Transcendental Inverse Eigenvalue Problem Associated with Longitudinal Vibrations in Rods

The dynamics of a continuous system is represented by a transcendental eigenvalue problem, whereas the associated discrete approximating model is characterized by an algebraic eigenvalue problem. The fact that the asymptotic behavior of the eigenvalues of a transcendental eigenvalue problem is different from the asymptotic behavior for an approximating algebraic eigenvalue problem forms an obstacle to solving the inverse problem of reconstructing the physical parameters of a continuous system based on a discrete model. To overcome this obstacle, a new mathematical model is presented where the transcendental eigenvalue problem associated with a continuous system with varying physical properties is approximated by a continuous system with piecewise constant physical parameters. An algorithm for solving the associated inverse eigenvalue problem is presented. Numerical examples demonstrate that the physical properties of a continuous system can be reconstructed by using this approach.[PUBLICATION ABSTRACT]