Path following using velocity-based approach in quasi-static analysis

Chandrashekhara, Sudhanva Kusuma; Zupan, Dejan

doi:10.1016/j.ijsolstr.2023.112292

Cited by 9 publications

(2 citation statements)

References 55 publications

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“…With more rotation degrees of freedom under consideration, up to three for full 3D motion, rotation matrix parametrization has some disadvantages, notably the presence of singularities at certain locations. Many alternative methods have therefore been proposed in the literature for the parametrization of rotations in 3D, some of which include Euler angles and Rodrigues parameters [37], unit quaternions [37,38,39,25,40,41,42,1,43] or a Lie group framework [32,33,44,34,24]. In this paper, we choose to parameterize the rotations in 3D using quaternions, in large part since they naturally lead to polynomial nonlinearities in the model, ideal for applying the ANM.…”

Section: Introductionmentioning

confidence: 99%

Quaternion-based finite element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

Debeurre,

Grolet,

Thomas

2024

Preprint

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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 in-plane (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.

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Section: Introductionmentioning

confidence: 99%

Quaternion-based finite element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

Debeurre,

Grolet,

Thomas

2024

Preprint

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“…Although the parametrisation of rotations therefore is no longer required, the application of boundary conditions as displacement constraints cannot be imposed explicitly which introduces further challenges to the method [18]. Other beam formulations based on the quaternion parametrisation of rotations [19], or based on velocities and angular velocities [20] have also been developed. Although these methods aim to avoid the difficulties of using spatial rotations as unknowns, they miss the opportunity to capitalise on the more appropriate description of the beam kinematics afforded by the Special Euclidean group.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Normal Modes of Highly Flexible Beam Structures Modelled under the SE(3) Lie Group Framework

Bagheri,

Sonneville,

Renson

2023

Preprint

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This work presents a shooting algorithm to compute the periodic responses of geometrically nonlinear structures modelled under the Special Euclidean (SE) Lie group formulation. The formulation is combined with a pseudo-arclength continuation method, while special adaptations are made to ensure compatibility with the $SE$ framework. Nonlinear normal modes (NNM) of various two-dimensional structures including a doubly-clamped beam, a shallow arch, and a cantilever beam are computed. Results are compared with a reference displacement-based FE model with von Kármán strains. Significant difference is observed in the dynamic response of the two models in test cases involving large degrees of beam deformation and rotation. Differences in the contribution of higher order modes substantially affect the frequency-energy dependence and the nonlinear modal interactions observed between the models. It is shown that the SE model, owing to its exact representation of the beam kinematics, is better suited at adequately capturing complex nonlinear dynamics compared to the von Kármán model.

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Quaternion-based finite-element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

Debeurre,

Grolet,

Thomas

2024

Multibody Syst Dyn

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Path following using velocity-based approach in quasi-static analysis

Cited by 9 publications

References 55 publications

Quaternion-based finite element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

Quaternion-based finite element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

Nonlinear Normal Modes of Highly Flexible Beam Structures Modelled under the SE(3) Lie Group Framework

Quaternion-based finite-element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

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