Nonlinear Dynamics

, Volume 34, Issue 1–2, pp 133–145 | Cite as

Strain Tensors in the Absolute Nodal Coordinate and the Floating Frame of Reference Formulation

  • J. Gerstmayr


The floating frame of reference (FFR) formulation and the absolute nodal coordinate(ANC) formulation are often used for the modeling of multibody systems. In the presentwork, a reduced strain model is derived for the ANC formulation which is equivalent tothe (small strain) FFR formulation. The reduced strain model is based on a co-rotatedreference configuration and the deformation is assumed to be small with respect to thisconfiguration. This configuration is described by a translation vector and a rotation matrix which are both determined from the motion of the body with respect to its fixedreference. The ANC formulation with reduced strain leads to a constant mass matrix. The stiffness matrix consists of two parts: The most significant part depends on the small-strain stiffness matrix of the body in the fixed reference configuration which is rotated by the rotation matrix and the second part is small and nonlinearly depending on the strain tensor. Both formulations represent displacements and deformations differently but lead to exactly thesame results in the case of equivalent floating reference configurations. Different aspects of both formulations are shown in a 2D example problem of a rotating hinged plate. A detailed description of the modeling in both cases as well as numerical results are presented.

floating frame of reference absolute nodal coordinates finite elements flexible multibody systems 


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  1. 1.
    Bonet, J. and Wood, R. D., Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge, 1997.Google Scholar
  2. 2.
    Crane III, C. D. and Duffy, J., Kinematic Analysis of Robot Manipulators, Cambridge University Press, Cambridge, 1998.Google Scholar
  3. 3.
    Gerstmayr, J., ‘Comparison of the absolute nodal coordinate and the floating frame of reference formulation by means of a simplified strain formulation’, in Proceedings of DETC'03 ASME Design Engineering Technical Conferences, September 3–9, Paper VIB-48306, Chicago, IL, 2003.Google Scholar
  4. 4.
    Gerstmayr, J. and Schöberl, J., ‘A 3D finite element approach to flexible multibody systems’, in Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), J. Eberhardsteiner, H. A. Mang, and F. G. Rammerstorfer (eds.), Vienna University of Technology, Austria, 2002.Google Scholar
  5. 5.
    Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer, Berlin, 1991.Google Scholar
  6. 6.
    Orden, J. C. G. and Goicolea, J. M., ‘Conserving properties in constrained dynamics of flexible multibody systems’, Multibody System Dynamics 4, 2000, 225–244.Google Scholar
  7. 7.
    Schwertassek, R., Wallrapp, O., and Shabana, A. A., ‘Flexible multibody simulation and choice of shape functions’, Nonlinear Dynamics 20, 1999, 361–380.Google Scholar
  8. 8.
    Shabana, A. A., Dynamics of Multibody Systems, 2nd edition, Cambridge University Press, Cambridge, 1998.Google Scholar
  9. 9.
    Shabana, A. A., ‘Flexible multibody dynamics: Review of past and recent developments’, Multibody System Dynamics 1, 1997, 189–222.Google Scholar
  10. 10.
    Shabana, A. A. and Schwertassek, R., ‘Equivalence of the floating frame of reference approach and finite elemnt formulations’, International Journal of Non-Linear Mechanics 33, 1998, 417–432.Google Scholar
  11. 11.
    Ziegler, F., Mechanics of Solids and Fluids, Springer, New York, 1991.Google Scholar
  12. 12.
    Zienkiewicz, O. C. and Taylor, R. L., Volume 2 – Solid Mechanics, Butterworth Heinemann, London, 2000.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • J. Gerstmayr
    • 1
  1. 1.Institute of Mechanics and Machine Design, Division of Technical MechanicsUniversity of LinzLinzAustria

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