Nonlinear response and stability analysis of beams using finite elements in time

Abstract

The dynamic response and stability analysis of beams undergoing large deflections and rotations is analyzed using the finite-element method in time. This formulation provides an efficient and consistent approach to predicting the dynamic response of nonlinear periodic systems as well as their stability boundaries based on Floquet's theory. This paper has two goals: 1) to present the formulation of the finite element in time equations for naturally curved and twisted beams undergoing large deflections and rotations, and 2) to discuss the predictions of this method when applied to several classical nonlinear beam problems for which analytical solutions exist and, in some cases, experimental results are available. These examples are 1) the natural vibration frequencies of a rotating beam, 2) the harmonic and superharmonic response of a clamped-clamped beam undergoing largeamplitude vibrations, 3) the dynamic instability of a cantilevered beam under a tip follower force, and 4) the parametric excitation of a beam under an axial pulsating load, with and without the presence of viscous damping forces. In all cases, close agreement is found between the analytical results and the predictions of the finite element in time approach, which appears to be an efficient and reliable technique for nonlinear dynamic response and stability analysis of periodic systems.