Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

Abstract

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates.

High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

“…Consequently, prior developments led in [17,18,21] with damping included were low-order approximations of the SSM, either with a graph style or a normal form style. Along the same lines, and as remarked in [49], the computations proposed in this contribution as well as those shown for example in [45][46][47], are approximations of the unique SSM, which is reached at a very high order only. Importantly, all lower order approximations of the SSM share the invariance property, up to the selected order, and can be thus used safely to provide accurate ROMs.…”

“…More recent progress focuses on working directly in the physical space, with in view application to structures modelled with the FEM. This has been realised using either a normal form approach [27,43,44], or the parametrisation method [45][46][47].…”

“…Here, we elaborate on the parametrisation method and introduce a different implementation to the approach in [45][46][47].…”

“…Under the parametrisation method framework, the theoretical foundations used in this contribution are equivalent to those already reported in [45][46][47], where arbitrary order expansions have already been shown, together with the possibility of using either graph or normal form style. The main differences can be listed as follows: (i) the focus here is on large FE models of mechanical systems discretised with 3D elements, in contrary to [45,48] where beam and plate elements are treated together with the possibility of including generic nonlinearities; (ii) the damping matrix is assumed to be diagonalised by the eigenvectors of the conservative system, which is a needed assumption when working with large FE models where different choices than proportional damping needs specific developments at the elementary level; (iii) thanks to the inherently second-order nature of the original equation of motion, displacement and velocity mappings can be treated separately allowing to show the relationship between the two at generic order and to retrieve homological equations in the sole displacement mapping; (iv) a number of implementation differences in the treatment of the direct computation are reported (e.g.…”

“…In the field of mechanical systems, elaborating on previous results on normal forms, a direct approach has been proposed in [43] and further developed in [27,44], allowing one to express the reduced dynamics with normal coordinates in an invariant-based span of the phase space. Leveraging on SSM, a direct approach has also been presented in [45][46][47], taking into account the damping and proposing arbitrary order approxima-tions in an automated framework. All these developments have been made publicly available in the code SSMtool 2.0, which embeds all the proposed feature to compute automatically high order approximations of the SSM [48].…”