The non-linear vibrations of simply supported column loaded by the mass element

Abstract

The stability and free vibrations of slender support systems were the subject of many scientific and research works. This paper presents the boundary problem of column vibrations in which the longitudinal inertia of the mass element, which loads the slender system, is taken into account. The column was analysed as a simply supported structure. The formulation of boundary problem was carried out by using kinetic criterion of stability. Taking into consideration the load from longitudinal inertia of mass element in the mathematical model makes the system non-linear. The non-linear components of the mathematical model were expanded into a power series of a small parameter. Numerical calculations allowed determination of the influence of the column oscillations amplitude on the non-linear component of the natural frequency of the system. It has been shown that in supporting systems loaded by a mass element, the vibrations amplitude considerably affect natural frequency at appropriately selected parameters of the system. Also experimental verification of the adopted mathematical model was carried out, which showed good agreement between numerical and experimental studies.