FINITE ELEMENT ANALYSIS OF A SOLAR SAIL AFFECTED BY CELESTIAL BODIES A Thesis Presented to the faculty of the Department of Mechanical Engineering California State University, Sacramento Submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in Mechanical Engineering by Jorell Salazar Gonzales SPRING 2013 © 2013 Jorell Salazar Gonzales ALL RIGHTS RESERVED ii FINITE ELEMENT ANALYSIS OF A SOLAR SAIL AFFECTED BY CELESTIAL BODIES A Thesis by Jorell Salazar Gonzales Approved by: ______________________________________, Committee Chair Illhan Tuzcu, Ph.D ______________________________________, Second Reader Akihiko Kumagai, Ph.D _______________________________ Date iii Student: Jorell Salazar Gonzales I certify that this student has met the requirements for format contained in the University format manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for the thesis. __________________________, Graduate Coordinator ___________________ Akihiko Kumagai, Ph.D Date Department of Mechanical Engineering iv Abstract of FINITE ELEMENT ANALYSIS OF A SOLAR SAIL AFFECTED BY CELESTIAL BODIES by Jorell Salazar Gonzales Solar Sails are spacecrafts that use light propulsion to push their large sails to accelerate forward. Just like how sailboats use the wind to move across the ocean, solar sails use the pressure from light particles emitted from the Sun to push them forward. In space, absent of friction, this tiny pressure provides a constant acceleration to the sails and ultimately gain large values of velocity. Solar sails rely heavily on their massively thin sails to remain flat. Unfortunately, these sails are prone to deformations due to external forces encountered in space. If deformations occur, the sail will not receive the maximum amount of solar pressure. Changes in the shape of the sail can also cause the spacecraft to change trajectories. This paper will show the deformation of the sail due to celestial bodies. Solidworks will be used to model and analyze these forces using the finite element method. An analytic approach will be used to find the vibration mode shapes and frequencies for the sail. The vibration mode shapes and frequencies will then be compared with the values obtained from Solidworks to validate the use of the finite v element method. Once verified, several loads such as solar pressure and gravitational forces from celestial bodies will be applied to the sail. Another test will be conducted to see the amount of deformation caused by changes in sail size. This thesis is focused completely on the analysis of the sails therefore, a simple quad-triangular solar sail configuration is used. The results show that gravitational forces caused by celestial bodies causes deformation to the sails depending on the orientation of the spacecraft. Increasing the sail size causes the deformations to grow exponentially. _______________________, Committee Chair Ilhan Tuzcu, Ph.D _______________________ Date vi ACKNOWLEDGMENTS I would like to take the time to express my fullest gratitude to all of those who have supported me. First I would like to thank my advisor, Dr. Ilhan Tuzcu, for his guidance. He has shown great patience and support into the completion of this thesis. Without his time, effort, and knowledge I would not have completed this paper. I would also like to thank my second advisor, Dr. Akihiko Kumagai, for taking the time to go through and revising my work. Lastly, I would like to thank my entire family and friends, but most of all my parents, for their countless encouragement and support. They have helped me through my academic achievements and I dedicate my work to them. vii TABLE OF CONTENTS Page Acknowledgments……………………………………………………………………….vii List of Tables……………………………………………………………………………...x List of Figures…………………………………………………………………………….xi Chapter 1. INTRODUCTION TO SOLAR SAILS..........................................................................1 1.1. Solar Sails................................................................................................................1 1.2. Previous Works on Solar Sailing.............................................................................2 1.3. Finite Element Method............................................................................................3 2. VIBRATION MODES...................................................................................................5 2.1. Newtonian Formulation...........................................................................................5 2.2. Solidworks.......................................................................................................…....7 3. SOLAR SAIL DEFORMATION................................................................................13 3.1. Solar Sail Construction..........................................................................................13 3.2. Solar Pressure........................................................................................................15 3.3. Celestial Bodies.....................................................................................................19 viii 3.4. Sail Sizing..............................................................................................................22 4. CONCLUSION.............................................................................................................24 Appendix...........................................................................................................................26 A.1. Mathematica Code for Vibration Modes..............................................................26 Bibliography......................................................................................................................27 ix LIST OF TABLES Tables Page 1. Table 2.1: Material Properties of Kapton................................................................8 2. Table 2.2: Five Mode Frequencies Between Newtonian Approach and FEM......10 3. Table 3.1: Material Properties................................................................................13 4. Table 3.2: Applied Forces on Solar Sail................................................................19 x LIST OF FIGURES Figures Page 1. Figure 2.1: Infinitesimal Element of a Membrane...................................................5 2. Figure 2.2: Right Isosceles Triangle........................................................................8 3. Figure 2.3: Mesh Convergence................................................................................9 4. Figure 2.4: Five Mode Shapes Using Newtonian Approach.................................11 5. Figure 2.5: Five Mode Shapes Using FEM...........................................................12 6. Figure 3.1: Core, Isometric View and Cross Sectional View Respectivly...…….14 7. Figure 3.2: Boom, Left: Isometric View and Right: Cross Sectional View..........15 8. Figure 3.3: 10 m Solar Sail Viewed in the x-y Plane. ………..…....................….15 10. Figure 3.4: Direction of Sun Light and Surface Normal of Sail............................17 11. Figure 3.5: Solar Sail with Applied Solar Pressure...............................................17 12. Figure 3.6: Deformed Solar Sail Due to Applied Solar Pressure.…...…………..18 13. Figure 3.7: Solar Sail Exposed to Gravitational Force and Solar Pressure ..........20 14. Figure 3.8: Deformed Solar Sail Due to Gravity and Solar Pressure....................21 15. Figure 3.9: Displacement Vs Distance..................................................................22 xi 16. Figure 3.10: Displacement Vs. Sail Size...............................................................23 xii 1 Chapter 1 INTRODUCTION TO SOLAR SAILS 1.1 Solar Sails Solar sails are spacecrafts that use light weight reflective sails that converts the Suns energy into momentum. The light emitted from the Sun applies a pressure, giving a constant force throughout the sail. The force applied to the sails of the spacecraft is proportional to its size. Newton's second law of motion dictates that a constant force applied to any mass will experience a constant acceleration. This would mean that the solar sails velocity would continue to increase due to the absence of drag in space. Without the limiting factor of fuel, solar sails are the ideal candidate for long term missions into space. The sails acceleration is dependent on the effective area that reflects light therefore, it is important that the sail remain a flat surface. Though ideal, solar sails have their limitations. If the sail is truly reflective, the amount of pressure it will experience is about 9.1x10-6 N/m2 at 1 AU [1]. Thus, in order for the solar sail to accelerate at .1 mm/s2 it would need to have a mass to area ratio of 91 g/m2. This amount of acceleration is small when applied for space exploration therefore, the sails are made to be very large relative to the weight of the cargo it brings. The sails must also be ultra thin compared to its area to reduce as much weight as possible. Unfortunately, this makes them extremely delicate. Due to this fact, sails are easily influenced by external forces that may exist in space. Deformations or wrinkling can result in severe consequences. Wrinkles increase intrinsic roughness which will reduce 2 their ability to reflect light. Light that is not reflected is absorbed, which will create hotspots throughout the sail. Deformations also result in the reduction of surface area. The spacecraft will not