This document summarizes a numerical analysis of rocket dynamics and structural components conducted by engineering students. MATLAB was used to simulate a rocket's flight to apogee using differential equations, and NX Nastran conducted finite element analysis of the rocket's shear pins and airframe under pressure. The rocket simulation results showed good agreement with a commercial software and provided insight into how dart rocket mass affects apogee. The FEA indicated the shear pins and airframe materials could withstand expected stresses from parachute deployment pressures. Further validation and code refinement is needed, but the analyses provided engineering understanding of rocket flight and component performance.

1. 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.

2. 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

3. 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.