Trade Study One

Operation Burnout (Team 4)

AE 442 S1

October 10th, 2019

Member Responsibility Aidan Dreher GNC

Destiny Fawley GNC Conor Hershey Avionics Elena Kamis Flight Software

Austin Lindell Structures Damian Markiewicz Propulsion

Adrian Metcalf Flight Software Aldo Montagner Structures

Joshua Super Avionics Jie Yang Propulsion

1. MISSION OBJECTIVE/SCOPE The goal of this mission is to vertically land a model rocket stage dropped from a UAV.

The scope of this trade study was to simulate how the thrust of a rocket engine could counteract the force of gravity to reach a velocity of 0 meters per second at an altitude of 0 meters. All simulations were done in Matlab and used the thrust curve of an F22 Aerotech engine, gravity, and drag of a flat circular plate. Values of rocket mass, length and diameter were adjusted to create 16 unique cases. Rocket body inertias were also calculated for every case. All results were checked using a “sanity check” to ensure values and results were physically possible.

2. ASSUMPTIONS The goal of these assumptions is to provide the simulation with parameters allowing the team to properly calculate the effects of each.

2.1 Altitude The rocket will be dropped from 20 meters above the landing pad.

2.2 Rocket Mass The rocket mass shall be evaluated at four cases, 1.0 kg, 1.25 kg, 1.5 kg, and 1.75kg. Each of these mass values shall be concentrated in the bottom 10 cm of the rocket body.

2.3 Rocket Length The rocket length shall be evaluated at two cases, 50 cm and 60 cm.

2.4 Rocket Diameter The rocket diameter shall be evaluated at two cases, 7.6 cm and 8.0 cm.

2.5 Thrust Curve The engine chosen and appropriate thrust curve used in the simulation will be the F22 Aerotech, which can be seen below in Figure 1. This thrust curve was taken directly from the Aerotech F22 data sheet.

Figure 1: F22 Thrust Curve

2.6 External Forces There will be no other aero or body forces applied other than drag and gravity.

2.7 Ignition Time Accuracy All ignition times chosen in the simulation are only given to an accuracy of two decimal places. This is due to the assumption that the onboard computing system will be unable to handle an accuracy greater than this. Thus, to prevent inaccurate simulations the ignition times used in our MATLAB simulation are kept to two significant figures.

2.8 Moment of Inertia The rocket body axes were created with the z axis in the axial direction and x and y axes in in the transverse directions. The rocket body consists of a thin-walled cylindrical body tube.

3. ANALYSIS The table below breaks down the sixteen separate simulations completed per the assumptions listed previously.

Table 1: Case Breakdowns

Case # Altitude (m) Rocket Mass (kg)

Rocket Length (cm)

Rocket Diameter (cm)

Thrust Curve Forces

1 20 1.00 50 7.6 F22 Aerotech Drag only 2 20 1.00 50 8.0 F22 Aerotech Drag only 3 20 1.00 60 7.6 F22 Aerotech Drag only 4 20 1.00 60 8.0 F22 Aerotech Drag only 5 20 1.25 50 7.6 F22 Aerotech Drag only 6 20 1.25 50 8.0 F22 Aerotech Drag only 7 20 1.25 60 7.6 F22 Aerotech Drag only 8 20 1.25 60 8.0 F22 Aerotech Drag only 9 20 1.50 50 7.6 F22 Aerotech Drag only 10 20 1.50 50 8.0 F22 Aerotech Drag only 11 20 1.50 60 7.6 F22 Aerotech Drag only 12 20 1.50 60 8.0 F22 Aerotech Drag only 13 20 1.75 50 7.6 F22 Aerotech Drag only 14 20 1.75 50 8.0 F22 Aerotech Drag only 15 20 1.75 60 7.6 F22 Aerotech Drag only 16 20 1.75 60 8.0 F22 Aerotech Drag only

Each of these cases were completed using a Matlab simulation. This simulation takes in four easily changeable parameters including mass, altitude, length and diameter as can be seen in Figure 2: Code Inputs.

Figure 2: Code Inputs

The motor thrust curve can also be adjusted, but for this simulation was kept as the F22 Aerotech throughout. In order to find the necessary ignition time, the team manually inputted values of ignition time in seconds out to two significant figures, rerunning the simulation until the flight path was that of the desired velocity. An example can be seen below in Figure 3: Ignition Time Input.

Figure 3: Ignition Time Input

The simulation then was set to print a graph relating velocity and altitude. The goal was to achieve 0 m/s at 0 m, but the team had varying levels of success achieving this. The graphs, resulting ignition times, and inertias were then saved off and are described further in the following sections.

The moments of inertia about the principle axes of the rocket were calculated with the assumptions stated in Section 2.8 Moment of Inertia. The rocket body was assumed to be a thin-walled cylinder with a 1cm thickness with the moment of inertia given by Eqs. (1) and (2):

𝐼 = 𝐼 =1

12𝑚(3(𝑟 + 𝑟 ) + ℎ ) (1)

𝐼 =1

2𝑚(𝑟 + 𝑟 ) (2)

where 𝐼 , 𝐼 , and 𝐼 are the moment of inertias about the body X, Y, and Z axes, m is the mass of the body tube, 𝑟 is the inner radius, 𝑟 is the outer radius, and h is the height of the length of the body tube. The rocket rotates around the center of gravity (c.g.), which is located in the center of the tube along the z axis by symmetry; therefore, the body inertias are simply the value calculated with Eqs. (1) and (2).

3.1 Case One Assumptions used for this set up are as stated in Table 1.

Figure 4: Case One

For Case One, the rocket nearly lands, however, due to the large force of the motor and low mass, the rocket is propelled upwards. The final velocity once landing on the ground is -30m/s. Therefore, case one is not an optimal set up of assumptions.

3.1.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0190kgm2, 0.0190kgm2, and 0.0013kgm2, respectively.

3.1.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case One is 1.58 seconds after rocket release.

3.2 Case Two Assumptions used for this set up are as stated in Table 1.

Figure 5: Case Two

For Case Two, the rocket nearly lands, however, due to the force of the motor and low mass, the rocket is propelled upwards. The final velocity once landing on the ground is -30m/s. Therefore, case one is not an optimal set up of assumptions. While like case one, case two reaches a higher altitude overall.

3.2.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0272 kgm2, 0.0272 kgm2, and 0.0013 kgm2, respectively.

3.2.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case One is 1.58 seconds after rocket release.

3.3 Case Three Assumptions used for this set up are as stated in Table 1.

Figure 6: Case Three

For Case Three, the rocket nearly lands, however, due to the force of the motor and low mass, the rocket is propelled upwards. The final velocity once landing on the ground is -30m/s. Therefore, case one is not a feasible solution. The similarity to case one suggests that the length has little impact on performance.

3.3.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0415 kgm2, 0.0415 kgm2, and 0.0014 kgm2, respectively.

3.3.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Three is 1.58 seconds after rocket release.

3.4 Case Four Assumptions used for this set up are as stated in Table 1.

Figure 7: Case Four

For Case Four, the rocket nearly lands, however, due to the force of the motor and low mass, the rocket is propelled upwards. The final velocity once landing on the ground is -30m/s. Therefore, case one is not a feasible solution. The similarity to case two suggests that the length has little impact on performance.

3.4.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0273 kgm2, 0.0273 kgm2, and 0.0014 kgm2, respectively.

3.4.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Four is 1.58 seconds after rocket release.

3.5 Case Five Assumptions used for this set up are as stated in Table 1.

Figure 8: Case Five

For Case Five, rocket ignition occurs at an altitude of around 9 meters. The rocket almost reaches the ground but begins ascending again. Eventually, the motor burns out and the rocket lands with a velocity of about -14 m/s.

3.5.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0244 kgm2, 0.0244 kgm2, and 0.0016 kgm2, respectively.

3.5.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Four is 1.46 seconds after rocket release.

3.6 Case Six Assumptions used for this set up are as stated in Table 1.

Figure 9: Case Six

For Case Six, rocket ignition occurs at an altitude of around 9 meters. The rocket reaches the ground at 0 m/s but begins ascending again. Eventually, the motor burns out and the rocket lands with a velocity of about -14 m/s.

3.6.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0245 kgm2, 0.0245 kgm2, and 0.0018 kgm2, respectively.

3.6.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Four is 1.47 seconds after rocket release.

3.7 Case Seven Assumptions used for this set up are as stated in Table 1.

Figure 10: Case Seven

For Case Seven, rocket ignition occurs at an altitude of around 9 meters. The rocket almost reaches the ground but begins ascending again. Eventually, the motor burns out and the rocket lands with a velocity of about -14 m/s.

3.7.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0348 kgm2, 0.0348 kgm2, and 0.0016 kgm2, respectively.

3.7.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Four is 1.46 seconds after rocket release.

3.8 Case Eight Assumptions used for this set up are as stated in Table 1.

Figure 11: Case Eight

For Case Eight, rocket ignition occurs at an altitude of around 9 meters (1.47 seconds after release). The rocket reaches the ground at 0 m/s but begins ascending again. Eventually, the motor burns out and the rocket lands with a velocity of about -14 m/s.

3.8.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0349 kgm2, 0.0349 kgm2, and 0.0018 kgm2, respectively.

3.8.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Four is 1.47 seconds after rocket release.

3.9 Case Nine Assumptions used for this set up are as stated in Table 1.

Figure 12: Case Nine

For Case Nine, the motor ignites at an altitude of about 11 meters. The rocket reached the ground with zero velocity and begins to ascend again. Then the motor burns out, and the rocket starts free fall. The rocket hits the ground with a velocity of about -4 m/s.

3.9.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0297 kgm2, 0.0297 kgm2, and 0.0020 kgm2, respectively.

3.9.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.34s after being dropped to reach a velocity of 0 m/s at 0m altitude.

3.10 Case Ten Assumptions used for this set up are as stated in Table 1.

Figure 13: Case Ten

For Case Ten, the motor ignites at an altitude of about 11 meters. The rocket reached the ground with zero velocity and begins to ascend again. Then the motor burns out, and the rocket starts free fall. The rocket hits the ground with a velocity of about -4 m/s.

3.10.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0298 kgm2, 0.0298 kgm2, and 0.0022 kgm2, respectively.

3.10.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.34s after being dropped to reach a velocity of 0 m/s at 0m altitude.

3.11 Case Eleven Assumptions used for this set up are as stated in Table 1.

Figure 14: Case Eleven

For Case Eleven, the motor ignites at an altitude of about 11 meters. The rocket reached the ground with zero velocity and begins to ascend again. Then the motor burns out, and the rocket starts to free fall. The rocket hits the ground with a velocity of about -4 m/s.

3.11.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0425 kgm2, 0.0425 kgm2, and 0.0020 kgm2, respectively.

3.11.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.34s after being dropped to minimize velocity at 0m altitude.

3.12 Case Twelve Assumptions used for this set up are as stated in Table 1.

Figure 15: Case Twelve

For Case Twelve, the motor ignites at an altitude of about 11 meters. The rocket reached the ground with zero velocity and begins to ascend again. Then the motor burns out, and the rocket starts to free fall. The rocket hits the ground with a velocity of about -4 m/s.

3.12.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0426 kgm2, 0.0426 kgm2, and 0.0022 kgm2, respectively.

3.12.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.34s after being dropped to minimize velocity at 0m altitude.

3.13 Case Thirteen Assumptions used for this set up are as stated in Table 1.

Figure 16: Case Thirteen

For Case Thirteen the motor ignites at altitude of 12 meters with a downward speed of 10m/s and starts decelerating to. The motor stops burning just before the rocket hits the ground. The rocket hits the ground with a downward speed of 2m/s. The simulation suggests the rocket can’t reach a velocity of 0 m/s at 0-meter altitude with the provided motor because the rocket is too heavy, and the motor cannot provide enough thrust.

3.13.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0351 kgm2, 0.0351 kgm2, and 0.0024 kgm2, respectively.

3.13.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.17s after being dropped to minimize velocity at 0m altitude.

3.14 Case Fourteen Assumptions used for this set up are as stated in Table 1.

Figure 17: Case Fourteen

For Case Fourteen, the motor ignites at an altitude of about 13 meters. The motor decelerates the rocket to about –2 m/s just before it hits the ground. The rocket does not reach 0 m/s before hitting the ground.

3.14.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0352 kgm2, 0.0352 kgm2, and 0.0026 kgm2, respectively.

3.14.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.17s after being dropped to minimize velocity at 0m altitude.

3.15 Case Fifteen Assumptions used for this set up are as stated in Table 1.

Figure 18: Case Fifteen

For Case Fifteen the motor ignites at an altitude of about 13 meters. The motor decelerates the rocket to about –2 m/s just before it hits the ground. The rocket does not reach 0 m/s before hitting the ground, because it is too heavy for the motor to slow it down before it hits the ground.

3.15.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0501 kgm2, 0.0501 kgm2, and 0.0024 kgm2, respectively.

3.15.2 Ignition Delay for v=0 and alt=0 The motor must ignite 1.17s after being dropped to minimize velocity at 0m altitude.

3.16 Case Sixteen Assumptions used for this set up are as stated in Table 1.

Figure 19: Case Sixteen

For Case Sixteen the motor ignites at an altitude of 13 meters and the rocket slows down to a velocity of –2 m/s. The rocket reaches the ground without the motor being able to slow it down to 0 m/s. It does, however, get very close at a value of approximately 2 m/s. This is because the rocket is too heavy and doesn’t have the necessary thrust to bring the rocket to a 0 m/s once it starts accelerating downwards.

3.16.1 Rocket Body Inertias The moment of inertias in the xx, yy, and zz directions using the simulation are 0.0502 kgm2, 0.0502 kgm2, and 0.0026 kgm2, respectively.

3.16.2 Ignition Delay for v=0 and alt=0 The optimal time of ignition for Case Sixteen is 1.17 seconds after rocket release.

4. FINDINGS 4.1 Impact of Mass on Ignition Timing The trend observed through these cases with respect to mass of the rocket is the following: As the mass of the rocket increases, the motor must ignite sooner for proper landing. In other words, mass of the rocket is inversely proportional to ignition time (measured from release).

4.2 Impact of Length on Ignition Timing Length did not impact ignition timing for the rocket.

4.3 Impact of Diameter on Ignition Timing Diameter did not impact ignition timing for the rocket except in the case when the mass of the rocket is 1.25kg. In cases 6 and 8, we can observe there is a 0.01 second increase in ignition timing as compared to a rocket with the same weight but with smaller diameter. The difference is minute and rare, so we can conclude that diameter has no overall effect on ignition timing.

5. TRADE STUDY CONCLUSION Based on the simulation one can see that overall, the best case to use would be cases 13-16 where the weight is a value of 1.75 kg. Even though the velocity never reaches 0 m/s, the final speed is the lowest of all cases provided at approximately 2 m/s. Preferably, the weight chosen would between 1.5 kg < x < 1.75 kg to achieve a final speed of 0 m/s. The length and diameter of the rocket, according to this simulation, doesn’t have much of an effect. Further trade studies will need to be completed to truly characterize the effects of these variables.

From this case study our team will continue to design according to the results found in cases 13-16, focusing on a weight of approximately 1.625 kgs, an altitude of 20 m, and an F22 Aerotech motor.