Stateczność Pewnych Niepryzmatycznych i Niejednorodnych Prętów Ściskanych Siłą Śledzącą - Gajewski, A. (Engineering Transactions, 1966)
On the basis of the familiar differential equation of vibration of a bar loaded by a compressive force, its stability is investigated if the compressive force is a non-conservative force preserving its position in relation to the bar and the bar, which is clamped at one end, is non-prismatic and nonhomogeneous. To calculate the critical force and the corresponding frequency the kinetic criterion of stability is used which leads, in our case, to the condition of the first derivative of the force with respect to the frequency becoming zero. The problem is solved by means of the perturbation method by expanding in power series (3.1) the deflection function and the compressive force (depending on the frequency). In the third part of the paper general expressions are given for three terms of the series (3.1) which expressing a relation between the compressive force and the vibration frequency of the bar. In the fourth part, taking into consideration the kinetic criterion of stability and expanding the steady frequency corresponding to the critical force, in a series of the small parameter, equations are obtained for the terms of the series expressing the critical force and the series expressing the frequency of vibration.