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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 21 May 2024 20:24:48 GMT2024-05-21T20:24:48ZFinite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS
http://hdl.handle.net/10985/8955
Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS
LAZARUS, Arnaud; THOMAS, Olivier; DEÜ, Jean-François
This article presents a finite element reduced order model for the nonlinear vibrations of piezoelectric layered beams with application to NEMS. In this model, the geometrical nonlinearities are taken into account through a von Kármán nonlinear strain–displacement relationship. The originality of the finite element electromechanical formulation is that the system electrical state is fully described by only a couple of variables per piezoelectric patches, namely the electric charge contained in the electrodes and the voltage between the electrodes. Due to the geometrical nonlinearity, the piezoelectric actuation introduces an original parametric excitation term in the equilibrium equation. The reduced-order formulation of the discretized problem is obtained by expanding the mechanical displacement unknown vector onto the short-circuit eigenmode basis. A particular attention is paid to the computation of the unknown nonlinear stiffness coefficients of the reduced-order model. Due to the particular form of the von Kármán nonlinearities, these coefficients are computed exactly, once for a given geometry, by prescribing relevant nodal displacements in nonlinear static solutions settings. Finally, the low-order model is computed with an original purely harmonic-based continuation method. Our numerical tool is then validated by computing the nonlinear vibrations of a mechanically excited homogeneous beam supported at both ends referenced in the literature. The more difficult case of the nonlinear oscillations of a layered nanobridge piezoelectrically actuated is also studied. Interesting vibratory phenomena such as parametric amplification or patch length dependence of the frequency output response are highlighted in order to help in the design of these nanodevices.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10985/89552012-01-01T00:00:00ZLAZARUS, ArnaudTHOMAS, OlivierDEÜ, Jean-FrançoisThis article presents a finite element reduced order model for the nonlinear vibrations of piezoelectric layered beams with application to NEMS. In this model, the geometrical nonlinearities are taken into account through a von Kármán nonlinear strain–displacement relationship. The originality of the finite element electromechanical formulation is that the system electrical state is fully described by only a couple of variables per piezoelectric patches, namely the electric charge contained in the electrodes and the voltage between the electrodes. Due to the geometrical nonlinearity, the piezoelectric actuation introduces an original parametric excitation term in the equilibrium equation. The reduced-order formulation of the discretized problem is obtained by expanding the mechanical displacement unknown vector onto the short-circuit eigenmode basis. A particular attention is paid to the computation of the unknown nonlinear stiffness coefficients of the reduced-order model. Due to the particular form of the von Kármán nonlinearities, these coefficients are computed exactly, once for a given geometry, by prescribing relevant nodal displacements in nonlinear static solutions settings. Finally, the low-order model is computed with an original purely harmonic-based continuation method. Our numerical tool is then validated by computing the nonlinear vibrations of a mechanically excited homogeneous beam supported at both ends referenced in the literature. The more difficult case of the nonlinear oscillations of a layered nanobridge piezoelectrically actuated is also studied. Interesting vibratory phenomena such as parametric amplification or patch length dependence of the frequency output response are highlighted in order to help in the design of these nanodevices.Nonlinear polarization coupling in freestanding nanowire/nanotube resonators
http://hdl.handle.net/10985/15158
Nonlinear polarization coupling in freestanding nanowire/nanotube resonators
VINCENT, Pascal; DESCOMBIN, Alexis; DAGHER, Samer; SEOUDI, Tarek; LAZARUS, Arnaud; THOMAS, Olivier; AYARI, Anthony; PURCELL, Stephen T.; PERISANU, Sorin
In this work, we study the nonlinear coupling between the transverse modes of nanoresonators such as nanotubes or nanowires in a singly clamped configuration. We previously showed that at high driving, this coupling could result in a transition from independent planar modes to a locked elliptical motion, with important modifications of the resonance curves. Here, we clarify the physical origins, associated with a 1:1 internal resonance, and study in depth this transition as a function of the relevant parameters. We present simple formulae that permit to predict the emergence of this transition as a function of the frequency difference between the polarizations and the nonlinear coefficients and give the “backbone curves” corresponding to the elliptical regime. We also show that the elliptical regime is associated with the emergence of a new set of solutions of which one branch is stable. Finally, we compare single and double clamped configurations and explain why the elliptical transition appears on different polarizations.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/151582019-01-01T00:00:00ZVINCENT, PascalDESCOMBIN, AlexisDAGHER, SamerSEOUDI, TarekLAZARUS, ArnaudTHOMAS, OlivierAYARI, AnthonyPURCELL, Stephen T.PERISANU, SorinIn this work, we study the nonlinear coupling between the transverse modes of nanoresonators such as nanotubes or nanowires in a singly clamped configuration. We previously showed that at high driving, this coupling could result in a transition from independent planar modes to a locked elliptical motion, with important modifications of the resonance curves. Here, we clarify the physical origins, associated with a 1:1 internal resonance, and study in depth this transition as a function of the relevant parameters. We present simple formulae that permit to predict the emergence of this transition as a function of the frequency difference between the polarizations and the nonlinear coefficients and give the “backbone curves” corresponding to the elliptical regime. We also show that the elliptical regime is associated with the emergence of a new set of solutions of which one branch is stable. Finally, we compare single and double clamped configurations and explain why the elliptical transition appears on different polarizations.A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems
http://hdl.handle.net/10985/22641
A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems
GUILLOT, Louis; LAZARUS, Arnaud; THOMAS, Olivier; VERGEZ, Christophe; COCHELIN, Bruno
Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill’s method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.
Tue, 01 Sep 2020 00:00:00 GMThttp://hdl.handle.net/10985/226412020-09-01T00:00:00ZGUILLOT, LouisLAZARUS, ArnaudTHOMAS, OlivierVERGEZ, ChristopheCOCHELIN, BrunoSince the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill’s method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.