Multibody System Dynamics

, 20:359 | Cite as

A geometrically exact beam element based on the absolute nodal coordinate formulation

  • Johannes Gerstmayr
  • Marko K. Matikainen
  • Aki M. Mikkola


In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.


Absolute nodal coordinate formulation Geometrically exact beam Large rotation vector formulation Finite elements Flexible multibody system 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Johannes Gerstmayr
    • 1
  • Marko K. Matikainen
    • 2
  • Aki M. Mikkola
    • 2
  1. 1.Linz Center of Mechatronics GmbHLinzAustria
  2. 2.Department of Mechanical Engineering, Institute of Mechatronics and Virtual EngineeringLappeenranta University of TechnologyLappeenrantaFinland

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