Inverse Eigenvalue Problem for a Modified Vibrating System

Abstract

Let $( \lambda _i )_1^n $ be the eigenvalues of a simply connected mass-spring system. Suppose that a simple oscillator of mass m and stiffness k is attached to the system and let $( \mu _i )_1^n $ be the eigenvalues of the modified system. The problem of constructing the physical elements of the system from $( \lambda _i )_1^n ,( \mu _i )_1^n ,k$, and m is considered.This problem is associated with a rank-two modification matrix. The analysis generalizes known results of the rank-one modification problem, where only a spring is added to the system. The necessary and sufficient conditions for the construction of a physically realizable system with positive mass and spring elements are established. If these conditions are satisfied and $k/m \ne \lambda _i $ for $i = 1,2, \ldots ,n$, then the construction is unique. When $k/m = \lambda _i $, there exists a continuous family of solutions, which is also characterized in the paper.