Transcendental Inverse Eigenvalue Problem Associated with Longitudinal Vibrations in Rods

Abstract

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. Nomenclature A, B = matrices representing transcendental eigenvalue problems Ai, Bi = arbitrary constants in general solutions of ordinary differential equations ci = speed of sound in the ith portion of the rod Fi = the ith equation defining the inverse problem h = length of the element in the finite difference model i = index (integer) J = Jacobian matrix of partial derivatives j = index (integer) K = stiffness matrix k = element stiffness M = mass matrix m = element mass m T = total mass of the rod n = model order P = a typical n × n matrix ˆ P = (n − 1) × (n − 1) leading principal submatrix of P pi = axial rigidity in the ith portion of the rod ri = normalized rigidity in the ith portion of the rod t = time variable ui = axial displacement in the ith portion of the rod vi = eigenfunction associated with the ith portion of the rod x = vector of unknown parameters x = position variable z = eigenvector of a transcendental matrix zA = eigenvector of A zB = eigenvector of B δ = vector of correction to the unknown parameters δi = correction to the ith unknown parameter e = error tolerance ηi = the ith lowest eigenvalue of the fixed-fixed rod λ = eigenvalue of a transcendental matrix λi = ith lowest eigenvalue of the fixed‐free rod μ = natural frequencies of the fixed‐fixed rod ρi = mass density in the ith portion of the rod