Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensiona
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ORIGINAL PAPER
Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements Alessandra Vizzaccaro1 · Arthur Givois2,3 · Pierluigi Longobardi1 · Yichang Shen4 · Jean-François Deü3 · Loïc Salles1 · Cyril Touzé4 · Olivier Thomas2 Received: 26 March 2020 / Accepted: 29 July 2020 © The Author(s) 2020
Abstract Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the modal expansion is observed when applying the method with 3D elements, because of nonlinear couplings occurring with very high frequency modes involving 3D thickness deformations. Focusing on the case of flat structures, we first show by computing all the modes of the structure that a converged solution can be exhibited by using either static condensation or normal form theory. We then show that static modal derivatives provide the same solution with fewer calculations. Finally, we propose a modified STEP, where the prescribed displacements are imposed solely on specific degrees of freedom of the structure, and show that this adjustment also provides efficiently a converged solution. Keywords Reduced order modeling · Geometric nonlinearities · Three-dimensional effect · Thickness modes · Modified STiffness Evaluation Procedure · Nonlinear modes · Modal derivatives
1 Introduction Geometrically nonlinear effects appear generally in thin structures such as beams, plates and shells, when the amplitude of the vibration is of the order of the thickness [26,37]. The von Kármán family of models for beams, plates and shells allows one to derive explicit partial differential equations (PDE) [1,8,18,37,39], showing clearly that a coupling between bending and longitudinal motions causes a non-
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Alessandra Vizzaccaro [email protected]
1
Imperial College London, Vibration University Technology Centre, London SW7 2AZ, UK
2
Arts et Metiers Institute of Technology, LISPEN, HESAM Université, 59000 Lille, France
3
Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Métiers, 2 Rue Conté, 75003 Paris, France
4
IMSIA, ENSTA Paris, CNRS, EDF, CEA, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France
linear restoring force of polynomial type in the equations of motion. This geometric nonlinearity is then at the root of complex behaviours, that also need dedicated computational strategies in order to derive quantitative predictions. On the phenomenological point of view, structural nonlinearities give rise to numerous nonlinear phenomena that have been analysed in a number of studies: frequency dependence on amplitude [14,20,25], hardening/softening behaviour [45,46], hysteresis and jump phenomena [24,36], mode couplin
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