Local and global instability and vibrations of a slender system consisting of two coaxial elements

The stability and free vibrations of a geometrically non-linear system built of a pipe and a rod are discussed in the paper. The pipe and the rod were connected between the mountings by an elastic element and the whole system was subjected to Euler’s load by external force whose direction is unchanging. The boundary problem (static and free vibrations) was formulated and solved, and numerical computations connected to the bifurcation load and natural frequency were carried out. The boundary problem was formulated using Hamilton’s principle and the straightforward expansion method. The calculations were carried out for different characteristic parameters of the considered system, such as rigidity, the placement of the elastic element, pre-stressing of the system as well as the flexural rigidity asymmetry factor of a column.