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Dynamics and Control of Magnetostatic Structures

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Dynamics and Control of Magnetostatic Structures Abstract Stock-Windsor, Jeffry Clifton Dynamics and Control of Magnetostatic Structures Under the direction of Larry Silverberg. The equations governing the dynamics of magnetostatic structures are formu- lated using Lagrangian mechanics. A potential energy function of gravitational, strain, and magnetostatic components is defined. The Lagrangian equations of motion are discretized and then linearized about equilibrium points created by the additional magnetostatic energy, leading to a linear system of ordinary differential equations. These equations are characterized by mass, stiffness, damping, gyroscopic, and circulatory effects. Four experiments are conducted. Using the one-degree-of-freedom magneto- static levitator, the measured static displacement is compared to those pre- dicted by the exact nonlinear solution and the discretized approximate solution. Three experiments are performed with the two-degree-of-freedom, spherical, magnetostatic pendulum: The natural frequencies of the pendulum are pre- dicted and compared with measurements; the pendulum is made to track a desired path using electromagnets to control the motion; and the pendulum’s oscillations about new equilibrium points are regulated using electromagnets and velocity feedback to control settling time. In the last experiment, the stabil- ity of the controlled system is proven by examining the eigenvalues about the new equilibrium position. ©1999 Jeffry Clifton Stock-Windsor Dynamics and Control of Magnetostatic Structures Jeffry Clifton Stock-Windsor A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy. Mechanical Engineering Raleigh 1999 Approved by: Dr. Ethelbert Chukwu Dr. Eric Klang Dr. Larry Royster Dr. Larry Silverberg, Chair of Advisory Committee Dedication To my wife, Christina Stock-Windsor. You are both my advocate and my inspiration. Vita I was born to John and Gilda Windsor on March 15, 1968 and grew up in the vil- lage of Oak Ridge, a cotton farming community in northeast Louisiana. After attending Riverfield Academy in Rayville through the tenth grade I was accepted at the Louisiana School for Math, Science, and the Arts in Nat- chitoches where I graduated from high school in 1986. With an academic schol- arship, I then attended Louisiana State University in Baton Rouge and in 1990 earned a Bachelor of Science degree in physics with a minor in mathemat- ics. In August 1991 I accepted a teaching assistantship in the Department of Mechanical and Aerospace Engineering at North Carolina State University where I concentrated in controls and minored in mathematics. During my second year of graduate work I received a research assistantship at the Mars Mission Research Center. In August 1993 I earned a Master of Science degree in mechanical engineering. That Fall I began work on my doctoral degree with Dr. Larry Silverberg. I examined the dynamics and control of magnetostatic structures with funding from the Mars Mission Research Center. The night of Thursday, September 26, 1996, under a total lunar eclipse I married Christina Sophia Stock, and we currently live in Raleigh, North Carolina. Acknowledgements I wish to thank my advisor Dr. Larry Silverberg for all of his guidance, support, and humor during the years finishing my degree. I also want to thank my gradu- ate committee, Dr. Ethelbert Chukwu from the Mathematics department and Drs. Eric Klang and Larry Royster from the Mechanical and Aerospace Engi- neering department, for their insightful questions. Thanks are also in order for the staff of the Mechanical and Aerospace Engineer- ing machine shop, Mike Breedlove and Skip Richardson. Their expert crafts- manship in constructing the magnetostatic spherical pendulum made my life much simpler. Finally, I would like to thank my family. My parents John R. Windsor, Sr. and Gilda Tyson Massingill have always supported and encouraged me in my efforts. My wife, Christina Stock-Windsor, in particular has been a constant source of reassurance and assistance while finishing this dissertation. Thank you. Table of Contents List of Tables.....................................................................................................vii List of Figures..................................................................................................viii Introduction........................................................................................................1 Governing Equations.........................................................................................4 Nonlinear Equations...................................................................................................... 4 Kinetic Energy 4 Potential Energy 5 Lagrangian Equations 7 Discretized Equations....................................................................................................7 Kinetic Energy 8 Potential Energy 8 Lagrangian Equations 12 Linearized Equations ...................................................................................................13 Magnetostatic Levitator ..................................................................................17 Apparatus.........................................................................................................................17 Equations of Motion.....................................................................................................18 Nonlinear Equations 18 Discretized Equations 21 Numerical Comparison 21 Static Displacement..................................................................................................... 22 Magnetostatic Pendulum ...............................................................................24 Apparatus........................................................................................................................24 Pendulum 25 Electromagnets 25 Power Supplies 26 Sensors & Computer 26 Calibration 27 Equations of Motion.................................................................................................... 29 Nonlinear Equations 29 Discretized Equations 34 Linearized Equations 34 Natural Frequencies......................................................................................................35 Apparatus 35 Equations of Motion 36 Experiment 36 Results 38 Tracking ............................................................................................................................41 Apparatus 41 Experiment 41 Results 44 Regulation....................................................................................................................... 45 Apparatus 45 Equations of Motion 45 Experiment 49 Results 49 Conclusions ...................................................................................................... 53 Works Consulted.............................................................................................. 55 Appendix: A Physics Primer for Magnetostatic Energy............................. 58 Classical Physics............................................................................................................ 58 Electromagnetism........................................................................................................ 59 Conservation of Charge 59 Maxwell’s Equations 60 Induction..........................................................................................................................61 Flux 62 Electromotive Force 63 Faraday’s Law 66 Filamentary Approximation ..................................................................................... 67 Magnetostatics.............................................................................................................. 68 Magnetic Vector Potential 69 Biot-Savart Law 70 Energy of Circuits ..........................................................................................................72 Power in Circuits 72 Work Done by Circuits 76 Energy and Virtual Work 79 Energy of a Magnetic Field 80 Field Energy and Induction Coefficients 82 Summary ......................................................................................................................... 85 Works Consulted for Appendix................................................................................ 86 List of Tables Table 3–1. Levitator’s experimental and predicted displacements................ 23 Table 4–1. Pendulum’s mean natural frequencies for 0.0A total current..... 38 Table 4–2. Pendulum’s mean natural frequencies for 0.6A total current..... 38 Table 4–3. Pendulum’s mean natural frequencies for 0.8A total current..... 39 Table 4–4. Pendulum’s mean natural frequencies for 1.0A total current..... 39 Table 4–5. Pendulum’s mean normalized tracking error for two paths........44 Table 4–6. Pendulum’s approximate settling times for various gains............ 50 Table A–1. Fundamental Equations of Classical Physics ................................... 58 Table A–2. Quantities, Symbols, and Units used in Table A–1 ......................... 59 vii List of Figures Figure 3–1. Magnetostatic levitator ..........................................................................17 Figure 3–2. Comparison of nonlinear and discretized simulated dynamic solutions..................................................................................................... 22 Figure 4–1. Magnetostatic pendulum......................................................................24 Figure 4–2. Close-up of magnetostatic pendulum’s electromagnetic coils.25
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