Influence of symmetries and imperfections on the non-linear vibration modes of archetypal structural systems

Abstract

The objective of the present work is to investigate how symmetries, initial geometric imperfections and energy level influence the number and stability of the non-linear normal modes and the existence of multimode solutions in structural systems liable to unstable buckling. To this aim, two archetypal models exhibiting interactive buckling phenomena, which lead to several unstable post-buckling paths and high imperfection sensitivity, are considered. As many structural elements, these models have several planes of symmetry. The inherent symmetries and post-buckling solutions have a marked influence on the underlying potential function and, consequently, on the non-linear dynamics of the system, whose stable solutions are limited by the energy level associated with the saddles lying on the boundary of the pre-buckling well. A detailed non-linear modal analysis is accomplished for increasing energy levels, by first considering the nominally perfect systems. Then, the symmetry breaking effect of initial geometric imperfections on the number and stability of the non-linear normal modes is investigated. Finally, some examples illustrate the influence of the superabundance of modes on the resonant forced behavior of the system.