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Modelling of flexible bodies with minimal coordinates by means of the corotational formulation

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Abstract

This paper is concerned with the extension of the minimal coordinates approach to flexible bodies. When using minimal coordinates, the number of configuration parameters corresponds exactly to the number of degrees of freedom and they can be chosen arbitrarily as far as there is a one-to-one relationship between the configuration of the system and the configuration parameters. In the rigid case, the equations of motion are obtained from the description of the translational and rotational motion of a frame attached to each body in terms of the chosen configuration parameters, and from the forces acting on each body. The extension to the simulation of flexible bodies naturally leads to a description of the motion of a flexible body from the one of its nodes. However, the relationship between the latter and the full internal motion of the body is not unique and is the subject of various developments. It was then proposed for the sake of generality to systematically treat flexible bodies as superelements, implemented according to the corotational approach, with a floating corotational frame. This allows to model any flexible body from its mass and stiffness matrices obtained from any available finite element code. Moreover, it doesn’t impose any restriction on the kinematics of the nodes which can then be expressed indifferently from absolute or relative coordinates as usually encountered with minimal coordinates. After a description of the adopted framework, the paper develops the equations of motion. Some test examples are presented, where the proposed approach will be compared to the ones obtained with the classical body reference frame approach and results from the literature. In some cases, the influence of the chosen corotational frame is analysed. The examples confirm that the corotational formulation should be restricted to flexible bodies involving only small deformations and rotational velocity. It is also shown that modelling can be adapted to improve the quality of the results.

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Notes

  1. While \(\boldsymbol{\mathrm{v}}\) denotes a vector specified by a magnitude, an orientation and a sense, \(\{\boldsymbol{\mathrm{v}}\}_{*}\) denotes the \(3\times1\) matrix gathering the coordinates of vector \(\boldsymbol{\mathrm{v}}\) in coordinate system *.

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Acknowledgements

The authors would like to acknowledge the Belgian National Fund for Scientific research (FNRS-FRS) for the FRIA grant allotted to H.N. Huynh.

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Authors and Affiliations

  1. Theoretical Mechanics, Dynamics and Vibrations, University of Mons, Place du Parc 20, 7000, Mons, Belgium

    Olivier Verlinden, Hoai Nam Huynh & Georges Kouroussis

  2. Machine Design and Production Engineering, University of Mons, Place du parc 20, 7000, Mons, Belgium

    Edouard Rivière-Lorphèvre

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  1. Olivier Verlinden
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  2. Hoai Nam Huynh
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  3. Georges Kouroussis
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  4. Edouard Rivière-Lorphèvre
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Correspondence to Olivier Verlinden.

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Verlinden, O., Huynh, H.N., Kouroussis, G. et al. Modelling of flexible bodies with minimal coordinates by means of the corotational formulation. Multibody Syst Dyn 42, 495–514 (2018). https://doi.org/10.1007/s11044-017-9609-0

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