Inverse problems in vibration

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 theo...

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Main Author: Gladwell, G.M.L.
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 1986, 1986
Edition:1st ed. 1986
Series:Mechanics: Dynamical Systems
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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