Dynamic analysis of centrally compressed inhomogeneous rods using the initial parameters method

The study considers an elastic rod within the framework of Bernoulli’s hypotheses, subjected to a time-independent longitudinal load. The problem formulation accounts for the squares of the section rotation angles, leading to the well-known stability equations for straight rods. Boundary conditions for both follower and dead loads at the ends are examined. Dimensionless state equations are presented, along with an analytical expression for the fundamental solution of the free transverse vibration problem. The dependencies of the first four natural frequencies on the compressive load are provided. Additionally, static and dynamic bending problems of rods under a time-independent longitudinal force are analyzed. The eigenstates are determined using the initial parameter method.