The free vibrations and optimization of the shape of a column subjected to a load by a follower force directed towards the positive pole applying the variational method

The results of numerical computations and theoretical research into optimization of the shape and the free vibrations of a cantilever column subjected to a load by the follower force directed towards the positive pole are presented in this paper. The systems of equations: of motion and of cross-sectional area of the considered column were determined on the basis of the total mechanical energy and Hamilton’s principle. Taking into account the formulated boundary conditions, optimized shapes of the systems were determined for the chosen values of geometrical parameter of a head subjected to a load. The values of natural frequency of the considered system were obtained taking into account distribution of flexural rigidity along the column length while the values of critical load of optimized column were obtained on the basis of curves of changes in the basic natural frequency in relation to external load. The results of numerical computations were presented in the case of one from three classes of the systems – versatile uniformly convergence column for the assumed criterion of constant volume of the system.