Inverse problems in vibration.

*(English)*Zbl 0646.73013
Mechanics: Dynamical Systems, 9. Dordrecht/Boston/Lancaster: Martinus Nijhoff Publishers, a member of the Kluwer Academic Publishers Group. X, 263 p. Dfl. 175.00; $ 79.50 £57.95 (1986).

Inverse problems in vibration are concerned with the construction of a model of a given type (e.g. mass-spring system, a string etc.), which has given eigenvalues and/or eigenvectors, i.e., given spectral data. This book is concerned exclusively with a special class of inverse problems, the so called reconstruction problems. Here the spectral data are such that there is one and only one vibrating system of the specified type which has the given spectral properties.

For these problems are always three questions of importance: 1. What spectral data is necessary and sufficient to ensure that the system, if it exists at all, is unique? 2. What are necessary and sufficient conditions which must be satisfied by this data to ensure that is does correspond to a realistic system, i.e., one with positive masses, lengths, cross-sectional areas etc.? 3. How can the (unique) system be reconstructed?

The book is divided into two parts; Chapters 1-7 are concerned with discrete systems; Chapters 8-10 with continuous ones. In each part there is alternation between theory and application. Thus Chapter 1 introduces matrices, which are then applied in Chapter 2. Chapter 3 studies Jacobian (i.e. tridiagonal) matrices, which are the class of matrices appearing in the inverse problems of Chapter 4. Chapter 5 draws largely on F. R. Gantmakher and M. G. Krein’s book: Oscillation matrices and kernels and small oscillations of mechanical systems. Akademie-Verlag (1964). On first reading, the reader is advised to note merely the principal results concerning oscillatory matrices; these are applied in Chapter 6, and particular in Chapter 7. Chapter 8 draws on Courant and Hilbert (1953) to give the basic properties of symmetric integral equations, and on Gantmakher and Krein to provide the extra properties of the eigenvalues and eigenfunctions of oscillatory kernels. The results of this Chapter are basic to the study of the Sturm-Liouville systems of Chapter 9, and particular for the Euler-Bernoulli problem of Chapter 10.

For these problems are always three questions of importance: 1. What spectral data is necessary and sufficient to ensure that the system, if it exists at all, is unique? 2. What are necessary and sufficient conditions which must be satisfied by this data to ensure that is does correspond to a realistic system, i.e., one with positive masses, lengths, cross-sectional areas etc.? 3. How can the (unique) system be reconstructed?

The book is divided into two parts; Chapters 1-7 are concerned with discrete systems; Chapters 8-10 with continuous ones. In each part there is alternation between theory and application. Thus Chapter 1 introduces matrices, which are then applied in Chapter 2. Chapter 3 studies Jacobian (i.e. tridiagonal) matrices, which are the class of matrices appearing in the inverse problems of Chapter 4. Chapter 5 draws largely on F. R. Gantmakher and M. G. Krein’s book: Oscillation matrices and kernels and small oscillations of mechanical systems. Akademie-Verlag (1964). On first reading, the reader is advised to note merely the principal results concerning oscillatory matrices; these are applied in Chapter 6, and particular in Chapter 7. Chapter 8 draws on Courant and Hilbert (1953) to give the basic properties of symmetric integral equations, and on Gantmakher and Krein to provide the extra properties of the eigenvalues and eigenfunctions of oscillatory kernels. The results of this Chapter are basic to the study of the Sturm-Liouville systems of Chapter 9, and particular for the Euler-Bernoulli problem of Chapter 10.

Reviewer: F.A.Emmerling

##### MSC:

74J25 | Inverse problems for waves in solid mechanics |

74H45 | Vibrations in dynamical problems in solid mechanics |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

70J99 | Linear vibration theory |