https://sam.ensam.eu:443 The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material. Wed, 11 Mar 2026 23:00:14 GMT 2026-03-11T23:00:14Z http://hdl.handle.net/10985/16775 On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models GIVOIS, Arthur; GROLET, Aurélien; THOMAS, Olivier; DEÜ, Jean-François 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. Tue, 01 Jan 2019 00:00:00 GMT http://hdl.handle.net/10985/16775 2019-01-01T00:00:00Z GIVOIS, Arthur GROLET, Aurélien THOMAS, Olivier DEÜ, Jean-François 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. http://hdl.handle.net/10985/16778 Dissipative solitons in forced cyclic and symmetric structures HOFFMANN, N.; FONTANELA, Francesco; GROLET, Aurélien; SALLES, Loïc; CHABCHOUB, Amin; CHAMPNEYS, Alan; PATSIAS, Sophoclis; HOFFMANN, Norbert The emergence of localised vibrations in cyclic and symmetric rotating struc-tures, such as bladed disks of aircraft engines, has challenged engineers in thepast few decades. In the linear regime, localised states may arise due to alack of symmetry, as for example induced by inhomogeneities. However, whenstructures deviate from the linear behaviour, e.g. due to material nonlinearities,geometric nonlinearities like large deformations, or other nonlinear elements likejoints or friction interfaces, localised states may arise even in perfectly symmet-ric structures. In this paper, a system consisting of coupled Duffing oscillatorswith linear viscous damping is subjected to external travelling wave forcing.The system may be considered a minimal model for bladed disks in turboma-chinery operating in the nonlinear regime, where such excitation may arise dueto imbalance or aerodynamic excitation. We demonstrate that near the reso-nance, in this non-conservative regime, localised vibration states bifurcate fromthe travelling waves. Complex bifurcation diagrams result, comprising stableand unstable dissipative solitons. The localised solutions can also be continuednumerically to a conservative limit, where solitons bifurcate from the backbonecurves of the travelling waves at finite amplitudes. Tue, 01 Jan 2019 00:00:00 GMT http://hdl.handle.net/10985/16778 2019-01-01T00:00:00Z HOFFMANN, N. FONTANELA, Francesco GROLET, Aurélien SALLES, Loïc CHABCHOUB, Amin CHAMPNEYS, Alan PATSIAS, Sophoclis HOFFMANN, Norbert The emergence of localised vibrations in cyclic and symmetric rotating struc-tures, such as bladed disks of aircraft engines, has challenged engineers in thepast few decades. In the linear regime, localised states may arise due to alack of symmetry, as for example induced by inhomogeneities. However, whenstructures deviate from the linear behaviour, e.g. due to material nonlinearities,geometric nonlinearities like large deformations, or other nonlinear elements likejoints or friction interfaces, localised states may arise even in perfectly symmet-ric structures. In this paper, a system consisting of coupled Duffing oscillatorswith linear viscous damping is subjected to external travelling wave forcing.The system may be considered a minimal model for bladed disks in turboma-chinery operating in the nonlinear regime, where such excitation may arise dueto imbalance or aerodynamic excitation. We demonstrate that near the reso-nance, in this non-conservative regime, localised vibration states bifurcate fromthe travelling waves. Complex bifurcation diagrams result, comprising stableand unstable dissipative solitons. The localised solutions can also be continuednumerically to a conservative limit, where solitons bifurcate from the backbonecurves of the travelling waves at finite amplitudes. http://hdl.handle.net/10985/25361 Quaternion-based finite-element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions DEBEURRE, Marielle; GROLET, Aurélien; THOMAS, Olivier In this paper, a novel method for computing the nonlinear dynamics of highly flexible slender structures in three dimensions (3D) is proposed. It is the extension to 3D of a previous work restricted to inplane (2D) deformations. It is based on the geometrically exact beam model, which is discretized with a finite-element method and solved entirely in the frequency domain with a harmonic balance method (HBM) coupled to an asymptotic numerical method (ANM) for continuation of periodic solutions. An important consideration is the parametrization of the rotations of the beam’s cross sections, much more demanding than in the 2D case. Here, the rotations are parametrized with quaternions, with the advantage of leading naturally to polynomial nonlinearities in the model, well suited for applying the ANM. Because of the HBM–ANM framework, this numerical strategy is capable of computing both the frequency response of the structure under periodic oscillations and its nonlinear modes (namely its backbone curves and deformed shapes in free conservative oscillations). To illustrate and validate this strategy, it is used to solve two 3D deformations test cases of the literature: a cantilever beam and a clamped–clamped beam subjected to one-to-one (1:1) internal resonance between two companion bending modes in the case of a nearly square cross section. Sat, 01 Jun 2024 00:00:00 GMT http://hdl.handle.net/10985/25361 2024-06-01T00:00:00Z DEBEURRE, Marielle GROLET, Aurélien THOMAS, Olivier In this paper, a novel method for computing the nonlinear dynamics of highly flexible slender structures in three dimensions (3D) is proposed. It is the extension to 3D of a previous work restricted to inplane (2D) deformations. It is based on the geometrically exact beam model, which is discretized with a finite-element method and solved entirely in the frequency domain with a harmonic balance method (HBM) coupled to an asymptotic numerical method (ANM) for continuation of periodic solutions. An important consideration is the parametrization of the rotations of the beam’s cross sections, much more demanding than in the 2D case. Here, the rotations are parametrized with quaternions, with the advantage of leading naturally to polynomial nonlinearities in the model, well suited for applying the ANM. Because of the HBM–ANM framework, this numerical strategy is capable of computing both the frequency response of the structure under periodic oscillations and its nonlinear modes (namely its backbone curves and deformed shapes in free conservative oscillations). To illustrate and validate this strategy, it is used to solve two 3D deformations test cases of the literature: a cantilever beam and a clamped–clamped beam subjected to one-to-one (1:1) internal resonance between two companion bending modes in the case of a nearly square cross section. http://hdl.handle.net/10985/24164 Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes DEBEURRE, Marielle; GROLET, Aurélien; THOMAS, Olivier In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically-oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross-sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever. Thu, 15 Jun 2023 00:00:00 GMT http://hdl.handle.net/10985/24164 2023-06-15T00:00:00Z DEBEURRE, Marielle GROLET, Aurélien THOMAS, Olivier In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically-oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross-sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever. http://hdl.handle.net/10985/24783 Finite element computation of nonlinear modes and frequency response of geometrically exact beam structures DEBEURRE, Marielle; GROLET, Aurélien; COCHELIN, Bruno; THOMAS, Olivier An original method for the simulation of the dynamics of highly flexible slender structures is presented. The flexible structures are modeled via a finite element (FE) discretization of a geometrically exact two-dimensional beam model, which entirely preserves the geometrical nonlinearities inherent in such systems where the rotation of the cross-section can be extreme. The FE equation is solved by a combination of harmonic balance (HBM) and asymptotic numerical (ANM) methods. The novel solving scheme is rooted entirely in the frequency domain and is capable of computing both the structure’s frequency response under periodic external forces as well as its nonlinear modes. An overview of the proposed numerical strategy is outlined and simulations are shown and discussed in detail for several test cases. Wed, 01 Mar 2023 00:00:00 GMT http://hdl.handle.net/10985/24783 2023-03-01T00:00:00Z DEBEURRE, Marielle GROLET, Aurélien COCHELIN, Bruno THOMAS, Olivier An original method for the simulation of the dynamics of highly flexible slender structures is presented. The flexible structures are modeled via a finite element (FE) discretization of a geometrically exact two-dimensional beam model, which entirely preserves the geometrical nonlinearities inherent in such systems where the rotation of the cross-section can be extreme. The FE equation is solved by a combination of harmonic balance (HBM) and asymptotic numerical (ANM) methods. The novel solving scheme is rooted entirely in the frequency domain and is capable of computing both the structure’s frequency response under periodic external forces as well as its nonlinear modes. An overview of the proposed numerical strategy is outlined and simulations are shown and discussed in detail for several test cases. http://hdl.handle.net/10985/25624 Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation BAYER, Fabia; LEINE, Remco I.; THOMAS, Olivier; GROLET, Aurélien In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves. Sun, 01 Dec 2024 00:00:00 GMT http://hdl.handle.net/10985/25624 2024-12-01T00:00:00Z BAYER, Fabia LEINE, Remco I. THOMAS, Olivier GROLET, Aurélien In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves. http://hdl.handle.net/10985/22535 Subharmonic centrifugal pendulum vibration absorbers allowing a rotational mobility MAHE, V.; RENAULT, Alexandre; GROLET, Aurélien; MAHE, Hervé; THOMAS, Olivier Rotating machines are often subjected to fluctuating torques, leading to vibrations of the rotor and finally to premature fatigue and noise pollution. This work addresses a new design of centrifugal pendulum vibration absorbers (CPVAs), used to reduce the vibrations in an automotive transmission line. These passive devices, composed of several masses oscillating along a trajectory relative to the rotor, are here tuned at a subharmonic of the targeted harmonic torque frequency. Thanks to the inherent non-linearities, a CPVA with two masses oscillating in phase opposition is able to efficiently counteract the input torque, with particular features such as saturation phenomena. This work particularly extends previous works to a new class of CPVA, whose peculiarity is that masses admit a significant rotation motion relative to the rotor, thus adding the benefit of their rotatory inertia. Results on the system’s subharmonic response and its stability are obtained thanks to an analytical perturbation method, and design guidelines are proposed. The validity of those results is also confirmed through comparisons with numerical solutions and the performance of this subharmonic system is compared to that of a classical CPVA tuned at the torque frequency. Thu, 01 Sep 2022 00:00:00 GMT http://hdl.handle.net/10985/22535 2022-09-01T00:00:00Z MAHE, V. RENAULT, Alexandre GROLET, Aurélien MAHE, Hervé THOMAS, Olivier Rotating machines are often subjected to fluctuating torques, leading to vibrations of the rotor and finally to premature fatigue and noise pollution. This work addresses a new design of centrifugal pendulum vibration absorbers (CPVAs), used to reduce the vibrations in an automotive transmission line. These passive devices, composed of several masses oscillating along a trajectory relative to the rotor, are here tuned at a subharmonic of the targeted harmonic torque frequency. Thanks to the inherent non-linearities, a CPVA with two masses oscillating in phase opposition is able to efficiently counteract the input torque, with particular features such as saturation phenomena. This work particularly extends previous works to a new class of CPVA, whose peculiarity is that masses admit a significant rotation motion relative to the rotor, thus adding the benefit of their rotatory inertia. Results on the system’s subharmonic response and its stability are obtained thanks to an analytical perturbation method, and design guidelines are proposed. The validity of those results is also confirmed through comparisons with numerical solutions and the performance of this subharmonic system is compared to that of a classical CPVA tuned at the torque frequency. http://hdl.handle.net/10985/22696 A comparison of robustness and performance of linear and nonlinear Lanchester dampers VAKILINEJAD, Mohammad; GROLET, Aurélien; THOMAS, Olivier In this paper, we study and compare performance and robustness of linear and nonlinear Lanchester dampers. The linear Lanchester damper consists of a small mass attached to a primary system through a linear dashpot, whereas the nonlinear Lanchester damper is linked to the primary mass through dry friction forces. In each case, we propose a semi-analytical method for computing the frequency response, for different values of the design parameters, in order to evaluate the performance and robustness of the two kinds of damper. Overall, it is shown that linear Lanchester dampers perform better than nonlinear damper both in terms of attenuation and robustness. Moreover, the nonlinear frequency response curves, that include the intrinsic non-smooth nature of the friction force, may serve as reference curve for further numerical studies. Sat, 01 Feb 2020 00:00:00 GMT http://hdl.handle.net/10985/22696 2020-02-01T00:00:00Z VAKILINEJAD, Mohammad GROLET, Aurélien THOMAS, Olivier In this paper, we study and compare performance and robustness of linear and nonlinear Lanchester dampers. The linear Lanchester damper consists of a small mass attached to a primary system through a linear dashpot, whereas the nonlinear Lanchester damper is linked to the primary mass through dry friction forces. In each case, we propose a semi-analytical method for computing the frequency response, for different values of the design parameters, in order to evaluate the performance and robustness of the two kinds of damper. Overall, it is shown that linear Lanchester dampers perform better than nonlinear damper both in terms of attenuation and robustness. Moreover, the nonlinear frequency response curves, that include the intrinsic non-smooth nature of the friction force, may serve as reference curve for further numerical studies. http://hdl.handle.net/10985/22661 Computation of dynamic transmission error for gear transmission systems using modal decomposition and Fourier series ABBOUD, Eddy; GROLET, Aurélien; MAHÉ, Hervé; THOMAS, Olivier In this paper, a method for computing the dynamics of a geared system excited by its static transmission error is proposed. The method is based on the iterative spectral method (ISM) and on the harmonic balance method (HBM). It is shown that the dynamic transmission error (DTE) can be obtained in the frequency domain by solving a linear system of equations, which in turn allows the computation of the modal and physical coordinates of the system. Mon, 01 Nov 2021 00:00:00 GMT http://hdl.handle.net/10985/22661 2021-11-01T00:00:00Z ABBOUD, Eddy GROLET, Aurélien MAHÉ, Hervé THOMAS, Olivier In this paper, a method for computing the dynamics of a geared system excited by its static transmission error is proposed. The method is based on the iterative spectral method (ISM) and on the harmonic balance method (HBM). It is shown that the dynamic transmission error (DTE) can be obtained in the frequency domain by solving a linear system of equations, which in turn allows the computation of the modal and physical coordinates of the system. http://hdl.handle.net/10985/22536 On the dynamic stability and efficiency of centrifugal pendulum vibration absorbers with rotating pendulums MAHÉ, V.; RENAULT, Alexandre; GROLET, Aurélien; MAHÉ, Hervé; THOMAS, Olivier The automotive industry uses centrifugal pendulum vibration absorbers (CPVAs) to reduce vibrations of the transmission system. These passive devices are made of several masses oscillating along a given path relative to a rotor. This work addresses a recent design of CPVA, in which the pendulums are allowed to rotate relatively to the rotor. The dynamic stability of this CPVA and the shifting of its operating point are investigated in this paper. These two aspects, crucial for an optimal vibration reduction, are assessed using an analytic dynamical model based on a perturbation method. The results obtained allow to propose new design guidelines. The validity of the model is confirmed through a comparison with a numerical resolution of the system’s dynamics. Sat, 01 Oct 2022 00:00:00 GMT http://hdl.handle.net/10985/22536 2022-10-01T00:00:00Z MAHÉ, V. RENAULT, Alexandre GROLET, Aurélien MAHÉ, Hervé THOMAS, Olivier The automotive industry uses centrifugal pendulum vibration absorbers (CPVAs) to reduce vibrations of the transmission system. These passive devices are made of several masses oscillating along a given path relative to a rotor. This work addresses a recent design of CPVA, in which the pendulums are allowed to rotate relatively to the rotor. The dynamic stability of this CPVA and the shifting of its operating point are investigated in this paper. These two aspects, crucial for an optimal vibration reduction, are assessed using an analytic dynamical model based on a perturbation method. The results obtained allow to propose new design guidelines. The validity of the model is confirmed through a comparison with a numerical resolution of the system’s dynamics.