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Development of an efficient calculation procedure for elastic forces in the ANCF beam element by using a constrained formulation

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Abstract

The absolute nodal coordinate formulation (ANCF) is often used for the large rotation and deformation analysis. In the ANCF, absolute nodal coordinates and global slopes are utilized as the element nodal coordinates. Accordingly, it gives a constant and symmetric mass matrix. On the other hand, elastic forces are generally expressed by highly nonlinear terms. Therefore, one of the important topics in the implementation of the ANCF is to derive mathematical descriptions of the elastic forces which can be calculated efficiently. This study focuses on development of efficient calculation procedures for the ANCF beam element under the assumption of the Euler–Bernoulli beam theory. In particular, we focus on the fact that the elastic forces can be described in simpler forms with the element coordinates. It implies that appropriate choices of the coordinate systems for the inertia and the elastic forces could contribute to the development of efficient calculation procedures. In the present formulation, the strain and the kinetic energies are expressed as functions of the element and the global coordinates, respectively. Then algebraic constraints regarding the relations between the global and the element coordinate systems are introduced by means of the Lagrange method of undetermined multipliers. These constraints are used for eliminating the redundant degrees of freedom from a standpoint of consistency in the formulation. Therefore, this method can be categorized as an augmented formulation technique. The equations of motions of this constrained system are derived by the method for constrained Hamiltonian system, namely Dirac’s method with the Poisson bracket formalism. In order to evaluate the proposed method, it is applied to the large deformation problem in the plane case. The equivalence of the previous approach and the proposed method is discussed by a comparison of the two. In addition, its computational performance is investigated.

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Acknowledgements

This study was partially supported by JSPS KAKENHI Grant Number 24760190.

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Authors and Affiliations

  1. Department of Mechanical Engineering, School of Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan

    Kensuke Hara

  2. Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa, Japan

    Masahiro Watanabe

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  1. Kensuke Hara
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  2. Masahiro Watanabe
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Correspondence to Kensuke Hara.

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Hara, K., Watanabe, M. Development of an efficient calculation procedure for elastic forces in the ANCF beam element by using a constrained formulation. Multibody Syst Dyn 43, 369–386 (2018). https://doi.org/10.1007/s11044-017-9594-3

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  • DOI: https://doi.org/10.1007/s11044-017-9594-3

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