New Results in Steady-State Analysis of a Transversally Excited, Buckled Beam

This paper considers the steady-state behaviour of transversally excited, buckled pinned-pinned beam, which is free to move axially on one side. This paper starts with a brief derivation of the partial differential equation of motion. Compared to other papers with similar research interest, this research is focused on higher order single-mode as well as multi-mode Galerkin discretizations of the beam's partial differential equation. The convergence of the static load-paths and eigenfrequencies (of the linearized system) of the various higher-order Taylor approximations is investigated. In the steady-state analyses of the semi-analytic models amplitude-frequency plots are presented based on 7th order approximations for the strains. These plots are obtained by solving two-point boundary value problems and by applying a path-following technique. Local stability and bifurcation analysis is carried out using Floquet theory. Dynamically interesting areas (bifurcation points, routes to chaos, snap-through regions) are analyzed using phase space plots and Poincaré plots. In addition, parameter variation studies are carried out. The accuracy of some smi-analytic results is verified by time expensive Finite Element analyses. This paper shows that the described semi-analytical approach is very useful for fast and extensive evaluation of the nonlinear dynamics of buckled beam systems. However, the use of higher order approximations of strains only will effectively improve the accuracy of the semi-analytic approach if axial mass effects will be modelled first.