https://hal-ensta-paris.archives-ouvertes.fr/hal-03882282Opreni, AndreaAndreaOpreniPOLIMI - Politecnico di Milano [Milan]Vizzaccaro, AlessandraAlessandraVizzaccaroUniversity of Bristol [Bristol]Touzé, CyrilCyrilTouzéIMSIA - UMR 9219 - Institut des Sciences de la mécanique et Applications industrielles - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - EDF R&D - EDF R&D - EDF - EDFUME - Unité de Mécanique - ENSTA Paris - École Nationale Supérieure de Techniques AvancéesENSTA Paris - École Nationale Supérieure de Techniques AvancéesIP Paris - Institut Polytechnique de ParisFrangi, AttilioAttilioFrangiPOLIMI - Politecnico di Milano [Milan]High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to generic forcing terms and parametrically excited systemsHAL CCSD2022[SPI.MECA.VIBR] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph][SPI.MECA.STRU] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph][NLIN] Nonlinear Sciences [physics][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS][SPI.NANO] Engineering Sciences [physics]/Micro and nanotechnologies/MicroelectronicsTouzé, Cyril2022-12-02 12:30:342023-03-24 14:53:292022-12-19 08:11:19enJournal articleshttps://hal-ensta-paris.archives-ouvertes.fr/hal-03882282/document10.1007/s11071-022-07651-9application/pdf1The direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary order of expansions are considered for the unknown mappings and the reduced dynamics, which are then solved sequentially through the homological equations, for both autonomous and time-dependent terms. It is emphasized that the two problems share a similar structure, which can be used for an efficient implementation of the non-autonomous added terms. A special emphasis is also put on the new resonance conditions arising due to the presence of the external forcing frequencies, which allows predicting phenomena such as parametric excitation and isolas formation. The method is then applied to structures of academic and industrial interest. First, the large amplitude vibrations of a forceddamped cantilever beam are studied. This example highlights that high order non-autonomous terms are compulsory to correctly estimate the maximum vibration amplitude experienced by the structure. The birth of isolated solutions is also illustrated on this example. The cantilever is then used to show how quadratic coupling creates conditions for the excitation of the parametric instability, all these features being correctly embedded in the reduction process. A shallow arch excited with multi-modal forcing is then studied to detail different forcing effects. Finally, the approach is validated on a structure of industrial relevance, i.e. a comb driven micro-electro-mechanical resonator. The accuracy and computational performance reported suggest that the proposed methodology can accurately predict the nonlinear dynamic response of a large class of nonlinear vibratory systems.