On the dynamic stability of elastic structures subjected to follower forces

Abstract

Abstract. Within the framework of the second-order theory, some classical stability problems, whose critical load corresponded to dynamic instability, were considered in the paper [1]. The main focus was on systems with just one lumped mass. This idealization, together with the assumption of negligible axial strain and the adoption of the second-order theory, reduced the considered systems to a single Lagrangian coordinate. In this way, static methods could be applied to derive the analytical expression of the stiffness coefficient and to study the dynamic stability, starting with a well-known example, namely, a cantilever beam with a lumped mass at the free end subjected to a follower load [2]. In this paper, a new lumped mass system is studied: a straight-axis beam with constant cross-sectional area and stiffness, mass-free, hinged at one end, simply supported at an intermediate point (with a sliding plane parallel to the beam axis) and with the other end free, where a lumped mass is present and a follower force is applied. As in the examples shown in [1], in this example the first asymptote of the stiffness coefficient corresponds to the critical load, due to divergence at infinity. It is shown that this critical load is equal to the buckling load due to divergence of an auxiliary structure, which differs from the original one in that the concentrated mass is replaced by a constraint that blocks the corresponding Lagrangian coordinate.