Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1-1-1995 Synthesis of launch vehicle design and trajectory optimization Veena Savithri Dorairajan Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Aerospace Engineering Commons Recommended Citation Dorairajan, Veena Savithri, "Synthesis of launch vehicle design and trajectory optimization" (1995). Retrospective Theses and Dissertations. 18155. https://lib.dr.iastate.edu/rtd/18155 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Synthesis of launch vehicle design and trajectory optimization by Veena Savithri Dorairajan A Thesis Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Department: Aerospace Engineering and Engineering Mechanics Major: Aerospace Engineering Signatures have been redacted for privacy Iowa State University Ames, Iowa 1995 11 TABLE OF CONTENTS NOMENCLATURE . ... VIll ACKNOWLEDGMENTS Xl CHAPTER 1. INTRODUCTION 1 Motivation and historical background . 1 Problem definition 2 Literature survey . 3 Aerodynamic methods for ascent 3 Propellant loading ....... 4 CHAPTER 2. MATHEMATICAL MODEL. 7 Assumptions 7 System model 9 Aerodynamic model 9 Pressure forces 11 Base pressure . 12 Skin friction force . 13 Lift and drag forces . 17 Vehicle Sizing ...... 17 Mass, propellant loading and payload . 18 III Transformation to a fixed end-time problem 21 Optimal control problem statement . 21 CHAPTER 3. SOLUTION METHOD 25 Direct approach . 25 Sequential quadratic programming 27 CHAPTER 4. VALIDATION OF AERODYNAMIC MODEL AND DISCUSSION OF RESULTS . 30 Validation of aerodynamic model 30 Hierarchy of problems 31 Class-I problems 35 Class-II problems 36 Discussion of results .. 37 CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS 63 Conclusions . 63 Recommendations 64 REFERENCES. 66 APPENDIX A. EARTH'S ROTATION . 69 APPENDIX B. VERTICAL RISE SEGMENT 70 APPENDIX C. VEHICLE DEFINITION . 71 APPENDIX D. BASE DRAG COMPARISON 72 APPENDIX E. PROPELLANT CHARACTERISTICS 73 APPENDIX F. ENGINE CHARACTERISTICS. 75 IV APPENDIX G. NUMERICAL INTEGRATION SCHEME 76 v LIST OF TABLES Table 4.1: Stage Lengths for the Various Cases ....... 50 Table 4.2: Effects of Imposing Dynamic Pressure Constraints 52 Table A.I: Comparison with and without earth rotation 69 Table C.I: Launch Vehicle Propulsion and Weight Data 71 Table D.l: Comparison with and without base-drag ... 72 VI LIST OF FIGURES Figure 2.1: Planar Ascent from Non-rotating Earth . .. 10 Figure 2.2: Effective Base Pressure Area For Calculation of Base Drag 14 Figure 2.3: Base-Pressure Coefficient vs. Mach Number. 15 Figure 2".4: Transition Point for Calculation of Skin Friction Drag 16 Figure 2.5: Vehicle Configuration . 18 Figure 3.1: Discrete Parameterization of u( t) . 28 Figure 4.1: Comparison of Cd vs IV! Profiles . 32 Figure 4.2: Drag Coefficient Cd vs Mach Number for a Vehicle of Length 160 ft Using the Assumed Aerodynamic Model . .. 33 Figure 4.3: Drag Coefficient Cd vs Mach Number for a Vehicle 'With 0"=25 deg ..................................... 34 Figure 4.4: Second-Stage Pitch Program for Class I Problems 39 Figure 4.5: Velocity History 43 Figure 4.6: Flight Path Angle 44 Figure 4.7: Altitude History . 45 Figure 4.8: Range vs Altitude 46 Figure 4.9: Pitch Angle Profile for Class II Problems 47 Vll Figure 4.10: Mass History for Class I and Class II Problems 48 Figure 4.11: Angle-of-attack History for Case 6 ..... 49 Figure 4.12: Application of Dynamic Pressure Constraints 51 Figure 4.13: Effect of Dynamic Pressure Constraint on Angle-of Attack History ................. 53 Figure 4.14: Velocity Profile Variation With Dynamic Pressure Constraint Levels. .. 55 Figure 4.15: Flight Path Angle Time History Variation vVith Dynamic Pressure Constraint Levels . 56 Figure 4.16: Comparison of Dynamic Pressure Histories 57 Figure 4.17: Pitch Program Variation \Vith Dynamic Pressure Constraints Levels. .. 58 Figure 4.18: Aerodynamic Heating Parameter History Variation With Dy- namic Pressure Constraint Levels 59 Figure 4.19: Aerodynamic Load Constraints. 60 Figure 4.20: Effect of Aerodynamic Load Constraints on Angle-of-Attack . 61 Figure 4.21: Effect of Dynamic pressure and Aerodynamic Load Constraints on Dynamic Pressure . " 62 Vlll NOMENCLATURE Cd drag coefficient; CI total skin~friction coefficient of the vehicle; CIL skin~friction coefficient for laminar flow; CIT skin~friction coefficient for turbulent flow; CL lift coefficient; CN normal~force coefficient; CNa aerodynamic normal~force~curve slope (variation of eN with a); D drag force, lbs; lsp specific impulse, sec; J performance index; J{ propellant sensitive mass fraction; L lift force, I bs; M Mach number; Re Reynolds number; 2 S reference area of vehicle, jt ; 2 SL wetted area over length 1, ft ; 2 Sx wetted area over length leone, ft ; T Thrust, lbs; IX a(h) speed of sound at altitude h, .1!.;sec d T slenderness ratio; d diameter of vehicle, It; g acceleration due to gravity, s!~; gs acceleration due to gravity on earth's surface, s£:2; h altitude, It; I vehicle length, It; leone nose-cone length, It; m vehicle instantaneous mass, slugs; mo initial mass, slugs; mstru stage structural mass, slugs; m cone structural mass of cone-cylinder that houses the payload, slugs; stage propellant mass, slugs; mpL payload mass, slugs; s surface range, It; first-stage burn time, sec; second-stage burn-out time, sec; v vehicle velocity, ~; angle-of-attack, rad; f3 propellant mass flow rate, slug/sec; 8 thrust steering angle, rad; flight-path-angle, rad; , , f' ~ VISCOSIty 0 aIr, jt-sec; II kinematic viscosity of air, ffi; x . f· slug p denslty 0 alr, ft3 ; nose cone half-angle, deg; T normalized time, T = f-; J () pitch angle, rad; Xl ACKNOWLEDGMENTS I am deeply indebted to my major professor Dr. Bion Pierson, for his invaluable guidance and advise in my research and graduate studies. Special thanks go to Dr. Jerry Vogel for his time and wisdom in sorting out some design problems I faced in course of my research. Most importantly, I would like to thank my husband Sugumar and my parents for their patience and encouragement. 1 CHAPTER 1. INTRODUCTION Motivation and historical background For years, launch vehicle designers and control personnel have worked in isolation without much interaction in designing the vehicle. Usually, the design group hands over a frozen vehicle configuration to the trajectory and controls group to find the optimal trajectory and best control design possible for that particular configuration. The purpose of this research is to show the advantages of combined trajectory and vehicle design optimization. This combination stresses the importance of controls personnel participating in vehicle design at a much earlier stage. Maximizing payload capability of a multistage launch vehicle flown to a pre­ scribed set of burnout conditions is a problem that frequently arises in trajectory optimization studies. If all the vehicle parameters are specified, the problem reduces to that of finding the optimum steering profile. In many cases, however, not all these parameters are specified, and those left unspecified can be varied to maximize pay­ load. A typical situation that occurs in the design of future launch vehicles is one in which the propulsion system (thrust and propellant flow rate) is specified, but some or all the propellant loading are left unspecified. The unspecified propellant loadings generally can be varied to achieve maximum payload capability for the vehicle. In this way, the trajectory person also participates in designing the vehicle. 2 Many authors [1-5] have treated the problem of optimizing the stage propellant loadings of multistage vehicles. None of these authors, however, have attempted to simultaneously optimize the steering program of these vehicles. Others [6-9] have used the calculus of variations approach to optimize the steering program for various rocket vehicles. In particular, Jurovics [9] treats the problem of optimizing the steering program for a multistage launch vehicle. He does not, however, consider the problem of optimizing the stage propellant loading or the kick angle. Tren and Spurlock [10] have considered the problem of simultaneously optimizing the steering program, propellant loading, and booster kick angle using a variational approach. But in the present study, we simultaneously optimize: (1) the vehicle design variables, nosecone half-angle, vehicle slenderness ratio, and stage propellant loading, and (2) the trajectory parameters, stage burn times, booster kick angle, and steering programs. Ruppe [11], in his paper on design considerations for future launch vehicles, discusses integrating the trajectory model with the vehicle model and propulsion model for optimization of future launch vehicles. Combined trajectory and vehicle optimization studies have been performed on applications other than launch vehicles also. \Vetzel and ~1oerder, optimize the ve­ hicle and trajectory for aerocapture at Mars [12]. Kluever and Pierson optimize the vehicle and trajectory for a nuclear electric spacecraft for lunar missions [13}. Problem definition The problem to be solved