A low order, torsion deformable spatial beam element based on the absolute nodal coordinate formulation and Bishop frame
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A low order, torsion deformable spatial beam element based on the absolute nodal coordinate formulation and Bishop frame Mehran Ebrahimi1 Hyunmin Cheong1
· Adrian Butscher1 ·
Received: 30 October 2019 / Accepted: 3 November 2020 © Springer Nature B.V. 2020
Abstract Heretofore, the Serret–Frenet frame has been the ubiquitous choice for analyzing the elastic deformations of beam elements. It is well-known that this frame is undefined at the inflection points and straight segments of the beam where its curvature is zero, leading to singularities and errors in their numerical analysis. On the other hand, there exists a lesserknown frame called Bishop which does not have the caveats of the Serret–Frenet frame and is well-defined everywhere along the beam center-line. Leveraging the Bishop frame, in this paper, we propose a new spatial, singularity-free low order beam element based on the absolute nodal coordinate formulation for both small and large deformation applications. This element, named ANCF14, has a constant mass matrix and can capture longitudinal, transverse (bending) and torsional deformations. It is a two-noded element with 7 degrees of freedom per node, which are global nodal coordinates, nodal slopes and their cross-sectional rotation about the center-line. The newly developed element is tested through four complex benchmarks. Comparing the ANCF14 results with theoretical and numerical results provided in other studies confirms the efficiency and accuracy of the proposed element. Keywords Torsion-deformable beam · Absolute nodal coordinate formulation · Bishop frame · Multibody dynamic systems
1 Introduction Multibody systems (MBSs) are mechanical assemblies consisting of interconnected rigid and flexible components that may undergo large rotations and displacements, as well as large deformations in their flexible parts. Simulation of MBSs has been an attractive subject
B M. Ebrahimi
[email protected] A. Butscher [email protected] H. Cheong [email protected]
1
Autodesk Research, 661 University Avenue, Toronto, ON M5G 1M1, Canada
M. Ebrahimi et al.
of interest in engineering literature and numerous studies have been devoted to improving the accuracy and efficiency of the numerical techniques for simulating the behavior of such systems [17, 31, 34, 36]. This paper presents the development of a new singularity-free three-dimensional beam element capable of handling longitudinal, bending and torsional deformations with a reduced number of degrees of freedom (DOF), compared to the original spatial elements introduced in [37] and [45], based on the absolute nodal coordinate formulation (ANCF) and Bishop frame. Several formulations have been proposed for analyzing the nonlinear deformation of beams in static and dynamic MBSs. The floating frame of reference [9], the incremental finite element [29, 32], the large rotation vector [39], the geometrically exact beam [2, 17, 39–41] and the absolute nodal coordinate formulation [19, 28, 33, 37, 45] are the most widely used. I
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