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Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models

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Abstract

We propose a direct method for computing modal coupling coefficients—due to geometrically nonlinear effects—for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the FE formulation, with specificities pertaining to the shell elements considered, namely, here elements of the “Mixed Interpolation of Tensorial Components” family. Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing—with a continuation procedure —the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes, and we find that the most important part of the dynamics can be predicted with a single oscillator equation.

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Authors and Affiliations

  1. Unité de Mécanique (UME), ENSTA-ParisTech, 828 Boulevard des maréchaux, 91762 , Palaiseau Cedex, France

    C. Touzé

  2. Inria/Reo and LJLL UMR 7958 UPMC, Rocquencourt, B.P. 105, 78153 , Le Chesnay, France

    M. Vidrascu

  3. Inria/MΞDISIM, 1 rue Honoré d’Estienne d’Orves, Campus de l’Ecole Polytechnique, 91120 , Palaiseau, France

    D. Chapelle

Authors
  1. C. Touzé
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  2. M. Vidrascu
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  3. D. Chapelle
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Correspondence to C. Touzé.

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Touzé, C., Vidrascu, M. & Chapelle, D. Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models. Comput Mech 54, 567–580 (2014). https://doi.org/10.1007/s00466-014-1006-4

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