https://sam.ensam.eu:443 The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material. Mon, 16 Mar 2026 02:04:21 GMT 2026-03-16T02:04:21Z 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/10985/25349 Phase resonance testing of highly flexible structures: Measurement of conservative nonlinear modes and nonlinear damping identification DEBEURRE, Marielle; BENACCHIO, Simon; GROLET, Aurélien; GRENAT, Clément; GIRAUD-AUDINE, Christophe; THOMAS, Olivier This article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure’s amplitude. Sat, 01 Jun 2024 00:00:00 GMT http://hdl.handle.net/10985/25349 2024-06-01T00:00:00Z DEBEURRE, Marielle BENACCHIO, Simon GROLET, Aurélien GRENAT, Clément GIRAUD-AUDINE, Christophe THOMAS, Olivier This article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure’s amplitude. http://hdl.handle.net/10985/25350 Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds MARTIN, Adrien; OPRENI, Andrea; VIZZACCARO, Alessandra; DEBEURRE, Marielle; SALLES, Loic; FRANGI, Attilio; THOMAS, Olivier; TOUZÉ, Cyril The direct parametrisation method for invariant manifolds is a nonlinear reduction technique which derives nonlinear mappings and reduced-order dynamics that describe the evolution of dynamical systems along a low-dimensional invariant-based span of the phase space. It can be directly applied to finite element problems. When the development is performed using an arbitrary order asymptotic expansion, it provides an efficient reduced-order modeling strategy for geometrically nonlinear structures. It is here applied to the case of rotating structures featuring centrifugal effect. A rotating cantilever beam with large amplitude vibrations is first selected in order to highlight the main features of the method. Numerical results show that the method provides accurate reduced-order models (ROMs) for any rotation speed and vibration amplitude of interest with a single master mode, thus offering remarkable reduction in the computational burden. The hardening/softening transition of the fundamental flexural mode with increasing rotation speed is then investigated in detail and a ROM parametrised with respect to rotation speed and forcing frequencies is detailed. The method is then applied to a twisted plate model representative of a fan blade, showing how the technique can handle more complex structures. Hardening/softening transition is also investigated as well as interpolation of ROMs, highlighting the efficacy of the method. Thu, 01 Jun 2023 00:00:00 GMT http://hdl.handle.net/10985/25350 2023-06-01T00:00:00Z MARTIN, Adrien OPRENI, Andrea VIZZACCARO, Alessandra DEBEURRE, Marielle SALLES, Loic FRANGI, Attilio THOMAS, Olivier TOUZÉ, Cyril The direct parametrisation method for invariant manifolds is a nonlinear reduction technique which derives nonlinear mappings and reduced-order dynamics that describe the evolution of dynamical systems along a low-dimensional invariant-based span of the phase space. It can be directly applied to finite element problems. When the development is performed using an arbitrary order asymptotic expansion, it provides an efficient reduced-order modeling strategy for geometrically nonlinear structures. It is here applied to the case of rotating structures featuring centrifugal effect. A rotating cantilever beam with large amplitude vibrations is first selected in order to highlight the main features of the method. Numerical results show that the method provides accurate reduced-order models (ROMs) for any rotation speed and vibration amplitude of interest with a single master mode, thus offering remarkable reduction in the computational burden. The hardening/softening transition of the fundamental flexural mode with increasing rotation speed is then investigated in detail and a ROM parametrised with respect to rotation speed and forcing frequencies is detailed. The method is then applied to a twisted plate model representative of a fan blade, showing how the technique can handle more complex structures. Hardening/softening transition is also investigated as well as interpolation of ROMs, highlighting the efficacy of the method.