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University of Southern California Dissertations and Theses
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Transient modeling, dynamic analysis, and feedback control of the Inductrack Maglev system
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Transient modeling, dynamic analysis, and feedback control of the Inductrack Maglev system
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TRANSIENT MODELING, DYNAMIC ANALYSIS, AND FEEDBACK CONTROL OF THE INDUCTRACK MAGLEV SYSTEM by Ruiyang Wang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) December 2021 Copyright 2021 Ruiyang Wang Acknowledgements I must convey my deepest gratitude to my advisor Dr. Bingen Yang, for enrolling me in his wonderful research group with plentiful resources. His passion, knowledge, and experience in various projects have been inspiring, teaching, and guiding me in the past six years. He is also a gracious and considerate person and ever helped me overcome many tough situations, in both academia and life. I would like to express my sincere appreciation to all the faculty members in my dissertation and qualifying exam committee: Dr. Henryk Flashner, Dr. Edmond A Jonckheere, Dr. Yan Jin, and Dr. Georey R Shiett. Their insightful views and suggestions on my research are so indispensable that endorsed and enriched my Ph.D. work. My extra gratefulness to Dr. Inna Abramova, for inexhaustibly requesting me as her TA to secure my nancial support. My earnest thanks also go to my great group members: Dr. Shibing Liu, Dr. Keyvan Noury, Dr. Sichen Yuan, Yichi Zhang, and Haowen Liu. Thank you all for your help, encouragement, and trust all the way through my memorable career at USC. My special thanks to Dr. Hao Gao, the excellent lab mate, engineer, and friend, for the creative research ideas he provided for my project. I owe my utmost gratitude to my family and friends, especially my parents Dr. Jun Wang and Zhengqiao Wang. Their endless love, understanding, and support are always the motivation that gets me through the most dicult time. This dissertation is dedicated to my paternal grandparents Gensheng Wang and Xueping Liu, and my maternal grandfather Yun’an Wang, in my loving memory. ii TableofContents Acknowledgements ii ListofTables v ListofFigures vi Abstract x Chapter1: Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Format of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter2: ProblemStatement 10 2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter3: ModelingoftheBenchmarkTransientInductrackSystem 15 3.1 Onboard Source Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Induced Coil Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Magnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.1 Equivalent Magnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.2 Magnetic Levitation and Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter4: ModelValidation 34 4.1 Magnetic Field by a Halbach Array Set of Finite Dimensions . . . . . . . . . . . . . . . . . 34 4.2 Time Response of Magnetic Forces and Induced Coil Currents . . . . . . . . . . . . . . . . 39 4.3 Comparison with the Original Steady-State Model . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.1 Steady-State Magnetic Forces at Dierent Traveling Speeds . . . . . . . . . . . . . 43 4.3.2 Steady-State Magnetic Forces with Dierent Levitation Gaps . . . . . . . . . . . . 45 4.4 Convergence of Magnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 iii Chapter5: TransientResponseoftheInductrackDynamicSystem 48 5.1 Vehicle Dynamics at Constant Traveling Speeds . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Vehicle Dynamics with Prescribed Propulsion Force . . . . . . . . . . . . . . . . . . . . . . 52 5.2.1 Vehicle in Free Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.2 Vehicle with Constant Propulsion Force . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.3 Vehicle in Acceleration and Deceleration . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter6: Model-BasedFeedbackControloftheInductrackSystem 60 6.1 Active Halbach Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 PID Control Based on a Simplied Linear Reference Model . . . . . . . . . . . . . . . . . . 62 6.3 Nonlinear Force-Current Mapping Function . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.4 Augmented State Equations for the Feedback Control System . . . . . . . . . . . . . . . . . 69 6.5 Numerical Examples of the Controlled Inductrack System . . . . . . . . . . . . . . . . . . . 71 6.6 Extension of the Feedback Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.6.1 Linear Controller Based on State Feedback . . . . . . . . . . . . . . . . . . . . . . . 79 6.6.2 Modied Nonlinear Mapping Function . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.6.3 Numerical Examples of State Feedback-Based Control . . . . . . . . . . . . . . . . 82 Chapter7: ModelExtension: A2DRigid-BodyInductrackSystem 87 7.1 Coordinate Systems and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2 Calculation of the Electromagnetic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3 Calculation of the Magnetic Force and Torque . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.4 Governing Equations and State-Space Representation . . . . . . . . . . . . . . . . . . . . . 96 7.5 Validation of the Rigid-Body Inductrack Model . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.5.1 The Internal Torque in Pseudo-Steady-State 2-DOF Motion . . . . . . . . . . . . . 98 7.5.2 Magnetic Force and Torque at Fixed Rotation Angle . . . . . . . . . . . . . . . . . 100 7.5.3 Transient Response of Pure Rotation Motion . . . . . . . . . . . . . . . . . . . . . . 103 7.6 Transient Response of the Rigid-Body Inductrack Model . . . . . . . . . . . . . . . . . . . 105 Chapter8: Conclusions 107 References 111 Appendices 118 A Analytical Expressions of the Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.1 Quadrature Results on Type 1 Integrals in Eq. (A.1) . . . . . . . . . . . . . . . . . . 120 A.2 Quadrature Results on Type 2 Integrals in Eq. (A.2) . . . . . . . . . . . . . . . . . . 122 A.3 Quadrature Results on Type 3 Integrals in Eq. (A.3) . . . . . . . . . . . . . . . . . . 124 B Proof of Equivalent Magnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 C Derivation of the Original Steady-State Inductrack Model . . . . . . . . . . . . . . . . . . . 132 iv ListofTables 4.1 List of the physical parameters used for numerical simulations . . . . . . . . . . . . . . . . 40 6.1 The parameters of the nonlinear mapping function used for numerical simulations . . . . 75 6.2 List of state feedback control parameters used for numerical simulations . . . . . . . . . . 83 v ListofFigures 2.1 Schematic of an Inductrack system: (a) in front view; (b) in side view; (c) the conguration of a Halbach array set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 Magnetization current model for a PM block in Case 1 . . . . . . . . . . . . . . . . . . . . . 16 3.2 Circuit model of thekth track coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Schematic of thekth circuit coil for Ampere force calculation . . . . . . . . . . . . . . . . . 25 3.4 The Inductrack system that has been modeled . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Timely updated track coils in the spatial window that moves with the vehicle . . . . . . . 30 4.1 A Halbach array set containing two wavelengths . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Distributions of magnetic eld components along longitudinal direction atx M = 0 with = 0:1d; 0:25d; 0:8d; 1:6d: (a) the longitudinal componentB y M ; (b) the vertical componentB z M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Magnetic eld distribution and the corresponding sinusoidal approximation: (a)B y M at = 0:8d; (b)B z M at = 0:8d; (c)B y M at = 1:6d; (d)B z M at = 1:6d . . . . . . . . . . 37 4.4 Distributions of magnetic eld components along crosswise direction when = 0:1d; 0:25d; 0:8d; 1:6d: (a) the longitudinal componentB y M aty M = 0:75; (b) the vertical componentB z M aty M = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.5 Time responses of maagnetic forces at = 0:8d under various traveling speeds: (a) magnetic levitation force; (b) magnetic drag force . . . . . . . . . . . . . . . . . . . . . . . 41 4.6 Time responses of the induced current of a specied coil at = 0:8d under various traveling speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.7 Magnetic levitation and drag forces calculated by truncated model and original steady-state model under various constant traveling speedsv at = 0:8d . . . . . . . . . . . . . . . . . 44 vi 4.8 Magnetic levitation and drag forces calculated by truncated model and original steady- state model under various xed levitation gaps : (a) at v = 2 m/s; (b) at v = 20 m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.9 Steady-state magnetic forces computed using dierent coil numbersN (in ratio Lc Lv ) at = 0:8d with multiple constant speeds: (a) the levitation force; (b) the drag force . . . . . 47 5.1 Time responses of the levitation gap of under various constant traveling speeds: (a) v = 1; 1:2; 1:5; 1:9 m/s; (b)v = 2; 2:5; 3 m/s; (c)v = 4; 5; 6; 8 m/s; (d)v = 10; 20; 50 m/s . 50 5.2 Dynamic responses of the vehicle in projectile motion whenv 0 = 5; 8; 10 m/s: (a) the levitation gap; (b) the displacement in the longitudinal direction . . . . . . . . . . . . . . . 53 5.3 Transient responses of the Inductrack system under a constant propulsion force F 0 = 475; 500; 600 N: (a) levitation gap; (b) longitudinal traveling speed; (c) magnetic levitation force; (d) magnetic drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4 A designed prole of the propulsion force, where: t 1 = 0:1 s,t 2 = 1:0 s,t 3 = 1:1 s, t 4 = 1:3 s,t 5 = 1:4 s,t 6 = 1:9 s,t 7 = 2:0 s, andF 1 = 250 N,F 2 =300 N . . . . . . . . 57 5.5 Dynamic responses of the Inductrack vehicle in acceleration and deceleration process: (a) longitudinal traveling speed; (b) levitation gap; (c) magnetic levitation force; (d) magnetic drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.1 The conguration of an active Halbach array set (front view) . . . . . . . . . . . . . . . . . 61 6.2 A linear reference model with a lumped-mass dynamic system . . . . . . . . . . . . . . . . 63 6.3 Block diagram for PID control based on the linear reference model . . . . . . . . . . . . . . 63 6.4 Results of PID tuning based on the linear reference model: (a) root locus plot; and (b) step response of closed-loop system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.5 Internal structure (the transient Inductrack model and a nonlinear mapping function), replacing the linear reference model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.6 Block diagram for the entire Inductrack system under feedback control . . . . . . . . . . . 66 6.7 Results of identication of the nonlinear force-current mapping function: (a) steady-state levitation force at dierent speeds and levitation gaps; and (b) identication of the speed-dependent parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.8 Transient responses of the levitation gap for the uncontrolled Inductrack system: (a) at v = 2 m/s; (b) atv = 5 m/s; (c) atv = 20 m/s; and (d) atv = 100 m/s . . . . . . . . . . . . 73 6.9 Transient responses of the levitation gap for the Inductrack system with feedback control: (a) atv = 2 m/s; (b) atv = 5 m/s; (c) atv = 20 m/s; and (d) atv = 100 m/s . . . . . . . . . 74 vii 6.10 The time-varying current densityJ C (t) in the active coils . . . . . . . . . . . . . . . . . . 76 6.11 The estimated and actual levitation forces in the Inductrack system with feedback control: (a) the estimated levitation forceF u ; and (b) the actual levitation force . . . . . . . . . . . 77 6.12 Transient responses of the levitation gap atv = 1:5 m/s, without and with feedback control 78 6.13 Transient responses of the controlled system atv = 60 m/s, with dierent reference levitation gaps: (a) levitation gap; and (b) active current densityJ C (t) . . . . . . . . . . . . 79 6.14 The block diagram of the Inductrack feedback control system based on state feedback . . . 80 6.15 Transient levitation gap atv = 2 andv = 20 m/s, under state feedback (SF)- and PID-based control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.16 Transient levitation gap at time-dependent traveling speed, subject to prescribed propulsion force, with and without control . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.17 Transient response of the equivalent current density of the active coils in the controlled Inductrack dynamic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.1 Schematic of a 2D, 3-DOF Inductrack system: (a) in front view; (b) in side view; and (c) the conguration of Halbach arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.2 Steady-state internal torque at dierent traveling speeds with xed levitation gap 0 = 0:8d = 0:02 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.3 Response of the steady-state internal torque at dierent xed levitation gaps with a traveling speedv = 1 m/s: (a) 0 = 0:2d = 0:005 m; (b) 0 = 0:4d = 0:01 m; (c) 0 = 0:6d = 0:015 m; (d) 0 = 0:8d = 0:02 m . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.4 Steady-state response under dierent xed rotation angles at constant traveling speed v = 1 m/s: (a) levitation force; (b) drag force . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.5 Steady-state response of the internal torque under dierent xed rotation angles at constant traveling speedv = 1 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.6 Transient responses in the pure rotation motion at two dierent speedsv = 1; 2; 5; 10 m/s: (a) rear levitation gapG 1 ; (b) front levitation gapG 2 ; (c) pitch rotation angle; (d) magnetic torqueT ; (e) levitation forceF l ; and (f) drag forceF d . . . . . . . . . . . . . . . . 104 7.7 Transient responses of the rigid-body Inductrack system with both vertical oscillation and pitch rotation atv = 1 m/s: (a) levitation gapG 3 ; (b) pitch rotation angle; (c)levitation forceF l ; and (d) magnetic torqueT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.1 Magnetic interaction between theith PM block and thekth track coil . . . . . . . . . . . . 127 B.2 Two closed independent paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 viii B.3 Increment of the distance vectorr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 C.4 Schematic of the original steady-state Inductrack model . . . . . . . . . . . . . . . . . . . . 132 ix Abstract As a new strategy for magnetic levitation envisioned in the 1990s, Inductrack systems with Halbach arrays of permanent magnets (PMs) have been applied to Maglev trains and intensively researched in various projects. In an Inductrack system, the magnetic interaction forces are coupled with the motion of the moving vehicle carrying Halbach arrays, which in many situations provokes complicated transient behaviors in the dynamic system. As a result, a steady-state-based Inductrack model, although having provided useful information for system design, cannot capture such transient features quantitatively with delity. In this research, a benchmark transient model of two degrees of freedom (2-DOF) for the Inductrack system is proposed. The highlight of this work lies in that the new model is derived from the rst laws of nature and is based on a complete transient scenario, without the assumption of any steady-state related quantities. It is found that the transient Inductrack system is governed by a set of nonlinear integro- dierential equations that fully describe the electro-magneto-mechanical couplings involved, which are solved by a combination of analytical and numerical approaches. Dynamic analysis of Inductrack systems with the new benchmark model consists of two stages. First, the proposed model is validated through comparison with the noted steady-state Inductrack model in the literature. Second, the transient response of the Inductrack system is obtained and analyzed with our model in several typical dynamic scenarios. From the numerical examples, it is believed that the newly-developed model can not only reproduce the well-known steady-state results but is also capable x of conducting in-depth transient analyses of Inductrack dynamic systems. Although only two degrees of freedom are initially considered, the proposed approach of modeling and analysis can be extended to general cases of multiple degrees of freedom, as enumerated with a 2D, 3-DOF rigid-body Inductrack system in this thesis. With the new transient model, it is discovered that an uncontrolled Inductrack system may be unstable even if the vehicle travels well below its operating speed and that instability can be persistent near and beyond the operating speed. It is, therefore, necessary to stabilize the system for safety and reliability. In this work, by taking advantage of the available 2-DOF benchmark transient model, a new feedback control method for Inductrack systems is proposed. In the control system development, active Halbach arrays are used as an actuator, and a feedback control law, which combines a properly tuned linear controller (either PID- or state feedback-based) and a nonlinear force-current mapping function, is created. The proposed control law is validated in numerical examples, where it eciently stabilizes the Inductrack system in a wide range of operating speeds. In the meantime, the controller renders a smooth system output (real-time levitation gap) with fast convergence to any prescribed reference step input (desired levitation gap). Keywords: Magnetic levitation, electrodynamic suspension, electro-magneto-mechanical coupling, Inductrack system, magnetic force, transient response, feedback control, nonlinear mapping function xi Chapter1 Introduction 1.1 Background Magnetic levitation or Maglev refers to the technology that utilizes magnetic forces to overcome the gravity and to suspend objects. Maglev has important engineering applications, especially in the eld of Maglev trains, where a high operating speed can be potentially achieved with the removal of frictions via non- contact motion. There are typically two Maglev strategies: (i) electromagnetic suspension (EMS), and (ii) electrodynamic suspension (EDS). The former is based on magnetic attraction forces between electromagnets and ferromagnetic guideways, while the latter is achieved by magnetic repulsive forces induced by the interaction between onboard magnets that are in motion and coils or conducting sheets that are stationary on the track. The onboard magnets in a traditional EDS system are superconducting magnets. As summarized by Lee [1] and many other researchers [2–7], the two Maglev strategies possess their own advantages and disadvantages. EDS systems are internally stable but can only achieve the levitation when the vehicle is in motion. In contrast, EMS systems can achieve the levitation even when the vehicle is stationary, but they are internally unstable and thus active feedback control is always needed. These two strategies have been both adopted by various Maglev train projects in dierent countries. 1 In 1990s, a novel EDS technique named "Inductrack" was proposed by researchers at the Lawrence Livermore National Laboratory (LLNL) [8, 9]. In this technique, the commonly used superconducting magnets are replaced by permanent magnets (PMs) that are aligned in a conguration called Halbach array. One important feature of Halbach array is that it can provide a single-sided strong magnetic eld on the track side. Like superconducting magnets, moving Halbach arrays interact with close-packed shorted circuit coils on the track, consequently generating magnetic repulsive forces that levitate the vehicle with Halbach arrays. The Inductrack technique has certain advantages over traditional EDS, such as weak magnetic eld on the vehicle side, cryogenic system free, and high lift-to-drag ratio at relatively low speeds [10]. Due to its low-cost and high-eciency potentials, the Inductrack concept has been applied to certain Maglev systems and considered in various research projects. The LLNL constructed a small model under NASA sponsorship to study the possibility of using Inductrack for rocket launching [11]. General Atomics (GA) developed low-speed urban Maglev trains based on an Inductrack conguration that contains double Halbach arrays with a ladder-track in between [12–15]. Proposed as the "fth mode" of transport, the Hyperloop system, in which passenger-carrying pods travel in vacuum tube at extremely high speed [16], has Inductrack as an option of its levitation strategy. Moreover, studies on Halbach arrays and Inductrack systems are also seen in applications like magnetic bearings, electrodynamic wheels, and aircraft take-o and landing assisting systems [17–19]. 1.2 Scope This research is concerned with mathematical modeling, dynamic analysis, and feedback control of Inductrack systems for Maglev transport. As shall be seen from the literature review later, although some work has been done on Inductrack systems, there is still a need to obtain an accurate and ecient model 2 for a better understanding of the transient behaviors and control designs for such kind of systems. The main purposes of this Ph.D. work include: 1. Derive a benchmark transient model for Inductrack systems based on the rst laws of nature; 2. Develop a solution method and a toolbox for simulation of the transient response of Inductrack systems; 3. Analyze the system stability and design model-based feedback control; 4. Extend the proposed model to a more general multi-degrees of freedom (M-DOF) case to address more realistic issues in an Inductrack system. To start with, a summary of the literature survey on the previous studies of Inductrack systems is reported in Sec. 1.3. 1.3 LiteratureSurvey Early research eorts on Inductrack systems were mainly focused on the characterization of magnetic elds and magnetic forces in steady-state circumstances. The original work can be found in an LLNL report by Post and Ryutov [8], in which the magnetic force predictions were partly veried by experiments on a small-scale demonstrative model [20]. The calculations therein are mainly based on the assumption of innite ideal sinusoidal magnetic eld, xed levitation gap and constant traveling speed of the vehicle, steady-state response of the induced coil currents, and averaged magnetic levitation and drag forces. The main results in Ref. [8] are agreed with those obtained by subsequent researchers. Other fundamental issues about Inductrack system characteristics have also been investigated. Murai and Hasegawa [21] presented numerical simulations of Inductrack magnetic levitation by a Fourier series analysis. Han et al. [22] optimized the geometry of Halbach arrays using scalar potential and Fourier series methods. Iniguez 3 and Raposo [23, 24] analyzed the low-speed force behavior of an Inductrack system and validated their results with a small-scale experimental device. Cho et al. [25] carried out a characteristic analysis of an EDS system with Halbach arrays based on a transfer relation theorem with magnetic vector potential, and they validated the results by using a high-speed rotary-type experimental facility. Ham et al. [26] studied a hybrid Halbach array mixed with PMs and electro-magnets, in which the control of magnetic eld is viable. Flankl et al. [27] derived power and loss scaling laws for EDS systems, and showed that their predictions were within 10% deviation compared to the 3D FEM simulations in terms of the drag coecient. Along with the characterization of magnetic elds and forces, mathematical modeling and dynamic analysis of Inductrack systems have also been performed. With an active magnet array conguration, Han [28, 29] developed a 6-DOF Inductrack dynamic model. It was found that the system is marginally stable, and that the motion of the Inductrack system is self-regulated in the lateral, roll, pitch and yaw directions. Kim and Ge [30, 31] investigated 1-, 2- and 4-DOF Inductrack systems through use of the LLNL’s small-scale Inductrack model for the NASA rocket launcher project. They concluded that the magnetic suspension system has no inherent damping in the lifting direction but can achieve a stable behavior in the traveling direction. Ko [32] applied wavelet transform to analyze the transient oscillatory response of an Inductrack system. It was found that such a system is subject to instability at low levitation height, which eventually led to a two-stage control design approach. Long et al. [33] used a linearized model to analyze the stability of a hybrid PM EDS-EMS system and applied a 1-DOF experimental mechanism to validate their controller design. Buth and Lu [34] studied the dynamic interaction between a exible guideway and an Inductrack Maglev vehicle, where the guideway was modeled as an Euler-Bernoulli beam and the vehicle as a 2-DOF spring-mass-damper system. Furthermore, a nite element model was created for numerical simulation, which showed that the magnetic force-motion coupling eect cannot be neglected when the vehicle moves fast. Pradhan and Katyayan [35] presented an analytical, simplied and linearized model and a numerical full-vehicle nonlinear model for Hyperloop capsules, with optimized parameters for design. 4 Other representative contributions in the areas of Halbach arrays and Inductrack systems are also seen in the literature throughout the recent two decades [36–58]. Nevertheless, while the previous investigations have provided good physical understanding and qualitative analysis of Inductrack systems, there still exist several key issues in modeling and analysis of this type of dynamic systems. First, most Inductrack models partly or completely make use of a steady-state assumption. A notable exception is with the 2-DOF innite-dimensional model devised by Storset and Paden [59, 60], although not much transient response computation based on this model has been reported. Steady-state-based models can be limited because transient responses are ignored, and in general they may not be used for stability analysis and feedback control. Second, as mentioned previously, the steady state of an Inductrack system is facilitated by an innite sinusoidal magnetic eld, steady-state coil currents and averaged magnetic forces. These presumed conditions are extremely dicult to maintain, if not impossible, especially in the case of transient motion. Third, the available expressions of magnetic forces often require a constant traveling speed and an invariant levitation gap for the moving vehicle, which do not reect the actual dynamic scenario when the vehicle experiences vibrations. As for the control application, although the feedback control of EMS systems [61–72] and traditional EDS systems [73–81] have been intensively studied, the feedback control problem of Inductrack systems has not been well addressed. Two main reasons contribute to this technical gap in the literature. First, the feedback control of an Inductrack system requires a truly transient model that can determine the time response of the system with delity. Such a transient model, however, was previously unavailable. Second, even with steady-state models, most control strategies need the linearization of the Inductrack dynamic system in consideration, which neglects some important nonlinear eects and complicated electro-magneto-mechanical couplings in the system. As matter of fact, no feedback control of Inductrack systems based on a completely transient model has been reported in the literature. 5 In summary of the above literature survey, it is necessary to develop accurate and computationally ecient transient models for in-depth understanding of transient behaviors and for stability analysis and feedback control of Inductrack systems. The development of a transient model should be based on the rst laws of nature with minimum assumptions and should cater for the actual engineering circumstances. To the best of the author’s knowledge, such a transient model for Inductrack systems has not yet been available. 1.4 MajorContributions This work is aimed to ll the above-mentioned technical gap of transient models and control methods of the Inductrack dynamic system. In this thesis, a new transient model with 2-DOF is proposed as a benchmark for modeling and analysis of Inductrack systems. The transient model is derived based on the fundamental principles of physics, without steady-state assumptions or presumed time-invariant quantities. This leads to a set of coupled nonlinear integro-dierential governing equations. After that, the governing equations of the transient model are cast into a state-space representation for solution. The transient model is validated with the previous results in the literature, and a numerical study exhibits the transient behaviors of Inductrack systems. Based on the newly-developed model, nonlinear feedback control of the Inductrack system is addressed and the model is extended to a 2D rigid body case with 3-DOF. At the end, concluding remarks and future research plans based on the presented work are mentioned. The work reported in this thesis is a summary and a natural extension of the precursor investigations by the author [82–86]. This research on Inductrack systems contains the following special features and new discoveries: (a) The model derivation starts from the rst laws of nature, without the initial assumption on an ideal sinusoidal magnetic eld. In the mathematical modeling, analytical formulas for magnetic elds and forces 6 are obtained and the non-sinusoidal distribution of the magnetic ux of the Halbach arrays due to their nite dimensions and geometries is precisely determined. (b) The determination of quantities like magnetic ux, induced current, and magnetic force is completely based on the most general transient dynamic scenarios, with no use of any steady-state condition. Instead of applying averaged forces, the magnetic interaction forces between a magnet block and a track coil are derived in detail, and they can be determined at any point of interest. (c) A state-space representation of the Inductrack system governed by a set of integro-dierential equations is established, which is followed by a procedure for numerical simulation of the system transient response. (d) A validation of the proposed model is done through comparisons with the noted steady-state results in the literature. To this end, the following steps are taken. First, the magnetic eld distribution created by Halbach arrays of nite dimensions is exhibited and analyzed. Second, a steady-state truncation of the transient model is made and the results in terms of steady-state magnetic forces are compared with those obtained from the steady-state models in literature. Third, the convergence characteristic of magnetic forces computed with dierent numbers of chosen coils is studied and veried. Meanwhile, from the steady-state validation, certain system parameters, such as critical speed and critical levitation gap, are estimated, which is very useful to system design and dynamic simulation. (e) As a major task of this work, the transient response of the Inductrack system is thoroughly investigated via numerical simulation. Four cases of dynamic scenarios are considered in the simulation: (i) a vehicle traveling at a constant speed; (ii) a vehicle in free motion; (iii) a vehicle subject to a constant prescribed propulsion force; and (iv) a vehicle in a process of acceleration and deceleration. In these cases, with dierent proles of the propulsion force, the corresponding motion of the vehicle in both the longitudinal and vertical directions is computed and discussed. As shall be seen from the simulation 7 results, an ideal steady-state response can hardly occur in these dynamic cases, which corroborates the necessity to develop a full transient model for Inductrack systems as presented in this work. (f) The model-based feedback control of the Inductrack dynamic system is addressed with the proposed transient model. In the control system conguration, active Halbach arrays are used as an actuator, which is installed in a transient Inductrack system for the rst time. In the development of the feedback control law, an eective two-step design approach is proposed. In the rst step, a properly tuned linear controller (with PID or state feedback) based on a linear reference model gives a control eort, resulting in an estimated levitation force. In the second step, a nonlinear mapping function, which takes the estimated levitation force as the input and computes the current density of the active Halbach arrays as the output, is generated based on the new transient Inductrack model. The active current density from the mapping function is the actual input to the transient Inductrack system, and it controls the actual output (levitation gap) of the moving vehicle. The so designed feedback controller ultimately stabilizes the Inductrack system in a wide range of traveling speed and enables the transient response (levitation gap) of the system to track an arbitrarily prescribed reference value. (g) The proposed modeling and simulation approaches are extended to the case of a 2D, 3-DOF rigid body Inductrack system, where more interesting results are obtained and analyzed. Evidently, the proposed transient model lays out a reliable foundation for modeling, analysis, and control of Inductrack systems. Although at most only three degrees of freedom are considered in this work, the methodology of modeling and analysis as presented can be further expanded to general Inductrack systems with multi-degrees of freedom. 1.5 FormatoftheDissertation The remainder of the thesis reports the main discoveries of this research and is organized as follows. The Inductrack system in consideration is described with basic assumptions stated and coordinate system 8 established in Chap. 2. The detailed modeling process and solution procedures of the transient Inductrack system are presented in Chap. 3. Based on the proposed model, a steady-state validation of the model is done in Chap. 4. With this validation, numerical examples of transient response of Inductrack systems are shown in Chap. 5, which corroborates that the proposed model can perform dynamic analysis of Inductrack systems. Based on the new transient model, the feedback control strategy of Inductrack dynaamic systems is discussed in Chap. 6. Furthermore, the model extension to rigid body conguration is conducted in Chap. 7. Lastly, the main contributions and future research directions of this work are summarized in Chap. 8. 9 Chapter2 ProblemStatement Prior to the model derivation, the schematic of the benchmark transient Inductrack system is proposed in this chapter along with the essential assumptions and the setup of the coordinates systems. 2.1 SystemDescription The Inductrack system in consideration is shown in Fig. 2.1, where a vehicle carrying Halbach arrays of PMs is moving above a track with an innite number of coils. Two identical Halbach array sets are mounted symmetrically beneath the uniform cuboid vehicle compartment by an inter-distancel M . The coils are window-framed with unied widths w and heights h, and are placed on the track along the longitudinal direction by equal center-to-center distancesd c . Figure 2.1(c) shows the conguration of a Halbach array set, in which each cuboid magnet block has widtha, lengthb and heightd. Assume thatb =d, which makes a square cross section of the PM blocks. As seen in Fig. 2.1(c), every four PM blocks line up a single wavelength () of Halbach array, namely, = 4b. Each array set can contain multiple wavelengths of arrays, denoted byN . An additional magnet block is attached to the end of the set for maintaining the property of symmetry. With this setting, there are totally 2N b PM blocks on the Inductrack system, whereN b = 4N + 1, withN being a positive integer chosen in design. 10 Figure 2.1: Schematic of an Inductrack system: (a) in front view; (b) in side view; (c) the conguration of a Halbach array set 2.2 BasicAssumptions In this work, the following basic assumptions are made for model developments and comparisons with the existing results in the literature. Assumption 1: 2-DOF motion. The goal of this eort is to investigate the transient dynamics of Inductrack systems and to understand the basic coupling mechanism between magnetic forces and vehicle motion in such a system. Thus, without loss of generality, the motion of a vehicle carrying Halbach arrays is primarily constrained in two directions - longitudinal and vertical, resulting in a benchmark 2-DOF 11 Inductrack model of a "volumetric lumped mass". Additionally, for the validation purposes, numerical simulation results based on the new transient model are readily compared with those original results that were also obtained by a lumped-mass conguration. The 2-DOF model can be extended to general M-DOF cases after the coupling mechanism in Inductrack systems is well understood. Assumption 2: Negligible size for induced coil currents. For eciency and accuracy in computation of transient response, the induced current circulating in each coil is considered dimensionless in thickness, or as a concentrated lamentary current. The negligible size for induced current has been used in most of the previous investigations, as well as in standard textbooks on electronics. Assumption3: Negligibleeectoftheinducedmagneticeldbycoil-coilinteractions. Com- pared to the source magnetic eld of the Halbach array set, the induced magnetic eld of a coil is signi- cantly small. As such, the eect of the induced magnetic eld on other track coils is ignored. Assumption 4: Prescribed propulsion force. The primary interest of this study is the dynamic behaviors of the Inductrack system in the levitation direction. Hence, the propulsion force acting on the vehicle along its traveling direction is simply treated as a known or given force. This assumption, of course, can be lifted once a model of electromagnetic propulsion force is used. 2.3 CoordinateSystems As mentioned previously, the magnetic forces in an Inductrack system, which are due to interaction between the source magnetic eld (produced by moving PMs) and the induced magnetic eld (produced by coils with induced currents), are motion-coupled. For convenience in modeling and analysis, dierent reference frames are needed for the description of moving source magnetic eld, changing locations of track coils measured from the vehicle, and for the time-varying electromagnetic quantities such as 12 magnetic ux, electromotive force (emf), induced current regarding each coil in order to calculate the magnetic forces. By the assumptions given in Sec. 2.2, two coordinate systems are dened: stationary inertial frame o I x I y I z I and moving magnet frameo M x M y M z M . As shown in Figs. 2.1(a) and 2.1(b), the origin of the inertial frameo I is located at the geometry center of the upper rim of a selected coil; the origin of the magnet frameo M is placed at the centroid of the far left PM block;l G ,h G denotes the horizontal and vertical distance between the origin of magnet frameo M and the center of mass (COM) of the vehicle G, respectively; andY (t) andZ(t) are the longitudinal and vertical displacements measured by the COM of the vehicle, respectively. Hereby, the transformation between the two dened coordinate systems is described by x M =x I y M =y I Y (t) +l G z M =z I Z(t) +h G (2.1) Furthermore, the parameter(t) in Fig. 2.1(b) measures the levitation gap and it is given by (t) =Z(t)h G 0:5d (2.2) To compute electromagnetic quantities regarding each coil, the location of each coil needs to be identied in real time under the moving magnet frame. The location of the kth coil is dened by the coordinates of the centroid of the upper rim of the coil, which are denoted by ( x k I ; y k I ; z k I ) in the inertial fame and by ( x k M ; y k M ; z k M ) in the magnet frame. Note that x k I = z k I = 0 because of the placement of the 13 coils along the track shown in Fig. 2.1(a). This, by Eq. (2.1), gives the location of the kth coil in the magnet frame as follows x k M = 0 y k M = y k I Y (t) +l G z k M =Z(t) +h G (2.3) As seen from Eq. (2.3), coordinates ( x k M ; y k M ; z k M ) are time-varying and they are coupled with the vehicle motion. Coordinates ( x k I ; y k I ; z k I ), on the other hand, are time-invariant. Therefore, the coupling mechanism of the Inductrack system originates from Eq. (2.3), which, as shall be seen in the subsequent derivations, eventually leads to convoluted time-varying electromagnetic quantities. The coordinate transformation presented herein can be extended to M-DOF models of Inductrack systems, for which both translation and rotation are involved in the vehicle motion. 14 Chapter3 ModelingoftheBenchmarkTransientInductrackSystem The establishment of a transient model for the Inductrack system in Fig. 2.1 takes four major steps. In the rst step, the analytical formulas for the source magnetic eld generated by the onboard Halbach arrays are derived. In the second step, the currents in the track coils that are induced by the motion of the vehicle are determined. In the third step, the interaction between the source magnetic eld and the induced magnetic eld is investigated, leading to the analytical expressions of magnetic levitation and drag forces that are applied to the vehicle. Finally, in the fourth step, application of Newton’s second law yields a set of nonlinear governing equations for the Inductrack model, which are then cast into state representations for solution. These steps are detailed in sequel. 3.1 OnboardSourceMagneticField In the Inductrack system, levitation of the vehicle is essentially caused by the moving source magnetic eld that is generated by the onboard PMs in Halbach arrays. Because these Halbach arrays in Fig. 2.1 are of nite length, an ideal sinusoidal distribution of the magnetic eld under the assumption of innite length of arrays, which is commonly adopted in most of the previous studies, is invalid. Indeed, to obtain an accurate transient model for the Inductrack system, a magnetic eld, which is produced by those Halbach 15 arrays with nite dimensions and which correspondingly has end or edge eects and decaying properties, must be derived. This is one key that dierentiates the current research from the previous investigations. It is known from physics that a PM block can be modeled as innite number of magnetization current loops circulating on its surface, with the direction of current circulation and the pole direction of the PM determined by the right-hand rule. The magnetic eld at any point in space caused by a current element on these loops can be formulated by the Biot-Savart law. Therefore, the magnetic eld of a PM block at a point can be expressed via a double integration, along the loop circulation direction and along the pole direction, respectively. Accordingly, the magnetic eld generated by Halbach arrays can be determined by superposition of the magnetic elds of all the PM blocks. It is also known that in a Halbach array conguration, there are four possibilities of the pole direction. Although a current loop method is available [87], due to the complexity of the problem in consideration, a coherent formulation of the source magnetic eld needs to be established for the calculation of time-varying electromagnetic quantities. This formulation is derived as follows. Figure 3.1: Magnetization current model for a PM block in Case 1 For the Halbach array conguration shown in Fig. 2.1(c), four possible cases of the pole direction of a PM are considered: Case 1 – the pole in the upward (+z M ) direction; Case 2 – the rightward (+y M ) direction; Case 3 – the downward (z M ) direction; and Case 4 – the leftward (y M ) direction. A schematic diagram of a PM in Case 1 is sketched in Fig. 3.1. Here for simplicity in presentation and derivation, 16 notationoxyz is used for the magnet frameo M x M y M z M , with the origino located at the centroid of the PM block. According to the Biot-Savart law [88], the magnetic ux densityB 1 at a pointP (x;y;z) outside the block is given by B 1 = 0 4 Z z I c 1 i ds 1 r 1 r 3 1 (3.1) wherei ds 1 represents a current element along the current loopc 1 , the pole is in the positivez direction, r 1 is the distance vector measured from this element to pointP withjr 1 j =r 1 , and 0 is the permeability of vacuum. The subscript "1" ofB 1 implies the rst case of the pole direction. For theith case,B i is used. On loopc 1 , four current segments of dierent directions are considered individually. For simplication, the magnetization current density distributed along the pole direction is treated as a constant J M , i.e., i =J M dz. By setting constantK = 0 J M 4 , Eq. (3.1) is reduced to B 1 = 4 X j=1 B 1j , withB 1j =K Z z Z c 1j ds 1j r 1j r 3 1j dz (3.2) where the second subscriptj ofB 1j represents the number of the segmented current along loopc 1 , as labeled in Fig. 3.1. For computational purposes, writeB 1j in Eq. (3.2) in the component form: B 1j =B x 1j (x;y;z)i +B y 1j (x;y;z)j +B z 1j (x;y;z)k (3.3) where scalarsB x 1j , B y 1j , B z 1j are components ofB 1j in thex-, y- andz-direction, withi, j, k being the corresponding unit vectors, respectively. These scalars have dierent analytical forms depending on the superscriptsx,y,z and the indexj of current segments. By performing the double integration in Eq. (3.2), 17 the analytical expressions of these components of the magnetic ux density are derived. Some of the expressions are given below: B y 11 = 0 B y 12 = K 2 [ x (x a 2 ;y b 2 ;zz 0 ) x (x + a 2 ;y b 2 ;zz 0 )]j z 0 = d 2 z 0 = d 2 B y 13 = 0 B y 14 = K 2 [ x (x a 2 ;y + b 2 ;zz 0 ) x (x + a 2 ;y + b 2 ;zz 0 )]j z 0 = d 2 z 0 = d 2 (3.4) and B z 11 =K[ x (x a 2 ;y b 2 ;zz 0 ) x (x a 2 ;y + b 2 ;zz 0 )]j z 0 = d 2 z 0 = d 2 B z 12 =K[ y (x a 2 ;y b 2 ;zz 0 ) y (x + a 2 ;y b 2 ;zz 0 )]j z 0 = d 2 z 0 = d 2 B z 13 =K[ x (x + a 2 ;y b 2 ;zz 0 ) x (x + a 2 ;y + b 2 ;zz 0 )]j z 0 = d 2 z 0 = d 2 B z 14 =K[ y (x a 2 ;y + b 2 ;zz 0 ) y (x + a 2 ;y + b 2 ;zz 0 )]j z 0 = d 2 z 0 = d 2 (3.5) Herez 0 is a dummy variable and the notationfj z 0 =z + 0 z 0 =z 0 meansf(z 0 = z + 0 )f(z 0 = z 0 ). Functions x , x , y in Eqs. (3.4) and (3.5) are of the form x ( x ; y ; z ) = ln x q 2 x + 2 y + 2 z x + q 2 x + 2 y + 2 z (3.6) x ( x ; y ; z ) = arctan 0 B @ y z x q 2 x + 2 y + 2 z 1 C A (3.7) y ( x ; y ; z ) = arctan 0 B @ x z y q 2 x + 2 y + 2 z 1 C A (3.8) 18 Due to limited space, the expressions forB x 1j are not listed here. For any 2D Inductrack model,B x 1j has no eect on the magnetic forces and thus the vehicle dynamics, which will be self-evident after going through the entire derivation. If the PM in Fig. 3.1 is replaced by a block in Case 2 with the pole in the positive y direction, the magnetic ux density can be obtained from slight revision of Eq. (3.2), and it is given by B 2 = 4 X j=1 B 2j , withB 2j =K Z y Z c 2j ds 2j r 2j r 3 2j dy (3.9) Similarly, with Eq. (3.9), the components ofB 2j are obtained in analytical form B y 21 =K[ x (x a 2 ;yy 0 ;z d 2 ) x (x a 2 ;yy 0 ;z + d 2 )]j y 0 = b 2 y 0 = b 2 B y 22 =K[ z (x a 2 ;yy 0 ;z + d 2 ) z (x + a 2 ;yy 0 ;z + d 2 )]j y 0 = b 2 y 0 = b 2 B y 23 =K[ x (x + a 2 ;yy 0 ;z d 2 ) x (x + a 2 ;yy 0 ;z + d 2 )]j y 0 = b 2 y 0 = b 2 B y 24 =K[ z (x a 2 ;yy 0 ;z d 2 ) z (x + a 2 ;yy 0 ;z d 2 )]j y 0 = b 2 y 0 = b 2 (3.10) and B z 21 = 0 B z 22 = K 2 [ x (x a 2 ;yy 0 ;z + d 2 ) x (x + a 2 ;yy 0 ;z + d 2 )]j y 0 = b 2 y 0 = b 2 B z 23 = 0 B z 24 = K 2 [ x (x a 2 ;yy 0 ;z d 2 ) x (x + a 2 ;yy 0 ;z d 2 )]j y 0 = b 2 y 0 = b 2 (3.11) where y 0 is a dummy variable, functions x and x are the same as given in Case 1, and z is a new function that is dened below z ( x ; y ; z ) = arctan 0 B @ x y z q 2 x + 2 y + 2 z 1 C A (3.12) 19 For PM blocks in Cases 3 and 4, due to the property of symmetry, B 3 = 4 X j=1 B 3j = 4 X j=1 (B 1j ) =B 1 (3.13) B 4 = 4 X j=1 B 4j = 4 X j=1 (B 2j ) =B 2 (3.14) which can be simply computed by switching the signs of Eqs. (3.4), (3.5), (3.10) and (3.11). With Eqs. (3.2) to (3.14), by superposition and coordinate transformation, the magnetic eld at pointP that is generated by all the PM blocks in Halbach arrays can be calculated. Thus, without approximation, the source magnetic eld is expressed analytically and the nite dimensions of the Halbach arrays on the vehicle are naturally considered. 3.2 InducedCoilCurrents In the proposed benchmark model, each track coil is modeled as a rst order inductive circuit with the equivalent lumped inductanceL eq and resistanceR eq . Thus, according to the assumptions in Sec. 2.2, the induced current circulating thekth coilI k caused by the relative motion between the vehicle and track is governed by the Kirchho’s voltage law [89]: L eq _ I k +R eq I k =e k (3.15) wheree k is the emf induced in thekth coil by the time-varying source magnetic eld at this coil location. 20 Figure 3.2: Circuit model of thekth track coil Figure 3.2 illustrates thekth circuit coil on the track measured from the moving magnet frameo M x M y M z M . As dened in Chap. 2, the location of this coil is ( x k M ; y k M ; z k M ). LetB k represent the magnetic ux density (also measured from magnet frame) in the planey M = y k M , where the coil stands. Denote the components ofB k in thex M -,y M -, andz M -direction byB x M k ,B y M k ,B z M k , respectively. According to the Faraday’s law [90], the emf induced in thekth coil by the source magnetic eld of the Halbach arrays on the moving vehicle is e k = Z S k @B k @t dS k (3.16) where vectorS k indicates the area enclosed by the frame of thekth coil, with its direction in the normal of the frame, i.e.,S k =S k j. It follows from Eq. (3.16) that e k = Z S k @B y M k @t dS k (3.17) Equation (3.17) indicates that the emf in each coil is only relevant to the component of the magnetic eld in the longitudinal (y M ) direction. Nevertheless,B y M k still depends on the vehicle’s motion in both the longitudinal and vertical directions, namely, Y (t) and Z(t). Thus, application of the chain rule to Eq. (3.17) gives e k = dy M dt Z S k @B y M k @y M dS k dz M dt Z S k @B y M k @z M dS k (3.18) 21 which, by the geometry of the coil and Eq. (2.1), leads to e k = _ Y (t) Z w 2 w 2 Z z k M z k M h @B y M k @y M (x 0 ; y k M ;z 0 ) dz 0 dx 0 + _ Z(t) Z w 2 w 2 B y M k (x 0 ; y k M ;z 0 )j z 0 = z k M z 0 = z k M h dx 0 (3.19) where x 0 and z 0 are dummy variables; y k M and z k M are the time-varying coil locations as described in Eq. (2.3). Equation (3.19) reveals the essential coupling mechanism in the 2-DOF Inductrack dynamic system: the induced emf (or the induced current from Eq. (3.15)) in each coil is coupled with both the displacement and velocity of the moving vehicle in both the longitudinal and vertical directions. This coupling consequentially introduces complexities and nonlinearities to this transient Inductrack model. Because the coils are rectangles, exact analytical forms for Eq. (3.19) can be obtained. The derivation involves evaluation of integrals like R x + x R z + z @B y @y dz 0 dx 0 and R x + x B y j z + z dx 0 . Following the derivation of Eqs. (3.2) to (3.14), the analytical expressions of the terms in Eq. (3.19) are derived and they are provided in Appx. A. The formulas in the appendix are useful in solution and dynamic analysis of the Inductrack system. In conclusion of this subsection, given a vehicle motion (position and velocity), the emfe k in each coil can be computed by Eq. (3.19), and with the knowledge ofe k , the induced currentI k can be determined by easy solution of Eq. (3.15). 3.3 MagneticForces In the above sections, the source magnetic eld generated by the Halbach arrays on the moving vehicle and the induced currents in the track coils are obtained. Physically, the currents induced in the track coils produce an induced magnetic eld, which in turn interacts with the source magnetic eld, resulting in magnetic forces applied onto the Halbach arrays and consequently at the vehicle. In the meantime, by the 22 law of action-reaction, magnetic forces are also applied to the coils by the magnets. In this section, these magnetic interaction forces are derived. The induced magnetic eld by the track coils can be obtained through use of the generalized Biot- Savart law, which is valid for time-varying currents. However, in this application, the coil currents hardly change within the time for light to travel from the coils to the onboard PMs. Because of this, the induced magnetic eld generated by the time-varying currents can be approximated by a quasi-static Biot-Savart law with adequate precision [91]. It means that the formula in Eq. (3.1) is still valid for calculating the induced magnetic eld, even though its "source" currentI k is time-varying. Thus, with the source and induced magnetic elds, their interaction forces can be estimated. 3.3.1 EquivalentMagneticForces The essence of magnetic forces in the Inductrack system is the appearance of Ampere forces due to the interaction between the PMs and the track coils of induced currents. With the PMs modeled as current loops earlier, Ampere forces acting on either a PM block or a coil can be derived in a unied way. For the convenience and clarity in the subsequent derivation, the following symbols are dened. F B ik : Ampere force acting on theith PM block, due to the induced magnetic eld produced by thekth coil;F C ik : Ampere force acting on thekth coil, due to the source magnetic eld produced by theith PM block; F B k : the summation of Ampere forces acting on all the PM blocks, due to the induced magnetic eld produced by thekth coil; F C k : the summation of Ampere forces acting on thekth coil, due to the source magnetic eld produced by all the PM blocks;F B : the resultant magnetic force acting on the entire Halbach arrays or on the vehicle; andF C : the resultant magnetic forces acting on all the track coils. 23 By the previous denitions, it is easy to see that F B = 1 X k=1 F B k = 1 X k=1 2N b X i=1 F B ik F C = 1 X k=1 F C k = 1 X k=1 2N b X i=1 F C ik (3.20) Theorem: The resultant magnetic forcesF B andF C are in the following equivalent relation F B = 1 X k=1 F B k = 1 X k=1 2N b X i=1 F B ik = 1 X k=1 2N b X i=1 (F C ik ) = 1 X k=1 (F C k ) =F C (3.21) Proof: As shown in Appx. B,F B ik =F C ik , which in double sum leads to Eq. (3.21). This proof can be done by considering the interaction force between two current loops [92] as PMs have been modeled as current loops. Equation (3.21) and Appx. B show that Newton’s third law (the law of action-reaction) still holds for the non-contact magnetic interaction forces in the Inductrack system. The theorem is useful in computing the resultant magnetic forceF B acting on the vehicle. For thekth coil, it is much simpler to computeF C k thanF B k because a coil only contains a single lamentary current loop. Moreover, the source magnetic eldB k at each coil location has already been dened and formulated in Secs. 3.1 and 3.2. Therefore, for eciency and accuracy in calculation of magnetic forces, it is preferred to computeF C k rst and then determineF B k by the equivalence relation (3.21). 3.3.2 MagneticLevitationandDragForce Figure 3.3 displays the schematic of the kth coil for magnetic force calculation, where I k dl k is the dierential current vector of a segment or element of the pathc k that is enclosed by the frame of thekth coil, and dF C k is an Ampere force acting on the current element. For the convenience in derivation, loop 24 c k is dissected into four individual pathsc k j , withj = 1; 2; 3; 4. The dierential current vector and the ux density corresponding to the path number are dened in the same manner, namely, byI k dl k j and B k j . Other notations are the same as they were given in Fig. 3.2. Figure 3.3: Schematic of thekth circuit coil for Ampere force calculation According to Ampere’s force law [93], the total Ampere force applied to thekth coilF C k is written as F C k =I k 4 X j=1 Z c k j dl k j B k j (3.22) whereB k j =B x M k j i +B y M k j j +B z M k j k, and vectors dl k j ofc k j are given by dl k 1 =i dx M , dl k 2 =k dz M , dl k 3 =i dx M , dl k 4 =k dz M (3.23) Substitute Eq. (3.23) into Eq. (3.22) to obtain Z c k 1 dl k 1 B k 1 = Z x M (B z M k 1 j +B y M k 1 k) dx M Z c k 2 dl k 2 B k 2 = Z z M (B y M k 2 iB x M k 2 j) dz M Z c k 3 dl k 3 B k 3 = Z x M (B z M k 3 jB y M k 3 k) dx M Z c k 4 dl k 4 B k 4 = Z z M (B y M k 4 i +B x M k 4 j) dz M (3.24) 25 By the property of symmetry ofB k j , there must beB x M k 2 =B x M k 4 andB y M k 2 =B y M k 4 , so the summation of the second and the fourth equations in Eq. (3.24) becomes zero; that is, Z c k 2 dl k 2 B k 2 + Z c k 4 dl k 4 B k 4 =0 (3.25) This implies that that neither the vertical coil rims nor the magnetic eld components along the crosswise direction (B x M k j ) contributes to the resultant magnetic force that eventually acts on the vehicle. With the coil geometry in consideration, substitution of Eqs. (3.24) and (3.25) into Eq. (3.22) yields F C k =jI k Z w 2 w 2 [B z M k 3 (x 0 ; y k M ; z k M h) [B z M k 1 (x 0 ; y k M ; z k M )] dx 0 +kI k Z w 2 w 2 [B y M k 1 (x 0 ; y k M ; z k M ) [B y M k 3 (x 0 ; y k M ; z k M h)] dx 0 (3.26) where y k M and z k M have been dened by Eq. (2.3). Finally, by the equivalence relation described in Eq. (3.21), the resultant magnetic force acting on the vehicle is F B =F l kF d j (3.27) whereF l andF d , from Eq. (3.26), are given by F l = 1 X k=1 I k Z w 2 w 2 [B y M k 3 (x 0 ; y k M ; z k M h) [B y M k 1 (x 0 ; y k M ; z k M )] dx 0 F d = 1 X k=1 I k Z w 2 w 2 [B z M k 3 (x 0 ; y k M ; z k M h) [B z M k 1 (x 0 ; y k M ; z k M )] dx 0 (3.28) The components F l and F d in the previous equation are the well-known magnetic levitation force and magnetic drag force, respectively. Computation of these forces by Eq. (3.28) involves the evaluation of integrals R x + x B y dx 0 and R x + x B z dx 0 , for which the formulas in Appx. A can be conveniently used. 26 3.4 GoverningEquations Assume that the vehicle is subject to a prescribed propulsion forceF p in the longitudinal direction, which can be any specied function of time. Let the total mass of the vehicle (including those PM blocks) beM. The equations of motion of the 2-DOF vehicle are established by Newton’s second law as follows M Y =F p F d M Z =F l Mg (3.29) whereY andZ are the longitudinal and vertical displacements of the vehicle as shown in Fig. 2.1, g is the gravitational acceleration, andF l andF d are the magnetic forces given in Eq. (3.28). Also, because the magnetic forces are dependent on the induced coil currents, the coil equations given in Sec. 3.2 are needed, namely, L eq _ I k +R eq I k =e k , k = 1; 2; 3;::: (3.30) where the emfe k can be computed by Eq. (3.19). Equations (3.29) and (3.30), along with the electro-magneto-mechanical coupling relations (3.19) and (3.28), completely describe the transient response of the Inductrack system. Note that Eqs. (3.19) and (3.28) involve integrands that are nonlinear functions of the vehicle displacements (Y andZ) by the coordinate transformation (2.3). This indicates that the governing equations (3.29) and (3.30) of the Inductrack system form a set of nonlinear integro-dierential equations. 3.5 State-SpaceRepresentation For convenience and clarity in presenting the dynamic analysis results, the 2-DOF transient model of Inductrack systems that has been developed so far is briey summarized rstly. Afterwards, a state-space 27 representation of the model is established in this section and a corresponding solution procedure for numerical simulation is given in the next section. Figure 3.4: The Inductrack system that has been modeled The Inductrack system in consideration is sketched in Fig. 3.4. The governing equations obtained are as follows M Y =F p F d M Z =F l Mg L eq _ I k +R eq I k =e k , k = 1; 2; 3;::: (3.31) whereY andZ are the longitudinal and vertical displacements of the vehicle;F p is a prescribed propulsion force;F l andF d are the magnetic levitation force and magnetic drag force, respectively; andI k ande k are the induced current and emf in thekth track coil. The rst two equations of Eq. (3.31) are about the motion of the vehicle and the third one is about the coil currents. The magnetic forces and the induced emf can be computed by F l = 1 X k=1 I k Z w 2 w 2 [B y M k 3 (x 0 ; y k M ; z k M h) [B y M k 1 (x 0 ; y k M ; z k M )] dx 0 (3.32) 28 F d = 1 X k=1 I k Z w 2 w 2 [B z M k 3 (x 0 ; y k M ; z k M h) [B z M k 1 (x 0 ; y k M ; z k M )] dx 0 (3.33) e k = _ Y (t) Z w 2 w 2 Z z k M z k M h @B y M k @y M (x 0 ; y k M ;z 0 ) dz 0 dx 0 + _ Z(t) Z w 2 w 2 B y M k (x 0 ; y k M ;z 0 )j z 0 = z k M z 0 = z k M h dx 0 (3.34) whereB y M k j andB z M k j are the horizontal and vertical components of the magnetic ux density of the source magnetic eld at thekth coil, respectively, wherej = 1 refers to the coil’s upper rim andj = 3 refers to its lower rim; and x k M ; y k M ; z k M are the coordinates in the magnet frame that are related to the displacements and the levitation gap of the vehicle by x M =x I y M =y I Y (t) +l G z M =z I Z(t) +h G (t) =Z(t)h G 0:5d (3.35) Refer to the previous derivations for the detail of all the system parameters. For numerical solution of the governing equations for the Inductrack system, a state-space represen- tation is established. To this end, select a nite numberN of the coils. According to Eq. (3.31), totalN + 4 state variables are dened as follows x 1 =I 1 , x 2 =I 2 , :::, x N =I N , x N+1 =Y , x N+2 = _ Y , x N+3 =Z, x N+4 = _ Z (3.36) where the rstN state variables are the currents of theN coils, and the last four are the displacements and velocities of the vehicle. 29 Introduction of these state variables (3.36) to Eq. (3.31) eventually yields the following N + 4 state equations: _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ) _ x N+1 =x N+2 _ x N+2 =f d (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ) + F p M _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 )g (3.37) wherek = 1; 2;:::;N andf k ,f l andf d are nonlinear functions that can be obtained from Eqs. (3.32)-(3.34). Functionsf l andf d are related to the magnetic drag and levitation forces according to Eqs. (3.32) and (3.33). Figure 3.5: Timely updated track coils in the spatial window that moves with the vehicle One special feature of the state space representation for the Inductrack system is that the rstN state variables (coil currents) are updated according to the real-time position of the vehicle. The purpose is to predict the magnetic forces accurately and eciently. In this work, theN coils are timely selected and updated within a prescribed spatial window that is xed at and moves with the vehicle; see Fig. 3.5. In other words, at any time, those coils in the window are in the closest neighborhood of the Halbach arrays on the vehicle. If the numberN of coils is large enough, due to the fast decay of magnetic interactions, those coils outside the window can be negligible. As the vehicle travels, some "new" coils enter the window 30 and an equal number of "old" coils exit it. Hence, in computation of the transient response of the Inductrack system, the rstN state equations in Eq. (3.37) must be updated timely. The state-space representation exhibits a mechanism of electro-magneto-mechanical coupling in the Inductrack system: the magnetic forces (F l andF d ) are coupled with the induced coil currents and the vehicle motion, while each coil current is dependent on both the displacement and velocity of the vehicle. Without the coupling mechanism as described by the state equations (3.37), it will be very dicult to investigate the transient response and stability of the Inductrack system with delity. 3.6 SolutionProcedure Because the state equations (3.37) are nonlinear and complicated, numerical solutions via Runge-Kutta method are perused. As mentioned in the previous subsection, theN coils corresponding to the rstN states need to be timely updated according to the real-time position of the vehicle in longitudinal direction, x N+1 (t). As a result, in numerical simulation, the rst N state variables in Eq. (3.37) will be updated according to the spatial window shown in Fig. 3.5. With this understanding, a solution procedure for the transient Inductrack model takes the six steps in the following. Step 1: Initialization. A set ofN track coils in sequence is initially selected. These coils are placed symmetrically with respect to the origin of the inertial frame, and they are in the spatial window as shown in Fig. 3.5. The integer N is such that the spanned length by the coils in the longitudinal direction is adequately greater than the length of vehicle (this will be discussed quantitively later). Without loss of generality, the initial conditions for Eq. (3.37) are set by x k (0) = 0, k = 1; 2;:::;N x N+1 (0) = 0, x N+2 (0) = _ Y (0) x N+3 (0) =Z(0), x N+4 (0) = 0 (3.38) 31 Also, set up a small time intervalT for numerical simulation at timest n =nT ,n = 0; 1; 2;:::. Step 2: Determination of the coil locations. With the vehicle displacements x N+1 (Y (t)) and x N+3 (Z(t)), the location of each coil in the magnet frame ( x k M ; y k M ; z k M ) are obtained by the coordinate transformation (3.35). Step 3: Input calculation. Results of input functions f k , f l and f d in Eq. (3.37) are computed through use of Eqs. (3.32)-(3.34), with the knowledge of the present values for all state variables and the coil locations in the magnet frame. Step 4: ODE solution. The state equations (3.37) are solved numerically by Runge-Kutta method in the current time stepT , which yields the values of the state variables att 1 . Step 5: Update of state variables. Evaluate x N+1 (t 1 ) and decide if a "shift" of the rst N state variables (coil update by the window in Fig. 3.5) is necessary. The condition for a shift of state variables is x N+1 (t 1 ) d c x N+1 (0) d c 1 (3.39) where the symbol "[ ]" refers the mathematical operation of oor rounding, andd c is the distance between two adjacent coils (Fig. 3.4). Under the condition (3.39), the rstN state variables are updated as follows x 1 (t 1 ) =x 2 (t 1 ), x 2 (t 1 ) =x 3 (t 1 ), :::, x N1 (t 1 ) =x N (t 1 ), x N (t 1 ) = (3.40) where is a small number due to the decay of the magnetic eld. Note that the result described by left- hand side of Eq. (3.39) reects the number of coils that the vehicle has passed during a time intervalT . In simulation,T is chosen adequately small so that the right-hand side of Eq. (3.39) always gives either 0 or 1, which thus guarantees the validity of the state shifting in Eq. (3.40). Step6: Iteration. Treat the updated values of the state variables as a new set of initial conditions. At the current timet n , this is done by settingt n ! 0 andt n+1 ! t 1 . Go back to Steps 2 to 5 to determine 32 the state variables at timet n+1 . This process of iteration continues until a preset nal simulation time is reached. With the benchmark 2-DOF transient model developed in Chaps. 2 and 3, a state-space representation in Sec. 3.5 and a solution procedure in Sec. 3.6 have been established. This allows numerical simulation of the transient response of the Inductrack system. 33 Chapter4 ModelValidation With the state-space representation and numerical solution procedure given in the previous chapter, validation of the proposed benchmark Inductrack model is presented in this chapter. Because transient response results for Inductrack systems are unavailable, the noted steady-state results in the literature are used for model validation. For comparison, in an assumed steady-state scenario, both levitation gap and traveling speed of the vehicle are time-invariant. This setup can easily be achieved by slightly manipulating Eq. (3.37). 4.1 MagneticFieldbyaHalbachArraySetofFiniteDimensions The calculation of the source magnetic eld presented in Chap. 3 starts from the fundamental principles of physics without pre-assumed steady-state approximations. Thus, with the new model, the determination of the magnetic eld generated by Halbach arrays naturally reveals a decaying property of the magnetic eld due to the nite dimensions of the PM blocks, which is known as the "end eect". 34 Figure 4.1: A Halbach array set containing two wavelengths To investigate the above-mentioned end eect, consider a set of Halbach arrays as shown in Fig. 4.1, where there are total nine PM blocks, and the origin of the magnet frame o M is placed at the centroid of the leftmost PM block. Note that the poles of the leftmost and rightmost PM blocks have the same orientation for maintaining the symmetric property of the source magnetic eld. For simulation purposes, the geometric parameters of the array set are temporarily chosen as follows:a = 0:15 m,b =d = 0:025 m, = 4b = 0:1 m, where is the wavelength of a Halbach array with four PM blocks. The total length of the Halbach array set then is 9b = 0:225 m. First, consider the distribution of the magnetic eld of the Halbach array set in the longitudinal (y M ) direction, at a test point that is a vertical distance or gap away from the bottom surface of the array set; see Fig. 4.1. The knowledge of the source magnetic eld is of essentially importance in design of an Inductrack system as it is related to the magnetic forces. By the formulas given in Chap. 3, the distributions of the magnetic ux density componentsB y M andB z M along they M -direction and atx M = 0 (a central line of the array set) are plotted in Fig. 4.2, for four dierent values of the gap: = 0:1d; 0:25d; 0:8d; 1:6d. 35 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 y M -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Flux density component along y M (Tesla) = 0.1d = 0.25d = 0.8d = 1.6d (a) -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 y M -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Flux density component along z M (Tesla) = 0.1d = 0.25d = 0.8d = 1.6d (b) Figure 4.2: Distributions of magnetic eld components along longitudinal direction atx M = 0 with = 0:1d; 0:25d; 0:8d; 1:6d: (a) the longitudinal componentB y M ; (b) the vertical componentB z M Figure 4.2 clearly exhibits the end eect due to the nite length of the Halbach arrays on the magnetic eld distribution. As can be seen from the gure, the magnetic ux density decays fast away from the region of the array set (-0.0125 m y M 0.2125 m), and it is almost vanished wheny M is out of the "eective region" [0:5; 2:5] (namely, -0.05 m y M 0.25 m), which has a length of 3. This decaying property of the magnetic ux density justies the selection ofN track coils in determination of the magnetic forces. Figure 4.2 also shows that for a relatively small gap (for instance, = 0:1d and = 0:25d), the magnetic ux density distribution cannot be approximated by a sinusoidal one, which is commonly adopted by mistake in the previous investigations. For a relatively large gap (say, = 0:8d and = 1:6d), a sinusoidal approximation of the magnetic eld seems viable in the middle part of the array set, where not much distortion of the magnetic eld distribution is seen. In such an approximation, the components of the magnetic ux density can be written as ~ B y M =B 0 e 2 sin 2y M ~ B z M =B 0 e 2 cos 2y M (4.1) 36 whereB 0 denotes the peak value of the magnetic eld at the lower surface of the Halbach array according to Ref. [20] and it isB 0 = 0:877 Tesla. -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 y M -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Flux density component along y M (Tesla) = 0.8d - Calculated = 0.8d - Approximated (a) -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 y M -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Flux density component along z M (Tesla) = 0.8d - Calculated = 0.8d - Approximated (b) -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 y M -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Flux density component along y M (Tesla) = 1.6d - Calculated = 1.6d - Approximated (c) -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 y M -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Flux density component along z M (Tesla) = 1.6d - Calculated = 1.6d - Approximated (d) Figure 4.3: Magnetic eld distribution and the corresponding sinusoidal approximation: (a)B y M at = 0:8d; (b)B z M at = 0:8d; (c)B y M at = 1:6d; (d)B z M at = 1:6d For comparison, the ux density distributions displayed in Fig. 4.2 for = 0:8d and = 1:6d and those by the sinusoidal approximation in Eq. (4.1) are plotted in Fig. 4.3. From Figs. 4.3(a) and (b), with = 0:8d, the sinusoidal approximation is agreed with the proposed model in the middle region of the array [0:25; 1:75] (that is, 0.025 m y M 0.175 m). However, with = 1:6d, the sinusoidal distributions are seen to shift even in the middle region [0:25; 1:75], as shown in Figs. 4.3(c) and (d). Furthermore, 37 the sinusoidally approximated magnetic eld cannot capture the end eect due to the nite length of the array set regardless of values, as demonstrated in Fig.4.2. It should be noted that the sinusoidal approximation can be used to determine the only unknown parameter in the analytical formulas for magnetic eld calculation presented in Chap. 3 – the magnetization current densityJ M along the pole direction of a PM block. This parameter can be identied by matching the peak values of magnetic eld calculated by the Biot-Savart formulation and sinusoidal approximation in its valid region. For instance, according to Eq. (4.1) with Figs. 4.3(a) and (b), B y M max j =0:8d =B 0 e 20:8d (4.2) where the left-hand side of Eq. (4.2) contains the unknown parameterJ M . WhenB 0 = 0:877 T [20], by trials, it is found thatJ M = 10 6 Ampere/meter. Next, consider the distributions of the ux density components (B y M andB z M ) in thex M -direction (across the array widtha). These distributions, which are important to the determination of the magnetic forces, are plotted againstx M in Fig. 4.4, for = 0:1d; 0:25d; 0:8d; 1:6d, and aty M = 0:75 andy M =. -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 x M -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Flux density component along y M (Tesla) = 0.1d = 0.25d = 0.8d = 1.6d (a) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 x M -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Flux density component along z M (Tesla) = 0.1d = 0.25d = 0.8d = 1.6d (b) Figure 4.4: Distributions of magnetic eld components along crosswise direction when = 0:1d; 0:25d; 0:8d; 1:6d: (a) the longitudinal componentB y M aty M = 0:75; (b) the vertical compo- nentB z M aty M = 38 In this example, the width of the Halbach array set is a = 0:15 m. Thus, it can be observed from Fig. 4.4 that bothB y M andB z M decay by almost half of their peak values at the edges of the array, where x M =0:5a =0:075 m. In other words, in calculating the magnetic forces, one cannot simply assume a uniform distribution of the source magnetic eld across the width of the array. For accurate results, the integrals in Eqs. (3.32) and (3.33) must be evaluated. In summary of this section, the source magnetic eld produced by a Halbach array set is computed through use of the new model developed in Chaps. 2 and 3, and the results are compared with those obtained by a sinusoidal approximation of the magnetic eld. The ux density distributions plotted in Figs. 4.2-4.4 show the end eect (decaying property) of the magnetic eld due to the nite length of the array set. This physical phenomenon, however, cannot be predicted by the sinusoidal approximation. Furthermore, for the ux density distributions along the longitudinal (y M ) direction, an eective region of 3 length is seen in Fig. 4.2, which provides some guide in selection of number N of track coils for computing the magnetic forces. Additionally, with the analytical formulas on exact quadrature given in Chap. 3 and Appx. A, the calculation of the magnetic eld distributions is highly ecient and accurate. 4.2 TimeResponseofMagneticForcesandInducedCoilCurrents Once the source magnetic eld is determined with delity, the proposed Inductrack model is rst validated in steady-state response. To this end, the proposed transient model described by Eq. (3.37) is truncated to a pseudo-steady-state model by xing both the levitation gap and longitudinal traveling speed. In this scenario, the responses of the magnetic levitation and drag forces and the induced current in a given coil are computed by the truncated model. The results obtained are then compared with the available results by the original steady-state model in the literature. 39 For numerical simulation, the parameters of the Inductrack system are selected based on an Inductrack demonstration model [20] in the previous publication and in the context of the conguration as described in Chap. 2. These parameters are listed in Table 4.1, in whichL v refers the vehicle length. Table 4.1: List of the physical parameters used for numerical simulations Parameter Value Unit Parameter Value Unit l M 0.2 m N 2 - h 0.09 m R eq 8:7 10 5 w 0.155 m L eq 8:4 10 7 H d c 0.016 m l G 0.3125 m a 0.15 m h G 0.0475 m b 0.025 m J M 10 6 A/m d 0.025 m M 20 kg 0.1 m g 9.8 m/s 2 L v 0.65 m The pseudo-steady-state truncation of Eq. (3.37) is obtained as _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ) _ x N+1 =v _ x N+2 = 0 _ x N+3 = 0 _ x N+4 = 0 (4.3) with x k (0) = 0, x N+1 (0) = 0, x N+2 (0) = v, x N+3 (0) = + 0:5d +h G , x N+4 (0) = 0, where v is the constant traveling speed of the vehicle in the longitudinal direction, and is the xed levitation gap. In this work, = 0:8d is chosen according to the experimental results claimed by Post [20], where the levitation height of the vehicle was about 0.02 m. Zero initial conditions for the longitudinal displacement, vertical velocity, and induced coil currents are assumed. In this case, the magnetic forces can be calculated by functionsf l andf d in Eq. (3.37). 40 For the convenience of computation and presentation, the numberN of track coils is chosen as an odd integer, and the coil in the middle (the N+1 2 th coil) is placed at the origin of inertial frame att = 0. For now, selectN = 101. This number is conservatively large for accurate computation because theN coils occupy a region ofd c (N 1) length that is much larger than the vehicle lengthL v . The eect ofN on simulation results will be further discussed in later sections. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) 0 100 200 300 400 500 600 Levitation force (N) v = 1 m/s v = 5 m/s v = 20 m/s v = 100 m/s (a) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) -200 -100 0 100 200 300 400 Drag force (N) v = 1 m/s v = 5 m/s v = 20 m/s v = 100 m/s (b) Figure 4.5: Time responses of maagnetic forces at = 0:8d under various traveling speeds: (a) magnetic levitation force; (b) magnetic drag force Numerical solution of Eq. (4.3) by the procedure given in Sec. 3.6 yields the response of magnetic forces and induced coil currents. Plotted in Fig. 4.5 are the time histories of the magnetic levitation forceF l and drag forceF d , at four dierent traveling speeds,v = 1; 5; 20; 100 m/s. As can be observed from the gure, the magnetic forces converge to their steady-state values within a short time. As the speedv increases, the nal value of the levitation force increases while that of the drag force (only for the four speeds in Fig. 4.5) decreases. Also, a higher speed renders the magnetic forces settling at their steady states faster, with some higher-frequency oscillations initially. The trends and nal values of the magnetic forces for dierent traveling speeds, as shown in Fig. 4.5, can provide guidance on system design, and they can be used to compare the proposed model with the original steady-state model [8]. 41 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (s) -800 -600 -400 -200 0 200 400 600 800 Induced current (A) v = 1 m/s v = 5 m/s v = 20 m/s v = 100 m/s Figure 4.6: Time responses of the induced current of a specied coil at = 0:8d under various traveling speeds Figure 4.6 shows the time response of the induced current in the coil that is the rightmost of 101 selected coils at the initial timet = 0. The location of the coil in the inertial frame is (0; 50d c ; 0). Again, four dierent traveling speeds,v = 1; 5; 20; 100 m/s, are considered in simulation. As seen from the gure, at a higher speed, the coil current is induced earlier and dies out sooner. This is simply because it takes a shorter time for the vehicle with onboard Halbach arrays to pass by the coil. At v = 1 m/s, in the simulation time window (0-0.8s), only half length of the vehicle (mounted with one Halbach array set) has passed by the coil. To see the current induced by the other array set which eventually goes o, a longer simulation time is needed. The results in Fig. 4.6 reveal that due to the nite dimensions of the Halbach array sets, the time response of the induced coil current is always transient, and in general it does not reach a sinusoidal steady state before it vanishes. This observation indicates that the development of a reliable transient model is necessary for dynamic analysis of Inductrack systems. 42 4.3 ComparisonwiththeOriginalSteady-StateModel With the truncated pseudo-steady-state model described by Eq. (4.3), the simulation results obtained in the previous subsection are compared with the agreed results claimed by researchers using the original steady-state model [8]. 4.3.1 Steady-StateMagneticForcesatDierentTravelingSpeeds As mentioned in Chap. 1, the original steady-state model of Inductrack systems was developed by the following assumptions: innite ideal sinusoidal magnetic eld in the longitudinal direction, constant traveling speed of the vehicle, time-invariant levitation gap, steady-state response of induced current, and averaged magnetic forces. Although this model does not capture transient behaviors, the results by the model in terms of steady-state magnetic forces can provide some intuitive and qualitative information for basic understanding of Inductrack systems. According to the parameters listed in Table 4.1, the total magnetic levitation force ~ F l and magnetic drag force ~ F d calculated by the original steady-state model are given by ~ F l = w 2 B 2 0 N L eq d c 1 1 + Req vLeq 2 e 2 ~ F d = w 2 B 2 0 N L eq d c Req vLeq 1 + Req vLeq 2 e 2 (4.4) where = 2 represents the wave number andB 0 = 0:877 T [20]. The detailed derivation of Eq. (4.4) is given in Appx. C. Figure 4.7 compares the steady-state magnetic levitation (lift) and drag forces predicted by the truncated model and by the original steady-state model, at dierent constant traveling speeds of the vehicle. The trends of the magnetic forces depicted by both models are in an agreement. As the traveling speedv increases, the levitation force keeps increasing and it converges to an asymptotic value around 43 v = 20 m/s. On the other hand, with v increasing, the drag force rst increases, peaks around 2 m/s, deceases afterwards, and then tends to vanish as the speed becomes very large. The result in the gure is in line with the agreed theory [9] in an Inductrack system: a large traveling speed leads to a large levitation force but a small drag force. The validated result on the magnetic forces shows great promises of Inductrack systems in high-speed transportation. 10 -1 10 0 10 1 10 2 Traveling speed (m/s) 0 50 100 150 200 250 300 350 400 Magnetic force (N) Lift (truncated model) Drag (truncated model) Lift (steady-state model) Drag (steady-state model) Figure 4.7: Magnetic levitation and drag forces calculated by truncated model and original steady-state model under various constant traveling speedsv at = 0:8d The comparison given in Fig. 4.7, however, shows a disparity between the magnetic levitation force by the proposed model and that by the steady-state model. This disparity is mainly caused by the assumption of a uniform magnetic eld distributionB 0 in the crosswise (x M ) direction in the derivation of Eq. (4.4), which for the Halbach array set of nite dimensions is physically untrue according to Fig. 4.4. As a result, with the original steady-state model, overestimation of the levitation force occurs, particularly at a high speed, at which the velocity-independent term of Eq. (4.4) becomes dominant. Such inaccuracy aects the reliability of the steady-state model in magnetic force estimation, even in the assumed steady-state scenario. 44 With the curves in Fig. 4.7, a critical speedv cr of the Inductrack system can be identied. For a given mass of the vehicle and a desired levitation height, the critical speed v cr is the minimum longitudinal speed at which the levitation force overcomes the gravity. For the current example withMg = 196 N by Table 4.1 and = 0:8d, the critical speedv cr 2:5 m/s from Fig. 4.7, where a levitation force of 200 N is used. Noticeably,v cr depends both on the weight of the vehicle and the targeted levitation gap, but it can be easily determined via simulations using the truncated model (4.3). 4.3.2 Steady-StateMagneticForceswithDierentLevitationGaps The proposed model and the original steady-state model are also compared in terms of steady-state magnetic forces with dierent levitation gaps. For this, two traveling speeds,v = 2 andv = 20 m/s, are selected, at which the maximum drag force and maximum levitation force are reached, respectively, according to Fig. 4.7. 0.4 0.6 0.8 1 1.2 1.4 1.6 Levitation gap divided by thickness of array 0 100 200 300 400 500 600 700 800 900 Magnetic force (N) Lift (truncated model) Drag (truncated model) Lift (steady-state model) Drag (steady-state model) (a) 0.4 0.6 0.8 1 1.2 1.4 1.6 Levitation gap divided by thickness of array 0 500 1000 1500 Magnetic force (N) Lift (truncated model) Drag (truncated model) Lift (steady-state model) Drag (steady-state model) (b) Figure 4.8: Magnetic levitation and drag forces calculated by truncated model and original steady-state model under various xed levitation gaps: (a) atv = 2 m/s; (b) atv = 20 m/s In Fig. 4.8, the magnet levitation (lift) and drag forces are plotted against the levitation gap. As seen from the gure, the trends of the magnetic forces predicted by the two models are in accordance with each other: the magnetic forces decay as the levitation gap increases, which is physically understandable. 45 However, disparities still exist, especially between the predicted levitation forces with a small levitation gap. This is mainly due to the assumption of an ideal sinusoidal magnetic eld distribution in the longitudinal (y M ) direction in the derivation of Eq. (4.4), which misses the distortion of the magnetic eld as observed in Fig. 4.2. Apparently, the steady-state force formula given in Eq. (4.4) may not be accurate when an onboard Halbach arrays get close to the track. Like in the previous subsection, with the curves in Fig. 4.8, a critical levitation gap cr for a given mass of the vehicle and traveling speed can be identied. The cr is the maximum gap at which the levitation force overcomes the gravity. In the current example, as seen from Fig. 4.8, cr = 0:75d forv = 2 m/s and cr = 0:75d forv = 20 m/s. Note that cr also tells the equilibrium levitation gap of the vehicle at a constant traveling speed, which is useful when the vertical vibration of the vehicle is considered in a dynamic scenario. Thus, the identication of the critical speed and critical levitation gap also provides important information for dynamic analysis and feedback control of Inductrack systems. 4.4 ConvergenceofMagneticForces As seen from Fig. 4.2, the magnetic eld produced by the Halbach array set of nite length has end eect in the longitudinal direction. Thus, it is important to show that the magnetic forces predicted by the truncated model (4.3) converge as the numberN of selected track coils in the spatial window of Fig. 3.5 increases. To this end, the magnetic forces are computed withN varying from 41 to 201. For more intuitive visualization, instead of the exact coil numberN, a ratio of coil length to vehicle length Lc Lv is used, where L v is the length of the vehicle, andL c is the expanded length of theN coils and it is dened by L c =d c (N 1) (4.5) In the simulation,L v = 6:5 = 0:65 m from Table 4.1. 46 1 1.5 2 2.5 3 3.5 4 4.5 5 L c / L v 50 100 150 200 250 300 350 Levitation force (N) v = 1 m/s v = 5 m/s v = 20 m/s v = 100 m/s (a) 1 1.5 2 2.5 3 3.5 4 4.5 5 L c / L v 0 20 40 60 80 100 120 140 Drag force (N) v = 1 m/s v = 5 m/s v = 20 m/s v = 100 m/s (b) Figure 4.9: Steady-state magnetic forces computed using dierent coil numbersN (in ratio Lc Lv ) at = 0:8d with multiple constant speeds: (a) the levitation force; (b) the drag force Plotted in Fig. 4.9 are the magnetic forces that are calculated for dierent values of the length ratio Lc Lv , with the traveling speedv = 1; 5; 20; 100 m/s. In the gure, as the length ratio increases, the trend of convergence of both the magnetic levitation and drag forces is clearly displayed. The results in the gure also suggest that a proper number of coils can be selected to guarantee both accuracy and eciency in computation. This number is a minimum one by which the computed magnetic forces are in small neighborhoods of their converged values. For the current example, the minimum length ratio is Lc Lv = 1:5, which by Eq. (4.5) and the data in Table 4.1 gives a minimum coil numberN = 61. This number is usually used as a standard in the subsequent simulations unless further specied. In this chapter, the benchmark model of Inductrack systems that is developed in Chaps. 2 and 3 is validated from the aspect of steady state, and the results obtained are compared with the available results in the literature. Also, critical speed, critical levitation gap, and minimum coil number for accuracy and eciency in computation are determined through use of the truncated model described by Eq. (4.3). These results are useful for physical understanding and dynamic analysis of Inductrack systems. 47 Chapter5 TransientResponseoftheInductrackDynamicSystem With the steady-state validation of the proposed Inductrack model in the previous chapter, dynamic analysis of the Inductrack system via numerical simulation of transient responses are ready to be presented in this chapter. For easy comparison, the same set of system parameters in Table 4.1 is used. In the simulation, two main categories of dynamic scenarios are considered: (i) the vehicle travels with a constant horizontal speed; and (ii) the vehicle travels when subject to a prescribed propulsion force. In each scenario, a truncated state-space model is reduced from Eq. (3.37) and the resulting state equations are solved by the numerical scheme presented in Sec. 3.6. Although valid transient response results on Inductrack systems are very limited in the literature, some comparisons and discussions are made based on the noted claims of both theoretical and experimental results to partially verify the proposed transient Inductrack model. 5.1 VehicleDynamicsatConstantTravelingSpeeds In a constant-speed case, the propulsion force is always assumed to be equal to the magnetic drag force so that the traveling speed of the vehicle remains unchanged. In this ideal situation, the motion of the 48 vehicle in the longitudinal (y I ) direction is certain, and only the dynamics of the vehicle in the vertical (z I ) direction needs to be studied. In this scenario, Eq. (3.37) is reduced to _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ) _ x N+1 =v _ x N+2 = 0 _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 )g (5.1) withx k (0) = 0,x N+1 (0) = 0,x N+2 (0) =v,x N+3 (0) = 0 + 0:5d +h G ,x N+4 (0) = 0. In the transient analysis, the initial levitation gap is chosen as 0 = 0:8d = 0:02 m, which is also used in the steady-state examples in Sec. 4.3.1. However, dierent from the steady-state truncation model described by Eq. (4.3), the transient model described by Eq. (5.1) allows a time-varying levitation gap(t), which is shown in Fig. 3.4 and related to the state variablex N+3 (t). Because the levitation gap is directly related to the vertical motion of the vehicle, terms "levitation gap" and "vertical motion" are alternatively used for the same meaning. Plotted in Fig. 5.1 are the time responses of the levitation gap (t), for the vehicle traveling at 14 dierent speeds, from 1 m/s to 50 m/s. This gure reveals some interesting stability features of the Inductrack system. As seen from Fig. 5.1(a), at a low speed v = 1 m/s, the levitation gap initially oscillates and then quickly settles at a steady-state value, (t) ! 0:01235 m, which represents an equilibrium position of the vehicle in the vertical direction. With a slightly higher speed (v = 1:2; 1:5; 1:9 m/s), the vehicle oscillates more, and it takes a longer time to settle to a steady-state value. Note that all the steady-state levitation gaps in Fig. 5.1(a) are smaller than the initial value 0 = 0:02 m, which means that at a lower traveling speed, the vehicle is closer to the track coils. This is 49 physically understandable: according to Fig. 4.7, a lower speed results in a smaller levitation force, which then requires a smaller levitation gap to suspend the vehicle. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Levitation gap (m) v = 1 m/s v = 1.2 m/s v = 1.5 m/s v = 1.9 m/s (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0.014 0.016 0.018 0.02 0.022 0.024 0.026 Levitation gap (m) v = 2 m/s v = 2.5 m/s v = 3 m/s (b) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 Levitation gap (m) v = 4 m/s v = 5 m/s v = 6 m/s v = 8 m/s (c) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 Levitation gap (m) v = 10 m/s v = 20 m/s v = 50 m/s (d) Figure 5.1: Time responses of the levitation gap of under various constant traveling speeds: (a) v = 1; 1:2; 1:5; 1:9 m/s; (b)v = 2; 2:5; 3 m/s; (c)v = 4; 5; 6; 8 m/s; (d)v = 10; 20; 50 m/s Starting fromv = 2 m/s, the levitation gap becomes persistently oscillatory and does not settle down, as shown in Fig. 5.1(b). In other words, with a speed exceeding a threshold value (between 1.9 m/s and 2 m/s), say v = 2; 2:5; 3 m/s, the vertical motion of the vehicle is oscillatory with an ever-increasing amplitude, indicating a trend of divergence instability. Also, for the speeds considered in Fig. 5.1(b), the rate of divergence increases at a higher speed. 50 However, the rate of divergence of the levitation gap decreases after the vehicle speed surpasses another threshold (between 3 m/s and 5 m/s), as observed from Fig. 5.1(c) for v = 5; 6; 8 m/s. Furthermore, at a much higher speed, sayv = 10; 20; 50 m/s, the rate of divergence reduces toward zero; see Fig. 5.1(d), where the vertical motion of the vehicle tends to settle at a steady-state value that is larger than the initial gap 0 = 0:02 m. Again, a larger steady-state gap is physically due to a larger levitation force generated at a higher speed, as seen from Fig. 4.7. In all the non-convergent cases of Figs. 5.1(b), (c), and even strictly of 5.1(d), a much longer simulation time only renders an ever-increasing amplitude of the vertical oscillation of the vehicle, which continues until the vehicle clashes with the track coils. Thus, the multiple equilibrium positions of the Inductrack system are viewed unstable. In summary of the simulation results presented in Fig. 5.1, the Inductrack system is a conditionally stable system, dependent upon its speed in the longitudinal direction. In the considered example, there are two critical transition speeds: (i)v cr1 between 1.9 m/s and 2 m/s, at which the vehicle switches from being stable to unstable; and (ii)v cr2 between 3 m/s and 5 m/s, at which the divergence rate of the levitation gap changes from increasing to decreasing. As the speed becomes very large (v!1), the vertical motion of the vehicle tends to be marginally stable. The development of the stability conditions for this nonlinear dynamic system in analytical form is dicult and is beyond the scope of the current eort, but it could be a subsequent research topic. The numerical results presented in this subsection are in accordance with the "negative damping eect" of EDS systems discussed by Ref. [94] and comprehensively summarized by Ref. [95], although the transition speedsv cr1 andv cr2 are not analytically determined. The results also reproduce the unstable heave oscillation discovered by Storset [60] with a periodic track model for Inductrack systems. Needless to say, for an Inductrack system, it is necessary to install damping mechanism and active control for stabilization of the vehicle motion in the vertical direction. 51 It is worthy to mention that the magnetic forces and the conditional stability phenomenon in the above dynamic scenario cannot be accurately predicted with the original steady-state model by simple assignment of a time-varying levitation gap. This is because the derivation of Eq. (4.4) only adopts a time-invariant levitation gap, and thus ignores the transient eects of the complicated electro-magneto-mechanical coupling mechanism in the Inductrack dynamic system. In the newly-proposed transient model, however, this coupling mechanism is carefully taken into consideration, as embedded in the state representation (3.37) and in its constant-speed truncation model as described by Eq. (5.1). 5.2 VehicleDynamicswithPrescribedPropulsionForce In this section, the full 2-DOF transient model governed by Eq. (3.37) will be used, with a prescribed propulsion forceF p in the longitudinal direction. In the transient response simulation, three sub-scenarios of this dynamic case are considered. 5.2.1 VehicleinFreeMotion A "free motion" describes the situation when no propulsion force is applied to the vehicle (F p 0). In this case, the Inductrack vehicle in the longitudinal direction is only subject to the magnetic drag force with a given initial velocityv 0 . The state-space model (3.37) is adjusted by _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ) _ x N+1 =x N+2 _ x N+2 =f d (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ) _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 )g (5.2) 52 withx k (0) = 0,x N+1 (0) = 0,x N+2 (0) =v 0 ,x N+3 (0) = 0 + 0:5d +h G ,x N+4 (0) = 0, where again, in the simulation, 0 = 0:8d = 0:02 m is used as the initial levitation gap for consistency. The following simulation imitates an early experiment on the LLNL Inductrack demonstration model [20]. In the experiment, the Inductrack vehicle was initially accelerated by the driving coils before entering the levitation zone, and then coasted freely in the longitudinal direction above the levitation coils. Due to the existence of physical constraints in real applications (e.g., supporting wheels on bottom of the vehicle), in the simulation, the vehicle shall be considered to have "landed on the track" when the levitation gap is small enough, say(t) 0:05d = 0:00125 m. 0 0.5 1 1.5 2 2.5 3 Time (s) 0.005 0.01 0.015 0.02 0.025 0.03 Levitation gap (m) v 0 = 5 m/s v 0 = 8 m/s v 0 = 10 m/s (a) 0 0.5 1 1.5 2 2.5 3 Time (s) 0 2 4 6 8 10 12 14 16 18 Longitudinal displacement (m) v 0 = 5 m/s v 0 = 8 m/s v 0 = 10 m/s (b) Figure 5.2: Dynamic responses of the vehicle in projectile motion whenv 0 = 5; 8; 10 m/s: (a) the levitation gap; (b) the displacement in the longitudinal direction Figure 5.2 plots the time responses of the levitation gap and the longitudinal displacement of the vehicle, with dierent initial longitudinal velocities, v 0 = 5; 8; 10 m/s. As seen from Fig. 5.2(a), while traveling, the vehicle oscillates up and down, and due to the ever-decreasing levitation force, it eventually lands on the track. Also seen from Fig. 5.2(a), a larger initial speed v 0 renders a longer traveling time before landing, which is translated into a farther "oating distance", as shown in Fig. 5.2(b). Furthermore, the simulation indicates that the oating distance of the vehicle atv = 10 m/s is about 16 meters, which 53 is very close to the experimental results reported by the LLNL [9, 20]. In the LLNL experiments, the vehicle coasted longitudinally for about 20 meters, including a 4.5-meter zone for acceleration. The results in Fig. 5.2 can be explained by the physical laws governing a free motion as follows. Without external energy input, the kinetic energy of the vehicle is gradually dissipated by the track coils through the magnetic drag force. As a result, the vehicle starting at an initial speedv 0 keeps slowing down, which in turn reduces the levitation force, then the vehicle levitation gap, and eventually renders the vehicle landing on the track. 5.2.2 VehiclewithConstantPropulsionForce The senario considered in this subsection is a more realistic one compared to the case of constant speed in Sec. 5.1 and the case of free motion in Sec. 5.2.1. When a constant propulsion forceF 0 is acting on the vehicle in the longitudinal direction, Eq. (3.37) becomes _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ) _ x N+1 =x N+2 _ x N+2 =f d (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ) + F 0 M _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 )g (5.3) withx k (0) = 0,x N+1 (0) = 0,x N+2 (0) = 0,x N+3 (0) = 0 + 0:5d +h G ,x N+4 (0) = 0, which describes the vehicle motion in both the vertical and longitudinal directions. In the simulation, the initial levitation gap is set to 0 = 0:8d. To achieve magnetic levitation, the propulsion forceF 0 needs to be large enough for two reasons: (i) to overcome a large magnetic drag force occurring at a low speed; and (ii) to produce a high enough traveling speed, which in turn generates an adequate magnetic levitation force for maintaining the vehicle over the 54 track with a safe clearance. Indeed, (i) and (ii) are correlated via the electro-magneto-mechanical coupling in the Inductrack system. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0.005 0.01 0.015 0.02 0.025 0.03 Levitation gap (m) F 0 = 475 N F 0 = 500 N F 0 = 600 N (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0 10 20 30 40 50 60 Longitudinal velocity (m/s) F 0 = 475 N F 0 = 500 N F 0 = 600 N (b) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0 100 200 300 400 500 600 Levitation force (N) F 0 = 475 N F 0 = 500 N F 0 = 600 N (c) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 0 100 200 300 400 500 600 700 800 Drag force (N) F 0 = 475 N F 0 = 500 N F 0 = 600 N (d) Figure 5.3: Transient responses of the Inductrack system under a constant propulsion force F 0 = 475; 500; 600 N: (a) levitation gap; (b) longitudinal traveling speed; (c) magnetic levitation force; (d) mag- netic drag force To show this physical phenomenon, the transient displacements and magmatic forces of the Inductrack system are depicted in Fig. 5.3, forF 0 = 475; 500; 600 N. As seen from Fig. 5.3(a), a 475 N propulsion force is not enough to lift the vehicle, and under this force, the vehicle keeps dropping down and lands on the track in about 0.4 second. For a propulsion force of 500 or 600 N, the vehicle initially drops down, but is eventually lifted as its traveling speed picks up to escape the low-speed zone with high drag force. 55 At this stage, an almost constant acceleration is achieved (see Fig. 5.3(b)), which is mainly due to the signicant reduction of the magnetic drag force at a high speed (see Fig. 5.3(d)). Note that the levitation gap oscillates in all the three propulsion force cases, which is due to the oscillatory levitation forces, as shown in Fig. 5.3(c). Although these levitation forces have a mean value around 200 N (about the vehicle weight), forF 0 = 475 N, however, the levitation force cannot prevent the vehicle from dropping down in the low-speed zone. On the other hand, forF 0 = 500 or 600 N, the levitation force can lift the vehicle as the vehicle starts gaining signicant speed outside the low-speed zone. Also, from Figs. 5.3(a) and (c), a large propulsion force can lead to an oscillatory levitation force of large amplitude and subsequently a large-amplitude levitation gap. In the examples ofF 0 = 500 and 600 N, the levitation gap and forces will settle to steady-state values if simulation time is long enough. This is because the vehicle will eventually have an innitely large speed, which will result in a marginally stable system from the experience obtained in Sec. 5.1. In general, depending on initial conditions and due to transient eects, the vertical motion of the vehicle may experience oscillations during the acceleration process. To resolve this issue, implementation of a damping mechanism and feedback control in the Inductrack system is necessary. The magnetic forces plotted in Figs. 5.3(c) and (d) are computed through use of the functionsf l and f d in Eq. (5.3). It is seen from these gures that both the levitation force and drag force are transient in nature, and they can vary irregularly due to the complicated electro-magneto-mechanical coupling in the Inductrack system. This indicates that in dynamic analysis, the magnetic forces can hardly be determined with delity by steady-state formulas, such as those given by Eq. (4.4). 5.2.3 VehicleinAccelerationandDeceleration With the proposed transient model, the propulsion force can be prescribed by an arbitrary function. In the following simulation, a prole of the propulsion force is specied to reect the vehicle in a process of 56 acceleration and deceleration. Without loss of generality, three initial conditions on the system parameters are assumed: (i) the vehicle initially has a speed that is high enough to avoid the high-drag zone, like those shown in Fig. 5.3(d); (ii) the system initially has a nonzero levitation gap 0 ; and (iii) the track coils initially have zero currents. With these conditions, the state equations of the Inductrack system are _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ) _ x N+1 =x N+2 _ x N+2 =f d (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ) + F p M _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 )g (5.4) with x k (0) = 0, x N+1 (0) = 0, x N+2 (0) = v 0 , x N+3 (0) = 0 + 0:5d +h G , x N+4 (0) = 0, and for simulation, setv 0 = 10 m/s and 0 = 0:8d = 0:02 m. Figure 5.4: A designed prole of the propulsion force, where:t 1 = 0:1 s,t 2 = 1:0 s,t 3 = 1:1 s,t 4 = 1:3 s, t 5 = 1:4 s,t 6 = 1:9 s,t 7 = 2:0 s, andF 1 = 250 N,F 2 =300 N With the steady-state results in Sec. 4.3 as the initial guide, a rough estimation of the levitation gap of the Inductrack vehicle traveling at a speed ofv = 20 m/s is = 0:95d = 0:02375 m; see Fig. 4.8(b). The goal of this simulation is letting the vehicle reach such speed (v = 20 m/s) and levitation gap ( = 0:02375 m) for a short time period, and then gradually land on the track. To this end, a prole of the propulsion forceF p is specied in Fig. 5.4, which implies that the vehicle is in a process of acceleration, deceleration, 57 and stop. Note that after timet 7 = 2 s, with no propulsion force, the vehicle is in "free motion" as described in Sec. 5.2.1. (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0 0.005 0.01 0.015 0.02 0.025 0.03 Levitation gap (m) t 7 (b) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0 50 100 150 200 250 300 350 400 450 Levitation force (N) t 7 (c) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) -50 0 50 100 150 200 250 300 350 400 Drag force (N) t 7 (d) Figure 5.5: Dynamic responses of the Inductrack vehicle in acceleration and deceleration process: (a) lon- gitudinal traveling speed; (b) levitation gap; (c) magnetic levitation force; (d) magnetic drag force Plotted in Fig. 5.5 are the time histories of the longitudinal speed, levitation gap and magnetic forces of the Inductrack system. Figure 5.5(a) displays the longitudinal speed of the vehicle in the acceleration- deceleration-stop process, and it shows that beforet 7 = 2 s the vehicle speed varies between 10 m/s and 20 m/s, and that the desired speed (v = 20 m/s) is reached betweent 3 andt 4 . In Fig. 5.5(b), the levitation gap beforet 7 is in a consistent oscillatory form, which is due to a similar pattern of the magnetic levitation 58 force, as shown in Fig. 5.5(c). However, when the vehicle is in the free motion aftert 7 , the magnetic drag force becomes more dominant with an increasing amplitude; see Fig 5.5(d). Also, it is seen from Figs. 5.5(c) and (d) that the magnetic forces do not have a steady state during the acceleration and deceleration process. This again proves the necessity of having a transient model for Inductrack systems for dynamic simulation and analysis. 59 Chapter6 Model-BasedFeedbackControloftheInductrackSystem As manifested by Eqs. (3.32)-(3.34), the transient Inductrack system is highly nonlinear and motion-coupled with time-varying electromagnetic quantities in complexity. Moreover, depending on the track coil numberN that is conservatively chosen, Eq. (3.37) may end up with numerous "shifting" state variables, as described in Sec. 3.6. These factors make it extremely dicult to linearize, or even to nd an equilibrium state of, the system with analytical expressions. As a result, direct application of linear or nonlinear theories to Eq. (3.37) may not be a realistic way to achieve the desired feedback control. Knowing that feedback control of the Inductrack system is necessary but dicult, in this chapter, we propose a straightforward and ecient model-based controller design approach. In this approach, feedback control is essentially achieved by assembling onboard active Halbach arrays with coils, which produce a controllable source magnetic eld. The controller design process takes two main steps. Firstly, a linear controller (PID, state feedback, etc.) is implemented and tuned based on a linear reference model, which outputs a control force. Secondly, according to the control force, the current in the active coils is computed by a nonlinear force-current mapping function that is pre-identied through simulations based on the truncated steady-state model (4.3). The resulting output current serves as the actual input to the full transient Inductrack model (3.37), by which, the transient response of the vehicle’s motion can be determined. The proposed nonlinear feedback control is detailed in sequel. 60 6.1 ActiveHalbachArrays For the Inductrack system illustrated in Fig. 2.1, it is more convenient to integrate a control system on the array side rather than on the coil side, mainly because the Halbach arrays are of nite length. In other words, a feedback controller is implemented on the moving vehicle, not on the coils on the stationary track. In the literature, active Halbach arrays have been shown to be able to produce a controllable onboard source magnetic eld in electromechanical systems [26, 33, 96]. Such an idea, however, has not been applied to feedback control of the Inductrack system with a transient dynamic model. In this work, the conguration of the active Halbach arrays developed in Ref. [96] is considered in an actuator design for implementation of feedback control. Figure 6.1: The conguration of an active Halbach array set (front view) Figure 6.1 sketches the conguration of an active Halbach array set, which consists of regular Halbach arrays in Fig. 2.1(c) and active coils winding around the PM blocks of the arrays. For each PM block in Fig. 6.1, the winding direction of the active coils (dashed arrows) and its original pole direction (solid arrows) follow the right-hand rule. In this preliminary investigation, to simplify the problem, two main assumptions are made as follows. Assumption 1: Each PM block is wound with n turns of close-packed active coils, with controllable current I C (t). For the entire array set, the current I C (t) circulates either along or opposite to the coil winding direction of each PM block simultaneously. 61 Assumption 2: The active coils have negligible dimension in thickness, namely,d a d in Fig. 6.1. In addition, the current densityJ C (t) produced by the active coils is uniformly distributed along the original pole direction of each PM. In other words,J C (t) andI C (t) are in the following linear relation J C (t) = n d I C (t) (6.1) The above assumptions provide convenience and eciency in computing the resulting source magnetic eld produced by the active Halbach array set in Fig. 6.1. The Biot-Savart formulation mentioned in Chap. 3 are still valid with a time-varying magnetization current densityJ [91], which satises J =J M +J C (t) (6.2) whereJ M is the time-invariant magnetization current density of the PMs in Eq. (3.2), andJ C (t) is the time-varying current density produced by the active coils. According to Assumption 1, the active current density J C (t) in Eq. (6.2) can be either positive or negative. Thus, the currents in active coils can either intensify or attenuate the original source magnetic eld provided by the regular Halbach arrays without altering the shape of distribution of the original source magnetic eld in space. And because of Eq. (6.1), the concepts of active current I C (t) and the corresponding active current densityJ C (t) are alternatively used in the subsequent derivations. 6.2 PIDControlBasedonaSimpliedLinearReferenceModel As mentioned earlier, due to the strong nonlinearities, complicated electromechanical coupling and a large number of state variables in an Inductrack dynamic system, the design of a feedback control law directly 62 based on the transient model (3.37) is currently unrealistic. To handle this dicult issue, an eective two- step controller design approach is proposed. As shall be seen, the newly developed transient Inductrack model plays an essentially important role in the feedback control system design. Figure 6.2: A linear reference model with a lumped-mass dynamic system In the rst step of the control design, a PID controller in rate-feedback conguration is set and tuned based on a simplied linear reference model. The reference model is a lumped-mass dynamic system as depicted in Fig. 6.2, whereM is the same as the total mass of the Inductrack vehicle. A block diagram for PID tuning with the reference model is drawn in Fig. 6.3, where an ideal sensor is considered. Figure 6.3: Block diagram for PID control based on the linear reference model Noticeably, the control eortu in Fig. 6.3 designates a control force. Indeed, it represents the resultant forceF z acting on the linear reference model in the vertical direction; see Fig. 6.2. According to Fig. 6.3, the PID controlled lumped-mass dynamic system has the following closed-loop transfer function: G cl (s) = K p s +K I Ms 3 +K D s 2 +K P s +K I (6.3) which indicates that the closed-loop system based on the reference model is a 3rd-order linear system. 63 The PID tuning process with the transfer functionG cl (s) can be performed by several methods, among which the root locus method is utilized in this section. Such PID tuning, being quite standard, is simply described as follows. By Eq. (6.3), the characteristic equation of the closed-loop system can be written as 1 +K P sz Ms 2 (sp) = 0 (6.4) wherep = K D M ,z = K I K P , andM = 20 kg. By properly assigning the locations of the open-loop pole p and the open-loop zeroz, a locus plot is generated in Fig. 6.4(a), where the three solid dots on the root loci mark the selected locations of the closed-loop poles and they determine the gainK P . The other two gains,K I andK D , are obtained with the values ofp,z,K P , andM. Figure 6.4(b) displays the unit step response of the closed-loop system with the tuned PID gains:K P = 4000 N/m,K I = 14000 N/(ms), and K D = 400 N/(ms −1 ), which are obtained to guarantee a maximum overshootM P < 40% and a settling timet s < 1 s. (a) (b) Figure 6.4: Results of PID tuning based on the linear reference model: (a) root locus plot; and (b) step response of closed-loop system There are two notable points regarding this PID gain tuning process. First, the control gains are independent of the traveling speed of the vehicle because they are tuned based on the linear reference 64 model that is also independent of the speed. As such, the nalized PID gains are applicable to all the numerical examples given in this work, as long as with the same vehicle massM = 20 kg. Second, the PID gains are "virtual" gains that do not need to be realized in hardware. In the proposed nonlinear feedback control, the control eort u in Fig. 6.3 only represents a computed input of a nonlinear force-current mapping function, which will be generated in the next step (Sec. 6.3). Thus, the PID gain tuning here is straightforward, leaving all the nonlinear speed-dependent eects to be handled later. It should be pointed out that the PID gain tuning described here is just one way to determine the control eortu. Besides, other feedback control methods are applicable to the linear reference model. For instance, the control eortu can be determined via eigenvalue assignment using state feedback, output feedback, etc., and such extension will be discussed briey in Sec. 6.6. Understanding of this concept makes it possible to extend the proposed nonlinear control method to general Inductrack systems with M-DOF models of the moving vehicle. 6.3 NonlinearForce-CurrentMappingFunction Although a simple linear model is used in the PID tuning (in the rst step), the newly-developed nonlinear model of the Inductrack system must be integrated in the feedback controller design (in the second step), in order to compute the transient response of the control system. To this end, with the determined PID control gains, the linear reference model in Fig. 6.3 is replaced by an internal structure as illustrated in Fig. 6.5, which includes the transient Inductrack model and a nonlinear mapping function as the authentic control law. After this replacement, a model-based nonlinear feedback control system with the transient Inductrack model as the control plant is constructed in Fig. 6.6. 65 Figure 6.5: Internal structure (the transient Inductrack model and a nonlinear mapping function), replacing the linear reference model Figure 6.6: Block diagram for the entire Inductrack system under feedback control By Figs. 6.5 and 6.6, the proposed nonlinear feedback control is explained as follows. As seen from Fig. 6.5, the summation of the control eortu of the PID controller (the control forceF z in Fig. 6.2) and the weight Mg of the Inductrack vehicle gives the estimated magnetic levitation force F u . By a pre- determined nonlinear mapping functionf NL , which takes the estimated levitation forceF u , vehicle speed v, and levitation gap as independent variables, the current density J in the active coils is computed. The current densityJ serves as the input to the transient Inductrack system (the plant), with the actual levitation gap being the output of the plant. The output is then fed back to compare with the reference levitation gap r, resulting in an error = r; see Fig. 6.6. With the error, the control eort u is determined by the PID gain tuning as described in Sec. 6.2. According to the previous discussion, the main eort in the second step of the control design is boiled down to the identication of the nonlinear force-current mapping functionf NL . Functionf NL , as shown in Fig. 6.5, maps the estimated levitation force F u to the total current density J given in Eq. (6.2). To 66 identifyf NL , the "steady-state" levitation force is calculated via numerical simulations through use of the truncated model (4.3). According to the previous investigations, the steady-state levitation forceF l;ss can be approximated as a speed-dependent function, which is proportional to the square of the source current density and in exponential decay with the levitation gap [8, 9, 97]. Therefore, the steady-state levitation force is constructed as F l;ss = K J (v) J 2 M J 2 ss e (v)ss (6.5) whereJ ss and ss are the current density and levitation gap, respectively, in steady state; andJ M is the magnetization current density of the PMs, which is a constant included in Eq. (6.5) for convenience in identication of the speed-dependent parametersK J (v) and(v). To identifyK J (v) and(v) in Eq. (6.5), chooseJ ss = J M , and thus numerical simulations based on the truncated model (4.3), which are without feedback control, can be performed with various values of the traveling speedv and the steady-state levitation gap ss . In each simulation, the steady-state levitation force F l;ss is computed by Eq. (3.32). Plotted in Fig. 6.7(a) are some simulation results on the F l;ss - ss correlation. At a traveling speed, sayv i , parametersK J (v i ) and(v i ) are identied through exponential curve tting of the simulation results, which yields theK J (v i ) and(v i ) curves as shown in Fig. 6.7(b). The physical parameters used in generating Figs. 6.7(a) and (b) are listed in Table 4.1. 67 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Steady-state levitation gap (m) 0 500 1000 1500 2000 2500 3000 Steady-state levitation force (N) v = 2 m/s v = 5 m/s v = 20 m/s v = 100 m/s (a) 10 0 10 1 10 2 Traveling speed (m/s) 1000 2000 3000 4000 K J 120 125 130 135 K J (tested) K J (fitted) (tested) (fitted) (b) Figure 6.7: Results of identication of the nonlinear force-current mapping function: (a) steady-state levita- tion force at dierent speeds and levitation gaps; and (b) identication of the speed-dependent parameters In Fig. 6.7(a), by setting the steady-state levitation force equal to the weight of vehicle, i.e. F l;ss = Mg = 196 N, the equilibrium levitation gap of the uncontrolled system (withJ ss =J M ) regarding each traveling speed e (v i ) can be measured from the horizontal axis. Asv increases, e (v) also increases but eventually will converge to an asymptotic value because the levitation force cannot increase unboundedly with the traveling speed in an EDS system. In the current example, e atv = 20 m/s is about 0:95d = 0:02375 m. Further increasingv cannot signicantly raise the value of e . Indeed, at higher speeds, say v 20 m/s, theF l;ss - ss curves are almost the same. From Fig. 6.7(b), it is observed that as v increases, parameter K J keeps increasing and eventually converges to its asymptote. This pattern is exactly the same as the trend of "levitation force versus traveling speed" in Inductrack systems as shown previously in Fig. 4.7, and this is becauseK J is closely related to the speed-dependent levitation force. The exponential coecient, on the other hand, only changes slightly in a wide range of traveling speed. Forv 5 m/s, the value of does not deviate much from 4 = 125:7, which was predicted by the original steady-state models [8, 9, 97]. Even at a low speed, sayv = 1 m/s, the deviation is less than 5.7%. Here, the small deviations are mainly due to the end eect of the magnetic eld 68 distribution generated by the Halbach arrays of nite length in Inductrack systems (see Fig. 4.2), which was not considered in the original steady-state models [8, 9, 97]. Finally, with the numerically identied parameters K J and, the nonlinear force-current mapping functionf NL in Fig. 6.5 is created by rst replacing the steady-state quantitiesF l;ss ,J ss , ss in Eq. (6.5) with the time-varying quantitiesF u ,J, and then solving Eq. (6.5) forJ. This leads to Jf NL (F u ;v;) =J M s jF u je (v) K J (v) (6.6) It follows from Eqs. (6.2) and (6.6) that the active current density is given by J C (t) =f NL (F u ;v;)J M (6.7) BecauseJ C (t) is the actual output of the proposed feedback control law (see Fig. 6.6), the control power can be measured by [J C (t)] 2 . The physical meaning of the nonlinear mapping function in Eq. (6.6) is explained as follows. The value of the function f NL is the total current density J needed for stable magnetic levitation of the vehicle at speedv and with the levitation gap. Although the functionf NL is generated based on steady-state quantities, it is a time-varying control input to the transient model of the Inductrack system. The eect of the so generated control input shall be validated in numerical examples in Sec. 6.5. 6.4 AugmentedStateEquationsfortheFeedbackControlSystem The transient model of the Inductrack system given in Chap. 3 was developed without consideration of feedback control. To apply the proposed nonlinear feedback control law to the transient Inductrack model, the following two issues regarding the original state-space representation (3.37) need to be addressed. First, the current density J in feedback control contains the time-varying component J C (t), and as a 69 result, it yields an extra term in the expression of the induced emfe k as given in Eq. (3.18). Second, the integral control action K I s as shown in Fig. 6.6 introduces a new state variable, which must be added in the control system formulation. In this section, these two issues are resolved in an augmented state-space representation. By applying the chain rule to Eq. (3.17) with time-varying current density J, the emf in Eq. (3.18) becomes e k = dy M dt Z S k @B y M k @y M dS k dz M dt Z S k @B y M k @z M dS k dJ dt Z S k @B y M k @J dS k (6.8) which by Eqs. (3.19) and (6.2) gives the modied expression of the induced emf as follows e k = _ Y (t) Z w 2 w 2 Z z k M z k M h @B y M k @y M (x 0 ; y k M ;z 0 ) dz 0 dx 0 + _ Z(t) Z w 2 w 2 B y M k (x 0 ; y k M ;z 0 )j z 0 = z k M z 0 = z k M h dx 0 _ J c (t) Z w 2 w 2 Z z k M z k M h @B y M k @J (x 0 ; y k M ;z 0 ) dz 0 dx 0 (6.9) The third term on the right-hand side of Eq. (6.9) is the extra term due to the rate of change of the active current density. This term represents the contribution of to the induced emf, which, while relatively smaller compared to those by the rst two vehicle motion-related terms, cannot be neglected. In calculation of this extra term, _ J c (t) can be approximated by J t at each time step and the double integral can also be evaluated by analytical formulas, which are obtainable by a similar approach as described in Chap. 3 and Appx. A. To deal with the integral control action, expand the state variables in Eq. (3.36) as follows x 1 =I 1 , x 2 =I 2 , :::, x N =I N , x N+1 =Y , x N+2 = _ Y , x N+3 =Z, x N+4 = _ Z, x N+5 =u (6.10) 70 Here a new state variable has been added:x N+5 =u, withu being the output of the PID control as shown in Fig. 6.6. Thus, assuming the vehicle travels with a constant speed, the state-space representation (3.37) is augmented as follows _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ;f NL (x N+2 ;x N+3 ;x N+5 )) _ x N+1 =x N+2 _ x N+2 = 0 _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;f NL (x N+2 ;x N+3 ;x N+5 ))g _ x N+5 =K P x N+4 +K I (r +h G + 0:5dx N+3 ) K D [f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;f NL (x N+2 ;x N+3 ;x N+5 ))g] (6.11) wherek = 1; 2;:::;N and the nonlinear mapping functionf NL gives the time-varying current density J. In Eq. (6.11), state variablex N+3 is related to the time-varying levitation gap through the coordinate transformation (3.35); r and 0 are the reference and initial levitation gap, respectively; the nonlinear functionsf k andf l are computed from Eqs. (6.9) and (3.32), respectively. 6.5 NumericalExamplesoftheControlledInductrackSystem In this section, the proposed nonlinear feedback control law is validated in numerical simulation. Without loss of generality, assume that the vehicle of the Inductrack system travels at a constant speedv in the longitudinal direction. For simplicity in the subsequent presentation, the word "system" means the Inductrack vehicle, the word "response" or "motion" refers to the vertical motion or levitation gap of the vehicle, and the word "stability" is about the boundedness of the vertical motion. 71 As discovered in Sec. 5.1, there exist two critical speeds (v cr1 andv cr2 , where 0<v cr1 <v cr2 ), which divide the entire speed range of the system into three stability regions. For convenince in discussing the numerical results, the stability regions are summrized as follows. (S1) For 0<v<v cr1 , the motion of the vehicle is stable; asv increases, the rate of the convergence of the levitation gap to a steady-state value keeps decreasing; and the system is marginally stable atv =v cr1 . (S2) Forv cr1 < v < v cr2 , the motion of the vehicle is unstable; asv increases, the rate of divergence of the levitation gap keeps increasing; and the system is most unstable with a highest divergence rate at v =v cr2 . (S3) Forv > v cr2 , the motion of the vehicle is always unstable; asv increases, however, the rate of divergence of the levitation gap decreases; and the system tends to be marginally stable asv!1. To validate the proposed control method, the transient responses of the moving vehicle without and with feedback control are compared. To determine the response of the uncontrolled system, the original state equations (5.1) are solved via numerical integration; to obtain the response of the controlled system, the augmented state equations (6.11) are solved. The initial conditions for Eq. (5.1) are the same as those for Eq. (6.11), but excludingx N+5 (0) = 0. The physical parameters of the Inductrack system used in the simulation are consistent with those listed in Table 4.1. Noticeably, the critical speedsv cr1 andv cr2 can be approximated by numerical simulations of Eq. (5.1) with multiple speedsv, and they arev cr1 1:9 m/s andv cr2 3:9 m/s, respectively. Consider four cases of the vehicle speed:v = 2; 5; 20; 100 m/s. Herev = 2 m/s is the jogging speed of a man, which is slightly higher thanv cr1 ;v = 5 m/s is the speed of a bicycle, which is greater thanv cr2 ; v = 20 m/s is the speed of a car moving on a paved county road; andv = 100 m/s is a possible operating speed of a Maglev train. Let the reference (desired) levitation gap ber = 0:95d = 0:02375 m, which is very close to the equilibrium levitation gap e atv = 20 m/s, and assume that the initial levitation gap is 0 = 0:8d = 0:02 m for each simulation. 72 0 1 2 3 4 5 6 7 8 9 10 Time (s) 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 Levitation gap (m) (a) 0 1 2 3 4 5 6 7 8 9 10 Time (s) 0 0.02 0.04 0.06 0.08 0.1 0.12 Levitation gap (m) (b) 0 1 2 3 4 5 6 7 8 9 10 Time (s) 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 Levitation gap (m) (c) 0 1 2 3 4 5 6 7 8 9 10 Time (s) 0.018 0.02 0.022 0.024 0.026 0.028 0.03 Levitation gap (m) (d) Figure 6.8: Transient responses of the levitation gap for the uncontrolled Inductrack system: (a) atv = 2 m/s; (b) atv = 5 m/s; (c) atv = 20 m/s; and (d) atv = 100 m/s The levitation gap responses of the uncontrolled vehicle in the four speed cases are plotted in Fig. 6.8. As can be seen, the system response in Fig. 6.8(a) follows the trend in the stability region (S2) forv cr1 < v <v cr2 ; Figs. 6.8(b)-(d) present the trend in the stability region (S3) forv >v cr2 ; and Fig. 6.8(b) shows a high divergence rate because the velocity (v = 5 m/s) is close tov cr2 . The instability phenomenon in the gures is well known as the "negative damping eect" discovered in EDS Maglev systems [94]. Apparently, in all the four speed cases, feedback control to stabilize the motion of the vehicle is essentially important for safe operation of the Inductrack system. 73 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026 0.027 Levitation gap (m) Transient Inductrack model Linear reference model (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026 Levitation gap (m) Transient Inductrack model Linear reference model (b) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026 Levitation gap (m) Transient Inductrack model Linear reference model (c) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026 Levitation gap (m) Transient Inductrack model Linear reference model (d) Figure 6.9: Transient responses of the levitation gap for the Inductrack system with feedback control: (a) atv = 2 m/s; (b) atv = 5 m/s; (c) atv = 20 m/s; and (d) atv = 100 m/s Now, with the proposed nonlinear feedback control law implemented, the levitation gap responses of the controlled vehicle in the four speed cases are exhibited in Figs. 6.9(a)-(d), by the solid curves. In the simulation, the gains of the PID controller, which are independent of the vehicle speed, are tuned as: K P = 4000 N/m,K I = 14000 N/(ms), andK D = 400 N/(ms −1 ); the parameters of the nonlinear mapping function in Fig. 6.7(b) are listed in Table 6.1. As seen from the gures, the feedback control law as described in the previous sections is able to eectively stabilize the response of the vehicle in all the four speed cases. Furthermore, the stabilized levitation gap in each speed case converges to the value r = 0:02375 m as desired. Also, other traveling speeds (v>v cr1 ) have been considered in the simulation 74 and similar results on stabilization and convergence to the desired levitation gap are obtained. Due to limited space, these results are not shown here. Table 6.1: The parameters of the nonlinear mapping function used for numerical simulations v (m/s) K J (Am −1 /N) (m −1 ) 1.5 1856 130.7 2 2193 129.9 5 3326 126.5 20 3695 125.1 60 3715 124.8 100 3717 124.8 For comparison purposes, the "preferred" levitation gap response, which is generated by the linear reference model in Figs. 6.2 and 6.3, is also added to Fig. 6.9; see the dashed curves. As can be seen, the actual levitation gap responses exhibit undershoots and overshoots in the rst two cases and longer settling times compared to those of the "preferred" response. Though the two responses eventually converge to the same reference value as time goes by. This is anticipated because the nonlinear mapping functionf NL in Fig. 6.5 is identied by steady-state quantities. It thus takes more eort for the PID controller, which is designed based on a simple linear reference model, to regulate the transient response of the Inductrack system with nonlinear eects and complicated electro-magneto-mechanical couplings. Therefore, in tuning the PID gains based on the linear reference model, the control specications such as overshoot and settling time should be set more stringently. Figure 6.10 depicts the time histories of the current densityJ C (t) in the active coils at the four dierent values of the vehicle speed, whose squared value (J 2 C ) is proportional to the control power. As can be seen from the gure, the steady-state value ofJ C (t), denoted byJ C;ss , can be either positive or negative, depending on the relation between the reference levitation gap r and the equilibrium levitation gap of the uncontrolled system e (v). Ifr > e (v), J C;ss > 0, which indicates that the source magnetic eld is intensied. Ifr < e (v), J C;ss < 0, which means that the source magnetic eld gets attenuated. In addition, the smaller the dierence betweenr and e (v), the lower the magnitude ofJ C;ss (in this example, 75 r e atv = 20 m/s). Accordingly, to minimize the control power consumed in the system, the reference gapr is to be set as close to the equilibrium gap e (v) at the operating speed as possible. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) -3 -2 -1 0 1 2 3 4 5 6 7 J C (t) (A/m) 10 5 v = 2 m/s v = 5 m/s v = 20 m/s v = 100 m/s Figure 6.10: The time-varying current densityJ C (t) in the active coils Figures 6.11(a) and (b) show the time histories of the estimated levitation force (denoted by F u in Fig. 6.6, withF u =u +Mg) and the actual levitation force calculated from the proposed transient model, respectively, regarding the same set of simulations as presented in Figs. 6.9 and 6.10. For clearer observation of the force variation trend, the displayed time in Figs. 6.11(a) and (b) is truncated to shorter spans, after which the responses almost converge. As foreseen, the actual levitation forces converge to the weight of the vehicle (Mg = 196 N) quickly to achieve stable suspension. However, depending on the traveling speed, the estimated levitation forces can converge to dierent values other than the vehicle’s weight; see Fig. 6.11(a). This is due to the fact that the transient responses of levitation gap, current density in active coils and estimated levitation force (Figs. 6.9-6.11) are in a dynamic balance, which is achieved by the proposed feedback control method with the tuned PID controller and the nonlinear mapping function. 76 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) 170 180 190 200 210 220 230 Estimated levitation force F u (N) v = 2 m/s v = 5 m/s v = 20 m/s v = 100 m/s (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (s) -50 0 50 100 150 200 250 300 350 400 Actual levitation force (N) v = 2 m/s v = 5 m/s v = 20 m/s v = 100 m/s (b) Figure 6.11: The estimated and actual levitation forces in the Inductrack system with feedback control: (a) the estimated levitation forceF u ; and (b) the actual levitation force The above-mentioned examples are all about the case whenv>v cr1 , in which the uncontrolled system is always unstable. When 0 < v < v cr1 , the response of the vehicle is bounded as it is in the stability region (S1). However, the levitation gap response is oscillatory, and it may not converge to the desired value (r = 0:02375 m in the current example). Under such circumstances, the proposed feedback control law is still useful for regulating system output. To show this, let the vehicle travel atv = 1:5 m/s, which is lower thanv cr1 . Figure 6.12 shows the levitation gap responses of the vehicle, without and with feedback control. As observed, with feedback control, the levitation gap quickly reaches the desired reference value and the unwanted oscillations are suppressed in a short time. 77 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 Levitation gap (m) Controlled system Uncontrolled system Figure 6.12: Transient responses of the levitation gap atv = 1:5 m/s, without and with feedback control In the last example, consider four cases of the reference levitation gap: r = 0:8d; 0:9d; 1:0d; 1:1d (r = 0:02; 0:0225; 0:025; 0:0275 m). In all the cases, the vehicle speed v = 60 m/s, the initial gap 0 = 0:8d = 0:02 m, and the PID gains are the same as those used to generate Figs. 6.9-6.12. Also, by Table 6.1, the parameters regarding the nonlinear mapping functions at v = 60 m/s are K J = 3715 Am −1 /N and = 124:8 m −1 . Plotted in Fig. 6.13 are the transient responses of the controlled system with the four dierent reference levitation gaps. As shown in Fig. 6.13(a), in each case, the Inductrack vehicle is eectively regulated by the proposed feedback control law such that the transient levitation gap smoothly converges to the corresponding reference value. Note that a larger dierence between r and 0 results in a longer setting time, which is dierent from a linear system. The time histories of the current density in the active coils are shown in Fig. 6.13(b). As can be speculated from Fig. 6.7(a), the equilibrium position e at v = 60 m/s is slightly higher than 0:95d = 0:02375 m. Therefore, the steady-state magnitude of the current density, J C;ss , is minimum whenr = 1:0d among the four cases, which indicates the minimum control power required for stable magnetic levitation. 78 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 Levitation gap (m) Reference gap r = 0.8d Reference gap r = 0.9d Reference gap r = 1.0d Reference gap r = 1.1d (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s) -3 -2 -1 0 1 2 3 4 5 J c (t) (A/m) 10 5 Reference gap r = 0.8d Reference gap r = 0.9d Reference gap r = 1.0d Reference gap r = 1.1d (b) Figure 6.13: Transient responses of the controlled system atv = 60 m/s, with dierent reference levitation gaps: (a) levitation gap; and (b) active current densityJ C (t) In this section, the proposed nonlinear feedback control method is validated through numerical simu- lations. The control method is shown to be able to eciently stabilize the vertical motion (levitation gap) of the Inductrack vehicle in a wide range of speed. Also, the feedback controller renders smooth and fast convergence of the system response to an arbitrary preset value of the desired levitation gap. 6.6 ExtensionoftheFeedbackControlStrategy It should be noted that the control method discussed in this chapter can be further extended. Specically, neither the linear portion is restrained in a PID control nor the usage of nonlinear mapping functionf NL requires time-invariant traveling speeds of the Inductrack vehicle. To manifest the author’s latest work, this section proposes a control stratagy based on state feedback for the Inductrack dynamic system, which is also able to handle the time-varying travling speed through a modication of the mapping function. 6.6.1 LinearControllerBasedonStateFeedback The control design takes partial state feedback, namely,x N+2 ,x N+3 , andx N+4 from the transient Induc- track model (3.37). These states correspond to the traveling speed, levitation gap, and vertical oscillation 79 rate of the vehicle, respectively. Three xed gainsk 1 ;k 2 ;k 3 and one varying gainf NL (determined by the nonlinear mapping function) are regulated to guarantee that the outputy (real-time levitation gap) tracks the step inputr (related to the reference levitation gap). For convenience,r is introduced from the output side this time, so that the goal of the controller congured in Fig. 6.14 is to attract the relevant states back to the origin. Figure 6.14: The block diagram of the Inductrack feedback control system based on state feedback The linear part of the proposed controller is set up by treating the assembly enclosed by the dashed box in Fig. 6.14 as a unit lumped mass. Dene statesz 1 ;z 2 ;z 3 for the corresponding linear system as follows z 1 =x N+3 r z 2 =x N+4 (6.12) andz 3 is the output of the integrator. The feedback gainsk 1 ,k 2 and integrator gaink 3 are tuned such that the control system is able to stabilize the above three states at the origin. Letz = [z 1 z 2 z 3 ] T and write the linear state equations of the closed-loop system in the matrix form: _ z =A cl z = 2 6 6 4 0 1 0 k 1 k 2 k 3 1 0 0 3 7 7 5 z (6.13) 80 with the characteristic equation 3 +k 2 2 +k 1 +k 3 = 0 (6.14) The response of Eq. (6.13) will behave like a second order system if the eigenvalues () of matrixA cl are properly assigned by tuning the gains based on Eq. (6.14). For instance, the desired eigenvalues can be set by ( + 10! n )( 2 + 2! n +! 2 n ) = 0 (6.15) where,! n denotes damping ratio and natural frequency of the corresponding second order system. A scalar of 10 is applied for the real eigenvalue that makes the complex eigenvalues of Eq. (6.15) dominant to obtain a faster response. In this way, the gains in Eq. (6.14) can be found through a term-by-term comparison with Eq. (6.15). 6.6.2 ModiedNonlinearMappingFunction To make the nonlinear force-current mapping function proposed in Sec. 6.3 valid for the time-varying traveling speed, one only needs to nd the value ofK J in Eq. (6.6) for all the speedsv. This requires a curve tting process forK J (v), which empirically has the following form: K J (v) = C 1 1 +C 2 v 2 (6.16) whereC 1 andC 2 purely depend on system parameters and they can be further identied via numerical simulations based on model (4.3). 81 With the similar state denition in Eq. (6.10), the state-space model of the controlled Inductrack system based on state feedback is revamped as _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ;f NL (x N+2 ;x N+3 ;x N+5 )) _ x N+1 =x N+2 _ x N+2 =f d (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;f NL (x N+2 ;x N+3 ;x N+5 )) + F p M _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;f NL (x N+2 ;x N+3 ;x N+5 ))g _ x N+5 =k 1 x N+4 k 3 (x N+3 r) k 2 [f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;f NL (x N+2 ;x N+3 ;x N+5 ))g] (6.17) wherek = 1; 2;:::;N. Although the expression in Eq. (6.17) resembles the one given in Eq. (6.11), this control design proposed in this section has twofold advantages. On one hand, the linear part of the controller constructed based on the state feedback can be generalized to pertain more complex dynamic scenarios, e.g., more states need to be used for control when the vehicle is in rotation. On the other, by adding a curve tting procedure of Eq. (6.16), as shall be seen later, the nonlinear mapping function in Eq. (6.6) is not only applicable to constant traveling speeds, but also valid for time-dependent speed when the vehicle travels in an acceleration or deceleration mode. 6.6.3 NumericalExamplesofStateFeedback-BasedControl In this subsection, numerical examples will be presented to validate the control design approach presented in Sec. 6.6. The proposed controller is implemented to the transient Inductrack model (6.17) with both constant traveling speeds and time-varying traveling speed, where the latter has not been addressed yet. 82 The physical and control parameters of the system used for numerical simulations in this work are listed in Tables 4.1 and 6.2. Table 6.2: List of state feedback control parameters used for numerical simulations Parameter Value k 1 347.4 k 2 48 k 3 1096 r(v = 2m=s) 0.05375 m r(v = 20m=s) 0.05875 m r(v =v(t)) 0.05875 m 0 0.02 m (v) 125.7 C 1 3716 C 2 2.78 The proposed controller based on state feedback is rstly utilized to stabilize the vibration of the Inductrack vehicle in the levitation direction (levitation gap) under constant traveling speeds, in order to make a comparison with Sec. 6.5. Two cases of traveling speeds,v = 2 andv = 20 m/s, are considered, and two cases of reference inputr (relevant to the reference levitation gap) are given in Table 6.2. With an initial levitation gap 0 = 0:8d = 0:02 m, the time histories of the levitation gap are shown in Fig. 6.15. For the comparison purposes, the time responses of the levitation gap regulated by a PID-based nonlinear controller (see Sec. 6.5) at the same traveling speeds are overplotted in Fig. 6.15. It is seen that at each speed, both controllers are able to make the levitation gap converged to the reference values. That is to say, the state feedback-based control algorithm investigated in this work can reproduce the function that has been achieved by the PID-based controller. With the state feedback gains carefully tuned using Eqs. (6.14) and (6.15), the proposed controller can achieve better performance, such as less overshoot and shorter settling time, as shown in Fig. 6.15. Furthermore, the state feedback structure in Fig. 6.14 will proactively bring convenience to the control design for more general M-DOF Inductrack dynamic systems, where more states or more advanced linear control algorithms are involved. 83 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) 0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026 Levitation gap (m) (t) v=2 m/s, SF v=2 m/s, PID v=20 m/s, SF v=20 m/s, PID Figure 6.15: Transient levitation gap atv = 2 andv = 20 m/s, under state feedback (SF)- and PID-based control Secondly, the proposed controller is implemented to stabilize the oscillating levitation gap when the Inductrack vehicle is traveling with a time-varying speed along the longitudinal direction. It also means that in the nonlinear part of the controller, the gainK J (v) in Eq. (6.6) is changing with time. To create a time-dependent traveling speed for the vehicle, let the propulsion forceF p in Eq. (6.17) be a square wave of amplitudeF 0 so that the vehicle dynamics in the longitudinal direction kicks in. In the simulation, the pulse width ofF p is set to 1.5 s and the value ofF 0 is carefully selected to accelerate the vehicle. Practically, F 0 should be large enough to guarantee that the vehicle escapes the high-drag region at low speeds within a short time; see Sec. 5.2.2. For the ecient use of the control energy, in the varying speed scenario, the control system will only be switched on when the following conditions are both satised: v(t)> 5 m=s;(t)> 0:4d = 0:01 m (6.18) 84 The initial gap is set as 0 = 0:8d = 0:02 m for consistency. With dierent amplitudes of the propulsion forceF 0 = 500; 600 N that were used in Sec. 5.2.2, the time responses of the levitation gap for the controlled and uncontrolled Inductrack system are exhibited in Fig. 6.16. In addition, to display the control energy, the time histories of the current density in the active coils for the controlled cases are shown in Fig. 6.17. 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) 0.005 0.01 0.015 0.02 0.025 0.03 Levitation gap (m) (t) F 0 =500 N, controlled F 0 =500 N, uncontrolled F 0 =600 N, controlled F 0 =600 N, uncontrolled Figure 6.16: Transient levitation gap at time-dependent traveling speed, subject to prescribed propulsion force, with and without control As can be observed from Fig. 6.16, during the accelerating stage and the cruising stage (in fact, the ve- hicle is decelerating slowly due to a low drag force at the cruising stage), the original vibrations in levita- tion gap are successfully suppressed by the implemented controller. This result conrms the applicability of the proposed control design in the scenario where there is time-varying traveling speed of the vehicle. 85 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Current Density in Active Coils (A/m) 10 5 J c (t) F 0 =500 N, controlled F 0 =600 N, controlled Figure 6.17: Transient response of the equivalent current density of the active coils in the controlled In- ductrack dynamic system Also, as seen in Fig. 6.17, the magnitude of the active coil current density, which is pertinent to the consumption of control energy, is always within 20% of the magnetization current densityJ M of the PMs. Noticeably, if the controller is switched on at the low speed region, the reference levitation gap should also be properly adjusted (refer to the decreased reference levitation gap atv = 2 m/s in Fig. 6.15). Otherwise, due to the limited levitation force that can be naturally generated by the moving vehicle at low speeds, the actuator will need to output signicantly large currents in the active coils, which may not be realistic. 86 Chapter7 ModelExtension: A2DRigid-BodyInductrackSystem As emphasized in previously, the benchmark transient Inductrack model derived in Chap. 3 can be extended to more general M-DOF cases. This chapter briey shows the derivation of 2D, 3-DOF rigid-body transient Inductrack model as an example. The approach can be further extended to the modeling of a full 3D, 6-DOF rigid-body Inductrack system. 7.1 CoordinateSystemsandTransformations Consider the schematic of a 2D rigid-body Inductrack model illustrated in Fig. 7.1, where the pitch rotation angle is denoted by(t). According to Fig. 7.1, three reference frames in such a system are dened: the inertial frame o I -x I y I z I , the body frameo B -x B y B z B , and the magnet frameo M -x M y M z M . The position vectors of a point measured from each of the frames are denoted byr I ,r B andr M , respectively. Note that they are dierent vectors, which satisfy the following transformation relation: r M =r B +o MB r I =r B +o IB r M =r I +o MI (7.1) 87 Figure 7.1: Schematic of a 2D, 3-DOF Inductrack system: (a) in front view; (b) in side view; and (c) the conguration of Halbach arrays On the other hand, for the same vectorr, it can be expressed in component form in any of those three dened frames. The component forms are subject to the following transformation: 2 6 6 4 x B y B z B 3 7 7 5 =R 2 6 6 4 x I y I z I 3 7 7 5 , 2 6 6 4 x I y I z I 3 7 7 5 =S 2 6 6 4 x B y B z B 3 7 7 5 , 2 6 6 4 x M y M z M 3 7 7 5 = 2 6 6 4 x B y B z B 3 7 7 5 (7.2) 88 whereR andS are passive and active rotation matrix, respectively. In a 2D space, the matrices are dened by R = 2 6 6 4 1 0 0 0 cos sin 0 sin cos 3 7 7 5 , S = 2 6 6 4 1 0 0 0 cos sin 0 sin cos 3 7 7 5 (7.3) withR T =S. To extend this derivation to general 3D rigid-body cases, matricesR andS should be replaced by 3D passive and active rotation matrices, which contain Euler angles describing pitch, roll and yaw motions. In order to calculate the electromagnetic quantities, it is crucial to determine the location of each coil in the magnet frame. Suppose the position vector of thekth coil is given by r k (the location of the centroid of the upper rim), all its possible component forms are dened as: r k I;I = h r k I;x I r k I;y I r k I;z I i T = h x k I y k I z k I i T r k M;M = h r k M;x M r k M;y M r k M;z M i T = h x k M y k M z k M i T (7.4) and r k I;M = h r k I;x M r k I;y M r k I;z M i T r k M;I = h r k M;x I r k M;y I r k M;z I i T (7.5) where the rst subscript of r k denotes from which frame the vector is measured, while the second subscript means based on which frame the components are dened. If the two subscripts are identical, the coordinates expression is allowed with a simplication as shown in Eq. (7.4). 89 From Eqs. (7.1)-(7.5), it follows that r k M;M =R r k M;I =R( r k I;I +o MI;I ) =R( r k I;I +o MB;I o IB;I ) =R( r k I;I +o MB;I r G;I ) =R( r k I;I r G;I ) +o MB;M (7.6) where according to Fig. 7.1, r k I;I ando MB;M are time-invariant.r G;I is the displacement vector measured by the COM of the vehicle from the inertial frame, that is, r G;I = h 0 Y (t) Z(t) i T (7.7) For a quick validation, let 0. Equation (7.6) will then reduce to 2 6 6 4 x k M y k M z k M 3 7 7 5 = 2 6 6 4 1 0 0 0 1 0 0 0 1 3 7 7 5 2 6 6 4 x k I y k I Y (t) z k I Z(t) 3 7 7 5 + 2 6 6 4 0 l G h G 3 7 7 5 = 2 6 6 4 x k I y k I Y (t) +l G z k I Z(t) +h G 3 7 7 5 (7.8) which yields exactly the same result as derived in Eq. (2.3). 7.2 CalculationoftheElectromagneticQuantities The magnetic ux density at thekth coilB k M = [B k x M B k y M B k z M ] T can be computed by the Biot-Savart law in the magnet frame. However, when the vehicle is in rotation, to compute the emfe k , components of the magnetic eld should be expressed in the inertial frame via B k I = h B k x I B k y I B k z I i T =SB k M (7.9) 90 This is because the coil location in inertial frame is time-invariant, which is convenient for the calculation ofe k . Note that in Eq. (7.9),B k M andB k I represent the same vector, but with dierent components in two dierent frames. According to the Faraday’s law, e k = @ @t Z S k SB k M dS k (7.10) by noting that dS k = [0 dS k 0] T with respect to the inertial frame, it can be derived that e k = @ @t Z S k S 2 6 6 4 B k x M B k y M B k z M 3 7 7 5 2 6 6 4 0 1 0 3 7 7 5 dS k = @ @t Z S k 0 B B B @ S 2 6 6 4 B k x M B k y M B k z M 3 7 7 5 1 C C C A T2 6 6 4 0 1 0 3 7 7 5 dS k (7.11) which follows that e k = @ @t Z S k 2 6 6 4 B k x M B k y M B k z M 3 7 7 5 T R 2 6 6 4 0 1 0 3 7 7 5 dS k = Z S k @ @t h (B k M ) T R 2 i dS k (7.12) whereR = [R 1 R 2 R 3 ], and in the 2D space,R 2 = [0 cos sin] T . Now evaluate the integrand with time derivative in Eq. (7.12) by the chain rule, @ @t h (B k M ) T R 2 i = @ @t (B k M ) T R 2 + (B k M ) T @R 2 @t (7.13) Recall that B k M = [B k x M B k y M B k z M ] T , r k M;M = [ x k M y k M z k M ] T , and each component of B k M is a function of vector r k M;M . Therefore, @B k M @t = @B k M @ r k M;M @ r k M;M @t (7.14) 91 Plug Eq. (7.6) in (7.14), @B k M @t = @B k M @ r k M;M @ @t h R( r k I;I r G;I ) +o MB;M i = @B k M @ r k M;M @ @t h R( r k I;I r G;I ) i (7.15) whereo MB;M can be omitted because it is time-invariant. Compute the time derivative term in the result of Eq. (7.15), @B k M @t = @B k M @ r k M;M @R @t ( r k I;I r G;I )R @r G;I @t (7.16) Plug Eq. (7.16) in Eq. (7.13), @ @t h (B k M ) T R 2 i = ( @B k M @ r k M;M @R @t ( r k I;I r G;I )R @r G;I @t ) T R 2 + (B k M ) T @R 2 @t (7.17) For the 2D, 3-DOF rigid-body model, Eq. (7.17) can be computed term by term as follows. Firstly, it is easy to show that the second term of Eq. (7.17) yields (B k M ) T @R 2 @t = _ (B k y M sin +B k z M cos) (7.18) Secondly, to compute the rst term of Eq. (7.17), some useful results are listed below in advance: @B k M @ r k M;M = 2 6 6 6 6 4 @B k x M @x M @B k x M @y M @B k x M @z M @B k y M @x M @B k y M @y M @B k y M @z M @B k z M @x M @B k z M @y M @B k z M @z M 3 7 7 7 7 5 , @R @t = 2 6 6 4 0 0 0 0 _ sin _ cos 0 _ cos _ sin 3 7 7 5 (7.19) r k I;I r G;I = 2 6 6 4 x k I y k I Y z k I Z 3 7 7 5 , R @r G;I @t = 2 6 6 4 0 _ Y cos + _ Z sin _ Y sin + _ Z cos 3 7 7 5 (7.20) 92 Note that only the location of the centroid of the coil’s upper rim is temporarily used as a locater in the above derivations. To evaluate the integrals later, the location of the coil’s lower rim is also used, and it is dened using a similar notationr k . Plug Eqs. (7.18)-(7.20) in Eq. (7.17), @ @t h (B k M ) T R 2 i = @B k y M @y M n [ _ ( y k I Y ) + _ Z] sin [ _ Y _ ( z k I Z)] cos o cos @B k z M @y M n [ _ ( y k I Y ) + _ Z] sin [ _ Y _ ( z k I Z)] cos o sin + @B k y M @z M n [ _ ( y k I Y ) + _ Z] cos + [ _ Y _ ( z k I Z)] sin o cos @B k z M @z M n [ _ ( y k I Y ) + _ Z] cos + [ _ Y _ ( z k I Z)] sin o sin B k y M _ sinB k z M _ cos (7.21) For a quick validation, suppose that there is no rotation, Eq. (7.21) reduces to @ @t h (B k M ) T R 2 i = @B k y M @y M _ Y @B k y M @z M _ Z (7.22) which exactly represents the case of lumped mass model; see Eqs. (3.18) and (3.19). Plugging Eq. (7.21) in Eq. (7.12) gives the following form for the emf: e k = Z S k @B k y M @y M f yy @B k z M @y M f zy + @B k y M @z M f yz @B k z M @z M f zz B k y M f y B k z M f z ! dS k (7.23) Notice that in Eq. (7.23), dS k is measured in the inertial frame, which is not consistent with the quadrature results (see Appx. A) that are derived in the magnet frame. Thus, consider the following transformation: dS k = dx I dz I = dx M d(y M sin +z M cos) = sindx M dy M + cosdx M dz M (7.24) 93 which expands Eq. (7.23) as follows: e k =f yy sin Z y k M y k M Z x + x @B k y M @y M dx M dy M f yy cos Z z k M z k M Z x + x @B k y M @y M dx M dz M +f zy sin Z y k M y k M Z x + x @B k z M @y M dx M dy M +f zy cos Z z k M z k M Z x + x @B k z M @y M dx M dz M f yz sin Z y k M y k M Z x + x @B k y M @z M dx M dy M f yz cos Z z k M z k M Z x + x @B k y M @z M dx M dz M +f zz sin Z y k M y k M Z x + x @B k z M @z M dx M dy M +f zz cos Z z k M z k M Z x + x @B k z M @z M dx M dz M +f y sin Z y k M y k M Z x + x B k y M dx M dy M +f y cos Z z k M z k M Z x + x B k y M dx M dz M +f z sin Z y k M y k M Z x + x B k z M dx M dy M +f z cos Z z k M z k M Z x + x B k z M dx M dz M (7.25) Note that each integrand in Eq. (7.25) is a function ofx M ,y M , andz M . Nevertheless, when6= 0,y M andz M are dependent of each other by y M = tan (z M z k M ) + y k M z M = cot (y M y k M ) + z k M (7.26) which guarantees that Eq. (7.25) outputs a scalar, although numerical integration is required. It should be noted that once = 0 (even if _ 6= 0), Eq. (7.25) will be highly simplied and analytical integration is assured to be applicable; refer to Appx. A. 7.3 CalculationoftheMagneticForceandTorque The Ampere force acting on thekth coil due to theith magnet blockF C i;k is given by F C i;k =I k 4 X j=1 Z c k j dl k j B i;k j I (7.27) 94 Taking account of the rotation transformation, Eq. (7.27) gives F C i;k =I k 4 X j=1 Z c k j dl k j SB i;k j M =I k 4 X j=1 Z c k j [dl k ] j SB i;k j M (7.28) where [dl k ] j is the skew matrix corresponding to vector dl k j . Namely, [dl k ] 1 = 2 6 6 4 0 0 0 0 0 dx I 0 dx I 0 3 7 7 5 , [dl k ] 2 = 2 6 6 4 0 dz I 0 dz I 0 0 0 0 0 3 7 7 5 [dl k ] 3 = 2 6 6 4 0 0 0 0 0 dx I 0 dx I 0 3 7 7 5 , [dl k ] 4 = 2 6 6 4 0 dz I 0 dz I 0 0 0 0 0 3 7 7 5 (7.29) Plugging Eq. (7.29) in (7.28) yields F C i;k =I k 2 6 6 6 4 0 R w 2 w 2 [(B i;k 3 y M B i;k 1 y M ) sin + (B i;k 3 z M B i;k 1 z M ) cos] dx M R w 2 w 2 [(B i;k 3 z M B i;k 1 z M ) sin (B i;k 3 y M B i;k 1 y M ) cos] dx M 3 7 7 7 5 (7.30) and by action-reaction, the total magnetic force acting on theith PM is F B i = 1 X k=1 I k 2 6 6 6 4 0 R w 2 w 2 [(B i;k 3 y M B i;k 1 y M ) sin (B i;k 3 z M B i;k 1 z M ) cos] dx M R w 2 w 2 [(B i;k 3 z M B i;k 1 z M ) sin + (B i;k 3 y M B i;k 1 y M ) cos] dx M 3 7 7 7 5 (7.31) To calculate the torque, the distance vectors measured from the COM of the vehicle and the geometry center of each magnet block should be found. This can be easily done in the magnet frame because they are time-invariant. Without loss of generality, dene theith distance vector asd i;M . Nevertheless, since magnetic forces are expressed by the component form under the inertial frame, the distance vectors should also be transformed to the inertial frame. In other words,d i;I =Sd i;M is used. 95 Givend i;I , the total torque applying on the vehicle is T = 2N b X i=1 d i;I F B i (7.32) where one should notice that in 2D space, both F B i and d i;I only contain components in longitudinal (y) and vertical (z) directions, so the cross product in Eq. (7.32) will only end up with a component inx direction, which follows the right-hand rule for rotation. 7.4 GoverningEquationsandState-SpaceRepresentation Based on the above derivations, the governing equations of the 3-DOF transient Inductrack model is for- mulated as follows: M Y =F p +F B Y M Z =F B Z Mg I G =T X L eq _ I k +R eq I k =e k , k = 1; 2;::: (7.33) whereF B = P 2N b i=1 F B i . Dene the following states: x k =I k , k = 1; 2;:::;N x N+1 =Y , x N+2 = _ Y , x N+3 =Z, x N+4 = _ Z x N+5 =, x N+6 = _ (7.34) 96 then similar to Eq. (3.37), the state-space representation of the 2D rigid-body transient Inductrack model is given by _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ;x N+5 ;x N+6 ) _ x N+1 =x N+2 _ x N+2 =f d (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;x N+5 ) + F p M _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;x N+5 )g _ x N+5 =x N+6 _ x N+6 =f r (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;x N+5 ) (7.35) wheref l andf d have been dened in Sec. 3.5. The functionf r , which describes the torque, essentially embraces bothf l andf d . 7.5 ValidationoftheRigid-BodyInductrackModel Before seeking for full transient response of the rigid-body Inductrack system, the derived model (7.35) will rstly go through a set of validations, which are based on steady-state or simpler dynamic scenarios. In each of these cases, it is assumed the vehicle travels with a constant horizontal speedv. 97 7.5.1 TheInternalTorqueinPseudo-Steady-State2-DOFMotion The setup of a pseudo-steady-state 1D, 2-DOF motion of the Inductrack system has been described in detail in Sec. 4.2. Based on Eqs. (4.3) and (7.35), the model of this scenario in state-space representation is simplied by _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ;x N+5 ;x N+6 ) _ x N+1 =v, _ x N+2 = 0, _ x N+3 = 0 _ x N+4 = 0, _ x N+5 = 0, _ x N+6 = 0 (7.36) where k = 1; 2;:::;N with x k (0) = 0, x N+1 (0) = 0, x N+2 (0) = v, x N+3 (0) = Z 0 , x N+4 (0) = 0, x N+5 (0) = 0 ,x N+6 (0) = 0. As usual, the model given in Eq. (7.36) is able to calculate the magnetic levitation force, magnetic drag force, as well as the magnetic torque applied onto the vehicle due to the magnetic forces. Prior to any discussion on the rigid-body motion, the internal torque in the 1D, 2-DOF lumped mass motion (namely, Eq. (7.36) with 0 = 0) is rstly investigated. This has not yet been studied in previous chapters. For consistency with previous simulations, set Z 0 = 0 +h G + 0:5d, where 0 = 0:8d = 0:02 m is the xed levitation gap. Run the simulation with dierent traveling speedsv and record the values of steady-state magnetic torque and this torque is named as the internal torque. Notice that, if a steady-state value of the response is not strictly converged but has a bounded oscillation, then its averaged value will be taken for the record. As such, the plot of the steady-state internal torque against dierent constant traveling speeds is depicted below. 98 10 -1 10 0 10 1 10 2 10 3 Traveling speed (m/s) -2 -1 0 1 2 3 4 5 6 7 Internal torque (N*m) Figure 7.2: Steady-state internal torque at dierent traveling speeds with xed levitation gap 0 = 0:8d = 0:02 m As can be observed in Fig. 7.2, in the low-speed range (say 0 < v < 2:5 m/s), the internal torque is negative, which induces a "nose down" rotation trend. In this range, as the speed increases, the amplitude of the internal torque rst increases and then decreases towards 0. After the traveling speed surpasses a threshold (about 2.5 m/s), the internal torque turns positive. As the speed further increases, the internal torque keeps increasing until it reaches an asymptotic value (about 6.5 Nm). In such case, the torque will cause a "nose up" rotation trend in the pitch direction. Now that there exists an internal torque even in a pseudo-steady state, the modeling of rigid-body Inductrack systems is of vital importance for further studies of the system dynamics. Another point we are interested in is that how the levitation gap will aect the internal torque. To investigate this eect, simulations are conducted with dierent xed levitation gaps with a traveling speed v = 1 m/s, and the time histories of the torque responses are plotted in Fig. 7.3. As seen from the gures, when the levitation gap shrinks, the response of the internal torque becomes more and more oscillatory, which will inevitably bring more diculties to the system control. This phenomenon is essentially due 99 to the irregularity of the magnetic eld distribution when it gets closer to bottom surface of the magnet arrays; see Fig. 4.2. Therefore, in a transient scenario of the rigid-body system, the reference levitation gap needs to be chosen appropriately to avoid the unwanted oscillations and our model can provide such guidance via simulations. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) -25 -20 -15 -10 -5 0 5 10 15 Internal torque (N*m) (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) -6 -5 -4 -3 -2 -1 0 1 Internal torque (N*m) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) -3 -2.5 -2 -1.5 -1 -0.5 0 Internal torque (N*m) (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 Internal torque (N*m) (d) Figure 7.3: Response of the steady-state internal torque at dierent xed levitation gaps with a traveling speed v = 1 m/s: (a) 0 = 0:2d = 0:005 m; (b) 0 = 0:4d = 0:01 m; (c) 0 = 0:6d = 0:015 m; (d) 0 = 0:8d = 0:02 m 7.5.2 MagneticForceandTorqueatFixedRotationAngle In this subsection, the circumstance in which the vehicle is tilted with a xed angle will be studied. From the derivations shown earlier in this chapter, it is convincing that the electro-magneto-mechanical interactions 100 in a rigid-body Inductrack system are much more involved. In addition, integrals that need to be evaluated in the rigid-body model do not have analytical expressions in general, which in most of the scenarios requires the application of numerical integration. With the rotation mechanism incorporated, the "levitation gap" is no longer a single value but it depends on the location of the measurement. For convenience, letG 1 be the rear levitation gap andG 2 be the front one. Dene a levitation gapG 3 as the algebraic average ofG 1 andG 2 , namely,G 3 = G 1 +G 2 2 . When the rotation angle is small, G 3 is approximately the levitation gap that was dened in the 1D, 2-DOF lumped-mass Inductrack model. In Eq. (7.36), considerv = 1 m/s and 0 = 0:8d = 0:02 m. Simulations are performed at dierent x rotation angles 0 = 0;1 ;2 . The response of levitation and drag forces, and that of the internal torque, are displayed in Figs. 7.4 and 7.5, respectively. 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) -20 0 20 40 60 80 100 120 Levitation force (N) 0 deg 1 deg -1 deg 2 deg -2 deg (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) 0 50 100 150 200 250 Drag force (N) 0 deg 1 deg -1 deg 2 deg -2 deg (b) Figure 7.4: Steady-state response under dierent xed rotation angles at constant traveling speedv = 1 m/s: (a) levitation force; (b) drag force From the results displayed in Figs. 7.4 and 7.5, it is concluded that increasing amplitude of the rotation angle will simultaneously cause increasing magnetic forces, because the levitation gap is no longer uniform. Moreover, signicant internal torques will be induced even if the vehicle is tilted by a relatively small angle in the pitch direction. When 0 > 0, the internal torque is negative, which reects 101 that the vehicle tends to have a "nose down" motion while its rear is closer to the track; on the other hand, when 0 < 0, the internal torque is positive, meaning that the vehicle tends to have a "nose up" motion while its front is closer to the track. 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) -60 -40 -20 0 20 40 60 Internal torque (N*m) 0 deg 1 deg -1 deg 2 deg -2 deg Figure 7.5: Steady-state response of the internal torque under dierent xed rotation angles at constant traveling speedv = 1 m/s. Nevertheless, there are two discrepancies discovered in Figs. 7.4 and 7.5. For one thing, the force responses when = 0 do not strictly overlap those when = 0 , although they persist the same prole; see Fig. 7.4. For another, the torque responses when = 0 and when = 0 are not strictly symmetric with respect to the x-axis, although each pair possesses a similar characteristic in terms of the oscillation; see Fig. 7.5. The reason for such discrepancies can be explained as follows: at the same moment, the array set mounted in front is always about to excited the track coils which originally had zero current, however, the array set mounted in rear can is always about to excite the track coils which have been excited by the front array set. Hence, through the complicated electro-magneto-mechanical interactions with transient responses, the such discrepancies eventually arised. 102 7.5.3 TransientResponseofPureRotationMotion In the last part of the validation, we move one step further and consider another pseudo steady-state scenario - the "rotation only" setting. Assume that the vertical oscillation of the vehicle has been stabilized by the controller, and then we want to observe how the transient response of the pitch rotation () looks like. In this case, the state space representation is formulated by _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ;x N+5 ;x N+6 ) _ x N+1 =v, _ x N+2 = 0, _ x N+3 = 0, _ x N+4 = 0, _ x N+5 =x N+6 _ x N+6 =f r (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;x N+5 ) (7.37) where k = 1; 2;:::;N with x k (0) = 0, x N+1 (0) = 0, x N+2 (0) = v, x N+3 (0) = Z 0 , x N+4 (0) = 0, x N+5 (0) = 0 ,x N+6 (0) = 0. Consider v = 1; 2; 5; 10 m/s, some preliminary results of the "rotation only" Inductrack model are presented in Fig. 7.6. The transient responses shown in Figs. 7.6(a)-(c) further validates the preliminary results obtained in Sec. 7.5.2, which indicates that a lower traveling speed renders a clockwise rotation trend while a higher traveling speed renders a counterclockwise rotation trend. The pitch rotation angles, although small, can be subject to a trend of divergence (see Fig. 7.6(c)), verifying the necessity of developing the rigid-body model. The transient responses of the levitation forces and torque can also be calculated by our proposed model as usual (see Figs. 7.6(d)-(f)), where the variation trend of the magnetic forces against traveling speed resembles those plotted in Fig. 4.5. 103 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) 0.018 0.0185 0.019 0.0195 0.02 0.0205 0.021 Rear levitation gap (m) v = 1 m/s v = 5 m/s v = 2 m/s v = 10 m/s (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) 0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 Front levitation gap (m) v = 1 m/s v = 5 m/s v = 2 m/s v = 10 m/s (b) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Pitch rotation angle (deg) v = 1 m/s v = 5 m/s v = 2 m/s v = 10 m/s (c) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -6 -4 -2 0 2 4 6 8 10 12 Internal torque (N*m) v = 1 m/s v = 5 m/s v = 2 m/s v = 10 m/s (d) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) 0 50 100 150 200 250 300 350 400 450 Levitation force (N) v = 1 m/s v = 5 m/s v = 2 m/s v = 10 m/s (e) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -50 0 50 100 150 200 250 300 Drag force (N) v = 1 m/s v = 5 m/s v = 2 m/s v = 10 m/s (f) Figure 7.6: Transient responses in the pure rotation motion at two dierent speedsv = 1; 2; 5; 10 m/s: (a) rear levitation gapG 1 ; (b) front levitation gapG 2 ; (c) pitch rotation angle; (d) magnetic torqueT ; (e) levitation forceF l ; and (f) drag forceF d 104 7.6 TransientResponseoftheRigid-BodyInductrackModel The last numerical example in this thesis shows a transient response of the rigid-body Inductrack model in which both vertical oscillation and rotation are involved. For simplication, again assume the vehicle travels with a constant horizontal speedv = 1 m/s. The state space model is written as _ x k =f k (x k ;x N+1 ;x N+2 ;x N+3 ;x N+4 ;x N+5 ;x N+6 ) _ x N+1 =v _ x N+2 = 0 _ x N+3 =x N+4 _ x N+4 =f l (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;x N+5 )g _ x N+5 =x N+6 _ x N+6 =f r (x 1 ;x 2 ;:::;x N ;x N+1 ;x N+3 ;x N+5 ) (7.38) where k = 1; 2;:::;N with x k (0) = 0, x N+1 (0) = 0, x N+2 (0) = v, x N+3 (0) = Z 0 , x N+4 (0) = 0, x N+5 (0) = 0,x N+6 (0) = 0; andZ 0 is in regard with 0 = 0:8d = 0:02 m. As observed in Fig. 7.7(a), the vertical oscillation has a trend of convergence that is similar to the case given in Fig. 5.1, and so as the levitation force in Fig. 7.7(c). This gives another validation for our 2-DOF benchmark model, which is very useful and ecient for predicting the transient response when the rotation motion is small. However, due to the existence of the magnetic torque (see Fig. 7.7(d)), generally, the pitch rotation angle can hardly converge to an asymptotic value (see Fig. 7.7(b)) even if the traveling speed of the vehicle is low. That is to say, when the simulation time is long enough, it is expected to see that the pitch angle of the vehicle will keep oscillating while the vehicle is almost sitting at a xed levitation gap. This transient feature will not be available without our proposed model. Also, the phenomenon seen in 105 Fig. 7.7(b) brings new challenges to the feedback control system presented in Chap. 6, where the oscillation in the rotation angle is required to be suppressed. This topic will be investigated in our future research. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Middle levitation gap (m) (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 Pitch rotation angle (deg) (b) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) 0 50 100 150 200 250 300 350 400 Levitation force (N) (c) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) -10 -8 -6 -4 -2 0 2 4 6 Magnetic torque (N*m) (d) Figure 7.7: Transient responses of the rigid-body Inductrack system with both vertical oscillation and pitch rotation atv = 1 m/s: (a) levitation gapG 3 ; (b) pitch rotation angle; (c)levitation forceF l ; and (d) magnetic torqueT In conclusion, Secs. 7.5 and 7.6 have shown and discussed some preliminary results about the newly- derived rigid-body transient Inductrack model. So far, all of the phenomena reected in simulations are in accordance with mathematical and physical rules. Based on these new discoveries, it will be interesting and convincing to proceed this study and consider the full transient scenarios in the 2D, 3-DOF rigid-body Inductrack dynamic system. This work is currently underway. 106 Chapter8 Conclusions In this dissertation, a novel transient model for the Inductrack system has been developed, followed by a model-based feedback control method. The main results are summarized as follows. (i) Based on the rst laws of nature, the benchmark transient Inductrack model is established and it is governed by a set of nonlinear integro-dierential equations. In the development, no assumption of steady-state quantities is smeared. Most importantly, the nite dimensions of the onboard Halbach arrays and the actual geometries of the track coils are clearly specied in system modeling, which is seen to have signicant eects in Sec. 4.1. (ii) Through a detailed description of induced coil currents and interactions between the source magnetic eld and the induced magnetic eld, the new transient model characterizes the generation of the magnetic levitation and drag forces in an Inductrack system. The electro-magneto-mechanical coupling mechanism is obtained in Eqs. (3.32)-(3.34). These results should help better understand the physical behaviors of the Inductrack dynamic system. (iii) The electromagnetic quantities of Inductrack systems, such as magnetic ux density, emf, and magnetic forces, are determined in analytical forms (Chap. 3 and Appx. A), which are useful for an in-depth physical understanding of Inductrack systems, as well as for ecient computation in dynamic analysis. 107 (iv) A theorem of equivalent magnetic forces is proven (Sec. 3.3 and Appx. B), which is a demonstration of Newton’s law of action-reaction on non-contact electromagnetic interaction forces in the Inductrack system. The theorem is also useful for ecient computation of the resultant magnetic forces applied at the moving vehicle. (v) The nonlinear governing equations of the transient Inductrack model are cast into a set of nonlinear state equations (3.37) for numerical simulations. The solution procedure for the state equations is designed with a specic moving window that enables the timely selection of a nite number of signicant coils during simulation. (vi) For validation purposes, the transient model is truncated into a pseudo-steady-state conguration by xing the levitation gap and traveling speed. The results in terms of "steady-state" magnetic forces are compared with those computed by the original steady-state model in the literature. It turns out that our truncated transient model can reproduce the noted steady-state results with even higher precision. Furthermore, a convergence study of the magnetic forces corresponding to dierent numbers of selected coils has been addressed. The converged magnetic forces not only give further validation of the proposed model but suggest the optimal number of coils that should be used in simulation as well. (vii) With the proposed 2-DOF benchmark model, the transient response of the Inductrack system in consideration is simulated in several dynamic scenarios, where the vehicle is prescribed with a propulsion force. The motion of the vehicle in both the longitudinal and vertical directions is analyzed in four typical cases. 1) When the propulsion force is equal to the magnetic drag force, the vehicle travels at a constant speed. It turns out that the stability behavior of the vehicle in its vertical motion changes at dierent constant speeds. Unstable oscillations are seen for most cases unless the traveling speed is very low, which has been known as the "negative damping eect" in EDS systems. 2) If there is no propulsion force, the vehicle with an initial speed is in a free motion and eventually lands on the track. The simulation results are in accordance with physical principles and the experimental results by the LLNL. 3) For a constant 108 propulsion force, it must be large enough to enforce the vehicle to escape the high-drag zone so as to achieve levitation. 4) For the given propulsion force prole in the vehicle’s acceleration and deceleration process, no steady-state behavior is observed in either the vehicle motion or the magnetic forces. Therefore, a transient model as presented in this work is essentially important to study the dynamic behaviors of Inductrack systems. (viii) An eective design method for nonlinear feedback control of Inductrack systems is proposed. In this method, a feedback controller consists of two components: a linear controller with either PID or state feedback, which is tuned based on a linear reference model; and a nonlinear force-current mapping function, which is generated based on the truncation of the proposed transient Inductrack model. The proposed control law is shown to be able to stabilize the system, at both constant and time-dependent traveling speed cases. Besides, the feedback controller also renders a smooth system output (real-time levitation gap) with fast convergence to any prescribed reference input (desired levitation gap). (ix) The utility of the new transient Inductrack model plays an essential role in the design and implementation of the proposed feedback controller. Indeed, the generation of the nonlinear mapping function and the calculation of the transient levitation gap of the controlled vehicle totally rely on this transient model. Without such a dynamic model that predicts the system transient response with delity, it would be impossible to develop an implementable feedback control law. (x) Although only a 2-DOF model is initially considered, the modeling technique, solution algorithm, and even control strategy presented in this work can be extended to general M-DOF transient models of Inductrack systems with both translation and rotation in motion. A good example has been enumerated in Chap. 7 with a 2D, 3-DOF rigid body Inductrack model. Indeed, the rotation induces much more complications to model derivation, but it addresses more realistic issues in this kind of engineering application. 109 The research presented in this dissertation oers a useful tool for dynamic analysis and control of Inductrack systems and for the in-depth understanding of the complicated electro-magneto-mechanical interactions in this type of dynamic system. Based on this work, many interesting research topics can be further explored in the future. 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The analytical results fori = 1; 2 are listed below. Then the results fori = 3; 4 can be simply obtained by changing the sign due to the property of symmetry. 119 A.1 QuadratureResultsonType1IntegralsinEq.(A.1) For convenience in expression, whenevery;z are constants, dene Y =y b 2 Y + =y + b 2 Z =z d 2 Z + =z + d 2 (A.4) Thus, Z x B y 11 dx = 0 Z x B y 12 dx = K 2 [(x a 2 ) x (x a 2 ;Y ;Z ) + 2D(x a 2 ;Y ;Z )] [(x a 2 ) x (x a 2 ;Y ;Z + ) + 2D(x a 2 ;Y ;Z + )] [(x + a 2 ) x (x + a 2 ;Y ;Z ) + 2D(x + a 2 ;Y ;Z )] +[(x + a 2 ) x (x + a 2 ;Y ;Z + ) + 2D(x + a 2 ;Y ;Z + )] j x + x Z x B y 13 dx = 0 Z x B y 14 dx = K 2 [(x a 2 ) x (x a 2 ;Y + ;Z ) + 2D(x a 2 ;Y + ;Z )] [(x a 2 ) x (x a 2 ;Y + ;Z + ) + 2D(x a 2 ;Y + ;Z + )] [(x + a 2 ) x (x + a 2 ;Y + ;Z ) + 2D(x + a 2 ;Y + ;Z )] +[(x + a 2 ) x (x + a 2 ;Y + ;Z + ) + 2D(x + a 2 ;Y + ;Z + )] j x + x (A.5) where x has been dened in Eq. (3.6), and D(d x ;d y ;d z ) = q d 2 x +d 2 y +d 2 z (A.6) 120 Besides, Z x B y 21 dx = K 2 [Y z (x a 2 ;Y ;Z ) +Z y (x a 2 ;Y ;Z ) + 2(x a 2 ) x (x a 2 ;Y ;Z )] [Y + z (x a 2 ;Y + ;Z ) +Z y (x a 2 ;Y + ;Z ) + 2(x a 2 ) x (x a 2 ;Y + ;Z )] [Y z (x a 2 ;Y ;Z + ) +Z + y (x a 2 ;Y ;Z + ) + 2(x a 2 ) x (x a 2 ;Y ;Z + )] +[Y + z (x a 2 ;Y + ;Z + ) +Z + y (x a 2 ;Y + ;Z + ) + 2(x a 2 ) x (x a 2 ;Y + ;Z + )] j x + x Z x B y 22 dx = K 2 [Z + y (x a 2 ;Y ;Z + ) 2(x a 2 ) z (x a 2 ;Y ;Z + )] [Z + y (x a 2 ;Y + ;Z + ) 2(x a 2 ) z (x a 2 ;Y + ;Z + )] [Z + y (x + a 2 ;Y ;Z + ) 2(x + a 2 ) z (x + a 2 ;Y ;Z + )] +[Z + y (x + a 2 ;Y + ;Z + ) 2(x + a 2 ) z (x + a 2 ;Y + ;Z + )] j x + x Z x B y 23 dx = K 2 [Y z (x + a 2 ;Y ;Z ) +Z y (x + a 2 ;Y ;Z ) + 2(x + a 2 ) x (x + a 2 ;Y ;Z )] [Y + z (x + a 2 ;Y + ;Z ) +Z y (x + a 2 ;Y + ;Z ) + 2(x + a 2 ) x (x + a 2 ;Y + ;Z )] [Y z (x + a 2 ;Y ;Z + ) +Z + y (x + a 2 ;Y ;Z + ) + 2(x + a 2 ) x (x + a 2 ;Y ;Z + )] +[Y + z (x + a 2 ;Y + ;Z + ) +Z + y (x + a 2 ;Y + ;Z + ) + 2(x + a 2 ) x (x + a 2 ;Y + ;Z + )] j x + x Z x B y 24 dx = K 2 [Z y (x a 2 ;Y ;Z ) 2(x a 2 ) z (x a 2 ;Y ;Z )] [Z y (x a 2 ;Y + ;Z ) 2(x a 2 ) z (x a 2 ;Y + ;Z )] [Z y (x + a 2 ;Y ;Z ) 2(x + a 2 ) z (x + a 2 ;Y ;Z )] +[Z y (x + a 2 ;Y + ;Z ) 2(x + a 2 ) z (x + a 2 ;Y + ;Z )] j x + x (A.7) where x and z have been dened in Eqs. (3.7) and (3.12), and y and z are given by y ( x ; y ; z ) = ln y q 2 x + 2 y + 2 z y + q 2 x + 2 y + 2 z (A.8) 121 z ( x ; y ; z ) = ln z q 2 x + 2 y + 2 z z + q 2 x + 2 y + 2 z (A.9) A.2 QuadratureResultsonType2IntegralsinEq.(A.2) Z x B z 11 dx = K 2 [Y z (x a 2 ;Y ;Z ) +Z y (x a 2 ;Y ;Z ) + 2(x a 2 ) x (x a 2 ;Y ;Z )] [Y z (x a 2 ;Y ;Z + ) +Z + y (x a 2 ;Y ;Z + ) + 2(x a 2 ) x (x a 2 ;Y ;Z + )] [Y + z (x a 2 ;Y + ;Z ) +Z y (x a 2 ;Y + ;Z ) + 2(x a 2 ) x (x a 2 ;Y + ;Z )] +[Y + z (x a 2 ;Y + ;Z + ) +Z + y (x a 2 ;Y + ;Z + ) + 2(x a 2 ) x (x a 2 ;Y + ;Z + )] j x + x Z x B z 12 dx = K 2 [Y z (x a 2 ;Y ;Z ) 2(x a 2 ) y (x a 2 ;Y ;Z )] [Y z (x a 2 ;Y ;Z + ) 2(x a 2 ) y (x a 2 ;Y ;Z + )] [Y z (x + a 2 ;Y ;Z ) 2(x + a 2 ) y (x + a 2 ;Y ;Z )] +[Y z (x + a 2 ;Y ;Z + ) 2(x + a 2 ) y (x + a 2 ;Y ;Z + )] j x + x Z x B z 13 dx = K 2 [Y z (x + a 2 ;Y ;Z ) +Z y (x + a 2 ;Y ;Z ) + 2(x + a 2 ) x (x + a 2 ;Y ;Z )] [Y z (x + a 2 ;Y ;Z + ) +Z + y (x + a 2 ;Y ;Z + ) + 2(x + a 2 ) x (x + a 2 ;Y ;Z + )] [Y + z (x + a 2 ;Y + ;Z ) +Z y (x + a 2 ;Y + ;Z ) + 2(x + a 2 ) x (x + a 2 ;Y + ;Z )] +[Y + z (x + a 2 ;Y + ;Z + ) +Z + y (x + a 2 ;Y + ;Z + ) + 2(x + a 2 ) x (x + a 2 ;Y + ;Z + )] j x + x Z x B z 14 dx = K 2 [Y + z (x a 2 ;Y + ;Z ) 2(x a 2 ) y (x a 2 ;Y + ;Z )] [Y + z (x a 2 ;Y + ;Z + ) 2(x a 2 ) y (x a 2 ;Y + ;Z + )] [Y + z (x + a 2 ;Y + ;Z ) 2(x + a 2 ) y (x + a 2 ;Y + ;Z )] +[Y + z (x + a 2 ;Y + ;Z + ) 2(x + a 2 ) y (x + a 2 ;Y + ;Z + )] j x + x (A.10) where y has been dened in Eq. (3.8). 122 Besides, Z x B z 21 dx = 0 Z x B z 22 dx = K 2 [(x a 2 ) x (x a 2 ;Y ;Z + ) + 2D(x a 2 ;Y ;Z + )] [(x a 2 ) x (x a 2 ;Y + ;Z + ) + 2D(x a 2 ;Y + ;Z + )] [(x + a 2 ) x (x + a 2 ;Y ;Z + ) + 2D(x + a 2 ;Y ;Z + )] +[(x + a 2 ) x (x + a 2 ;Y + ;Z + ) + 2D(x + a 2 ;Y + ;Z + )] j x + x Z x B z 23 dx = 0 Z x B z 24 dx = K 2 [(x a 2 ) x (x a 2 ;Y ;Z ) + 2D(x a 2 ;Y ;Z )] [(x a 2 ) x (x a 2 ;Y + ;Z ) + 2D(x a 2 ;Y + ;Z )] [(x + a 2 ) x (x + a 2 ;Y ;Z ) + 2D(x + a 2 ;Y ;Z )] +[(x + a 2 ) x (x + a 2 ;Y + ;Z ) + 2D(x + a 2 ;Y + ;Z )] j x + x (A.11) 123 A.3 QuadratureResultsonType3IntegralsinEq.(A.3) Z A B y 0 11 dA = 0 Z A B y 0 12 dA = K 2 [Y z (x a 2 ;Y ;z + d 2 ) 2(x a 2 ) y (x a 2 ;Y ;z + d 2 )] [Y z (x a 2 ;Y ;z d 2 ) 2(x a 2 ) y (x a 2 ;Y ;z d 2 )] [Y z (x a 2 ;Y ;z + + d 2 ) 2(x a 2 ) y (x a 2 ;Y ;z + + d 2 )] +[Y z (x a 2 ;Y ;z + d 2 ) 2(x a 2 ) y (x a 2 ;Y ;z + d 2 )] [Y z (x + a 2 ;Y ;z + d 2 ) 2(x + a 2 ) y (x a 2 ;Y ;z + d 2 )] +[Y z (x + a 2 ;Y ;z d 2 ) 2(x + a 2 ) y (x a 2 ;Y ;z d 2 )] +[Y z (x + a 2 ;Y ;z + + d 2 ) 2(x + a 2 ) y (x a 2 ;Y ;z + + d 2 )] [Y z (x + a 2 ;Y ;z + d 2 ) 2(x + a 2 ) y (x a 2 ;Y ;z + d 2 )] j x + x Z A B y 0 13 dA = 0 Z A B y 0 14 dA = K 2 [Y + z (x a 2 ;Y + ;z + d 2 ) 2(x a 2 ) y (x a 2 ;Y + ;z + d 2 )] [Y + z (x a 2 ;Y + ;z d 2 ) 2(x a 2 ) y (x a 2 ;Y + ;z d 2 )] [Y + z (x a 2 ;Y + ;z + + d 2 ) 2(x a 2 ) y (x a 2 ;Y + ;z + + d 2 )] +[Y + z (x a 2 ;Y + ;z + d 2 ) 2(x a 2 ) y (x a 2 ;Y + ;z + d 2 )] [Y + z (x + a 2 ;Y + ;z + d 2 ) 2(x + a 2 ) y (x a 2 ;Y + ;z + d 2 )] +[Y + z (x + a 2 ;Y + ;z d 2 ) 2(x + a 2 ) y (x a 2 ;Y + ;z d 2 )] +[Y + z (x + a 2 ;Y + ;z + + d 2 ) 2(x + a 2 ) y (x a 2 ;Y + ;z + + d 2 )] [Y + z (x + a 2 ;Y + ;z + d 2 ) 2(x + a 2 ) y (x a 2 ;Y + ;z + d 2 )] j x + x (A.12) 124 Besides, Z A B y 0 21 dA = K 2 [(z + d 2 ) z (x a 2 ;Y ;z + d 2 ) + 2D(x a 2 ;Y ;z + d 2 )] [(z d 2 ) z (x a 2 ;Y ;z d 2 ) + 2D(x a 2 ;Y ;z d 2 )] [(z + d 2 ) z (x a 2 ;Y + ;z + d 2 ) + 2D(x a 2 ;Y + ;z + d 2 )] +[(z d 2 ) z (x a 2 ;Y + ;z d 2 ) + 2D(x a 2 ;Y + ;z d 2 )] [(z + + d 2 ) z (x a 2 ;Y ;z + + d 2 ) + 2D(x a 2 ;Y ;z + + d 2 )] +[(z + d 2 ) z (x a 2 ;Y ;z + d 2 ) + 2D(x a 2 ;Y ;z + d 2 )] +[(z + + d 2 ) z (x a 2 ;Y + ;z + + d 2 ) + 2D(x a 2 ;Y + ;z + + d 2 )] [(z + d 2 ) z (x a 2 ;Y + ;z + d 2 ) + 2D(x a 2 ;Y + ;z + + d 2 )] j x + x Z A B y 0 22 dA = K 2 [(x a 2 ) x (x a 2 ;Y ;z + + d 2 ) + 2D(x a 2 ;Y ;z + + d 2 )] [(x a 2 ) x (x a 2 ;Y ;z + d 2 ) + 2D(x a 2 ;Y ;z + d 2 )] [(x a 2 ) x (x a 2 ;Y + ;z + + d 2 ) + 2D(x a 2 ;Y + ;z + + d 2 )] +[(x a 2 ) x (x a 2 ;Y + ;z + d 2 ) + 2D(x a 2 ;Y + ;z + d 2 )] [(x + a 2 ) x (x + a 2 ;Y ;z + + d 2 ) + 2D(x + a 2 ;Y ;z + + d 2 )] +[(x + a 2 ) x (x + a 2 ;Y ;z + d 2 ) + 2D(x + a 2 ;Y ;z + d 2 )] +[(x + a 2 ) x (x + a 2 ;Y + ;z + + d 2 ) + 2D(x + a 2 ;Y + ;z + + d 2 )] [(x + a 2 ) x (x + a 2 ;Y + ;z + d 2 ) + 2D(x + a 2 ;Y + ;z + d 2 )] j x + x (A.13) 125 Z A B y 0 23 dA = K 2 [(z + d 2 ) z (x + a 2 ;Y ;z + d 2 ) + 2D(x + a 2 ;Y ;z + d 2 )] [(z d 2 ) z (x + a 2 ;Y ;z d 2 ) + 2D(x + a 2 ;Y ;z d 2 )] [(z + d 2 ) z (x + a 2 ;Y + ;z + d 2 ) + 2D(x + a 2 ;Y + ;z + d 2 )] +[(z d 2 ) z (x + a 2 ;Y + ;z d 2 ) + 2D(x + a 2 ;Y + ;z d 2 )] [(z + + d 2 ) z (x + a 2 ;Y ;z + + d 2 ) + 2D(x + a 2 ;Y ;z + + d 2 )] +[(z + d 2 ) z (x + a 2 ;Y ;z + d 2 ) + 2D(x + a 2 ;Y ;z + d 2 )] +[(z + + d 2 ) z (x + a 2 ;Y + ;z + + d 2 ) + 2D(x + a 2 ;Y + ;z + + d 2 )] [(z + d 2 ) z (x + a 2 ;Y + ;z + d 2 ) + 2D(x + a 2 ;Y + ;z + + d 2 )] j x + x Z A B y 0 24 dA = K 2 [(x a 2 ) x (x a 2 ;Y ;z + d 2 ) + 2D(x a 2 ;Y ;z + d 2 )] [(x a 2 ) x (x a 2 ;Y ;z d 2 ) + 2D(x a 2 ;Y ;z d 2 )] [(x a 2 ) x (x a 2 ;Y + ;z + d 2 ) + 2D(x a 2 ;Y + ;z + d 2 )] +[(x a 2 ) x (x a 2 ;Y + ;z d 2 ) + 2D(x a 2 ;Y + ;z d 2 )] [(x + a 2 ) x (x + a 2 ;Y ;z + d 2 ) + 2D(x + a 2 ;Y ;z + d 2 )] +[(x + a 2 ) x (x + a 2 ;Y ;z d 2 ) + 2D(x + a 2 ;Y ;z d 2 )] +[(x + a 2 ) x (x + a 2 ;Y + ;z + d 2 ) + 2D(x + a 2 ;Y + ;z + d 2 )] [(x + a 2 ) x (x + a 2 ;Y + ;z d 2 ) + 2D(x + a 2 ;Y + ;z d 2 )] j x + x (A.14) The analytical formulas given in Eqs. (A.5) to (A.14) are useful for computing the electromagnetic quantities in dynamic analysis of the Inductrack system. 126 B ProofofEquivalentMagneticForces It is easy to see that Eq. (3.21) is true if F B ik =F C ik (B.1) HereF B ik is the Ampere force at theith PM block by the magnetic eld of the induced current in the kth track coil, andF C ik is the Ampere force at thekth coil by the source magnetic eld of theith PM block. Physically, Eq. (B.1) is a demonstration of Newton’s law of action-reaction on the non-contact magnetic forces in the Inductrack system. To prove Eq. (B.1), consider the illustration of the magnetic interaction between the ith PM block and thekth track coil in Fig. B.1, whereB S ik denotes the source magnetic eld by the PM,B I ik denotes the induced magnetic eld by the coil;r ik is a distance vector measured from the magnetization current elementi ds i to the induced current elementI k dl k ; andc i andc k are the closed paths of the currenti on the PM block andI k along the coil, respectively. Also, in the gure,p refers to the general pole direction of the PM block. Figure B.1: Magnetic interaction between theith PM block and thekth track coil According to the Ampere force formula [93], the incremental Ampere force acting on a single loopc i of the PM block is dF B ik = I c i i ds i B I ik (B.2) 127 where the inducedB I ik , by quasi-static Biot-Savart law, is given by B I ik = 0 4 I c k I k dl k (r ik ) r 3 ik (B.3) Perform integration of dF B ik along the pole directionp and use Eqs. (B.2) and (B.3), to obtain the Ampere force acting on the whole PM block F B ik = 0 iI k 4 Z p I c i ds i I c k dl k r ik r 3 ik ! dp (B.4) Because ds i on the PM block is independent of loopc k of the coil, Eq. (B.4) is rewritten as F B ik = 0 iI k 4 Z p I c i I c k ds i (dl k r ik ) r 3 ik dp (B.5) Similarly, the Ampere force at the single-looped coil is given by F C ik = I c k I k dl k B S ik (B.6) where the source magnetic eldB S ik is represented by B S ik = 0 4 Z p I c i i ds i r ik r 3 ik dp (B.7) withp representing the pole direction as shown in Fig. B.1. Substituting Eq. (B.7) into Eq. (B.6) yields the Ampere force acting on the coil F B ik = 0 iI k 4 I c k dl k Z p I c i ds i r ik r 3 ik dp ! (B.8) 128 Again, becausec k and dl k are independent of the directionp and pathc i , Eq. (B.8) is reduced to F B ik = 0 iI k 4 Z p I c i I c k dl k (ds i r ik ) r 3 ik dp (B.9) According to Eqs. (B.5) and (B.9), if the following relation I c i I c k ds i (dl k r ik ) r 3 ik = I c i I c k dl k (ds i r ik ) r 3 ik (B.10) holds, Eq. (B.1) is true. By the property of vector triple product, Eq. (B.10) can be written as I c i I c k dl k (ds i r ik )r ik (ds i dl k ) r 3 ik = I c i I c k ds i (dl k r ik )r ik (dl k ds i ) r 3 ik (B.11) Lemma: Consider two directed and closed pathsC s andC l in 3D space as shown in Fig. B.2, where ds and dl are directed incremental vectors of the two paths, respectively; andr is a distance vector from pointA onC s to pointP onC l . Assume that the paths are independent of each other, and that they are not connected at any point. The following two equations hold I Cs I C l dl(dsr) r 3 =0 (B.12) I Cs I C l ds(dlr) r 3 =0 (B.13) wherer =jrj = ! AP . 129 Figure B.2: Two closed independent paths Figure B.3: Increment of the distance vectorr Proof: Consider the dot product dsr in Eq. (B.12). In Fig. B.3, a particle travels an innitesimal distance on pathC s from pointA to pointB, which is represented by ds. In the travel, the distance vector changes fromr tor 0 . Let pointQ be a projection of pointB on lineAP . The dot product thus can be written as dsr =rjdsj cos =r ! AQ (B.14) where is the angle between ds andr. Also, from Fig. B.3, the increment of lengthr is dr = r 0 jrj = ! BP ! AQ + ! QP (B.15) 130 where vectors ! BP , ! AQ and ! QP can be identied from the gure. Because ds is innitesimally small, the angle between and ds is the same as, namely, 0 = . This implies that angle\BPQ is innitesimally small. Thus, ! BP = ! QP , which by Eq. (B.15) gives ! AQ =dr (B.16) It follows from Eqs. (B.14) and (B.16) that dsr =r dr (B.17) Now, substitute Eq. (B.17) into the left-hand side of Eq. (B.12) to obtain I Cs I C l dl(dsr) r 3 = I Cs I C l dl(dr) r 2 = I C l dl I Cs dr r 2 (B.18) where the interchange of the order of integration is legitimate because dl is independent of path C s . Because H Cs dr r 2 = 0 for any closed path, Eq. (B.18) warrants Eq. (B.12). With a similar argument, it can be shown that the dot product dlr =r dr, which leads to I Cs I C l ds(dlr) r 3 = I Cs ds I C l dr r 2 =0 (B.19) This proves Eq. (B.13). Q.E.D. 131 C DerivationoftheOriginalSteady-StateInductrackModel In the derivation of Eq. (4.4), the following assumptions are made: (i) an ideal innite sinusoidal magnetic eld is produced by Halbach arrays; (ii) the vehicle travels at a constant speed; (iii) the vehicle maintains a xed (constant) levitation gap over the track; (iv) the induced coil currents are in steady state; and (v) averaged magnetic forces are considered. The establishment of the original steady-state model described by Eq. (4.4), in the context of the Inductrack conguration in Fig. 2.1, takes three steps: (a) for a single coil moving below the innite length of Halbach arrays, calculate the magnetic forces acting on it; (b) average the magnetic forces within the time it takes the coil to pass by a single wavelength of Halbach arrays; and (c) determine the total steady-state magnetic forces based on the averaged forces calculated in (b). These steps are detailed in sequel. Figure C.4: Schematic of the original steady-state Inductrack model A schematic of the original steady-state model is shown in Fig. C.4, where a coil of indexk is moving beneath the Halbach arrays, in the leftward (y M ) direction of a stationary magnet frameo M x M y M z M . The coil maintains a constant speedv and a xed levitation gap. By a sinusoidal approximation [8], the 132 2D magnetic ux density at the upper rim of this coil measured from the stationary magnet frame is represented by ~ B y M k =B 0 sin [(y 0 vt)]e ~ B z M k =B 0 cos [(y 0 vt)]e (C.1) wherey 0 is an arbitrary initial position of the coil, and is the wave number given by = 2 . From Eq. (C.1), the magnetic ux going through the coil is obtained by integrating ~ B y M k within the area the coil encloses. With the assumption of uniform distribution of the magnetic eld in thex M -direction, the magnetic ux is obtained as ~ k =wB 0 sin [(y 0 vt)] Z +h e d = wB 0 sin [(y 0 vt)]e (e h 1) wB 0 sin [(y 0 vt)]e (C.2) Heree h is negligible because physically and mathematicallye h 1. Noticeably, comparison between Eq. (C.1) and (C.2) indicates that the ux is in phase with the magnetic eld. According to Faraday’s law and Kirchho’s voltage law, the induced current ~ I k in the coil is governed by the dierential equation L eq _ ~ I k +R eq ~ I k = d ~ k dt (C.3) which, by Eq. (C.2), becomes _ ~ I k + R eq L eq ~ I k = vwB 0 L eq cos [(y 0 vt)]e (C.4) 133 Without loss of generality, assume zero initial induced current. Solution of Eq. (C.4) yields ~ I k = vwB 0 L eq e e Req Leq t Z t 0 e Req Leq cos [(y 0 v)] d (C.5) which, through integration, gives the induced coil current in the following analytical form ~ I k = wB 0 Leq e 1 + Req vLeq 2 ( sin [(y 0 vt)] R eq vL eq cos [(y 0 vt)] sin (y 0 )e Req Leq t + R eq vL eq cos (y 0 )e Req Leq t ) (C.6) Now, neglecting the transient terms in Eq. (C.6) yields the steady-state coil current ~ I k;ss = wB 0 Leq e 1 + Req vLeq 2 ( sin [(y 0 vt)] R eq vL eq cos [(y 0 vt)] ) (C.7) Apply the Ampere force formulation with right-hand rule, to obtain the components of the magnetic force acting on the coil as follows ~ f z M k = ~ B y M k ~ I k;ss w ~ f y M k = ~ B z M k ~ I k;ss w (C.8) where forces acting on the lower rim of the coil have been neglected because of the exponential decay of the magnetic eld as described by Eq. (C.1). Substituting Eqs. (C.1) and (C.7) into (C.8) gives ~ f z M k = w 2 B 2 0 2Leq e 2 1 + Req vLeq 2 ( R eq vL eq sin [2(y 0 vt)] + cos [2(y 0 vt)] 1 ) ~ f y M k = w 2 B 2 0 2Leq e 2 1 + Req vLeq 2 ( sin [2(y 0 vt)] R eq vL eq cos [2(y 0 vt)] + R eq vL eq ) (C.9) 134 By the law of action-reaction, the magnetic levitation force ~ F k l and magnetic drag force ~ F k d applied to the Halbach arrays by this moving coil satisfy ~ F k l = ~ f z M k ~ F k d = ~ f y M k (C.10) Consider a time period,t 1 t t 1 + v , during which the coil with speedv passes a wavelength of Halbach array. Thus, averaging ~ F k l and ~ F k d within this time period results in F k l = v Z t 1 + v t 1 ~ F k l dt = w 2 B 2 0 2L eq 1 1 + Req vLeq 2 e 2 F k d = v Z t 1 + v t 1 ~ F k l dt = w 2 B 2 0 2L eq Req vLeq 1 + Req vLeq 2 e 2 (C.11) According to the system conguration shown in Fig. 2.1, there are 2N wavelengths of Halbach arrays and the coil number per wavelength is dc . Thus, the total magnetic forces calculated by the steady-state Inductrack model are obtained as follows ~ F l = F k l 2N d c = w 2 B 2 0 N L eq d c 1 1 + Req vLeq 2 e 2 ~ F d = F k d 2N d c = w 2 B 2 0 N L eq d c Req vLeq 1 + Req vLeq 2 e 2 (C.12) which are the same as those given in Eq. (4.4). It should be noted that Eq. (4.4) is only valid for a constant . When is time-varying, which is inevitable in transient motion of the vehicle, Eq. (C.4) no longer holds, which disqualies the subsequent steady-state derivations. Evidently, a transient model needs to be considered in this case. 135
Abstract (if available)
Abstract
As a new strategy for magnetic levitation envisioned in the 1990s, Inductrack systems with Halbach arrays of permanent magnets (PMs) have been applied to Maglev trains and intensively researched in various projects. In an Inductrack system, the magnetic interaction forces are coupled with the motion of the moving vehicle carrying Halbach arrays, which in many situations provokes complicated transient behaviors in the dynamic system. As a result, a steady-state-based Inductrack model, although having provided useful information for system design, cannot capture such transient features quantitatively with fidelity. ❧ In this research, a benchmark transient model of two degrees of freedom (2-DOF) for the Inductrack system is proposed. The highlight of this work lies in that the new model is derived from the first laws of nature and is based on a complete transient scenario, without the assumption of any steady-state related quantities. It is found that the transient Inductrack system is governed by a set of nonlinear integro-differential equations that fully describe the electro-magneto-mechanical couplings involved, which are solved by a combination of analytical and numerical approaches. ❧ Dynamic analysis of Inductrack systems with the new benchmark model consists of two stages. First, the proposed model is validated through comparison with the noted steady-state Inductrack model in the literature. Second, the transient response of the Inductrack system is obtained and analyzed with our model in several typical dynamic scenarios. From the numerical examples, it is believed that the newly-developed model can not only reproduce the well-known steady-state results but is also capable of conducting in-depth transient analyses of Inductrack dynamic systems. Although only two degrees of freedom are initially considered, the proposed approach of modeling and analysis can be extended to general cases of multiple degrees of freedom, as enumerated with a 2D, 3-DOF rigid-body Inductrack system in this thesis. ❧ With the new transient model, it is discovered that an uncontrolled Inductrack system may be unstable even if the vehicle travels well below its operating speed and that instability can be persistent near and beyond the operating speed. It is, therefore, necessary to stabilize the system for safety and reliability. In this work, by taking advantage of the available 2-DOF benchmark transient model, a new feedback control method for Inductrack systems is proposed. In the control system development, active Halbach arrays are used as an actuator, and a feedback control law, which combines a properly tuned linear controller (either PID- or state feedback-based) and a nonlinear force-current mapping function, is created. The proposed control law is validated in numerical examples, where it efficiently stabilizes the Inductrack system in a wide range of operating speeds. In the meantime, the controller renders a smooth system output (real-time levitation gap) with fast convergence to any prescribed reference step input (desired levitation gap).
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Asset Metadata
Creator
Wang, Ruiyang
(author)
Core Title
Transient modeling, dynamic analysis, and feedback control of the Inductrack Maglev system
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Degree Conferral Date
2021-12
Publication Date
09/18/2023
Defense Date
08/26/2021
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
electrodynamic suspension,electro-magneto-mechanical coupling,feedback control,Inductrack system,magnetic force,magnetic levitation,nonlinear mapping function,OAI-PMH Harvest,transient response
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Flashner, Henryk (
committee chair
), Jonckheere, Edmond (
committee chair
), Yang, Bingen (
committee chair
)
Creator Email
ruiyangw@usc.edu,wry040614@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-oUC15919475
Unique identifier
UC15919475
Legacy Identifier
etd-WangRuiyan-10077
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Wang, Ruiyang
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Repository Email
cisadmin@lib.usc.edu
Tags
electrodynamic suspension
electro-magneto-mechanical coupling
feedback control
Inductrack system
magnetic force
magnetic levitation
nonlinear mapping function
transient response