Inverse Problems in Vibration

Abstract

This article concerns infinitesimal free vibrations of undamped elastic systems of finite extent. A review is made of the literature relating to the unique reconstruction of a vibrating system from natural frequency data. The literature is divided into two groups—those papers concerning discrete systems, for which the inverse problems are closely related to matrix inverse eigenvalue problems, and those concerning continuous systems governed either by one or the other of the Sturm–Liouville equations or by the Euler–Bernoulli equation for flexural vibrations of a thin beam.

Assessing how changes to a structure can create gaps in the natural frequency spectrum

“…When effects are specified, and possible causes sought, the formulation is classified as inverse. Inverse structural vibration problems have been reviewed by Gladwell (1986), updated in Gladwell (1996), Ellishakoff (2000), and recently by Mottershead and Ram (2006) in a work concentrating on vibration absorption.…”

Non-rotating beams isospectral to a given rotating uniform beam

“…An important class of problems for such vibrating systems can be broadly classified into inverse problems and isospectral problems. In inverse problems [2,3], one tries to determine the material properties of a system for a given frequency spectrum. In general, more than one frequency spectrum is required for a reconstruction procedure to determine the material properties of the system [4].…”

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

“…Structured matrices, such as centrosymmetric matrices, orthogonal-symplectic matrices, pseudo-centrosymmetric matrices, mirrorsymetric matrices, and generalized reflexive matrices are being used to solve practical problems in optimized control, electron theory, graph theory, and so on [4][5][6][7][8]. The inverse eigenvalue problem has many applications, for example, in control design for second-order systems, control design, quantum mechanics, neuron transport theory, finite element model updating for damped or gyroscopic systems and system identification [9,10]. One of the most basic control tasks is the static output feedback pole placement.…”

“…Diele et al studied some inverse eigenvalue problems for matrices with Toeplitz-related structure [13]. Inverse eigenvalue problems with Jacobi, Toeplitz, and nonnegative matrices occur in a large number of applications ranging from applied mechanics and physics to numerical analysis, for more details see [9][10][11]. The eigenvalues and the corresponding eigenvectors x of the quadratic eigenvalue problem Q( )x ∶= ( 2 A + B + C)x = 0 can interpret the dynamical behavior of the second order differential system Aẍ(t) + Bẋ(t) + Cx(t) = f (t) with mass, damping and stiffness matrices which arises in the acoustic simulation of poro-elastic materials, the elastic deformation of anisotropic materials, and finite element discretization in structural analysis [14][15][16].…”