PhD Dissertations and Master's Theses Spring 4-2021 A Comparative Study Between 6 Degree-of-Freedom Trajectory Model and Modified Point Mass Trajectory Model of Spinning Projectiles Ange Du [email protected] Follow this and additional works at: https://commons.erau.edu/edt Part of the Acoustics, Dynamics, and Controls Commons, and the Aerodynamics and Fluid Mechanics Commons Scholarly Commons Citation Du, Ange, "A Comparative Study Between 6 Degree-of-Freedom Trajectory Model and Modified ointP Mass Trajectory Model of Spinning Projectiles" (2021). PhD Dissertations and Master's Theses. 594. https://commons.erau.edu/edt/594 This Thesis - Open Access is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in PhD Dissertations and Master's Theses by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. A Comparative Study Between 6 Degree-of-Freedom Trajectory Model and Modified Point Mass Trajectory Model of Spinning Projectiles by Ange (Phil) Du A Thesis Submitted to the College of Engineering Department of Mechanical Engineering in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mechanical Engineering Embry-Riddle Aeronautical University Daytona Beach, Florida April 2021 A Comparative Study Between 6 Degree-of-Freedom Trajectory Model and Modified Point Mass Trajectory Model on Spinning Projectiles by Ange (Phil) Du This thesis was prepared under the direction of the candidate’s Thesis Committee Chair, Dr. Jean-Michel Dhainaut, Professor, Daytona Beach Campus, and Thesis Committee Members Eduardo Divo, Professor, Daytona Beach Campus, and Mr. Bryan Litz, Chief Ballistician at Applied Ballistics and Berger Bullets, and has been approved by the Thesis Committee. It was submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering. Thesis Review Committee: ____________________________________ Jean-Michel Dhainaut, Ph.D. Committee Chair _________________________________ _______________________________ Eduardo. Divo, Ph.D. Bryan Litz Committee Member Chief Ballistician ________________________________ _______________________________ Jean-Michel Dhainaut, Ph.D. Eduardo. Divo, Ph.D. Graduate Program Coordinator, Department Chair, Mechanical Engineering Mechanical Engineering ________________________________ _______________________________ Maj Mirmirani, Ph.D. Christopher Grant, Ph.D. Dean, College of Engineering Associate Vice President of Academics ___________ Date i ACKNOWLEDGEMENTS I would like to express my sincere gratitude toward Dr. Jean-Michel Dhainaut, for giving me the opportunity to work on such an intriguing topic, and his continuous support and mentoring along this journey. I would also like to appreciate the time and valuable advises by my committee members, Dr. Eduardo Divo and Mr. Bryan Litz, and the support by the Department of Mechanical Engineering at Embry-Riddle. For me as an enthusiast in simulation type video games, this is a topic that comes close to my heart, as there is a unique satisfaction of seeing physics come to life in a virtual world. My friends and clanmates at GTRC would certainly understand such feelings. Thank you for being my happy escape despite our physical distances. Lastly, but certainly not least, I would like thank my parents, my family for always standing behind my decisions, and providing the best help they can. I would never be where I am without you. ii ABSTRACT Researcher: Ange (Phil) Du Title: A Comparative Study Between 6 Degree-of-Freedom Trajectory Model and Modified Point Mass Trajectory Model on Spinning Projectiles Institution: Embry-Riddle Aeronautical University Degree: Master of Science in Mechanical Engineering Year: 2021 For spinning projectiles, the 6 Degree-of-Freedom model can closely capture their trajectories with high accuracy and details. However, it comes with the drawbacks of long computation time and needing many aerodynamic coefficients which can be hard to obtain. The Modified Point Mass Trajectory Model was introduced as a simplified solution. This work compares them with each other, and with some more conventional methods. A program is developed to simulate and visualize trajectories. With some augmentations, the Modified Point Mass model is able to generate comparable results in most cases, but not including special cases such as when the projectile is unstable. It also allows for simulation time step as much as 60 times while retaining accuracy. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................................................................................ ii ABSTRACT ....................................................................................................................... iii LIST OF FIGURES ........................................................................................................... vi LIST OF TABLES ............................................................................................................. ix ABBREVIATIONS ............................................................................................................ x NOMENCLATURE .......................................................................................................... xi CHAPTER 1: INTRODUCTION ................................................................................. 1 CHAPTER 2: LITERATURE REVIEW AND FORMULATION .............................. 4 2. 1. Classical Approach to Trajectory Computing ......................................................... 4 2. 2. Trajectory Models Formulation .............................................................................. 6 2. 2. 1. Vacuum Trajectory ...........................................................................................6 2. 2. 2. Point Mass Trajectory .......................................................................................7 2. 2. 3. Point Mass Trajectory with Effect of Wind ....................................................10 2. 2. 4. Point Mass Trajectory with Coriolis Effect ....................................................10 2. 2. 5. 6 Degree-of-Freedom Trajectory ....................................................................11 2. 2. 6. Modified Point Mass Trajectory .....................................................................18 2. 2. 7. Aerodynamic Jump Due to Crosswind ...........................................................19 2. 2. 8. Spin Drift ........................................................................................................20 2. 3. Related Concepts ................................................................................................... 22 2. 3. 1. Time-Accurate CFD .......................................................................................22 2. 3. 2. CFD for Generating Aerodynamic Coefficients .............................................22 CHAPTER 3: METHODS .......................................................................................... 24 3. 1. Programming Overview ........................................................................................ 24 3. 1. 1. 6DOF Model ODE Function ..........................................................................30 3. 1. 2. MPM Model ODE Function ...........................................................................31 iv 3. 2. Aerodynamic Jump Compensation ....................................................................... 32 3. 3. Measurement of Trajectory Differences ............................................................... 33 3. 4. Design of Experiment............................................................................................ 34 3. 5. Projectile used and Aerodynamic Coefficients ..................................................... 36 CHAPTER 4: RESULTS AND COMPARISON ....................................................... 43 4. 1. Validation of 6DOF Model ................................................................................... 43 4. 2. Baseline Case: Flat Firing ..................................................................................... 46 4. 3. Simulation Time Step ............................................................................................ 53 4. 4. Uphill/Downhill Firing .......................................................................................... 54 4. 5. Crosswind and Aerodynamic Jump....................................................................... 56 4. 6. Spin Drift ............................................................................................................... 60 4. 7. Rifling Twist Rate and Stability ............................................................................ 63 CONCLUSION ................................................................................................................. 68 REFERENCES ................................................................................................................. 70 APPENDIX ....................................................................................................................... 74 A. Raw Simulation Results in form of Range Card ................................................ 74 v LIST OF FIGURES Figure 2.1: Standard G1 and G7 drag coefficient curves and their projectile shapes ......... 4 Figure 2.2: Drag coefficient of a .243 bullet using experimentally measured data vs. using G1 and