Numerical Methods Laboratory 5

Rocket Simulation

MEEN 3260

Lab Section 402 and 404

December 4th, 2014

By: Michael Prokop and William Kassel

Instructor: Dr. Phil Voglewede

TA: Mr. Shaoli Wu

Abstract:

In order to determine the validation of a rocket design an engineering analysis is needed.

This project analyzes the dynamics of a rocket’s flight to apogee in MATLAB while testing

several structural components in NX Nastran concurrently. Though all results out of a computer

are wrong the interpretation of the data is left for engineering judgment due model mismatch or

error truncation in the analysis as will hopefully be made apparent in this following report.

Introduction:

Rocket design requires the combination of multiple engineering areas. The engineering areas include mechanics of materials, dynamics, statics, thermodynamics, and fluid dynamics. The depth of this

report includes the understanding of the dynamics of a rocket and the mechanics of materials of the

structure of the rocket when ejection of two parachutes occur. The two programs used to solve these

problems were MATLAB and NX Nastran. MATLAB was used to simulate a 3 degrees of freedom DOF rocket flight to apogee by implementing an ordinary differential equation (ODE) solver, Runge Kutta

Fehlberg was chosen as the ideal ODE solver. Using NX8.5 CAD program a rocket was roughly modeled

to meet the 2015 High Powered Rocket Competition requirements of being a dual deployment (two parachutes) as well as being boosted dart rocket. NX Nastran, a finite element analysis (FEA) solver was

used due to the familiarity of the software over using the ANYSYS program.

For this project a rocket with a dart section will be looked into. A dart rocket is a smaller rocket that detaches in flight after thrust from the motor seizes, which is 1.1 seconds for the Cesaroni I445 motor

used. The purpose of the dart is to achieve a higher altitude than normally would be meet with a single

large rocket. The MATLAB code will attempt to determine an unknown altitude the rocket will achieve

along with the uncertainty of how much weight to put into the dart section to optimize the apogee (See Appendix A). The FEA will look at the shear pins of the rocket used for a parachute ejection system and

the effect it has on the airframe (See Appendix B).

Assumptions:

Rocket Simulation:

Using a 4th order Runge Kutta to solve three second order initial value problem (IVP) differential equations multiple assumptions were made. The start time (t0), final time (tf), auxiliary conditions were

base of engineer judgment. Most assumptions took place in the function file (Figure 6A-7A). Thrust curve

was interpolated, mass moment of inertia was approximated with RockSim. Coefficient of drag was based

off RockSim and dart was assumed to be 30% lower than the main rocket. Coefficient of lift was based off engineering judgment. All areas were approximated as close as possible to the model. Density of air is

a function of height. There is a constant wind velocity of 4 MPH. The final large assumption of the

simulation is that the dart instantaneously separates as the motor burns out. Rocket FEA Shear Pin:

Shear pins were assumed to be made out of nylon 6/6 with a yield stress of 69*10^3 Pa. Total of

two shear pins which are both 2-56 X 1/4" each. Case I is for the dart parachute bay and Case II is for the

main rocket parachute bay. Airframe is made out of Phenolic Cardboard with a yield stress of 58*10^6 Pa.

Analysis: Rocket Simulation: The analysis of the rocket began with researching articles already published

in the area of rocket dynamics. Due to the familiarity with ODE solvers a Runge Kutta Fehlberg was able

to be implemented or else the simulation would have very difficult to solve. However, with the ability to program, results can be collected to analysis the system. Figure 1A through 3A show the concept to the

trick method being implemented to reduce the 2nd order ODE IVP into several 1st order ODE IVP in

which Runge Kutta Fehlberg can solve and iterate. The code was challenging with hurdles needing to be

overcome. Some of these hurdles were getting re-familiar on how ODE45 works and how to implement it, applying if loops, and being able to interpolate data during iterations. The final code for this project can

be seen in Figure 4A through 7A. The main rocket design, which was quickly thrown together due to the

scope of this project, can be seen in Figure 8A and the dart section of the rocket in Figure 9A. Interpolation of the results will be discussed in the results section.

Rocket FEA Shear Pin: The analysis of the shear pin began by applying mechanics of materials to

the problem which can be seen in Figure 1B. With knowing the force in order to shear the shear pin one could determine the pressure on the wood bulkhead plates in order to achieve the shear force required as

seen in Figure 2B. The next step was to determine the radial pressure acting on the airframe. This is

because in the past competition the group had an abnormally in the airframe during flight due to the

ejection. The radial pressure was calculated using pressure vessel equations from mechanics of materials as seen in Figure 3B. Due to the complex nature of the Case I section and the Case II section a scaled

down version of the shear pin was made as seen in Figure 4B. Case I again deals with the dart rocket

shear pins and Case II deals with the main rocket shear pins. Scaled down version of the shear pin was

meshed with a meshing smaller than the automatic choice in order for a better resolution. The two cases for the airframe can be seen in Figure 6B through 7B. Due to the difficulty of not being able to apply a

pressure on an interval that the actual airframe will experience the entire inner section was subjected to

the pressure and the sections not exposed to the pressure were ignored. Take for instance Figure 7B, the highlighted yellow section does not see the radial pressure. All FEA modeling had the challenge of

overcoming the issue with no results being produced from the simulation. This error was fixed by

revisiting the constraints of the assembly along with trying different methods of fixing the model.

Results:

Rocket Simulation: The results from the simulation were great to see, as this was a direct

application of applying numerical methods to solve a complex system which would be otherwise hard to solve analytically. Knowing all results out of a computer are wrong it is up to the engineer to decipher the

results. Unfortunately there is no experimental data available to test the validity of the MATLAB code,

however, we were able to compare it to a commercial rocket simulation package called RockSim. Figure 10A shows the simulation of just the entire rocket going to apogee (RockSim was not able to apply the

dart rocket simulation). Though there are discrepancies between Figure 10A and Figure 11A (MATLAB

code simulation) such as the angle of attack. They also had some commonalities to it as well such as the altitude achieved. With a comparison to see how well the MATLAB code simulation works now it can be

applied to the dart rocket separation flight. Figure 13A through 15A are plots of the dart rocket with

different percentages of weight of the entire rocket. It is seen from the results that the dart rocket with the

largest mass actually goes the highest. Yes, that is correct the heaviest design goes the highest. Without this simulation our engineering judgment would have guessed that the lighter rocket would have achieved

a higher apogee due to mass*gravity always acting down, however this is not the case due to conservation

of energy contribution. Granted the code still needs to be validated against experimental data and debugged further. The final comment on the simulation was the lack of apogee gain from the dart rocket.

From experience, the dart rocket should have had a more effect of apogee gain, showing there is room for

improvement in the code.

Rocket FEA Shear Pin: This section has results for the shear pin and the two cases of the airframes. The results of the shear pin can be seen in Figure 5B, which is a figure of stresses. It is

important to note that the stress in the shear pin is greater than the nylon yield stress, showing that the

shear pin potential gave away with the hand calculated force. Case I, represents the airframe of the dart rocket pressurized. Though the result of Figure 6B is unrealistic as the pressure isn’t applied throughout

the entire airframe it is still a nice depiction of how the airframe will handle the pressure. For Case I the

yield stress in the airframe elements is below the yield stress of the phenolic cardboard, meaning the airframe potentially survives the pressurization of 478*10^3 Pa. Case II, represents the airframe of the of

the main rocket pressurized. Again this isn’t a true depiction of the airframe, as seen in Figure 7B. For

this case we are only concerned with the airframe not highlighted in red. With this in mind one can see

that the stress in the elements of concern are below the yield stress of the phenolic cardboard, demonstrating that it has the potential to survive the pressurization of 356*10^3Pa.