The subject of the work are the free vibrations of a slender column subjected to a follower force directed towards the positive pole. The analyzed system is characterized by flexural stiffness variable in steps, modelled under the condition of constant volume. The model takes into account the flexibility of structural nodes by supporting the loaded end of the column with a spring with linear characteristics.

The issue was formulated on the basis of the Bernoulli-Euler’s theory and then solved using the variational method (Hamilton’s principle). Determination of differential equations o motion of individual segments of the column and their solutions, taking into account the boundary conditions, made it possible to determine the transcendental equation giving the vibration frequency values.

As a part of the numerical analysis, an influence of the variable shape of the column on the dynamics was determined. The influence of internal damping and structural damping in the mounting on the change in natural vibration frequency was considered in detail. For comparative purposes, the first two frequencies of the tested system and the comparative system (without damping), with different column geometry were analyzed. Based on the presented results, the impact of damping on the dynamics of the system was determined and it was indicated as one of the ways to control the dynamic properties.

Keywords: free vibrations, internal damping, constructional damping