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Optimal Aeroelastic Trim for Rotorcraft With OPTIMAL AEROELASTIC TRIM FOR ROTORCRAFT WITH CONSTRAINED, NON-UNIQUE TRIM SOLUTIONS A Thesis Presented to The Academic Faculty by Troy C. Schank In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Aerospace Engineering Georgia Institute of Technology April 2008 i Optimal Aeroelastic Trim for Rotorcraft with Constrained, Non-Unique Trim Solutions Approved By Professor Dimitri N. Mavris, Advisor Professor Dewey H. Hodges School of Aerospace Engineering School of Aerospace Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Daniel P. Schrage, Co-advisor Professor J.V.R. Prasad School of Aerospace Engineering School of Aerospace Engineering Georgia Institute of Technology Georgia Institute of Technology Professor David A. Peters Date Approved: January 16, 2008 Department of Mechanical, Aerospace and Structural Engineering Washington University, St. Louis ii ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Dimitri Mavris, for providing me the opportunity to pursue my educational goals at Georgia Tech. I would also like to thank my co-advisor Dr. Schrage for providing many unique research opportunities in the area of rotorcraft performance and systems design. In addition, Dr. Dewey Hodges and Dr. David Peters provided invaluable guidance in the area of rotorcraft dynamics and aeroelasticity both in person and through their published work. Their examples of sincere kindness and technical excellence have been inspirational. Finally, I appreciate the gracious feedback on this work from Dr. J.V.R. Prasad. I would also like to extend gratitude to my parents. My mother has a love for books and learning that set an example to me to ask questions and seek answers. I recall our home always having a fresh supply of books, replenished every couple weeks from the library. Likewise, my father’s work ethic has been an example of getting up every day, rolling up your sleeves and accomplishing something worthwhile. I must also acknowledge the love and support of my three boys, Randall, Quintan and Rhett. No matter what kind of day I have, their cheerful smiles greet me home each evening. Their unconditional love and support help to keep things in perspective. Lastly, I want to express gratitude to my life long friend and wife, Megan. Without her patience and support, I simply would not have completed this work. The journey we have experienced in this endeavor has stretched us both to a new understanding of life and of ourselves. Thank you. Troy C. Schank Georgia Institute of Technology January 2008 iii TABLE OF CONTENTS ACKNOWLEDGEMENTS...............................................................................................iii LIST OF TABLES............................................................................................................ vii LIST OF FIGURES .........................................................................................................viii LIST OF SYMBOLS ......................................................................................................... xi SUMMARY.....................................................................................................................xiii CHAPTER 1 INTRODUCTION ........................................................................................ 1 1.1 Rotorcraft Trim ......................................................................................................... 4 1.1.1 Multidisciplinary Nature of Trim ....................................................................... 4 1.1.2 Governing Equations .......................................................................................... 6 1.1.3 Periodicity Conditions ........................................................................................ 7 1.1.4 Trim Conditions and Constraints........................................................................ 8 1.2 Optimal Trim ............................................................................................................ 9 1.2.1 Optimal Trim Formulation ............................................................................... 10 1.2.2 Nonlinear Programming Methods .................................................................... 11 1.2.3 Optimal Trim Constraints................................................................................. 11 1.2.4 Optimal Rotorcraft Trim Considerations.......................................................... 12 1.3 Objectives and Scope of Work ............................................................................... 13 1.4 Research Questions and Hypotheses ...................................................................... 14 1.4.1 Primary Research Question and Hypothesis .................................................... 15 1.4.2 Supporting Question and Hypothesis No. 1 ..................................................... 15 1.4.3 Supporting Question and Hypothesis No. 2 ..................................................... 15 CHAPTER 2 LITERATURE REVIEW ........................................................................... 16 2.1 Rotorcraft Trim Methods ........................................................................................ 16 2.1.1 Direct Numerical Integration............................................................................ 17 2.1.2 Harmonic Balance ............................................................................................ 17 2.1.3 Periodic Shooting ............................................................................................. 19 2.1.4 Autopilot........................................................................................................... 20 2.1.5 Neural Net Function Approximation................................................................ 21 iv 2.2 Trim Optimization .................................................................................................. 22 2.2.1 Previous Applications....................................................................................... 22 2.2.2 Nonlinear Programming Methods .................................................................... 24 2.3 Variable Rotor speed Autorotation ......................................................................... 27 2.3.1 Background....................................................................................................... 27 2.3.2 Variable Rotor Speed Trim Methods................................................................ 29 2.4 Rotor Blade Stress and Strain ................................................................................. 30 2.5 Chapter Summary ................................................................................................... 32 CHAPTER 3 GENERALIZED REDUCED GRADIENT OPTIMAL TRIM ................. 33 3.1 Mathematical Formulation...................................................................................... 33 3.1.1 Reduced Gradient Determination ..................................................................... 33 3.1.2 Inequality Constraints....................................................................................... 36 3.1.3 Search Direction ............................................................................................... 38 3.1.4 Optimality Conditions ...................................................................................... 40 3.2 Optimal Trim Algorithm Implementation .............................................................. 40 3.2.1 Implementation Considerations........................................................................ 40 3.2.2 Algorithm Description...................................................................................... 42 3.3 Method Validation .................................................................................................. 46 3.3.1 Validation Case: Hover .................................................................................... 48 3.3.2 Validation Case: Forward Flight ...................................................................... 51 3.3.3 Optimal Trim Results Discussion..................................................................... 56 3.4 Chapter Summary ................................................................................................... 58 CHAPTER 4 AUTOROTATIVE TRIM WITH VARIABLE ROTOR SPEED.............. 59 4.1 Physics of Autorotation........................................................................................... 59 4.2 Numerical Method for Variable Rotor speed.......................................................... 62 4.3 Rotor speed Correlation .......................................................................................... 65 4.3.1 Isolated Rotor Trim .......................................................................................... 66 4.3.2 6 DOF Vehicle Trim......................................................................................... 74 4.4 Optimal Trim Application to Autorotation............................................................. 78 4.5 Chapter Summary ................................................................................................... 80 CHAPTER 5 SURROGATE BLADE STRESS AND STRAIN...................................... 81 v 5.1 Rotor Aeroelastic and Structural Analysis Framework .......................................... 82 5.2 Surrogate Stress/Strain Method .............................................................................. 83 5.3
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