An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .