Extremal inverse eigenvalue problem for bordered diagonal matrices

This paper investigates in-line spring-mass systems (An), fixed at one end and free at the other, with n-degrees of freedom (d.f.). The objective is to find feasible in-line systems (B(n)) that are isospectral to a given system. The spring-mass systems, A(n) and B(n), are represented by Jacobi matrices. An error function is developed with the help of the Jacobi matrices A(n) and B(n). The problem of finding the isospectral systems is posed as an optimization problem with the aim of minimizing the error function. The approach for creating isospectral systems uses the fact that the trace of two isospectral Jacobi matrices A(n) and B(n) should be identical. A modification is made to the diagonal elements of the given Jacobi matrix (A(n)), to create the isospectral systems. The optimization problem is solved using the firefly algorithm augmented by a local search procedure. Numerical results are obtained and resulting isospectral systems are shown for 4 d.f. and 10 d.f. systems.

Exploring isospectral spring–mass systems with firefly algorithm

We solve an inverse eigenvalue problem arising from interior mass grounding of a spring - mass system. This generalizes a result of Ram, which dealt with the grounding of an end mass. Our method is to recast this generalized problem in the framework of a different problem previously solved by the authors and others.

Inverse eigenvalue problem: existence of special spring - mass systems

Given the sequences of real numbers (ii) n ; (ni) nA1 1 ; satisfying the interlacing property ii < ni < ii+1; 1 i i i n A 1; we present a new numerical procedure to construct a spring-mass system with eigenvalues (ii) n ; where the interlaced spectrum (ni) nA1 1 corresponds to the modiÞed system whose mass mr+1; 1 i r i n A 2; is Þxed. The method, which is a modiÞcation of the fast orthogonal reduction technique, appears to be computationally less expensive than other in the literature.

/pdf/on-the-numerical-reconstruction-of-a-spring-mass-system-from-5ctk76o1fw.pdf

On the numerical reconstruction of a spring-mass system from its natural frequencies

On the Reconstruction of a Damped Vibrating System from Two Complex Spectra, Part 1: Theory

It is known that a simple spring-mass system may be reconstructed uniquely (apart from a single scaling factor) from the poles and zeros of the frequency response function corresponding to sinusoidal forcing at an end. The (squares of the) poles yield the eigenvalues of a tridiagonal matrix, A, while the zeros yield the eigenvalues of the matrix, A*, with the last row and column deleted. There are proven numerical methods for reconstructing A. The authors show that, if forcing is applied at an interior point, then the reconstruction is unique if that point is not a node of any eigenmode; if it is, there is a family of systems with the required properties. In either case the system may be constructed using modifications of proven techniques.

The reconstruction of a tridiagonal system from its frequency response at an interior point

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The reconstruction of a tridiagonal system from its frequency response at an interior point