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This paper considers the boundary value problem concerning the natural vibrations of a single-rod column, in which two stiffness discontinuities occur at a given position. The column is subjected to a load of a follower force directed towards the negative pole. The load is generated by loading heads made of circular elements. The stiffness discontinuities of the column rod were modeled using rotational springs with a given stiffness. The boundary value problem of column stability was formulated based on the Hamilton's principle. In the case of stability tests, the kinetic stability criterion was used to determine the critical load. Based on the mathematical model, numerical calculations were performed regarding the stability problem. Based on the calculations, the critical force and natural frequencies of the system was determined. Numerical calculations were performed for various values ​​of the system parameters, which include: the stiffness value of the rotational springs defining the stiffness discontinuities of the column rod, the location of the stiffness discontinuities along the rod, the parameters of the loading system including the radiuses of the loading heads and the length of the bolt connecting the head of the loading system with the column rod.