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

This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM) is obtained by an expansion onto the eigenmode basis of the associated linearized pro...

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Veröffentlicht in:	Nonlinear dynamics Jg. 97; H. 2; S. 1747 - 1781
Hauptverfasser:	Givois, Arthur, Grolet, Aurélien, Thomas, Olivier, Deü, Jean-François
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Dordrecht Springer Netherlands 01.07.2019 Springer Nature B.V Springer Verlag
Schlagworte:	Asymptotic methods Automotive Engineering Beams (structural) Circular plates Classical Mechanics Computation Computer simulation Control Dynamical Systems Engineering Engineering Sciences Finite element method Forced vibration Frequency response Harmonic balance method Mathematical models Mechanical Engineering Mechanics Nonlinear phenomena Nonlinear response Numerical methods Original Paper Oscillators Perforated plates Reduced order models Structural mechanics Thickness Transverse oscillation Vibration Geometric nonlinearities Non-intrusive stiffness evaluation procedure Continuation method Reduced-order finite element model
ISSN:	0924-090X, 1573-269X
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Abstract	This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM) is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify the ROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.
AbstractList	This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM)is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify theROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators. This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM) is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify the ROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.
Author	Thomas, Olivier Givois, Arthur Deü, Jean-François Grolet, Aurélien
Author_xml	– sequence: 1 givenname: Arthur orcidid: 0000-0003-4327-1671 surname: Givois fullname: Givois, Arthur email: arthur.givois@ensam.eu organization: Laboratoire d’Ingénierie des Systèmes Physiques et Numériques (LISPEN EA 7515), Arts et Métiers ParisTech, Laboratoire de Mécanique des Structures et des Systèmes Couplés (LMSSC EA 3196), Conservatoire National des Arts et Métiers – sequence: 2 givenname: Aurélien surname: Grolet fullname: Grolet, Aurélien organization: Laboratoire d’Ingénierie des Systèmes Physiques et Numériques (LISPEN EA 7515), Arts et Métiers ParisTech – sequence: 3 givenname: Olivier orcidid: 0000-0001-7240-5259 surname: Thomas fullname: Thomas, Olivier organization: Laboratoire d’Ingénierie des Systèmes Physiques et Numériques (LISPEN EA 7515), Arts et Métiers ParisTech – sequence: 4 givenname: Jean-François orcidid: 0000-0002-0107-8024 surname: Deü fullname: Deü, Jean-François organization: Laboratoire de Mécanique des Structures et des Systèmes Couplés (LMSSC EA 3196), Conservatoire National des Arts et Métiers
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Keywords	Geometric nonlinearities Non-intrusive stiffness evaluation procedure Continuation method Reduced-order finite element model
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Publisher	Springer Netherlands Springer Nature B.V Springer Verlag
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Snippet	This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations,...
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SubjectTerms	Asymptotic methods Automotive Engineering Beams (structural) Circular plates Classical Mechanics Computation Computer simulation Control Dynamical Systems Engineering Engineering Sciences Finite element method Forced vibration Frequency response Harmonic balance method Mathematical models Mechanical Engineering Mechanics Nonlinear phenomena Nonlinear response Numerical methods Original Paper Oscillators Perforated plates Reduced order models Structural mechanics Thickness Transverse oscillation Vibration
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Title	On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models
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