked third at 0.259. Using these numbers, the optimal rocket configuration can be found. D. Data Collection Data Collection began with isolating values of interest from existing launch vehicles and orbital systems. Datum for launch vehicles were selected from the United Launch Alliance’s Atlas V(8-10), Delta IV(11-13), and Delta II(13-15),

SpaceX’s Falcon9(16), the European Space Agency’s Vega(17) and Ariane V(18-19), and Russia’s Soyuz(20) and Rockot(21). Datum for orbital systems were selected from Orbital ATK’s Star 48[], Orion 38[], and Pegasus(25),

Russia’s Volga(26), Briz-M(27), Fregat, and APB(28)upper stages, and the European Space Agency’s AVUM(29). Values of interest include propellant type, specific impulse (Isp), thrust to mass ratio, reliability, payload mass, inert mass,

and propellant mass .

Because the rocket is being stored inside the airplane instead of beneath or above, stages requiring cryogenic fuel are eligible for consideration because the coolant tanks can be stored alongside the rocket in the cargo bay. Isp and thrust were always taken at vacuum as the launch altitude pressure is closer to that of sea-level. Reliability was assessed by locating each vehicle’s percent of missions completed successfully. While this may not represent specific stage failure, isolating specific failures for each loss of vehicle is not always possible, making this approach the most practical and expedient. These values were further analyzed to determine if correlation exists between them. Calculating the inert mass fraction (finert) of each stage and plotting it against nondimensionalized Isp and stage payload yields the plot in Fig. 2. For interest of clarity, the data for SpaceX’s Falcon 9 was not included in this plot as its unique design behaves as an outlier from the rest of the finert vs. payload data.

9 Trinity Aerospace

Given the wide scattering of the data in both sets, it appears as if neither payload nor Isp have a clear relationship with finert. As such, it was decided that selecting an finert value could not be done directly from either value. As such, it was decided that each stage must be used as a whole and could not be decomposed into separate systems, but instead scaled as needed for the purposes of modeling the USDS. It is assumed that the fuel to oxidizer ratio stays constant for each engine, regardless of the scaled size. Provided this, stages were divided into two groups, lower and upper stages, to better represent the change in propulsion requirements as the rocket leaves the atmosphere and achieves orbit. Three stage rockets were initially considered, but due to the added complexity, three stages were rejected in favor of two. Solid boosters in parallel staging with first main stages were considered first stages for the purposes of this analysis. Furthermore, interstage masses and fairing masses were studied. Given that many payloads are constrained by fairing size rather than launch vehicle mass restrictions(30), fairings with similar payloads to this mission were selected. These masses of these fairings range from ~800 kg(13) and ~500 kg(17), and were outfitted on launch vehicles with larger diameters than that of the anticipated design. Thus, the estimated fairing mass was 500 kg. The masses of interstages had a much greater range, from non-existent(17) to several thousand kilograms(9). As such, an interstage mass of 300kg based off of the first component of the Atlas V interstage was included to approximate the need for a structural connection between stages. Both systems are modeled as solid-fueled inert mass, being significantly less complex than liquid-fueled inert mass, but still retaining significant complexity in electronics and explosive decoupling systems to contribute to cost significantly. E. Design and Optimization Once data collection was complete, the design process began. Factors of safety for the launch vehicle and the OMS ΔV provisions were determined as 1.1 and 1.5 respectively. With these additional provisions, each mission stage will be able to compensate for potential failures, making vehicle loss and mission failure from shortened burn times or failed engine starts less problematic. Furthermore, this additional ΔV can be used to orient the first stage after it has been dropped by the plane over the ocean and deorbit the second stage once it deploys the payload, preventing the accumulation of hazardous space material in low earth orbit (LEO). While a factor of safety results in a larger and more expensive rocket, it improves the performance of each vehicle, making the program more successful and sustainable in the long run. The initial thrust to weight ratio was also determined to be an important factor to consider for enabling a successful launch. In order to counteract gravity, the active engine of the first stage must have a thrust to weight ratio greater than one. Some vehicles, such as the Falcon 9, achieve ratios over ten(16) by using extremely high thrust engines, while others have lower ratios, such as the Atlas V, with a ratio of about two(10). However, these must also

10 Trinity Aerospace

counter the significant force of drag from the thick, sea-level atmosphere. Thus, a lower thrust system should be able to perform adequately at altitude, and a minimum thrust to weight ratio of 1.1 for the first stage was decided. Data was then compiled into an optimization code (Appendix C) where each first stage, second stage, and OMS model as well as the ΔV distribution were each iterated through twice. In the first iteration cycle, maximum and minimum values for system mass, cost, and reliability were found. System mass was calculated using the rocket equation, Eq. (9), where m0 represents the stage mass, mpay represents the stage payload, and g0 represents gravitational acceleration at sea-level.

1 )/(1 )m0 = mpay * eΔV /(g I )0* sp * ( − f inert − f inert * eΔV /(g I )0* sp (12)

Stages with non-positive masses were discarded, and the total cost was calculated by multiplying the stage mass by finert and 1-finert to produce the inert mass and propellant mass. These were then cost estimated using their listed prices. Total system reliability was found by multiplying each stage’s success rate to produce a total expected success rate. After total launch vehicle mass was determined, the thrust to weight was calculated (engine thrust scaling is described further under sizing). If this did not exceed specification, the system was discarded. Once a system was assessed, mass, cost, and reliability were compared to the current stored maximum and minimum mass, cost, and reliability. If new data exceeded old extrema, the old values were replaced by the new ones. Once the final extrema were found, they could be used in conjunction with the following equation, Eq. (10) to nondimensionalize results and score the designs for comparison. Nondim represents the nondimensionalized value, dim represents the original value, and dimmin and dimmax represent the minimum and maximum values respectively.

ondim (dim im ) / (dim dim )n = − d min max − min

(13) Thus, during the second iteration cycle, each system mass, cost, and reliability could be nondimensionalized and the system could be assessed using the scoring weights determined from the AHP in Eq. (11):

core .475 ass .259 ost .267 eliability s = − 0 * m − 0 * c + 0 * r (14)

Negative weightings indicate the demand for lower valued terms, thus mass and cost both contribute negative scores to the total score while reliability contributes positively. Systems with a current high score were stored and later output for further analysis and sizing. F. Component Sizing Using the system selected, each of the stages were modeled after their existing base stage. If it is assumed that stage mass scales linearly, approximations for final volume can be made using the following equations. Objects with the model subscript represent the specifications of the existing stage, and len_rad represents the ratio of length to radius.

olume volume / mass massv = model model *

(15) adius (volume / ( π en rad ))r = * l − model

1/3 (16)

ength en rad adius l = l − * r (17)

Thus final dimensions for the USDS stages can be found. If it is assumed that the fairing holds the same relative dimensions as the Vega fairing and has the same radius as the upper stage, fairing length can be found to approximate the total launch vehicle length. Engines were scaled in a similar manner to represent lowered capability of smaller scale. As each stage was scaled, so was each engine’s specifications, including thrust. This type of scaling resulted in large amounts of ΔV being placed in the lower and generally less efficient, stage, making the whole system more costly, but also much


more representative of reality. Upper stages do not experience this ΔV inflation as they are not constrained to providing a thrust to weight ratio greater than one. Instead of high thrust characteristics, upper stage engines favor high efficiency to reduce overall mass and expenditure. G. Results and Conclusion Through this analysis, a final space system was produced:

Stage 1 Stage 2 OMS

Model Atlas V Core Stage Atlas V Centaur Fregat - M

Mass 33800 kg 5940 kg 99 kg

Engine Scaled RD-180 Engine (2 Chambers) Scaled RL-10C-1 Scaled S5.92

Engine Weight 608 kg 43 kg 1 kg

Thrust 460 kN 26 kN 238 N

Isp 338s 450s 326s

Fuel Rocket Propellant - 1 (Kerosene) Liquid Hydrogen Unsymmetrical Dimethylhydrazine

Oxidizer Liquid Oxygen Liquid Oxygen Nitrogen Tetroxide

Propellant Mass 31500 kg 5360 kg 85 kg

Fuel Mass* 18200 kg 8120 kg 58 kg

Oxidizer Mass* 13300 kg 1480 kg 26 kg

Inert Mass 2330 kg 1100 kg 14 lg

Length 15.6 m 8.1 m 0.4 m

Diameter 1.9 m 1.9 m 0.8 m

Cost $2,960,000 $684,000 $15,700

Main Stage ΔV Fraction 0.505 0.495

Interstage Fairing Total

Model Atlas V Vega N/A

Mass 300 kg 500 kg 40,800 kg

Cost $150,000 $250,000 $4,060,000

Length N/A 5.6 m 29.3 m

Diameter 1.9 m 1.9 m 1.9 m

Reliability 100% 100% 100%**

* Mixture ratios were obtained by engine and used to calculate the required fuel and oxidizer masses ** Each stage has a 100% success rate in the field When compared against similar system configurations, this result scored the highest in every metric.


It should also be noted that this case study represents the worst case scenario for the space system. Use of the USDS targeting locations at lower latitudes will result in lower costs required as the space system would require less propellant and therefore less expense (seen below).

In the future, the use of scaled engines could be replaced with engines with the required operational characteristics so as to reduce costs for engine development. Additionally, the use of an interstage could be eliminated entirely by including the structure on the first stage eg. Rockot(21), Vega(17). This would reduce the modularity of the launch vehicle stages, but ultimately reduce costs for the project. Similarly, the use of a fairing would not be necessary if the payload system were shaped appropriately and utilized appropriate material to withstand the drag forces acting on the launch vehicle nose. This would further reduce vehicle mass, lowering cost and allowing more volume within the payload shell to be dedicated to cargo. If selected, Trinity Aerospace will begin implementing these changes to further improve the USDS performance. Much of the space system’s efficiency arises from the use of high Isp, cryogenic propellant in the upper stage. This significantly reduces mass from other, larger systems, bringing down expenses in the process. However, this adds significant complexity to both the launch vehicle and the aircraft design. Provided to short life cycle of the rocket, it can utilize lower complexity, passive cooling systems if the propellant if sufficiently cold at launch. As


such, the aircraft would need to include an active cooling tank to store liquid hydrogen and oxygen at launch temperature. Thus, the most expensive and complex components could be reused with the aircraft rather than with the single use launch vehicle, keeping costs affordable.

IV. Results and Conclusions

Thus, the project comes to an initial buy-in price of $680,480,000, making each successive launch have a cost of $5,480,000. Based on the promise of reliability, the Trinity Aerospace USDS serves as a viable, efficient system for delivering cargo into inaccessible regions of the world within the 30 hours of takeoff. If given the bid for this project, we at Trinity Aerospace will immediately begin looking for new ways to further develop the USDS, making it more affordable for successive launches and introduce further technological improvements to increase efficiency.


V. Appendices

A. Air Force Base Runways

Air Force Base Location Longest Runway

Length at AFB (m) Elevation (m)

Maxwell Air Force Base Montgomery, Alabama, USA 2442 52

Luke Air Force Base Glendale, Arizona, USA 3052 330.7

Davis Monthan Air Force Base Tucson, Arizona, USA 4158 824

Little Rock Air Force Base Jacksonville, Arkansas, USA 3658 94.5

Edwards Air Force Base Edwards, California, USA 4579 704

Travis Air Force Base Fairfield, California, USA 3353 19.3

Vandenberg Air Force Base Lompoc, California, USA 4572 112.2

Beale Air Force Base Marysville, California, USA 3658 34.4

Buckley Air Force Base Aurora, Colorado, USA 3355 1726

Dover Air Force Base Dover, Delaware, USA 3933 9

MacDill Air Force Base Auxiliary Field Avon Park, Florida, USA 2438 20.7

Patrick Air Force Base Cocoa Beach, Florida, USA 2744 2.4

Tyndall Air Force Base Panama City, Florida, USA 3050 5.3

MacDill Air Force Base Tampa, Florida, USA 3481 4.3

Eglin Air Force Base Valparaiso, Florida, USA 3654 25.6

Moody Air Force Base Valdosta, Georgia, USA 2835 71

Robins Air Force Base Warner Robins, Georgia, USA 3658 90

Mountain Home Air Force Base Mountain Home, Idaho, USA 4118 913.3

Scott Air Force Base/MidAmerica Airport Belleville, Illinois, USA 3048 139.9

Grissom Air Reserve Base Peru, Indiana, USA 3810 247.1

Mc Connell Air Force Base Wichita, Kansas, USA 3660 418

Barksdale Air Force Base Bossier City, Louisiana, USA 3584 50.4

Keesler Air Force Base Biloxi, Mississippi, USA 2326 10

Columbus Air Force Base Columbus, Mississippi, USA 3659 66

Whiteman Air Force Base Knob Noster, Missouri, USA 3780 265.5

Offutt Air Force Base Omaha, Nebraska, USA 3567 319.6

Creech Air Force Base Indian Springs, Nevada, USA 2744 955.3

Nellis Air Force Base Las Vegas, Nevada, USA 3085 570

Holloman Air Force Base Alamogordo, New Mexico, USA 3935 1247.5

Cannon Air Force Base Clovis, New Mexico, USA 3049 1309.1


Seymour Johnson Air Force Base

Goldsboro, North Carolina, USA 3584 33

Grand Forks Air Force Base Grand Forks, North Dakota, USA 3765 278

Minot Air Force Base Minot, North Dakota, USA 4023 508

Wright-Patterson Air Force Base Dayton, Ohio, USA 3840 250.9

Altus Air Force Base Altus, Oklahoma, USA 4097 421

Vance Air Force Base Enid, Oklahoma, USA 2809 398.4

Tinker Air Force Base Oklahoma City, Oklahoma, USA 3383 393.4

Charleston Air Force Base Charleston, South Carolina, USA 2744 14

Shaw Air Force Base Sumter, South Carolina, USA 3052 73.5

Ellsworth Air Force Base Rapid City, South Dakota, USA 4114 999

Dyess Air Force Base Abilene, Texas, USA 4115 545.7

Laughlin Air Force Base Del Rio, Texas, USA 2698 329

Randolph Air Force Base Universal City, Texas, USA 2546 232

Sheppard Air Force Base Wichita Falls, Texas, USA 3049 310.6

Hill Air Force Base Ogden, Utah, USA 4115 1459.7

Langley Air Force Base Hampton, Virginia, USA 3049 2.4

Fairchild Air Force Base Spokane, Washington, USA 4236 750 B. Analytic Hierarchy Process CONSISTENCY MATRIX FOR COST Least Massive Least Cost Most Reliable

Least Massive 1 0.1428571429 0.2

Least Cost 7 1 3

Most Reliable 5 0.33333 1

Sum 13 1.476187143 4.2 NORMALIZED

Least Massive Least Cost Most Reliable AVERAGE

Least Massive 0.07692307692 0.0967744121 0.04761904762 0.07377217888

Least Cost 0.5384615385 0.6774208845 0.7142857143 0.6433893791

Most Reliable 0.3846153846 0.2258047034 0.2380952381 0.282838442 CONSISTENCY MATRIX FOR MASS Least Massive Least Cost Most Reliable

Least Massive 1 7 5


Least Cost 0.1428571429 1 1

Most Reliable 0.2 1 1

Sum 1.342857143 9 7 NORMALIZED


Least Massive 0.744680851 0.7777777778 0.7142857143 0.7455814477

Least Cost 0.1063829788 0.1111111111 0.1428571429 0.1201170776

Most Reliable 0.1489361702 0.1111111111 0.1428571429 0.1343014747 CONSISTENCY MATRIX FOR RELIABILITY Least Massive Least Cost Most Reliable

Least Massive 1 0.2 0.1111111

Least Cost 5 1 0.1111111

Most Reliable 9 9 1

Sum 15 10.2 1.2222222 NORMALIZED


Least Massive 0.06666666667 0.01960784314 0.09090908347 0.05906119776

Least Cost 0.3333333333 0.09803921569 0.09090908347 0.1740938775

Most Reliable 0.6 0.8823529412 0.8181818331 0.7668449247 OPTIMIZED SCORING MATRIX Criteria Weights Mass Cost Reliability

Cost 0.25 0.07377217888 0.6433893791 0.282838442

Mass 0.6 0.7455814477 0.1201170776 0.1343014747

Reliability 0.15 0.05906119776 0.1740938775 0.7668449247 Weighted Ratings 0.474651093 0.259031673 0.266317234


C. Rocket Design Script """

This code determines the optimal configuration for a two-stage launch vehicle and deorbit

stage.

MAIN SCRIPT:

Noah Marquand

DELTAV:

Kyle Duckering

"""

from scipy import *

from numpy import *

def deltaV(i = 70, lamda = 35):

# Constants

pi = 4 * arctan(1.0)

w_earth = 7.2921159 * 10 **-5 #rad/s

r_earth = 6378.0 #km

mu_earth = 3.986*10**5

# Fixed Parameters

h0 = 49000.0 #ft

h0 = h0 / 3.2808 / 1000.0 #km

#h0 = 14935.2 m = 14.9 km

hf = 400.0 #km

v_plane = 0 #km/s

i = i * pi / 180.0

lamda = lamda * pi / 180.0

betta = arcsin((cos(i)/cos(lamda)))

#Calculate ideal dV from earth (assuming launched in direction of inclination?)

v_0 = w_earth * r_earth * cos(lamda) * sin(betta) + v_plane / 1000.0 #km/s

v_final = (mu_earth / (hf + r_earth)) ** 0.5 #km/s

deltaV_ideal = v_final - v_0 #km/s

# Launch losses

deltaV_drag = (150 - 0.0075 * h0 / 1000.0) / 1000.0 #km/s

deltaV_gravity = (150 - 0.075 * h0 / 1000.0) / 1000.0 #km/s

deltaV_steering = .200 #km/s (Given)

deltaV_effective = deltaV_ideal + deltaV_drag + deltaV_gravity + deltaV_steering

return deltaV_effective

def stage(m_pay = 10000, dV = 3000, Isp = 325, Finert = 0.08):

g0 = 9.81

m0 = m_pay * (e**(dV/(g0*Isp))) * (1-Finert)/(1 - Finert * (e**(dV / (g0 * Isp))))

return m0

def score(mass, cost, rel):

points = (mass * -0.475) + (cost * -0.26) + (rel * 0.27) # Optimal System

#points = -1 * mass # Optimal mass

#points = -1 * cost # Optimal cost

#points = (-1 * mass * 0.5) + (-1 * cost * 0.5) # Cost and Mass Equal Weighting

#points = rel # Optimal reliability

return points

def nondim(val, max_val, min_val):

free = 1


if (max_val - min_val) != 0:

free = (val - min_val) / (max_val - min_val)

return free

### Main script

## Instance data

# Factor of Safety

FOS_OMS = 1.5

FOS_main = 1.1

FOS_TtoW = 1.1

# Mission Data

dV_OMS = 0.09 * 1000 * FOS_OMS # km/s

dV_mission = deltaV() * 1000 * FOS_main # km/s

g0 = 9.81 # m/s^2

# Cost

prop_cost = 20 # per kg

sInert_cost = 500 # per kg

lInert_cost = 1000 # per kg

# Mass

m_pay = 2000 # kg

istage_mass = 300 # kg

fairing_mass = 500 # kg

# Lower Stage

'''

Atlas V Core

Atlas V SRM

Delta IV Core

Delta IV SRM

Delta II Core

Delta II SRM

Falcon 9 Core

Soyuz Booster

Soyuz Core

Vega Stage 1

Vega Stage 2

Ariane V Core

Rokot Core

'''

prop1 = ['l','s','l','s','l','s','l','l','l','s','s','l','l']

Isp1 = [338,275,414,274,301,278,311,313,315,280,288,431,310]

Finert1 =

[0.06877,0.1229,0.1156,0.1139,0.0558,0.08564,0.05142,0.08777,0.06583,0.08885,0.09452,0.07959,0

.07388]

rel1 = [1,1,0.97,0.97,0.987,0.987,0.929,0.972,0.972,1,1,0.955,0.933]

Ttomass1 = [13.6,36.1,15.6,15.6,10.4,50,15.4,23.5,10,31.3,42.6,7.3,26.8]

# Upper Stage

'''

Atlas V Upper Stage

Delta IV Upper Stage

Delta II Upper Stage

Falcon 9 Upper Stage

Soyuz Stage 3

Vega Stage 3

Ariane V Stage 3

Rokot Stage 2

'''

prop2 = ['l','l','l','l','l','s','l','l']

Isp2 = [450,466,319,345,326,296,391,320]

Finert2 = [0.09721,0.1179,0.1367,0.03587,0.0964,0.1194,0.2335,0.581]

rel2 = [1,0.97,0.987,0.929,0.972,1,0.955,0.933]


# OMS

'''

Rokot Stage 3

Soyuz-U Volga

Soyuz-U Fregat

Soyuz-U Fregat-M

Strela APB Upper Stage

Vega Stage 4

Zenit-3S Fregat SB

Minotaur Orion 38

Minotaur V Star 48BV

Proton Briz-M

'''

propOMS = ['l','l','l','l','l','l','l','l','s','l']

IspOMS = [326,307,331,331,200,314.6,331,287,288,329]

FinertOMS = [0.2095,0.4972,0.1505,0.1431,0.6591,0.5439,0.112,0.117,0.07138,0.1069]

relOMS = [0.933,0.973,1,1,1,1,0.889,0.929,1,0.929]

# Nondimensionalizing data

max_mass_tot = -1

min_mass_tot = -1

max_cost_tot = -1

min_cost_tot = -1

max_rel_tot = -1

min_rel_tot = -1

# Optimum values

op_score_tot = -1

op_mass_tot = -1

op_cost_tot = -1

op_rel_tot = -1

op_prop_dist = -1

op_inert1 = -1

op_inert2 = -1

op_inertOMS = -1

op_prop1 = -1

op_prop2 = -1

op_propOMS = -1

op_config = [0,0,0]

# Other

x_list = linspace(0, 1, 10**3)

indexOMS = 0

index2 = 0

index1 = 0

''' Begin Analysis '''

### Determine data ranges

## Subassemblies

fairing_cost = fairing_mass * sInert_cost # approximation

istage_cost = istage_mass * sInert_cost # approximation

# Loop through dV distributions

for x in x_list:

dV1 = x * dV_mission

dV2 = (1 - x) * dV_mission

## Search sample data

while index1 < len(prop1):



while indexOMS < len(propOMS):

massOMS = stage(m_pay, dV_OMS, IspOMS[indexOMS], FinertOMS[indexOMS]) - m_pay

if massOMS > 0:

if propOMS[indexOMS] == 'l':

costOMS = (massOMS * FinertOMS[indexOMS]) * lInert_cost + (massOMS *

(1-FinertOMS[indexOMS])) * prop_cost

else:

costOMS = (massOMS * FinertOMS[indexOMS]) * sInert_cost + (massOMS *


# Upper Stage

m_pay2 = m_pay + massOMS + fairing_mass

mass2 = stage(m_pay2, dV2, Isp2[index2], Finert2[index2]) - m_pay2

if (mass2 > 0):

if prop2[index2] == 'l':

cost2 = (mass2 * Finert2[index2]) * lInert_cost + (mass2 *

(1-Finert2[index2])) * prop_cost

else:

cost2 = (mass2 * Finert2[index2]) * sInert_cost + (mass2 *


# Lower Stage

m_pay1 = m_pay2 + mass2 + istage_mass


if (mass1 > 0):

# Check thrust to weight

thrust1 = Ttomass1[index1] * mass1

mass_tot = mass1 + istage_mass + mass2 + massOMS + fairing_mass

if (thrust1 > (FOS_TtoW * (mass_tot + m_pay) * 9.8)):




else:



# Sum results

cost_tot = cost1 + istage_cost + cost2 + costOMS +

fairing_cost

rel_tot = rel1[index1] * rel2[index2] * relOMS[indexOMS]

# Store Extrema

if mass_tot > max_mass_tot:

max_mass_tot = mass_tot

if (mass_tot < min_mass_tot) or (min_mass_tot == -1):

min_mass_tot = mass_tot

if cost_tot > max_cost_tot:

max_cost_tot = cost_tot

if (cost_tot < min_cost_tot) or (min_cost_tot == -1):

min_cost_tot = cost_tot

if rel_tot > max_rel_tot:

max_rel_tot = rel_tot

if (rel_tot < min_rel_tot) or (min_rel_tot == -1):

min_rel_tot = rel_tot

# Iterate OMS

indexOMS += 1

# Iterate stage 2

indexOMS = 0

index2 += 1

# Iterate stage 1

indexOMS = 0

index2 = 0

index1 += 1

# Reset

indexOMS = 0


index2 = 0

index1 = 0

### Optimize System

## Mainstages

# Loop through dV distributions

for x in x_list:


dV2 = (1-x) * dV_mission

# Search sample data



while indexOMS < len(propOMS):

massOMS = stage(m_pay, dV_OMS, IspOMS[indexOMS], FinertOMS[indexOMS]) - m_pay

if massOMS > 0:

if propOMS[indexOMS] == 'l':

costOMS = (massOMS * FinertOMS[indexOMS]) * lInert_cost + (massOMS *


else:

costOMS = (massOMS * FinertOMS[indexOMS]) * sInert_cost + (massOMS *


prop_massOMS = massOMS / (1 + (FinertOMS[indexOMS] / (1 -

FinertOMS[indexOMS])))

# Upper Stage






else:



prop_mass2 = mass2 / (1 + (Finert2[index2] / (1 - Finert2[index2])))

# Lower Stage

if (mass2 > 0):



if (mass1 > 0):




else:



prop_mass1 = mass1 / (1 + (Finert1[index1] / (1 -

Finert1[index1])))

# Check thrust to weight

thrust1 = Ttomass1[index1] * mass1

mass_tot = mass1 + istage_mass + mass2 + massOMS + fairing_mass

if (thrust1 > (FOS_TtoW * (mass_tot + m_pay) * 9.8)):

# Score system

cost_tot = cost1 + istage_cost + cost2 + costOMS +

fairing_cost

rel_tot = rel1[index1] * rel2[index2] * relOMS[indexOMS]

nonMass = nondim(mass_tot, max_mass_tot, min_mass_tot)

nonCost = nondim(cost_tot, max_cost_tot, min_cost_tot)

nonRel = nondim(rel_tot, max_rel_tot, min_rel_tot)

score_tot = score(nonMass, nonCost, nonRel)


# Compare with existing system

if (score_tot > op_score_tot) or (op_score_tot == -1):

op_score_tot = score_tot

op_mass_tot = mass_tot

op_cost_tot = cost_tot

op_prop_dist = x

op_inert1 = mass1 - prop_mass1

op_inert2 = mass2 - prop_mass2

op_inertOMS = massOMS - prop_massOMS

op_prop1 = prop_mass1

op_prop2 = prop_mass2

op_propOMS = prop_massOMS

op_rel_tot = rel_tot

op_config[0] = index1

op_config[1] = index2

op_config[2] = indexOMS

# Iterate OMS

indexOMS += 1

# Iterate stage 2

indexOMS = 0

index2 += 1

# Iterate stage 1

indexOMS = 0

index2 = 0

index1 += 1

# Reset

indexOMS = 0

index2 = 0

index1 = 0

''' Output results '''

print("Optimal Score: %f" % op_score_tot)

print(" Mass: %d kg\n Cost: $ %d\n Reliability: %.3f" % (op_mass_tot, op_cost_tot,

op_rel_tot))

print(" Configuration: %d, %d, %d\n Delta-V Distribution: %f" % (op_config[0],

op_config[1], op_config[2], op_prop_dist))

print(" Stage 1 Inert Mass: %d kg\n Stage 1 Propellant Mass: %d kg\n" % (op_inert1,

op_prop1))

print(" Stage 2 Inert Mass: %d kg\n Stage 2 Propellant Mass: %d kg\n" % (op_inert2,

op_prop2))

print("OMS Inert Mass: %d kg\n OMS Propellant Mass: %d kg" % (op_inertOMS, op_propOMS))

D. Cost due to Target Latitude Code """

This code confirms the increase in cost with target latitude

MAIN SCRIPT:

Noah Marquand

DELTAV:

Kyle Duckerling

"""

from scipy import *

from numpy import *

import matplotlib.pyplot as plt

def deltaV(i = 70, lamda = 35):

# Constants

pi = 4 * arctan(1.0)

w_earth = 7.2921159 * 10 **-5 #rad/s

r_earth = 6378.0 #km


mu_earth = 3.986*10**5

# Fixed Parameters

h0 = 49000.0 #ft

h0 = h0 / 3.2808 / 1000.0 #km

#h0 = 14935.2 m = 14.9 km

hf = 400.0 #km

v_plane = 0 #km/s

i = i * pi / 180.0

lamda = lamda * pi / 180.0

betta = arcsin((cos(i)/cos(lamda)))

#Calculate ideal dV from earth (assuming launched in direction of inclination)

v_0 = w_earth * r_earth * cos(lamda) * sin(betta) + v_plane / 1000.0 #km/s

v_final = (mu_earth / (hf + r_earth)) ** 0.5 #km/s

deltaV_ideal = v_final - v_0 #km/s

# Launch losses

deltaV_drag = (150 - 0.0075 * h0 / 1000.0) / 1000.0 #km/s

deltaV_gravity = (150 - 0.075 * h0 / 1000.0) / 1000.0 #km/s

deltaV_steering = .200 #km/s (Given)

deltaV_effective = deltaV_ideal + deltaV_drag + deltaV_gravity + deltaV_steering

return deltaV_effective

### Main script

## Instance data

# Factor of Safety

FOS_OMS = 1.5

FOS_main = 1.1

# Mission Data

dV_OMS = 0.09 * 1000 * FOS_OMS # km/s

dV_mission = deltaV() * 1000 * FOS_main # km/s

g0 = 9.81 # m/s^2

# Cost

prop_cost = 20 # per kg

sInert_cost = 500 # per kg

lInert_cost = 1000 # per kg

# Mass

m_pay = 2000 # kg

istage_mass = 300 # kg

fairing_mass = 500 # kg

# Optimal rocket configuration

Isp1 = 338

inert1 = 2326

Isp2 = 450

inert2 = 1103

IspOMS = 331

inertOMS = 14

# Other

i_list = linspace(35, 70, 10**3)

cost_list = []

x = 0.505


''' Begin Analysis '''

## Subassemblies

fairing_cost = fairing_mass * sInert_cost # approximation

istage_cost = istage_mass * sInert_cost # approximation

## Loop through target latitudes

for i in i_list:

dV_mission = deltaV(i, 35) * 1000 * FOS_main # km/s


dV2 = (1 - x) * dV_mission

FinertOMS = ((e ** (dV_OMS / (g0 * IspOMS))) * ((m_pay / inertOMS) + 1) - (m_pay /

inertOMS)) ** -1

costOMS = inertOMS * lInert_cost + ((inertOMS / FinertOMS) - inertOMS) * prop_cost

massOMS = inertOMS + ((inertOMS / FinertOMS) - inertOMS)

# Upper Stage


Finert2 = ((e ** (dV2 / (g0 * Isp2))) * ((m_pay2 / inert2) + 1) - (m_pay / inert2)) ** -1

cost2 = inert2 * lInert_cost + ((inert2 / Finert2) - inert2) * prop_cost

mass2 = inert2 + ((inert2 / Finert2) - inert2)

# Lower Stage


Finert1 = ((e ** (dV1 / (g0 * Isp1))) * ((m_pay / inert1) + 1) - (m_pay1 / inert1)) ** -1

cost1 = inert1 * lInert_cost + ((inert1 / Finert1) - inert1) * prop_cost

mass1 = inert1 + ((inert1 / Finert1) - inert1)

# Sum results

cost_tot = cost1 + istage_cost + cost2 + costOMS + fairing_cost

# Store data

cost_list.append(cost_tot)

### Output results

plt.plot(i_list, cost_list)

plt.xlabel("Target Latitude")

plt.ylabel("Cost ($)")

plt.title("Cost vs. Target Latitude")


Appendix E. CAD Models

E.1 Carrier aircraft

E.2 Launch vehicle exploded view


E.3 Launch vehicle assembled

E.4 Carrier aircraft with launch vehicle stowed


Appendix F. Standard Atmosphere

#PROGRAMMER: NICHOLAS TURO-SHIELDS

def standard_atmosphere(h = 0, units = 'SI'):

T_set = [288.16, 216.66, 216.66, 282.66, 282.66, 165.66, 165.66, 225.66] # list (array) of

temperature points that define endpoints of each layer (starting at the ground), K

h_set = [0, 11000, 25000, 47000, 53000, 79000, 90000, 105000] # list (array) of

geopotential altitude points that define endpoints of each layer (starting at the ground), m

a_set = [-6.5*10**-3, 0, 3*10**-3, 0, -4.5*10**-3, 0, 4*10**-3] # list (array) of gradient

layer slopes (starting at the ground), K/m

p = [101325, 0, 0, 0, 0, 0, 0, 0] #Atmospheric pressure (Pa)

rho = [1.225, 0, 0, 0, 0, 0, 0, 0] #Atmospheric density (kg/m^3)

R = 287.04 # Gas Constant [J/(kg*K)]

g0 = 9.80665 # m/s^2

if units == 'eng':

h *= 0.3048

layers = 1

while h > h_set[layers]: #calculates the layer of the atmosphere the specified altitude

falls within

layers += 1

i = 0

while i != layers: #iterates through gradient and isothermal layers until the top layer is

reached

if i % 2 == 0: #gradient layer

if (layers == 1) | (i == layers - 1):

T = T_set[i] + (a_set[i] * (h - h_set[i]))

else:

T = T_set[i] + (a_set[i] * (h_set[i + 1] - h_set[i]))

p[i + 1] = p[i] * (T / T_set[i]) ** (-g0 / (a_set[i] * R))

rho[i + 1] = rho[i] * (T / T_set[i]) ** -((g0 / (a_set[i] * R)) + 1)

if i % 2 != 0: #isothermal layer

T = T_set[i]

if i == layers - 1:

p[i + 1] = p[i] * exp(-(g0 / (R*T)) * (h - h_set[i]))

else:

p[i + 1] = p[i] * exp(-(g0 / (R*T)) * (h_set[i + 1] - h_set[i]))

rho[i + 1] = rho[i] * p[i + 1] / p[i]

i += 1

if units == 'eng':

T *= 1.8

p[i] *= 0.020885434

rho[i] *= 0.00194032

return T, rho[i], p[i]

Appendix G. Runway Calculations


% 'username':'nturoshi','assignment':'Takeoff and Landing Distance Intermediate

Mastery','course':'fall-2016-aae-251','variables':

% 'W': 'weight of aircraft during takeoff, newtons'

% 'T': 'thrust of aircraft during takeoff, newtons'

% 'rho': ` 'freestream density, kg/m^3'

% 'S': 'wing area, m^2'

% 'CL_max': 'maximum lift coefficient, nondimensional'

% 'CL': 'lift coefficient during takeoff, nondimensional'

% 'CD': 'drag coefficient during takeoff, nondimensional'

% 'mu_r': 'rolling friction coefficient, nondimensional'

% 's_lo': 'liftoff distance, m'

clear variables;

runwaydata = csvread('Runways.csv');

runway_length = runwaydata(:,1);

runway_elev = runwaydata(:,2);

clear runwaydata

rE = 6378e3; % radius of Earth, m

mu_E = 3.986e14; % Gravitational Parameter [m^3/s^2]

m = 350000; % mass [kg]

S = 576; % wing area [m^2]

CL_max = 1.5;

CL = 0.6;

CD = 0.04;

mu_r = 0.02;

datacount = length(runway_elev);

thrust_minimum = zeros(datacount, 1);

safety_factor = 1.2; %takeoff velocity multiplier of stall speed

for i = 1:datacount

[~, ~, rho] = standard_atmosphere(AltConvert(runway_elev(i), 'SI'), 'SI');

thrust = 0;

g = (mu_E / (rE + runway_elev(i))^2);

W = m * g;

V_lo(i) = safety_factor * sqrt((2*W) / (rho * S * CL_max));

V_infty = 0.7 * V_lo(i);

q_infty = 0.5 * rho * V_infty^2;

L = CL * q_infty * S;

D = CD * q_infty * S;

s_lo = (safety_factor^2 * W^2) / (g * rho * S * CL_max * (thrust - (D + mu_r*(W - L))));

while (s_lo > runway_length(i)) || (s_lo < 0)

thrust = thrust + 1;

s_lo = (safety_factor^2 * W^2) / (g * rho * S * CL_max * (thrust - (D + mu_r*(W -

L))));

end

thrust_minimum(i) = thrust;

end

location = find(max(thrust_minimum))

min_req_thrust = max(thrust_minimum);

fprintf('Total Thrust Required: %d N', min_req_thrust);

nEngines = 4;

fprintf('\nThrust Required by each of the %d engines: %.0f N\n', nEngines, min_req_thrust /

nEngines);

Appendix H. Range and Endurance Code


#PROGRAMMER: NICHOLAS TURO-SHIELDS

import numpy as np


from scipy import *

import standard_atmosphere as atmo

import math

TSFC = 0.324 #[lb/(lb*hr)]

c_t = TSFC / 3600 #[lb/(lb*sec)]

nEngines = 4

b = 67.88 #span [m]

S = 576 # [m]

AR = b**2/ S

e = 0.9

CD0 = 0.04

g = 9.81

max_req_range = 3500 * 1.5 #maximum required range [km] (safety factor 1.5)

m_pay = 42000 #[kg]

empty_mass = (m_pay + 20857) / 0.9314

empty_wt = empty_mass * g #empty plane weight (with no fuel) (Newtons)

fuel_mass_max = 193620 / 1000 * 840

fuel_mass_max = 145000

fuel_mass = fuel_mass_max

T, rho, p = atmo.standard_atmosphere(15240, 'SI') #density at altitude

speed_sound = sqrt(1.4 * 287.04 * T) #speed of sound (m/s)

V_lower = 1 # lower bound of velocity, (m/s)

V_upper = 0.8 * speed_sound # upper bound of velocity, (m/s)

flt_range = []

fuel_wt = fuel_mass * g #total fuel (Newtons)

full_mass = empty_mass + m_pay + fuel_mass #gross mass of the plane filled with fuel (empty

weight + fuel_wt + pay_wt) (Newtons)

full_wt = full_mass * g

payload_wt = m_pay * g

max_mass = 2.1438 * empty_mass + 15078

V_set = np.linspace(V_lower, V_upper, 5001)

#pretest run

W_0 = full_wt

W_1 = full_wt - fuel_wt

for v in V_set:

q_inf = 0.5 * rho * (v**2)

CL = W_0 / (q_inf * S)

CD = CD0 + (CL**2 / (pi * e * AR))

flt_range.append(2 * sqrt(2/(rho * S)) / (nEngines * c_t) * sqrt(CL) / CD * (sqrt(W_0) -

sqrt(W_1))/1000)

#end pretest

while (max(flt_range) >= max_req_range):

flt_range = []


fuel_wt = fuel_mass * g #total fuel (Newtons)

full_mass = empty_mass + m_pay + fuel_mass #gross mass of the plane filled with fuel

(empty weight + fuel_wt + pay_wt) (Newtons)

full_wt = full_mass * g

W_0 = full_wt

W_1 = full_wt - fuel_wt

for v in V_set:

q_inf = 0.5 * rho * (v**2)

CL = W_0 / (q_inf * S)

CD = CD0 + (CL**2 / (pi * e * AR))

flt_range.append(2 * sqrt(2/(rho * S)) / (nEngines * c_t) * sqrt(CL) / CD * (sqrt(W_0)

- sqrt(W_1))/1000) #calculating ranges in km

fuel_mass -= 100

full_mass = empty_mass + m_pay + fuel_mass #total gross mass of the plane (Newtons)

print('Total Mass', full_mass, 'kg')

print('Min. Fuel to Complete Worst-Case scenario', fuel_mass, 'kg')

#calculating max values for Range

i_optimal_R = [i for i, item in enumerate(flt_range) if item == max(flt_range)]

i_optimal_R = i_optimal_R[0]

v_optimal_R = float(V_set[[i for i, item in enumerate(flt_range) if item == max(flt_range)]])

plt.plot(V_set, flt_range, 'g', label = 'Range')

plt.plot(v_optimal_R, flt_range[i_optimal_R], 'go', label = "Maximum Range")

plt.plot([v_optimal_R, v_optimal_R], [0, flt_range[i_optimal_R]], 'k')

plt.title('Range')

plt.xlabel('$V_\infty$ (m/s)')

plt.ylabel('Range (km)')

plt.xlim([0, .9*speed_sound])

plt.legend(loc = 'lower right')

print('m_inert cost', 10000 * empty_mass)

print('Speed that Maximizes Range at Sea-Level:', v_optimal_R, 'm/s')

print('Maximum Range:', flt_range[i_optimal_R], 'km')


Appendix I. Thrust Required to Reach Service Ceiling #PROGRAMMER: NICHOLAS TURO-SHIELDS

import numpy as np


from scipy import *

from scipy import integrate

import standard_atmosphere as atmo

def getGeometric(h):

return h * rE / (rE - h)

def AltConvert(hG):

#Converts a geometric altitude (above sea level) to a geopotential altitude

h = rE/(rE+hG)*hG

return h

S = 576 # wing area, (m^2)

b = 67.88 # wingspan, (m)

AR = (b**2) / S

e = 0.9 # Oswald efficiency factor (dimensionnless)

m = 250000 # mass, (kg)

W = m * 9.81

C_D_0 = .04 # zero-drag coefficient

rE = 6.371e6 # radius of earth [m]

TA_0 = 1380000 # Newtons

V_lower = 10 # lower bound of velocity, (m/s)

V_upper = 700 # upper bound of velocity, (m/s)

V_set = np.linspace(V_lower, V_upper, 1000)

h_lower = 0

h_upper = 20000

height_pts = 601

h_set = np.linspace(h_lower, h_upper, height_pts)

T, rho0, p = atmo.standard_atmosphere(0, 'SI')

del T, p

loop_count = 0

max_RC = []

RC_inv = []

geometric_altitudes = []

for h in h_set:

excess_power = []

PR_0 = []

PA = []

TR_0 = []

TA = []

geometric_altitudes.append(getGeometric(h))

T, rho, p = atmo.standard_atmosphere(h, 'SI')

del T, p

for v in V_set:

v *= sqrt(rho0/rho) #velocity scaling for altitude

q_inf = 0.5*rho*v**2

CL = W / (q_inf*S)

CD = C_D_0 + (CL**2/(pi * e * AR))

pwr_req = W * v / (CL/CD)

TR_0.append(W / (CL/CD)) #PR scaling for altitude

PR_0.append(pwr_req * sqrt(rho0/rho)) #PR scaling for altitude

pwr_avail = TA_0 * v * rho/rho0 #PA scaling for altitude


PA.append(pwr_avail)

excess_power.append(pwr_avail - pwr_req)

max_RC.append(60*max(excess_power)/W) #appends in m/min

if max_RC[loop_count] > 30.48: #m/min

serv_ceil = h

if loop_count == 0:

for i, item in enumerate(excess_power):

if item == max(excess_power):

max_RC0_index = i

plt.figure(1)

plt.title('Thrust Curves at Sea-Level')

plt.xlabel('$V_\infty$ (m/s)')

plt.ylabel('Thrust (Newtons)')

plt.plot([V_lower, V_upper], [TA_0, TA_0], label = 'Thrust Available')

plt.plot(V_set, TR_0, label = 'Thrust Required')

plt.plot([V_set[max_RC0_index], V_set[max_RC0_index]], [PR_0[max_RC0_index],

PA[max_RC0_index]], 'r-', label = 'Max Excess Power')

plt.ylim([0, 2000000])

plt.legend()

loop_count += 1

geom_serv_ceil = getGeometric(serv_ceil)

print('Service Ceiling (geopotential):', serv_ceil, 'm')

print('(c) Service Ceiling:', geom_serv_ceil, 'm')


VI. Acknowledgments

The employees of Trinity Aerospace would like to thank Professor Grant and DARPA for the opportunity and resources to develop the Undenied Supply Delivery System for the United States and for humanity.

VII. References

(1)“AirNav: 1MS8 - Columbus Air Force Base Auxiliary Field,” AirNav: 1MS8 - Columbus Air Force Base Auxiliary Field Available: http://www.airnav.com/airport/1MS8. (2)“AirNav: KHIF - Hill Air Force Base,” AirNav: KHIF - Hill Air Force Base Available: http://www.airnav.com/airport/KHIF. (3)“Military Turbofan Engine Data,” Military Turbofan Engine Data Available: http://www.aircraftenginedesign.com/TableB2.html. (4) “TURBOKART,” TURBOKART Available: http://www.turbokart.com/about_ge90.htm. (5)“The CF6 Engine,” The CF6 Engine | Engines | Commercial | GE Aviation Available: http://www.geaviation.com/commercial/engines/cf6/. (6)“Aerospaceweb.org | Aircraft Museum - Military Transports,” Aerospaceweb.org | Aircraft Museum - Military Transports Available: http://www.aerospaceweb.org/aircraft/transport-m/. (7) ==Chernyshova, E., “What a Real Russian Winter Looks Like,” NBCNews.com Available: http://www.nbcnews.com/news/photo/what-real-russian-winter-looks-n21251. (8)“Delta IV,” ulalaunch.com Available: http://www.ulalaunch.com/uploads/docs/Launch_Vehicles/AV_DIV_product_card.pdf. (9)“Atlas V 551 – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/atlas-v-551/. (10)“Atlas 5 Data Sheet,” Atlas 5 Data Sheet Available: http://www.spacelaunchreport.com/atlas5.html. (11)“Delta IV Heavy – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/delta-iv-heavy/. (12)“Delta IV Data Sheet,” Delta IV Data Sheet Available: http://www.spacelaunchreport.com/delta4.html. (13)“Delta II,” ulalaunch.com Available: http://www.ulalaunch.com/uploads/docs/Launch_Vehicles/Delta_II_Product_Card.pdf. (14)“Delta II 7320 – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/delta-ii-7320/. (15)“Delta II Data Sheet,” Delta II Data Sheet Available: http://www.spacelaunchreport.com/delta2.html. (16)“Falcon 9 v1.1 & F9R – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/falcon-9-v1-1-f9r/. (17)“Vega – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/vega/. (18)“Ariane 5 Data Sheet,” Ariane 5 Data Sheet Available: http://www.spacelaunchreport.com/ariane5.html.


http://www.aircraftenginedesign.com/TableB2.html

http://www.turbokart.com/about_ge90.htm

http://www.geaviation.com/commercial/engines/cf6/

(19)“Ariane 5 ECA – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/ariane-5-eca/. (20)“Soyuz U – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/soyuz-u/. (21)“Rockot – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/rockot/. (22)“Minotaur V – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/minotaur-v/. (23)“Space Launch Report ... Minotaur Data Sheet,” Space Launch Report ... Minotaur Data Sheet Available: http://www.spacelaunchreport.com/mintaur4.html. (24)“Minotaur I – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/minotaur-i/. (25)“Pegasus XL – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/pegasus-xl/. (26)“Soyuz 2-1v – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/soyuz-2-1v/. (27)“Proton-M/Briz-M – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/proton-m-briz-m/. (28)“Soyuz FG/Fregat – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/soyuz-fg-fregat/. (29)“Strela – Rockets,” Rockets Available: http://spaceflight101.com/spacerockets/strela/. (30)“Atlas V,” Atlas V - United Launch Alliance Available: http://www.ulalaunch.com/products_atlasv.aspx. (31)“Earth,” Google Earth Available: http://earth.google.com/.


Contributions

Kyle Duckering: delta V code, OMS data collection, CAD modeling for rocket, AHP, code debugging Noah Marquand: Rocket data collection, rocket design code, rocket scaling, cost due to target latitude code Nicholas Turo-Shields: Coded the aircraft analysis functions, completed the CAD design for the aircraft system. Gathered AFB data and helped format the report Sree Vellanki: Engine data collection, Aircraft performance,


undenied supply delivery system - career account web service

Documents