Improvement of the dynamic properties of a column with step-variable flexural stiffness in terms of structural mounting damping and internal damping

The subject of the work are the free vibrations of a slender columns subjected to a force directed towards the positive pole. This column can model structures found in the engineering, mining or construction industries, machine parts, bridge elements or support structures. The analyzed system is characterized by bending stiffness variable in steps, modelled taking into account the constant volume condition. 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 problem 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. 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 directions of external load and column geometry (variable cross-sections), were compared. 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. The presented analysis of vibrations of damped beams is a starting point for further research on other shapes of slender systems, ultimately systems with continuous change of cross-section.