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http://hdl.handle.net/10985/22642
Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques
TOUZÉ, Cyril; VIZZACCARO, Alessandra; THOMAS, Olivier
This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by
their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These
developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures discretised with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.
Thu, 01 Jul 2021 00:00:00 GMThttp://hdl.handle.net/10985/226422021-07-01T00:00:00ZTOUZÉ, CyrilVIZZACCARO, AlessandraTHOMAS, OlivierThis paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by
their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These
developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures discretised with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements
http://hdl.handle.net/10985/19598
Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements
VIZZACCARO, Alessandra; GIVOIS, Arthur; LONGOBARDI, Pierluigi; SHEN, Yichang; DEÜ, Jean-François; SALLES, Loïc; TOUZÉ, Cyril; THOMAS, Olivier
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.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10985/195982020-01-01T00:00:00ZVIZZACCARO, AlessandraGIVOIS, ArthurLONGOBARDI, PierluigiSHEN, YichangDEÜ, Jean-FrançoisSALLES, LoïcTOUZÉ, CyrilTHOMAS, OlivierNon-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.Nonlinear dynamics of coupled oscillators in 1:2 internal resonance: effects of the non-resonant quadratic terms and recovery of the saturation effect
http://hdl.handle.net/10985/22697
Nonlinear dynamics of coupled oscillators in 1:2 internal resonance: effects of the non-resonant quadratic terms and recovery of the saturation effect
SHAMI, Zein Alabidin; SHEN, Yichang; GIRAUD-AUDINE, Christophe; TOUZÉ, Cyril; THOMAS, Olivier
This article considers the nonlinear dynamics of coupled oscillators featuring strong coupling in 1:2 internal resonance. In forced oscillations, this particular interaction is the source of energy exchange, leading to a particular shape of the response curves, as well as quasi-periodic responses and a saturation phenomenon. These main features are embedded in the simplest system which considers only the two resonant quadratic monomials conveying the 1:2 internal resonance, since they are the proeminent source allowing one to explain these phenomena. However, it has been shown recently that those features can be substantially modified by the presence of non-resonant quadratic terms. The aim of the present study is thus to explain the effect of the non-resonant
quadratic terms on the dynamics. To that purpose, the normal form up to the third order is used, since the effect of the non-resonant quadratic terms will be transferred into the resonant cubic terms. Analytical solutions are detailed using a second-order mutliple scale expansion. A thorough investigation of the backbone curves, their stability and bifurcation, and the link to the forced–damped solutions, is detailed, showing in particular interesting features that had not been addressed in earlier studies. Finally, the saturation effect is investigated, and it is shown how to correct the detuning effect of the cubic terms thanks to a specific tuning of non-resonant quadratic terms and resonant cubic terms. This choice, derived analytically, is shown to extend the validity of the saturation effect to larger amplitudes, which can thus be used in all applications where this effect is needed e.g. for control.
Mon, 01 Aug 2022 00:00:00 GMThttp://hdl.handle.net/10985/226972022-08-01T00:00:00ZSHAMI, Zein AlabidinSHEN, YichangGIRAUD-AUDINE, ChristopheTOUZÉ, CyrilTHOMAS, OlivierThis article considers the nonlinear dynamics of coupled oscillators featuring strong coupling in 1:2 internal resonance. In forced oscillations, this particular interaction is the source of energy exchange, leading to a particular shape of the response curves, as well as quasi-periodic responses and a saturation phenomenon. These main features are embedded in the simplest system which considers only the two resonant quadratic monomials conveying the 1:2 internal resonance, since they are the proeminent source allowing one to explain these phenomena. However, it has been shown recently that those features can be substantially modified by the presence of non-resonant quadratic terms. The aim of the present study is thus to explain the effect of the non-resonant
quadratic terms on the dynamics. To that purpose, the normal form up to the third order is used, since the effect of the non-resonant quadratic terms will be transferred into the resonant cubic terms. Analytical solutions are detailed using a second-order mutliple scale expansion. A thorough investigation of the backbone curves, their stability and bifurcation, and the link to the forced–damped solutions, is detailed, showing in particular interesting features that had not been addressed in earlier studies. Finally, the saturation effect is investigated, and it is shown how to correct the detuning effect of the cubic terms thanks to a specific tuning of non-resonant quadratic terms and resonant cubic terms. This choice, derived analytically, is shown to extend the validity of the saturation effect to larger amplitudes, which can thus be used in all applications where this effect is needed e.g. for control.Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures
http://hdl.handle.net/10985/22695
Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures
SHEN, Yichang; VIZZACCARO, Alessandra; KESMIA, Nassim; SALLES, Loïc; THOMAS, Olivier; TOUZÉ, Cyril
The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam).
Mon, 01 Mar 2021 00:00:00 GMThttp://hdl.handle.net/10985/226952021-03-01T00:00:00ZSHEN, YichangVIZZACCARO, AlessandraKESMIA, NassimSALLES, LoïcTHOMAS, OlivierTOUZÉ, CyrilThe aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam).Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances
http://hdl.handle.net/10985/8952
Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances
MONTEIL, Mélodie; TOUZÉ, Cyril; THOMAS, Olivier; BENACCHIO, Simon
This paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω1, ω2, and ω3 such that ω3 = 2ω2 = 4ω1, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω3 ω2 2ω1. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamics
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10985/89522014-01-01T00:00:00ZMONTEIL, MélodieTOUZÉ, CyrilTHOMAS, OlivierBENACCHIO, SimonThis paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω1, ω2, and ω3 such that ω3 = 2ω2 = 4ω1, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω3 ω2 2ω1. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamicsBackbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification
http://hdl.handle.net/10985/22645
Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification
GIVOIS, Arthur; TAN, Jin-Jack; TOUZÉ, Cyril; THOMAS, Olivier
A system composed of two cubic nonlinear oscillators with close natural frequencies, and thus displaying a 1:1 internal resonance, is studied both theoretically and experimentally, with a special emphasis on the free oscillations and the backbone
curves. The instability regions of uncoupled solutions are derived and the bifurcation scenario as a function of the parameters of the problem is established, showing in an exhaustive manner all possible solutions. The backbone curves are then experimentally
measured on a circular plate, where the asymmetric modes are known to display companion configurations with close eigenfrequencies. A control system based on a Phase-Locked Loop (PLL) is used to measure the backbone curves and also the frequency response function in the forced and damped case, including unstable branches. The model is used for a complete
identification of the unknown parameters and an excellent comparison is drawn out between theoretical prediction and measurements.
Sat, 01 Feb 2020 00:00:00 GMThttp://hdl.handle.net/10985/226452020-02-01T00:00:00ZGIVOIS, ArthurTAN, Jin-JackTOUZÉ, CyrilTHOMAS, OlivierA system composed of two cubic nonlinear oscillators with close natural frequencies, and thus displaying a 1:1 internal resonance, is studied both theoretically and experimentally, with a special emphasis on the free oscillations and the backbone
curves. The instability regions of uncoupled solutions are derived and the bifurcation scenario as a function of the parameters of the problem is established, showing in an exhaustive manner all possible solutions. The backbone curves are then experimentally
measured on a circular plate, where the asymmetric modes are known to display companion configurations with close eigenfrequencies. A control system based on a Phase-Locked Loop (PLL) is used to measure the backbone curves and also the frequency response function in the forced and damped case, including unstable branches. The model is used for a complete
identification of the unknown parameters and an excellent comparison is drawn out between theoretical prediction and measurements.Identification of mode couplings in nonlinear vibrations of the steelpan
http://hdl.handle.net/10985/8943
Identification of mode couplings in nonlinear vibrations of the steelpan
MONTEIL, Mélodie; THOMAS, Olivier; TOUZÉ, Cyril
The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying 1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.
The authors are grateful to Bertrand David (Telecom-ParisTech) for computing the code allowing the STFT filtering procedure used in Section 5.1. The filter has been designed in the framework of the PAFI project (Plateforme d’Aide la facture Instrumentale, www.pafi.fr) which is also thanked.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/89432015-01-01T00:00:00ZMONTEIL, MélodieTHOMAS, OlivierTOUZÉ, CyrilThe vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying 1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.An upper bound for validity limits of asymptotic analytical approaches based on normal form theory
http://hdl.handle.net/10985/7473
An upper bound for validity limits of asymptotic analytical approaches based on normal form theory
LAMARQUE, Claude-Henri; TOUZÉ, Cyril; THOMAS, Olivier
Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10985/74732012-01-01T00:00:00ZLAMARQUE, Claude-HenriTOUZÉ, CyrilTHOMAS, OlivierPerturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.Conservative Numerical Methods for the Full von Kármán Plate Equations
http://hdl.handle.net/10985/9876
Conservative Numerical Methods for the Full von Kármán Plate Equations
BILBAO, Stefan; THOMAS, Olivier; TOUZÉ, Cyril; DUCCESCHI, Michele
This article is concerned with the numerical solution of the full dynamical von Kármán plate equations for geometrically nonlinear (large-amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von Kármán system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semidiscrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretizations.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/98762015-01-01T00:00:00ZBILBAO, StefanTHOMAS, OlivierTOUZÉ, CyrilDUCCESCHI, MicheleThis article is concerned with the numerical solution of the full dynamical von Kármán plate equations for geometrically nonlinear (large-amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von Kármán system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semidiscrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretizations.Nonlinear vibrations of steelpans: analysis of mode coupling in view of modal sound synthesis.
http://hdl.handle.net/10985/10115
Nonlinear vibrations of steelpans: analysis of mode coupling in view of modal sound synthesis.
MONTEIL, Mélodie; TOUZÉ, Cyril; THOMAS, Olivier
Steelpans are musical percussions made from steel barrels. During the manufacturing, the metal is stretched and bended, to produce a set of thin shells that are the differents notes of the instrument. In normal playing, each note is struck, and the sound reveals some nonlinear characteristics which give its peculiar tone to the instrument. In this paper, an experimental approach is first presented in order to show the complex dynamics existing in steelpan’s vibrations. Then two models, based on typical modal interactions, are proposed to quantify these nonlinearities. Finally, one of them is observed in free oscillations simulations, in order to compare the internal resonance model to the steelpan vibrations behaviour in normal playing. The aim is to identify the important modes participating in the vibrations in view of building reduced-order models for modal sound synthesis.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10985/101152013-01-01T00:00:00ZMONTEIL, MélodieTOUZÉ, CyrilTHOMAS, OlivierSteelpans are musical percussions made from steel barrels. During the manufacturing, the metal is stretched and bended, to produce a set of thin shells that are the differents notes of the instrument. In normal playing, each note is struck, and the sound reveals some nonlinear characteristics which give its peculiar tone to the instrument. In this paper, an experimental approach is first presented in order to show the complex dynamics existing in steelpan’s vibrations. Then two models, based on typical modal interactions, are proposed to quantify these nonlinearities. Finally, one of them is observed in free oscillations simulations, in order to compare the internal resonance model to the steelpan vibrations behaviour in normal playing. The aim is to identify the important modes participating in the vibrations in view of building reduced-order models for modal sound synthesis.