<p>We study harmonic vibrations of a linearly elastic rod of finite length with small-scale inhomogeneities with different characteristics (elastic, viscoelastic, plastic, inhomogeneous) described by linear equations of state. We construct a mathematical model that takes into account the influence of these defects by placing of a point singularity of infinite order at the center of this inhomogeneity. On this basis, we formulate a boundary-value problem for the differential equation with hypersingular right-hand side whose solution is equivalent to the solution of the original problem. We propose a procedure for the evaluation of the coefficients of a hypersingular series with a point support simulating the defect. The procedure is based on the decompositions in infinite series in a small parameter with coefficients that are hypersingular generalized functions. The proposed procedure is used to solve both the direct problem of determination of frequencies and modes of natural vibrations of the rod with given characteristics of defects, and the inverse problem of determination of the integral characteristics of defectiveness of the rod for the known shifts of the eigenfrequencies of vibrations. The suggested approach is based on the recursive solving of the boundary-value problems, which allows one to describe the degree of defectiveness of the rod with the required accuracy.</p>