Free vibration analysis of a column with tip mass subjected to an axial load

In this study, free vibration of an elastically supported column under the influence of an axial load with singular mass at the free end was investigated in the framework of the Euler-Bernoulli beam theory. The system has the weight of the end mass and the weight of the column as the axial force. The weight of the column is considered as an axial force that varies linearly along the axis. The elastic support at the lower end of the column is formed with translation and rotation springs. The differential equation of the transverse free vibrations of the column has a variable coefficient and it was solved by the Frobenius force series method. The differential equation was solved under the relevant boundary conditions and the frequency equation was obtained and the natural frequencies of the system were obtained by calculating the roots of the equation numerically. The natural frequencies obtained for the selected values of parameters such as axial force, eccentricity and rotational inertia of the end mass were given in the tables in comparison with the results in the literature and the effect of these parameters on the frequencies was explained. If the stiffnesses of the springs representing the elastic support are too great, with the assumption that this support will be a fixed support, the problem has also been solved for the case where the column is fixedly supported, and it has been shown that this is achieved by the obtained results.