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Lagrangian Mechanics Modeling of Free Surface-Affected Marine Craft Lagrangian Mechanics Modeling of Free Surface-Affected Marine Craft Thomas Battista Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Craig A. Woolsey, Chair Francis Valentinis, Co-Chair Stefano Brizzolara Eric G. Paterson March 13, 2018 Blacksburg, Virginia Keywords: Lagrangian Mechanics, Potential Flow Hydrodynamics, Fluid-Body Interactions Copyright 2018, Thomas Battista Lagrangian Mechanics Modeling of Free Surface-Affected Marine Craft Thomas Battista (ABSTRACT) This dissertation presents the first nonlinear unified maneuvering and seakeeping model for marine craft derived from first principles. Prior unified models are based on linearity as- sumptions. Linear models offer a concise representation of the system dynamics and perform reasonably well under the modeling assumptions, but they neglect nonlinear hydrodynamic forces and moments incurred from more aggressive vehicle maneuvers. These nonlinear ef- fects are amplified in a natural seaway where the vehicle is subject to random, and sometimes significant, wave forcing. Lagrangian mechanics modeling techniques are used in this dissertation to model the motion of marine craft which are affected by a free surface. To begin the modeling effort, the system Lagrangian is decomposed into a rigid body Lagrangian, a fluid body Lagrangian, and a free surface Lagrangian. Together, these Lagrangian functions characterize the energy of a closed system containing the vehicle, the ambient fluid, and the free surface. These La- grangian functions are first used to construct a free surface-affected nonlinear maneuvering model which characterizes the hydrodynamic forces and moments due to arbitrary vehicle motions in a semi-infinite volume of perfect fluid. The effects due to incident waves are then incorporated using a fourth function, the disturbance Lagrangian. The disturbance La- grangian accounts for the fluid energy due to externally generated fluid motions. Combining the four Lagrangian functions enables one to the derive equations of motion for a marine craft undergoing arbitrary maneuvers in a seaway. The principal contribution of this modeling effort is that it yields ODE representations of the potential flow hydrodynamic effects for nonlinear vehicle motions. These nonlinear expressions alleviate the small-motion assumptions inherent to prior unified models, enabling higher model fidelity for a number of critical applications including rapid design optimization, training simulator design, nonlinear control and observer design, and stability analysis. This work was supported by the Office of Naval Research under grant numbers N00014-14- 1-0651 and N00014-16-1-2749. Lagrangian Mechanics Modeling of Free Surface-Affected Marine Craft Thomas Battista (GENERAL AUDIENCE ABSTRACT) Although ships have been used for thousands of years, modeling the dynamics of marine craft has historically been restricted by the complex nature of the hydrodynamics. The principal challenge is that the vehicle motion is coupled to the ambient fluid motion, effectively requir- ing one to solve an infinite dimensional set of equations to predict the hydrodynamic forces and moments acting on a marine vehicle. Additional challenges arise in parametric mod- eling, where one approximates the fluid behavior using reduced-order ordinary differential equations. Parametric models are typically required for model-based state estimation and feedback control design, while also supporting other applications including vehicle design and submarine operator training. In this dissertation, Lagrangian mechanics is used to derive nonlinear, parametric motion models for marine craft operating in the presence of a free surface. In Lagrangian mechanics, one constructs the equations of motion for a dynamic system using a system Lagrangian, a scalar energy-like function canonically defined as the system kinetic energy minus the system potential energies. The Lagrangian functions are identified under ideal flow assumptions and are used to derive two sets of equations. The first set of equations neglects hydrodynamic forces due to exogenous fluid motions and may be interpreted as a nonlinear calm water maneuvering model. The second set of equations incorporates effects due to exogenous fluid motion, and may be interpreted as a nonlinear, unified maneuvering and seakeeping model. Having identified the state- and time-dependent model parameters, one may use these models to rapidly simulate surface-affected marine craft maneuvers, enabling model-based control design and state estimation algorithms. It may be worth while, however, to remark that the hydrodynamical systems afford extremely interesting and beautiful illustrations of the Principle of Least Action, the Reciprocal Theorems of Helmholtz, and other general dynamical theories. - Sir Horace Lamb, Hydrodynamics v To my grandfather, Ralph. vi Acknowledgements I am incredibly thankful for my time in Blacksburg and all the wonderful people that made these eight years so memorable. I could not have completed my PhD if it wasn't for the camaraderie and support of a number of people. I would first like to thank my undergrad- uate friends who helped me to fall in love with Blacksburg, including my hall mates Brian Chase, Andrew Hemmen, and Dan Naiman, my roommates Kenny Maben, Dan Mead and Andrew Renner, and my senior design team David Black, Garrett Hehn, Caroline Kirk, Peter Marquis, Brian McCarthy, Khanh Ngo, Lindsay Wolff, and Will Woltman. I also owe thanks to my graduate school friends for all the enlightening and entertaining conversations in both bars and classrooms. In particular, I have to thank Dr. David Allen, Dr. Matt Chan, Danielle Cristino, Javier Gonzalez-Rocha, Allie Guettler, Seyong Jung, Dr. Ben Lutz, Hunter McClelland, Dr. Shibabrat Naik, Gary Nave, Dr. Andrew Rogers, Klajdi Sinani, Dylan Thomas, and Will Tyson. I must also thank several professors for helping me to realize my passion for mathematics, mechanics, and control theory, including Dr. Mazen Farhood, Dr. Serkan Gugercin, Dr. Shane Ross, Dr. Cornel Sultan and my former co-advisor Dr. Leigh McCue-Weil. I am also thankful for my conversations with Dr. Tristan Perez at Queensland University of Technology, whose expertise in ship dynamics helped to guide me throughout my graduate career. None of this would be possible without the sponsorship of the Office of Naval Research. I am also thankful for all the assistance I received from the staff in the Kevin T. Crofton Department of Aerospace and Ocean Engineering at Virginia Tech. My committee members played a tremendous role throughout the course of my PhD pro- gram. Dr. Eric Paterson's and Dr. Stefano Brizzolara's extensive knowledge of naval archi- tecture and fluid mechanics helped me to keep my research grounded in the \real world". I am also thankful for the support of my co-advisor, Dr. Francis Valentinis, who often con- tributed valuable remarks all the way from Australia. I am also grateful for the year that vii Dr. Valentinis spent with us at Virginia Tech. My constant exposure to such a passionate and knowledgable person had an invaluable impact on so many aspects my graduate studies. I am also indebted to my co-advisor Dr. Craig Woolsey, who recruited me straight from my undergraduate program at Virginia Tech. His consistent mentoring and dedication to my academic development helped me to grow as both a researcher and a person over these four years. I am incredibly thankful for the Blacksburg Community Band and the Summer Musical Enterprise for giving me the opportunity to immerse myself in music on a regular basis. I cannot overstate my appreciation for the therapeutic relief that music has brought me over the years. I must also thank the people in my hometown of Colts Neck for helping me grow into the person I am today, including my best friends Mike Aragones, Will Feng, Eric Gutierrez, Justin Geringer, Ryan McDonough, Andrew Oh, Nick Reardon, and Rob Romano. I would also like to thank Mr. Bryan Parks and Mr. Joe Santonacita for being great teachers who taught me how exciting and powerful education can be. Finally, I have to thank my high school band director, Jack Maniaci, for his character advice and for showing me that I would always have an outlet in music. It should go without saying (but I'll say it anyway) that I would not be where I am today without the love and support of my family. My parents Joe and Michele were always there for me, despite being eight hours away, and for that I am sincerely grateful. I would also like to thank my grandfather, Ralph, who gave up his dream of pursuing a PhD to support his family, but who always hoped that one of his decedents would earn the title of \Doctor". He instilled his passion for knowledge in me at a young age and this played a major part in my choice to pursue a PhD. Finally, I would like to thank my loving girlfriend Liz for keeping me sane during my final year of grad school and for believing in me every step of the way. viii Contents List of Figures xiii List of Tables xvi Chapter 1 Introduction 1 1.1 Background and Motivation . 1 1.2 International Collaboration: Hypothesis, Objectives, and Approach . 6 1.3 Summary of Contributions . 8 1.4 Overview . 10 Chapter 2 Rigid Body Dynamics 12 2.1 Mathematic Preliminaries . 13 2.2 Kinematics . 15 2.2.1 Kinematic Relations using Euler Angles . 16 2.2.2 Kinematic Relations in SE(3) . 17 2.3 Kinetics . 17 2.3.1 Hamilton's Principle . 19 2.3.2 Lagrangian Mechanics . 20 2.3.3 Extensions of the Euler-Lagrange Equations . 22 2.3.4 Hamiltonian Mechanics . 25 2.3.5 Non-Canonical Hamiltonian Systems . 26 2.4 Rigid Body Equations of Motion . 27 ix Chapter 3 Fluid Mechanics 31 3.1 Common Forces . 32 3.1.1 Primary Effects . 33 3.1.2 Secondary Effects . 36 3.2 Standard Classes of Models . 37 3.2.1 Maneuvering Models .
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