Physical parameters reconstruction of a free-free mass-spring system from its spectra

Consider a linear, n degree-of-freedom, free–free, vibratory system. Suppose that the system under consideration is a multiply connected system, in the sense that each of its concentrated masses may be connected through a linear spring to each of its other masses. Let $\lambda _1^{(i)} , \cdots ,\lambda _i^{(i)} $, be the eigenvalues of the system in the case where $n - i$ masses of the system are pinned to the ground. Suppose that for $i = 1,2, \cdots ,n$ the sets $\lambda _1^{(i)} , \cdots ,\lambda _i^{(i)} $ and the total mass of the system are given. An algorithm, which reconstructs the mass and stiffness matrices of the system from the data, is presented. The multiplicity of the solutions and the sensitivity of the problem to perturbation are investigated.