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Modelling and Integration Concepts of Multibody Systems on Lie Groups

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Multibody Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 35))

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Abstract

Lie group integration schemes for multibody systems (MBS) are attractive as they provide a coordinate-free and thus singularity-free approach to MBS modeling. The Lie group setting also allows for developing integration schemes that preserve motion integrals and coadjoint orbits. Most of the recently proposed Lie group integration schemes are based on variants of generalized alpha Newmark schemes. In this chapter constrained MBS are modeled by a system of differential-algebraic equations (DAE) on a configuration space being a subvariety of the Lie group \(SE(3)^{n}\). This is transformed to an index 1 DAE system that is integrated with Munthe-Kaas (MK) integration scheme. The chapter further addresses geometric integration schemes that preserve integrals of motion. In this context, a non-canonical Lie-group Störmer-Verlet integration scheme with direct \(SO(3)\) rotational update is presented. The method is 2nd order accurate and it is angular momentum preserving for a free-spinning body. Moreover, although being fully explicit, the method achieves excellent conservation of the angular momentum of a free rotational body and the motion integrals of the Lagrangian top. A higher-order coadjoint-preserving integration scheme on \(SO(3)\) is also presented. This method exactly preserves spatial angular momentum of a free body and it is particularly numerically efficient.

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Acknowledgments

The authors thank Dario Zlatar, Ph.D. student at University of Zagreb, for programming part of the numerical experiments presented in the chapter.

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Authors and Affiliations

  1. University of Michigan–Shanghai Jiao Tong University Joint Institute, Shanghai, China

    Andreas Müller

  2. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

    Zdravko Terze

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  1. Andreas Müller
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  2. Zdravko Terze
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Correspondence to Andreas Müller .

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Editors and Affiliations

  1. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

    Zdravko Terze

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Müller, A., Terze, Z. (2014). Modelling and Integration Concepts of Multibody Systems on Lie Groups. In: Terze, Z. (eds) Multibody Dynamics. Computational Methods in Applied Sciences, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-07260-9_6

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  • DOI: https://doi.org/10.1007/978-3-319-07260-9_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07259-3

  • Online ISBN: 978-3-319-07260-9

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