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Proquest Dissertations RICE UNIVERSITY Magnetic Damping of an Elastic Conductor by Jeffrey M. Hokanson A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Master of Arts APPROVED, THESIS COMMITTEE: IJU&_ Mark Embread Co-chair Associate Professor of Computational and Applied M/jmematics Steven J. Cox, (Co-chair Professor of Computational and Applied Mathematics David Damanik Associate Professor of Mathematics Houston, Texas April, 2009 UMI Number: 1466785 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI® UMI Microform 1466785 Copyright 2009 by ProQuest LLC All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ii ABSTRACT Magnetic Damping of an Elastic Conductor by Jeffrey M. Hokanson Many applications call for a design that maximizes the rate of energy decay. Typical problems of this class include one dimensional damped wave operators, where energy dissipation is caused by a damping operator acting on the velocity. Two damping operators are well understood: a multiplication operator (known as viscous damping) and a scaled Laplacian (known as Kelvin-Voigt damping). Paralleling the analysis of viscous damping, this thesis investigates energy decay for a novel third operator known as magnetic damping, where the damping is expressed via a rank-one self-adjoint operator, dependent on a function a. This operator describes a conductive monochord embedded in a spatially varying magnetic field perpendicular to the monochord and proportional to a. Through an analysis of the spectrum, this thesis suggests that unless a has a singularity at one boundary for any finite time, there exist initial conditions that give arbitrarily small energy decay at any time. Contents List of Illustrations v List of Tables vi 1 Introduction 1 2 A Menagerie of Wave Operators 5 2.1 Undamped wave operator 6 2.2 Kelvin-Voigt damping 7 2.3 Viscous damping 8 2.4 Magnetic damping . 13 3 Basic Results for Magnetic Damping 16 3.1 Explicit form for the resolvent 17 3.1.1 Inverting A — A via triangular factorization 17 3.1.2 Green's function 19 3.2 Discrete spectra 19 3.3 Shooting function 25 4 Bounded Variation Fields 27 4.1 Spectral abscissa for bounded variation a 27 4.2 Bounds on the spectrum . 29 4.3 Resolvent bound in terms of Fourier coefficients 32 4.4 A sinusoidal damping field 33 4.5 Finite bandwidth a . 35 5 Singular Damping Fields 37 5.1 Rate of leaving the axis 37 5.1.1 Example when a(x) = ~f/x 39 5.2 Resolvent upper bound 40 5.3 Resolvent lower bound 42 5.4 Is a(x) = 7/z optimal? 46 6 Conclusions 48 A Derivation of Magnetic Damping 50 iv B Spectral Methods 54 B.l Chebyshev differentiation matrices 54 B.2 Approximating the undamped wave operator 55 B.3 Approximating the magnetic damping operator 57 Bibliography 61 Illustrations 2.1 Spectrum of the Kelvin-Voigt damping with constant a(x) = 7 = 2/5. 9 2.2 Spectrum of the viscous damping operator with constant a(x) =7 = 2. 11 2.3 Spectrum of the magnetic damping operator with constant a(x) = 7 = 0.8745 . 15 3.1 Green's function for a(x) = 1 20 4.1 Bounds provided by Theorem 5 31 4.2 Eigenvalues of A when a(x) = 7^2(2;) 34 5.1 Upper bound on the resolvent for a singular field 43 5.2 Mollification lower bound on the resolvent 45 5.3 Spectrum of A when a(x) = 2/(nx) 46 A.l Physical setting for magnetic damping 51 B.l Code generating an approximation of the undamped wave operator. 57 B.2 Code generating an approximation of the magnetic damping operator. 59 Tables 5.1 Illustration of Theorem 7 40 5.2 Numerically optimal finite Laurent series 47 A. 1 Variables required for derivation of magnetic damping equations of motion 51 B.l Convergence of spectral methods for the undamped wave operator . 56 B.2 Eigenvalue convergence for spectral approximation of the magnetic damping operator when a € BV(0,n) 59 B.3 Eigenvalue convergence for spectral approximation of the magnetic damping operator when a(x) = 2/(nx) 60 1 Chapter 1 Introduction Minimization of energy is a common design objective for physical systems. Frequently, engineers attempt to design systems such that the energy introduced by perturbations is quickly removed. For example, a car designer might want a door to close silently. A key strategy problem would then be to place damping material in the door to maximize the rate of energy decay. Design problems such as these motivate the study of damped wave operators, which manifest many characteristics of harder problems yet yield to rigorous analysis. Two of these operators have been subjected to thorough study: the Kelvin-Voigt and viscous damping operators. In this thesis I analyze a novel third damped wave operator associated with magnetic damping to find the conditions for maximizing energy decay. Magnetic damping was first studied by Leibowitz and Ackerberg in 1963 [14]. After rescaling variables, they found that a conductive string embedded in a magnetic field string obeyed the integro-differential equation d2u d2u r a = a{x) w-w - h «(%«&*)<* (11) u(0,t)=u(ir,t) = 0, where u(x, t) measures transverse displacement and a(x) is proportional to the mag­ netic field intensity. A derivation of the equations of motion is given in Appendix A. In this thesis I demonstrate how the choice of a affects the rate of energy decay. Magnetic damping can be understood in a similar manner .to other damped wave 2 operators. Following the approach of Cox and Zuazua [7] for viscous damping, I study energy decay by converting the second order partial differential equation (1.1) into the first order form • " u u d_ = A dt du du at dt where A is an operator posed on a space X endowed with an inner product measur­ ing system energy. Under these conditions the spectrum of A reveals energy decay properties. For magnetic damping, A takes the form 0 / (1.2) A —a(-,a) Here and for the remainder of this thesis, (•, •) is the standard L2(0, n) inner product (f,9)= / f(x)g(x)dx. Jo Similarly, the L2(0,7r) norm is denoted || -|| , where ||/|| — (/, f)1?2. The operator A is posed on the space X = HQ(0,IT) X L2(0,TT) with energy inner product T T 7 ([«, v] ,[f, g] )x= [ u'(xjf (?)dx+ f v(x)g~&)dx (1.3) Jo Jo and norm ||V||x = (V>V)x • The norm || -|| x is a measure of the total energy in the string: the first term in (1.3) measures potential energy; the second kinetic. Let L(X) denote the set of bounded linear operators on X. For T G L(X) the energy norm induces the operator norm \\T\\nx)= sup ||7V||x. \\v\\x=i The operator A maps Dom(A) = {[u,vf e 7#(0, vr) x L2(0, TT) : u" - a(v,a) e L2(0, TT)} 3 to X. This operator also generates a Co semigroup; thus for an initial condition Vo G X, the energy in the system is written as tA 2 £(t) = \\e V0\\ x. (Here we understand etA in the sense of [22, §15].) The asymptotic rate of energy decay, u(A), is the smallest scalar such that tA 2 t2 £(t)<\\e \\l(X)\\Vo\\ x<Ce "M (1.4) for some C > 0 and for all Vo £ Dom(A). Let cr(A) denote the spectrum of A, that is, the set of all A € C such that A — A does not have a bounded, densely defined inverse. The spectral abscissa, Oi{A) = sup Re (A), \£<T(A) is always a lower bound on energy decay: a (A) < u(A). Under certain restrictions on A, the reverse inequality can be shown, and thus a(A) = OJ(A). Theorem 4 in chapter 4 will show that if a is a function of bounded variation, a G BV(0, -K), then a(A) = u(A). Further, when a 6 BV(0,7r), Theorem 3 demonstrates that a(A) — 0. Thus for bounded variation magnetic fields at any finite time r, there exists initial conditions that manifest arbitrarily small energy decay at t = r. The lack of energy decay will motivate my study of singular fields in chapter 5, i.e., those a for which there exist constants C > 0 and.p > —3/2 such that \a(x)\<Cxp(n-x)p for all x G (0,7r). Even when a is singular, the spectrum of A still contains only eigenvalues, as demonstrated in Theorem 1 in chapter 3. Analysis and computations for such singular damping are considerably complicated by the unbounded nature of 4 a. No proofs establishing a(A) = u(A) are demonstrated here for singular fields, nor any results establishing a{A) < 0. However, several bounds on the resolvent and eigenvalue estimates obtained in chapter 5 suggest a(A) < 0. The final chapter points to directions in which results presented here might be extended to prove both a(A) < 0 and a(A) = u{A).
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