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

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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.

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References

  1. Eberhard, P., Schiehlen, W.: Computational dynamics of multibody systems: history, formalisms, and applications. J. Comput. Nonlinear Dyn. 1(1), 3–12 (2006)

    Article  Google Scholar 

  2. Wasfy, T.M., Noor, A.K.: Computational strategies for flexible multibody systems. Appl. Mech. Rev. 56(2), 553–613 (2003)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  5. Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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. Géradin, M., Cardona, A.: Flexible Multibody System: A Finite Element Approach. Wiley, New York (2001)

    Google Scholar 

  10. Hughes, T.J.R.: The Finite Element Method. Prentice Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  11. Bathe, K.J.: Finite Element Procedures. Prentice Hall, Englewood Cliffs (1996)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  13. Borri, M., Merlini, T.: A large displacement formulation for anisotropic beam analysis. Meccanica 21, 30–37 (1986)

    Article  MATH  Google Scholar 

  14. Danielson, D.A., Hodges, D.H.: Nonlinear beam kinematics by decomposition of the rotation tensor. J. Appl. Mech. 54(2), 258–262 (1987)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  16. Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1, 339–348 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Bauchau, O.A., Craig, J.I.: Structural Analysis with Application to Aerospace Structures. Springer, Dordrecht (2009)

    Google Scholar 

  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)

    Article  Google Scholar 

  20. Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. 43, 125–131 (1921)

    Article  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  22. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

  23. Reissner, E.: On one-dimensional finite-strain beam theory: the plane problem. Z. Angew. Math. Phys. 23, 795–804 (1972)

    Article  MATH  Google Scholar 

  24. Reissner, E.: On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math. 52, 87–95 (1973)

    MATH  Google Scholar 

  25. Reissner, E.: On finite deformations of space-curved beams. Z. Angew. Math. Phys. 32, 734–744 (1981)

    Article  MATH  Google Scholar 

  26. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs (1969)

    Google Scholar 

  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)

    Article  Google Scholar 

  28. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123, 606–613 (2001)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  30. Bauchau, O.A.: Flexible Multibody Dynamics. Springer, Dordrecht (2011)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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 

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Authors and Affiliations

  1. University of Michigan-Shanghai Jiao Tong University Joint Institute, 800 Dong Chuan Road, Shanghai, 200240, China

    Olivier A. Bauchau & Shilei Han

  2. Department of Mechanical Engineering, Lappeenranta University of Technology, Skinnarilankatu 34, SF-53851, Lappeenranta, Finland

    Aki Mikkola & Marko K. Matikainen

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  1. Olivier A. Bauchau
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  2. Shilei Han
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  3. Aki Mikkola
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  4. Marko K. Matikainen
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Correspondence to Olivier A. Bauchau.

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Bauchau, O.A., Han, S., Mikkola, A. et al. Comparison of the absolute nodal coordinate and geometrically exact formulations for beams. Multibody Syst Dyn 32, 67–85 (2014). https://doi.org/10.1007/s11044-013-9374-7

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  • DOI: https://doi.org/10.1007/s11044-013-9374-7

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