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In this paper one considers the boundary problem of natural vibrations of a single rod column in which stiffness discontinuity occurs in a specific position. The column is loaded with a follower force directed towards the negative pole. The load is being generated with the use of loading heads with the circular outlines. The stiffness discontinuity of the column rod is modeled by means of the rotational spring with a given stiffness. The boundary problem of the natural vibrations of the column has been formulated on the basis of Hamilton's principle. In order to estimate the magnitude of the critical load one uses the kinetic stability criterion during simulations. The created mathematical model allowed one to perform both vibrations and stability simulations. In the case of vibration studies the characteristic curves have been determined (curves representing relationship between natural frequencies and external load). During investigations into the stability phenomenon the critical force of the system has been estimated. Both simulations were performed at different magnitudes of the key parameters of the considered system such as: rotational spring stiffness determining the stiffness discontinuity, the location of the stiffness discontinuity, and the parameters of the loading system including radiuses of the loading heads and the length of the transom used to connect the loading head with the column’s main rod.