An Inverse Mode Problem for the Continuous Model of an Axially Vibrating Rod

An inverse mode problem for the continuous model of an axially vibrating rod is formulated and solved. It is shown that the density and axial rigidity functions are determined by two eigenvalues, their corresponding eigenfunctions, and the total mass of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive parameters are given. It is shown that when these conditions are satisfied, the solution is unique. Since the data required for the construction may be available from measurements, the presented approach is well-suited for modal analysis applications.