Nonlinear Normal Modes of Highly Flexible Beam Structures Modelled under the SE(3) Lie Group Framework

Abstract

This work presents a shooting algorithm to compute the periodic responses of geometrically nonlinear structures modelled under the Special Euclidean (SE) Lie group formulation. The formulation is combined with a pseudo-arclength continuation method, while special adaptations are made to ensure compatibility with the $SE$ framework. Nonlinear normal modes (NNM) of various two-dimensional structures including a doubly-clamped beam, a shallow arch, and a cantilever beam are computed. Results are compared with a reference displacement-based FE model with von Kármán strains. Significant difference is observed in the dynamic response of the two models in test cases involving large degrees of beam deformation and rotation. Differences in the contribution of higher order modes substantially affect the frequency-energy dependence and the nonlinear modal interactions observed between the models. It is shown that the SE model, owing to its exact representation of the beam kinematics, is better suited at adequately capturing complex nonlinear dynamics compared to the von Kármán model.