Numerical computation of nonlinear normal modes in a modal derivative subspace

Abstract

Nonlinear normal modes (NNMs) offer a solid theoretical framework for interpreting a wide class of nonlinear dynamic phenomena. However, the computation of NNMs for large-scale models can be time consuming, particularly when nonlinearities are distributed across the degrees of freedom. In this paper, the NNMs 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 NNMs 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.

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

“…9 and 12), for which the natural frequencies of the companion modes are naturally tuned and fulfill Eq. (42). In this case, well documented in [19,66], the backbone curve of one of the companion modes becomes unstable after a pitchfork bifurcation (denoted by PF in Fig.…”

“…A last family of method is based on the addition of a modal derivative subspace to the transverse linear basis [41]. It enables to capture efficiently the relevant in-plane displacement associated with a transverse linear mode [42]. Since the size of the subspace grows quadratically with the number of investigated eigenmodes, the concept of quadratic manifold has been proposed to minimize the number of unknowns [43].…”

“…The so-called nonlinear normal modes-(NNMs) can be represented as frequency energy plots (or amplitude-frequency plots, also called "backbone curves" [12]) and computed by various analytical [48] or numerical [49] methods. Among these last, a method combining shooting and pseudo-arclength continuation proposed in [50] enables to compute NNMs from reduced-order finite element models: with a homemade finite element beam formulation [42,51] or some finite element commercial codes [52], the energy is computed by injecting the result of the ROM in the full FE model. Among the huge literature described above, the present article addresses the computation of the nonlinear frequency response of thin structures using a continuation method and finite element modal reduced-order models, obtained by the STEP.…”

A Novel Derivation for Modal Derivatives Based on Volterra Series Representation and Its Use in Nonlinear Model Order Reduction

“…Particularly in nonlinear structural dynamics, the concept of basis augmentation with so-called modal derivatives (MDs) has been successfully applied in numerous applications [8,1,25,28,27,19], in order to reduce the computational effort involved in the numerical analysis. Recently, MDs have been also applied for the efficient computation of nonlinear normal modes (NNMs) [10] and the nonlinear frequency response function (NLFRF) of structural systems [26,22,23].…”

Nonlinear Moment Matching for the Simulation-Free Reduction of Structural Systems

“…Farhat et al [2014]). Known in the context of nonlinear frequency response/modal analysis (Kerschen et al [2009]), the Harmonic Balance Method (HBM) and the Nonlinear Normal Modes (NNMs) have lately become very popular also in the model reduction field (Weeger et al [2014], Sombroek et al [2018]). These procedures, however, rely on either numerical continuation methods or shooting techniques, making them also fairly computational expensive in comparison to a numerical integration needed e.g.…”