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Multibody System Dynamics

, Volume 32, Issue 1, pp 67–85 | Cite as

Comparison of the absolute nodal coordinate and geometrically exact formulations for beams

  • Olivier A. Bauchau
  • Shilei Han
  • Aki Mikkola
  • Marko K. Matikainen
Article

Abstract

The modeling of flexibility in multibody systems has received increase scrutiny in recent years. The use of finite element techniques is becoming more prevalent, although the formulation of structural elements must be modified to accommodate the large displacements and rotations that characterize multibody systems. Two formulations have emerged that have the potential of handling all the complexities found in these systems: the absolute nodal coordinate formulation and the geometrically exact formulation. Both approaches have been used to formulate naturally curved and twisted beams, plate, and shells. After a brief review of the two formulations, this paper presents a detailed comparison between these two approaches; a simple planar beam problem is examined using both kinematic and static solution procedures. In the kinematic solution, the exact nodal displacements are prescribed and the predicted displacement and strain fields inside the element are compared for the two methods. The accuracies of the predicted strain fields are found to differ: The predictions of the geometrically exact formulation are more accurate than those of the absolute nodal coordinate formulation. For the static solution, the principle of virtual work is used to determine the solution of the problem. For the geometrically exact formulation, the predictions of the static solution are more accurate than those obtained from the kinematic solution; in contrast, the same order of accuracy is obtained for the two solution procedures when using the absolute nodal coordinate formulation. It appears that the kinematic description of structural problems offered by the absolute nodal coordinate formulation leads to inherently lower accuracy predictions than those provided by the geometrically exact formulation. These observations provide a rational for explaining why the absolute nodal coordinate formulation computationally intensive.

Keywords

Flexible multibody systems Finite element procedures 

References

  1. 1.
    Eberhard, P., Schiehlen, W.: Computational dynamics of multibody systems: history, formalisms, and applications. J. Comput. Nonlinear Dyn. 1(1), 3–12 (2006) CrossRefGoogle Scholar
  2. 2.
    Wasfy, T.M., Noor, A.K.: Computational strategies for flexible multibody systems. Appl. Mech. Rev. 56(2), 553–613 (2003) CrossRefGoogle Scholar
  3. 3.
    Shabana, A.A., Wehage, R.A.: A coordinate reduction technique for dynamic analysis of spatial substructures with large angular rotations. J. Struct. Mech. 11(3), 401–431 (1983) CrossRefGoogle Scholar
  4. 4.
    Agrawal, O.P., Shabana, A.A.: Application of deformable-body mean axis to flexible multibody system dynamics. Comput. Methods Appl. Mech. Eng. 56(2), 217–245 (1986) CrossRefMATHGoogle Scholar
  5. 5.
    Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    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) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    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, May 2012 Google Scholar
  9. 9.
    Géradin, M., Cardona, A.: Flexible Multibody System: A Finite Element Approach. Wiley, New York (2001) Google Scholar
  10. 10.
    Hughes, T.J.R.: The Finite Element Method. Prentice Hall, Englewood Cliffs (1987) MATHGoogle Scholar
  11. 11.
    Bathe, K.J.: Finite Element Procedures. Prentice Hall, Englewood Cliffs (1996) Google Scholar
  12. 12.
    Simo, J.C., Vu-Quoc, L.: A three dimensional finite strain rod model. Part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58(1), 79–116 (1986) CrossRefMATHGoogle Scholar
  13. 13.
    Borri, M., Merlini, T.: A large displacement formulation for anisotropic beam analysis. Meccanica 21, 30–37 (1986) CrossRefMATHGoogle Scholar
  14. 14.
    Danielson, D.A., Hodges, D.H.: Nonlinear beam kinematics by decomposition of the rotation tensor. J. Appl. Mech. 54(2), 258–262 (1987) CrossRefMATHGoogle Scholar
  15. 15.
    Danielson, D.A., Hodges, D.H.: A beam theory for large global rotation, moderate local rotation, and small strain. J. Appl. Mech. 55(1), 179–184 (1988) CrossRefGoogle Scholar
  16. 16.
    Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1, 339–348 (1997) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    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) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Bauchau, O.A., Craig, J.I.: Structural Analysis with Application to Aerospace Structures. Springer, Dordrecht (2009) Google Scholar
  19. 19.
    Timoshenko, S.P.: On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section. Philos. Mag. 41, 744–746 (1921) CrossRefGoogle Scholar
  20. 20.
    Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. 43, 125–131 (1921) CrossRefGoogle Scholar
  21. 21.
    Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. Z. Angew. Math. Phys. 12, A-69–A-77 (1945) MathSciNetGoogle Scholar
  22. 22.
    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18, 31–38 (1951) MATHGoogle Scholar
  23. 23.
    Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. Z. Angew. Math. Phys. 23, 795–804 (1972) CrossRefMATHGoogle Scholar
  24. 24.
    Reissner, E.: On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math. 52, 87–95 (1973) MATHGoogle Scholar
  25. 25.
    Reissner, E.: On finite deformations of space-curved beams. Z. Angew. Math. Phys. 32, 734–744 (1981) CrossRefMATHGoogle Scholar
  26. 26.
    Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs (1969) Google Scholar
  27. 27.
    Gerstmayr, J., Irschik, H.: On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach. J. Sound Vib. 318(3), 461–487 (2008) CrossRefGoogle Scholar
  28. 28.
    Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123, 606–613 (2001) CrossRefGoogle Scholar
  29. 29.
    Yakoub, R.Y., Shabana, A.A.: Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. J. Mech. Des. 123, 614–621 (2001) CrossRefGoogle Scholar
  30. 30.
    Bauchau, O.A.: Flexible Multibody Dynamics. Springer, Dordrecht (2011) CrossRefMATHGoogle Scholar
  31. 31.
    Crisfield, M.A., Jelenić, G.: Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc. R. Soc., Math. Phys. Eng. Sci. 455(1983), 1125–1147 (1999) CrossRefMATHGoogle Scholar
  32. 32.
    Bauchau, O.A., Han, S.L.: Interpolation of rotation and motion. Multibody Syst. Dyn. (2013). doi: 10.1007/s11044-013-9365-8 Google Scholar
  33. 33.
    Shabana, A.A., Mikkola, A.M.: Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity. J. Mech. Des. 125(2), 342–350 (2003) CrossRefGoogle Scholar
  34. 34.
    Shabana, A.A., Maqueda, L.G.: Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements. Multibody Syst. Dyn. 20, 239–249 (2008) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Maqueda, L.G., Shabana, A.A.: Numerical investigation of the slope discontinuities in large deformation finite element formulations. Nonlinear Dyn. 58, 23–37 (2009) CrossRefMATHGoogle Scholar
  36. 36.
    Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 6th edn. Elsevier, Butterworth-Heinemann, Amsterdam (2005) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Olivier A. Bauchau
    • 1
  • Shilei Han
    • 1
  • Aki Mikkola
    • 2
  • Marko K. Matikainen
    • 2
  1. 1.University of Michigan-Shanghai Jiao Tong University Joint InstituteShanghaiChina
  2. 2.Department of Mechanical EngineeringLappeenranta University of TechnologyLappeenrantaFinland

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