A slender system, built as a geometrically non-linear system and subjected to specific load by a follower force directed towards the positive pole, is discussed. The load is induced by heads composed of circular-profile elements. The problem of stability and free vibration was formulated on the basis of Hamilton's principle and then, owing to its non-linearity, solved using the straightforward expansion method. This paper provides example results of numerical computations concerning the influence of selected parameters characterizing the considered column (including pre-stressing) on the stability and free vibration. The accuracy of the adopted mathematical model is also proved on the basis of experimental studies.

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The boundary value problem of the stability and free vibration of a hydraulic cylinder has been formulated and solved in this paper. The considered hydraulic cylinder has been elastically fixed at both ends and loaded by Euler force. An elastical fixation has been modelled by rotational springs. The mentioned above hydraulic cylinder consists of a piston rod and cylinder replaced by rods. There are conditions of continuity between the rods. The boundary value problem has been formulated on the basis of minimum potential energy (static problem) and on the basis of Hamilton's principle (free vibration problem). In this paper example results of numerical calculations of the stability and free vibration have been presented. Experimental research has been performed in order to verify the correctness of the assumed mathematical model. Professional measuring apparatus and special stand for research into the slender systems have been used in experiment. Natural frequencies have been measured in dependence on the values of an external load.

Considered herein is the vibration and stability problems of a slender column subjected to generalized load with a force directed toward the positive pole. The load is developed by heads composed of circular profile elements. The geometrically nonlinear problem of stability and free vibrations is formulated on the basis of Hamilton's principle, and due to nonlinearity, the problem is solved by applying the small parameter method. Vibration and stability results show the influence of chosen parameters that characterize the considered column (including initial prestressing). The assumed mathematical model is validated by experimental results.

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