Inverse Problems in Vibration SOLID MECHANICS AND ITS APPLICATIONS Volume 119 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, andHow m The aim of this series is to provide lucid accounts written by authoritative res giving vision and insight in answering these questions on the subject of mecha relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it the foundation of mechanics; variational formulations; computational me statics, kinematics and dynamics of rigid and elastic bodies: vibrations of so structures; dynamical systems and chaos; the theories of elasticity, plasti viscoelasticity; composite materials; rods, beams, shells and membranes; s control and stability; soils, rocks and geomechanics; fracture; tribology; expe mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts ar graphs defining the current state of the field; others are accessible to final yea graduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages. Inverse Problems in Vibration Second Edition by Graham M.L. Gladwell University of Waterloo, Department of Civil Engineering, Waterloo, Ontario, Canada KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 1-4020-2721-4 Print ISBN: 1-4020-2670-6 ©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://ebooks.springerlink.com and the Springer Global Website Online at: http://www.springeronline.com All appearance indicates neither a total exclusion nor a manifest presence of divinity, but the presence of a God who hides himself. Everything bears this character. Pascal’s Pensées, 555. Contents 1 Matrix Analysis 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic definitions and notation . . . . . . . . . . . . . . . . . . . 1 1.3 Matrix inversion and determinants . . . . . . . . . . . . . . . . 6 1.4 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . 13 2 Vibrations of Discrete Systems 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Vibration of some simple systems . . . . . . . . . . . . . . . . . 19 2.3 Transverse vibration of a beam . . . . . . . . . . . . . . . . . . 24 2.4 Generalised coordinates and Lagrange’s equations: the rod . . . 26 2.5 Vibration of a membrane and an acoustic cavity . . . . . . . . . 30 2.6 Natural frequencies and normal modes . . . . . . . . . . . . . . 35 2.7 Principal coordinates and receptances . . . . . . . . . . . . . . . 38 2.8 Rayleigh’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9 Vibration under constraint . . . . . . . . . . . . . . . . . . . . . 43 2.10 Iterative and independent definitions of eigenvalues . . . . . . . 46 3 Jacobi Matrices 49 3.1 Sturm sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Eigenvectors of Jacobi matrices . . . . . . . . . . . . . . . . . . 57 3.4 Generalised eigenvalue problems . . . . . . . . . . . . . . . . . . 61 4 Inverse Problems for Jacobi Systems 63 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 An inverse problem for a Jacobi matrix . . . . . . . . . . . . . . 65 4.3 Variants of the inverse problem for a Jacobi matrix . . . . . . . 68 4.4 Reconstructing a spring-mass system; by end constraint. . . . . 74 4.5 Reconstruction by using modification . . . . . . . . . . . . . . . 81 4.6 Persymmetric systems . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Inverse generalised eigenvalue problems . . . . . . . . . . . . . . 86 4.8 Interior point reconstruction . . . . . . . . . . . . . . . . . . . . 87 vii viii Contents 5 Inverse Problems for Some More General Systems 93 5.1 Introduction: graph theory . . . . . . . . . . . . . . . . . . . . . 93 5.2 Matrix transformations . . . . . . . . . . . . . . . . . . . . . . . 98 5.3 The star and the path . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 Periodic Jacobi matrices . . . . . . . . . . . . . . . . . . . . . . 103 5.5 The block Lanczos algorithm . . . . . . . . . . . . . . . . . . . . 105 5.6 Inverse problems for pentadiagonal matrices . . . . . . . . . . . 108 5.7 Inverse eigenvalue problems for a tree . . . . . . . . . . . . . . . 110 6 Positivity 118 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3 A general representation of a symmetric matrix . . . . . . . . . 125 6.4 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.5 Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.6 Totally non-negative matrices . . . . . . . . . . . . . . . . . . . 133 6.7 Oscillatory matrices . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.8 Totally positive matrices . . . . . . . . . . . . . . . . . . . . . . 143 6.9 Oscillatory systems of vectors . . . . . . . . . . . . . . . . . . . 145 6.10 Eigenproperties of TN matrices . . . . . . . . . . . . . . . . . . 148 6.11 u-line analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7 Isospectral Systems 153 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.2 Isospectral flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.3 Isospectral Jacobi systems . . . . . . . . . . . . . . . . . . . . . 160 7.4 Isospectral oscillatory systems . . . . . . . . . . . . . . . . . . . 166 7.5 Isospectral beams . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.6 Isospectral finite-element models . . . . . . . . . . . . . . . . . . 175 7.7 Isospectral flow, continued . . . . . . . . . . . . . . . . . . . . . 180 8 The Discrete Vibrating Beam 185 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.2 The eigenanalysis of the cantilever beam . . . . . . . . . . . . . 186 8.3 The forced response of the beam . . . . . . . . . . . . . . . . . . 189 8.4 The spectra of the beam . . . . . . . . . . . . . . . . . . . . . . 190 8.5 Conditions on the data for inversion . . . . . . . . . . . . . . . . 193 8.6 Inversion by using orthogonality . . . . . . . . . . . . . . . . . . 196 8.7 A numerical procedure for the inverse problem . . . . . . . . . . 199 9 Discrete Modes and Nodes 202 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 9.2 The inverse mode problem for a Jacobi matrix . . . . . . . . . . 203 9.3 The inverse problem for a single mode of a spring-mass system . 206 9.4 The reconstruction of a spring-mass system from two modes . . 209 9.5 The inverse mode problem for the vibrating beam . . . . . . . . 211 Contents ix 9.6 Courant’s nodal line theorem. . . . . . . . . . . . . . . . . . . . 214 9.7 Some properties of FEM eigenvectors . . . . . . . . . . . . . . . 217 9.8 Strong sign graphs . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.9 Weak sign graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9.10 Generalisation to M(cid:62)K problems . . . . . . . . . . . . . . . . . 229 10 Green’s Functions and Integral Equations 231 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 10.2 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 237 10.3 Some functional analysis . . . . . . . . . . . . . . . . . . . . . . 240 10.4 The Green’s function integral equation . . . . . . . . . . . . . . 251 10.5 Oscillatory properties of Green’s functions . . . . . . . . . . . . 255 10.6 Oscillatory systems of functions . . . . . . . . . . . . . . . . . . 259 10.7 Perron’s Theorem and compound kernels . . . . . . . . . . . . . 266 10.8 The interlacing of eigenvalues . . . . . . . . . . . . . . . . . . . 271 10.9 Asymptotic behaviour of eigenvalues and eigenfunctions . . . . 276 10.10 Impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . . 284 11 Inversion of Continuous Second-Order Systems 289 11.1 A historical review . . . . . . . . . . . . . . . . . . . . . . . . . 289 11.2 Transformation operators . . . . . . . . . . . . . . . . . . . . . . 294 11.3 The hyperbolic equation for (cid:78)((cid:123)(cid:62)(cid:124)) . . . . . . . . . . . . . . . . 296 11.4 Uniqueness of solution of an inverse problem . . . . . . . . . . . 303 11.5 The Gel’fand-Levitan integral equation . . . . . . . . . . . . . . 305 11.6 Reconstruction of the Sturm-Liouville system . . . . . . . . . . 312 11.7 An inverse problem for the vibrating rod . . . . . . . . . . . . . 315 11.8 An inverse problem for the taut string . . . . . . . . . . . . . . 319 11.9 Some non-classical methods . . . . . . . . . . . . . . . . . . . . 321 11.10 Some other uniqueness theorems . . . . . . . . . . . . . . . . . . 326 11.11 Reconstruction from the impulse response . . . . . . . . . . . . 331 12 A Miscellany of Inverse Problems 335 12.1 Constructing a piecewise uniform rod from two spectra . . . . . 335 12.2 Isospectral rods and the Darboux transformation . . . . . . . . 344 12.3 The double Darboux transformation. . . . . . . . . . . . . . . . 351 12.4 Gottlieb’s research . . . . . . . . . . . . . . . . . . . . . . . . . 355 12.5 Explicit formulae for potentials . . . . . . . . . . . . . . . . . . 361 12.6 The research of Y.M. Ram et al. . . . . . . . . . . . . . . . . . . 364 13 The Euler-Bernoulli Beam 368 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 13.2 Oscillatory properties of the Green’s function . . . . . . . . . . 373 13.3 Nodes and zeros for the cantilever beam . . . . . . . . . . . . . 381 13.4 The fundamental conditions on the data . . . . . . . . . . . . . 383 13.5 The spectra of the beam . . . . . . . . . . . . . . . . . . . . . . 386 13.6 Statement of the inverse problem . . . . . . . . . . . . . . . . . 391 x Contents 13.7 The reconstruction procedure . . . . . . . . . . . . . . . . . . . 393 13.8 The total positivity of matrix P is su(cid:33)cient . . . . . . . . . . . 399 14 Continuous Modes and Nodes 402 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 14.2 Sturm’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 403 14.3 Applications of Sturm’s Theorems . . . . . . . . . . . . . . . . . 407 14.4 The research of Hald and McLaughlin . . . . . . . . . . . . . . . 411 15 Damage Identification 417 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 15.2 Damage identification in rods . . . . . . . . . . . . . . . . . . . 419 15.3 Damage identification in beams . . . . . . . . . . . . . . . . . . 422 Index 426 Bibliography 432 Preface The last thing one settles in writing a book is what one should put in first. Pascal’s Pensées, 19 In 1902 Jacques Hadamard introduced the term well-posed problem. His definition,anabstractionfromtheknownpropertiesoftheclassicalproblemsof mathematical physics, had three elements: Existence: the problem has a solution Uniqueness: the problem has only one solution Continuity: the solution is a continuous function of the data. Much of the research into theoretical physics and engineering before and after1902hasconcentratedonformulatingproblems,withproperlychoseninitial and/orboundaryconditions,sothattheirsolutionsdohavethesecharacteristics: the problems are well posed. Over the years it began to be recognized that there were important and apparently sensible questions that could be asked that did not fall into the categoryofwell-posedproblems. Theywereeventuallycalledill-posedproblems. Many of these problems looked like a classical problem except that the roles of known and unknown quantitites had been reversed: the data, the known, were related to the outcome, the solution of a classical problem; while the unknowns were related to the data for the classical problem: they were thus called inverse problems,incontrasttothedirect classicalproblems. (Laterreflectionsuggested that the choice of which to be called direct and which to be called inverse was partly a historical accident.) For completeness, one should add that not all such inverse problems are ill-posed, and not all ill-posed problems are inverse problems! This book is about inverse problems in vibration, andmany of these problems are ill-posed because they fail to satisfy one or more of Hadamard’s criteria: theymaynothaveasolutionatall,unlessthedataareproperlychosen; they may have many solutions; the solution may not be a continuous function of the data, in particular, as the data are varied by small amounts, it can leave the feasible region in which there is one or more solutions, and enter the region where there is no solution. xi