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Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

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Multibody Dynamics

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

Abstract

For the elastodynamic simulation of a geometrically exact beam [1], an asynchronous variational integrator (AVI) [2] is derived from a PDE viewpoint. Variational integrators are symplectic and conserve discrete momentum maps and since the presented integrator is derived in the Lie group setting (\(SO(3)\) for the representation of rotational degrees of freedom), it intrinsically preserves the group structure without the need for constraints [3]. The discrete Euler-Lagrange equations are derived in a general manner and then applied to the beam. A decrease of computational cost is to be expected in situations, where the time steps have to be very low in certain parts of the beam but not everywhere, e.g. if some regions of the beam are moving faster than others. The implementation allows synchronous as well as asynchronous time stepping and shows very good energy behavior, i.e. there is no drift of the total energy for conservative systems.

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Notes

  1. 1.

    The symbol ‘:’ represents a scalar matrix-matrix product with the summation over two indices, i.e. \(A{:}B=A_{ij}B_{ij}\in \mathbb {R}\) for two matrices of matching dimension.

  2. 2.

    Note that \(\dfrac{\partial \left( \cdot \right) }{\partial q}\) denotes \(\dfrac{\partial \left( \cdot \right) }{\partial x}\) and 2\(\left[ \left( \Lambda ^T\dfrac{\partial \left( \cdot \right) }{\partial \Lambda }\right) ^{(A)}\right] ^{\vee }\), respectively.

  3. 3.

    Of course, other evaluations of the Lagrangian depending on a different number or combinations of nodes are possible, as e.g. an evaluation at midpoints of nodes in the first argument of \(L\) (cf. [18]).

  4. 4.

    The unit of the Lagrangian density is \(\mathrm {\dfrac{J}{m}} = N\), i.e. energy per length. Therefore, the units for the conjugate momenta are

    $$\begin{aligned} \left[ \Pi \right]&= \mathrm {\dfrac{Ns}{m}} = \mathrm {\dfrac{\dfrac{kgm^2}{s}}{m}}&\left[ \Gamma \right]&= \mathrm {\dfrac{Ns}{m}} = \mathrm {\dfrac{\dfrac{kgm}{s}}{m}}&\left[ \Sigma \right]&= \mathrm {Nm}&\left[ \sigma \right]&= \mathrm {N} .\end{aligned}$$

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Authors and Affiliations

  1. Applied Dynamics, University of Erlangen-Nuremberg, Haberstrasse 1, 91058, Erlangen, Germany

    Thomas Leitz & Sigrid Leyendecker

  2. Computational Dynamics and Optimal Control, Department of Mathematics, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Sina Ober-Blöbaum

Authors
  1. Thomas Leitz
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  2. Sina Ober-Blöbaum
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  3. Sigrid Leyendecker
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Correspondence to Sigrid Leyendecker .

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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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Leitz, T., Ober-Blöbaum, S., Leyendecker, S. (2014). Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration. 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_8

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  • DOI: https://doi.org/10.1007/978-3-319-07260-9_8

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