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 |  موضوع: كتاب Inverse Problems in Vibration - Graham M.L. Gladwell الأحد 28 أبريل 2013, 7:28 pm | |
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Inverse Problems in Vibration - Graham M.L. Gladwell
ويتناول الموضوعات الأتية :
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 normalmodes 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 Sturmsequences 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 problemfor a Jacobimatrix 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 positivematrices 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 inversemode problemfor a Jacobimatrix 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�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 (� �) 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 problemfor the taut string 319
11.9 Some non-classicalmethods 321
11.10 Some other uniqueness theorems 326
11.11 Reconstruction fromthe 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 sucient 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
Bibliography
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