A Symmetric Inverse Vibration Problem

This paper considers the inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices. The inverse problem of interest here is that of determining real symmetric coefficient matrices, assumed to represent the mass, damping, and stiffness matrices, given the natural frequencies and damping ratios of the structure (i.e., the system eigenvalues). The approach presented here allows for repeated eigenvalues, whether simple or not, and for rigid body modes. The method is algorithmic and results in a computer code for determining mass normalized damping, and stiffness matrices for the case that each mode of the system is underdamped.