Numerical computation of nonlinear normal modes in a modal derivative subspace

Abstract

Nonlinear normal modes offer a solid theoretical framework for interpreting a wide class of nonlinear dynamic phenomena. However, their computation for large-scale models can be time consuming, particularly when nonlinearities are distributed across the degrees of freedom. In this paper, the nonlinear normal modes of systems featuring distributed geometric nonlinearities are computed from reduced-order models comprising linear normal modes and modal derivatives. Modal derivatives stem from the differentiation of the eigenvalue problem associated with the underlying linearised vibrations and can therefore account for some of the distortions introduced by nonlinearity. The cases of the Roorda’s frame model, a doubly-clamped beam, and a shallow arch discretised with planar beam finite elements are investigated. A comparison between the nonlinear normal modes computed from the full and reduced-order models highlights the capability of the reduction method to capture the essential nonlinear phenomena, including low-order modal interactions.