The last thing one settles in writing a book is what one should put in first. Pascal's Pensees Classical vibration theory is concerned, in large part, with the infinitesimal (i. e. , linear) undamped free vibration of various discrete or continuous bodies. One of the basic problems in this theory is the determination of the natural frequencies (eigen­ frequencies or simply eigenvalues) and normal modes of the vibrating body. A body which is modelled as a discrete system' of rigid masses, rigid rods, massless springs, etc. , will be governed by an ordinary matrix differential equation in time t. It will have a finite number of eigenvalues, and the normal modes will be vectors, called eigenvectors. A body which is modelled as a continuous system will be governed by a partial differential equation in time and one or more spatial variables. It will have an infinite number of eigenvalues, and the normal modes will be functions (eigen­ functions) of the space variables. In the context of this classical theory, inverse problems are concerned with the construction of a model of a given type; e. g. , a mass-spring system, a string, etc. , which has given eigenvalues and/or eigenvectors or eigenfunctions; i. e. , given spec­ tral data. In general, if some such spectral data is given, there can be no system, a unique system, or many systems, having these properties.

1 - Elementary Matrix Analysis.- 1.1 Introduction.- 1.2 Basic definitions and notations.- 1.3 Matrix inversion and determinants.- 1.4 Eigenvalues and eigenvectors.- 2 - Vibration of Discrete Systems.- 2.1 Introduction.- 2.2 Vibration of some simple systems.- 2.3 Transverse vibration of a beam.- 2.4 Generalized coordinates and Lagrange's equations.- 2.5 Natural frequencies and normal modes.- 2.6 Principal coordinates and receptances.- 2.7 Rayleigh's Principle.- 2.8 Vibration under constraint.- 2.9 Iterative and independent definitions of eigenvalues.- 3 - Jacobian Matrices.- 3.1 Sturm sequences.- 3.2 Orthogonal polynomials.- 3.3 Eigenvectors of Jacobian matrices.- 4 - Inversion of Discrete Second-Order Systems.- 4.1 Introduction.- 4.2 An inverse problem for a Jacobian matrix.- 4.3 Variants of the inverse problem for a Jacobian matrix.- 4.4 Inverse eigenvalue problems for spring-mass system.- 5 - Further Properties of Matrices.- 5.1 Introduction.- 5.2 Minors.- 5.3 Further properties of symmetric matrices.- 5.4 Perron's theorem and associated matrices.- 5.5 Oscillatory matrices.- 5.6 Oscillatory systems of vectors.- 5.7 Eigenvalues of oscillatory matrices.- 5.8 u-Line analysis.- 6 - Some Applications of the Theory of Oscillatory Matrices.- 6.1 The inverse mode problem for a Jacobian matrix.- 6.2 The inverse problem for a single mode of a spring-mass system.- 6.3 The reconstruction of a spring-mass system from two modes.- 6.4 A note on the matrices appearing in a finite element model of a rod.- 7 - The Inverse Problem for the Discrete Vibrating Beam.- 7.1 Introduction.- 7.2 The eigenanalysis of the clamped-free beam.- 7.3 The forced response of the beam.- 7.4 The spectra of the beam.- 7.5 Conditions of the data.- 7.6 Inversion by using orthogonality.-7.7 The block-Lanczos algorithm.- 7.8 A numerical procedure for the beam inverse problem.- 8 - Green's Functions and Integral Equations.- 8.1 Introduction.- 8.2 Sturm-Liouville systems.- 8.3 Green's functions.- 8.4 Symmetric kernels and their eigenvalues.- 8.5 Oscillatory properties of Sturm-Liouville kernels.- 8.6 Completeness.- 8.7 Nodes and zeros.- 8.8 Oscillatory systems of functions.- 8.9 Perron's theorem and associated kernels.- 8.10 The interlacing of eigenvalues.- 8.11 Asymptotic behaviour of eigenvalues and eigenfunctions.- 8.12 Impulse responses.- 9 - Inversion of Continuous Second-Order Systems.- 9.1 Introduction.- 9.2 A historical overview.- 9.3 The reconstruction procedure.- 9.4 The Gel'fand-Levitan integral equation.- 9.5 Reconstruction of the differential equation.- 9.6 The inverse problem for the vibrating rod.- 9.7 Reconstruction from the impulse response.- 10 - The Euler-Bernoulli Beam.- 10.1 Introduction.- 10.2 Oscillatory properties of Euler-BernouUi kernels.- 10.3 The eigenfunctions of the cantilever beam.- 10.4 The spectra of the beam.- 10.5 Statement of the inverse problem.- 10.6 The reconstruction procedure.- 10.7 The positivity of matrix P is sufficient.- 10.8 Determination of feasible data.