Multibody System Dynamics

, Volume 37, Issue 1, pp 29–48 | Cite as

Validation of flexible multibody dynamics beam formulations using benchmark problems

  • Olivier A. Bauchau
  • Peter Betsch
  • Alberto Cardona
  • Johannes Gerstmayr
  • Ben Jonker
  • Pierangelo Masarati
  • Valentin Sonneville
Article
First Online:
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Accepted:

Abstract

As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed, but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism, focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity, resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available.

Keywords

Multibody dynamics Beam models Benchmark problems 
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Notes

Conflict of interest

The authors declare that they have no conflict of interest. This article does not contain any studies with human participants or animals performed by any of the authors. Informed consent was obtained from all individual participants included in the study.

References

  1. 1.
    Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49(1), 55–70 (1985) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Belytschko, T., Hsieh, B.J.: Nonlinear transient finite element analysis with convected coordinates. Int. J. Numer. Methods Eng. 7, 255–271 (1973) CrossRefMATHGoogle Scholar
  4. 4.
    Shabana, A.A., Hussien, H.A., Escalona, J.L.: Application of the absolute nodal coordinate formulation to large rotation and large deformation problems. J. Mech. Des. 120, 188–195 (1998) CrossRefGoogle Scholar
  5. 5.
    Escalona, J.L., Hussien, H.A., Shabana, A.A.: Application of the absolute nodal co-ordinate formulation to multibody system dynamics. J. Sound Vib. 214(5), 833–851 (1998) CrossRefGoogle Scholar
  6. 6.
    Romero, I.: A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations. Multibody Syst. Dyn. 20, 51–68 (2008) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gerstmayr, J., Sugiyama, H., Mikkola, A.: An overview on the developments of the absolute nodal coordinate formulation. In: Proceedings of the Second Joint International Conference on Multibody System Dynamics, Stuttgart, Germany (2012) Google Scholar
  8. 8.
    Bauchau, O.A., Han, S.L., Mikkola, A., Matikainen, M.K.: Comparison of the absolute nodal coordinate and geometrically exact formulations for beams. Multibody Syst. Dyn. 32(1), 67–85 (2014) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bauchau, O.A., Han, S.L., Mikkola, A., Matikainen, M.K., Gruber, P.: Experimental validation of flexible multibody dynamics beam formulations. Multibody Syst. Dyn. 34(4), 373–389 (2015) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hegemier, G.A., Nair, S.: A nonlinear dynamical theory for heterogeneous, anisotropic, elastic rods. AIAA J. 15(1), 8–15 (1977) CrossRefMATHGoogle Scholar
  11. 11.
    Hodges, D.H.: A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int. J. Solids Struct. 26(11), 1253–1273 (1990) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cardona, A., Géradin, M.: A beam finite element non-linear theory with finite rotation. Int. J. Numer. Methods Eng. 26, 2403–2438 (1988) CrossRefMATHGoogle Scholar
  13. 13.
    Hilber, H.M., Hughes, T.J.R., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5, 283–292 (1977) CrossRefGoogle Scholar
  14. 14.
    Betsch, P., Steinmann, P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191, 467–488 (2001) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Celledoni, E., Owren, B.: Lie group methods for rigid body dynamics and time integration on manifolds. Comput. Methods Appl. Mech. Eng. 192(3–4), 421–438 (2003) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bauchau, O.A., Han, S.L.: Three-dimensional beam theory for flexible multibody dynamics. J. Comput. Nonlinear Dyn. 9(4), 041011 (2014), 12 pp. CrossRefGoogle Scholar
  17. 17.
    Han, S.L., Bauchau, O.A.: Nonlinear three-dimensional beam theory for flexible multibody dynamics. Multibody Syst. Dyn. 34(3), 211–242 (2015) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dowell, E.H., Traybar, J.J.: An experimental study of the nonlinear stiffness of a rotor blade undergoing flap, lag, and twist deformations. Aerospace and Mechanical Science Report 1257, Princeton University (1975) Google Scholar
  19. 19.
    Dowell, E.H., Traybar, J.J., Hodges, D.H.: An experimental-theoretical correlation study of non-linear bending and torsion deformations of a cantilever beam. J. Sound Vib. 50(4), 533–544 (1977) CrossRefGoogle Scholar
  20. 20.
    Bauchau, O.A., Craig, J.I.: Structural Analysis with Application to Aerospace Structures. Springer, Dordrecht/Heidelberg/London/New-York (2009) Google Scholar
  21. 21.
    Bauchau, O.A., Bottasso, C.L., Nikishkov, Y.G.: Modeling rotorcraft dynamics with finite element multibody procedures. Math. Comput. Model. 33(10–11), 1113–1137 (2001) CrossRefMATHGoogle Scholar
  22. 22.
    Bauchau, O.A.: Flexible Multibody Dynamics. Springer, Dordrecht/Heidelberg/London/New-York (2011) CrossRefMATHGoogle Scholar
  23. 23.
    Nachbagauer, K., Gruber, P., Gerstmayr, J.: Structural and continuum mechanics approaches for a 3D shear deformable ANCF beam finite element: application to static and linearized dynamic examples. J. Comput. Nonlinear Dyn. 8(2), 021004 (2013) CrossRefGoogle Scholar
  24. 24.
    Masarati, P., Morandini, M., Mantegazza, P.: An efficient formulation for general-purpose multibody/multiphysics analysis. J. Comput. Nonlinear Dyn. 9(4), 041001 (2014) CrossRefGoogle Scholar
  25. 25.
    Ghiringhelli, G.L., Masarati, P., Mantegazza, P.: A multi-body implementation of finite volume beams. AIAA J. 38(1), 131–138 (2000) CrossRefGoogle Scholar
  26. 26.
    Cardona, A.: An integrated approach to mechanism analysis. PhD thesis, Université de Liège, Belgium (1989) Google Scholar
  27. 27.
    Betsch, P., Steinmann, P.: Frame-indifferent beam element based upon the geometrically exact beam theory. Int. J. Numer. Methods Eng. 54, 1775–1788 (2002) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Leyendecker, S., Betsch, P., Steinmann, P.: The discrete null space method for the energy consistent integration of constrained mechanical systems. Part III: flexible multibody dynamics. Multibody Syst. Dyn. 19(1–2), 45–72 (2008) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lens, E.V., Cardona, A.: A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics. Comput. Struct. 86(1–2), 47–63 (2008) CrossRefGoogle Scholar
  30. 30.
    Cardona, A., Klapka, I., Géradin, M.: Design of a new finite element programming environment. Eng. Comput. 11, 365–381 (1994) CrossRefMATHGoogle Scholar
  31. 31.
    Sonneville, V., Brüls, O.: A formulation on the special Euclidean group for dynamic analysis of multibody systems. J. Comput. Nonlinear Dyn. 9(4) (2014) Google Scholar
  32. 32.
    Sonneville, V., Cardona, A., Brüls, O.: Geometrically exact beam finite element formulated on the special Euclidean group \(\mathit{SE}(3)\). Comput. Methods Appl. Mech. Eng. 268(1), 451–474 (2014) MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Jonker, J.B.: A finite element dynamic analysis of spatial mechanisms with flexible links. Comput. Methods Appl. Mech. Eng. 76, 17–40 (1989) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Jonker, J.B., Meijaard, J.P.: A geometrically non-linear formulation of a three-dimensional beam element for solving large deflection multibody system problems. Int. J. Non-Linear Mech. 53, 63–74 (2013) CrossRefGoogle Scholar
  35. 35.
    Eugster, S.R., Hesch, C., Betsch, P., Glocker, Ch.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int. J. Numer. Methods Eng. 97(2), 111–129 (2014) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Brüls, O., Cardona, A., Arnold, M.: Lie group generalized-\(\alpha\) time integration of constrained flexible multibody systems. Mech. Mach. Theory 48, 121–137 (2012) CrossRefGoogle Scholar
  37. 37.
    Besseling, J.F.: Non-linear theory for elastic beams and rods and its finite element representation. Comput. Methods Appl. Mech. Eng. 12, 205–220 (1982) CrossRefMATHGoogle Scholar
  38. 38.
    Reissner, E.: On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math. 52, 87–95 (1973) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Olivier A. Bauchau
    • 1
  • Peter Betsch
    • 2
  • Alberto Cardona
    • 3
  • Johannes Gerstmayr
    • 4
  • Ben Jonker
    • 5
  • Pierangelo Masarati
    • 6
  • Valentin Sonneville
    • 7
  1. 1.University of MarylandCollege ParkUSA
  2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.CIMEC (UNL/Conicet)Santa FeArgentina
  4. 4.Leopold-Franzens Universität InnsbruckInnsbruckAustria
  5. 5.University of TwenteEnschedeThe Netherlands
  6. 6.Politecnico di MilanoMilanoItaly
  7. 7.Université de LiègeLiègeBelgium

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