Natural Frequencies and Eigenfunctions of a Composite Rod

The problem of finding the natural frequencies and eigenfunctions of the fuel element of a nuclear reactor is considered. A rod with a piecewise constant material density in the longitudinal direction is used as a model. The frequency equation is explicitly derived, and the natural frequencies of the first twelve harmonics are determined numerically and analytically. Analytical expressions for eigenfunctions are obtained, which are in good agreement with calculations made using the Galerkin method and numerical calculations in the ANSYS program. The results will be used to simulate the dynamics of the NEPTUN pulsed reactor for examining the stability of its operation.