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Numerical investigation of the slope discontinuities in large deformation finite element formulations

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Abstract

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.

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References

  1. Dibold, M., Gerstmayr, J., Irschik, H.: On the accuracy and computational cost of the absolute nodal coordinate formulation and the floating frame of reference formulation in deformable multibody systems. In: Proceedings of ASME International Design Engineering Technical Conferences and Computer Information in Engineering Conference (DETC2007/ MSNDC-34756), Las Vegas, NV, USA (2007)

  2. Dimitrochenko, O., Mikkola, A.M.: Large deformation triangular plate elements for multibody problems. In: Proceedings of ASME International Design Engineering Technical Conferences and Computer Information in Engineering Conference (DETC2007/ MSNDC-34741), Las Vegas, NV, USA (2007)

  3. Dufva, K., Shabana, A.A.: Analysis of thin plate structures using the absolute nodal coordinate formulation. In: Proceeding of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, vol. 219, pp. 345–355 (2005)

  4. Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dyn. 45, 109–130 (2006)

    Article  MATH  Google Scholar 

  5. Hughes, T.J.R., Winget, J.: Finite rotation effects in numerical integration of rate constitutive equation arising in large deformation analysis. Int. J. Numer. Methods Eng. 15, 1862–1867 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  6. Iwai, R., Kobayashi, N.: A new flexible multibody beam element based on the absolute nodal coordinate formulation using the global shape function and the analytical mode shape function. Nonlinear Dyn. 34, 207–232 (2003)

    Article  MATH  Google Scholar 

  7. Kim, H.W., Yoo, W.S.: Experimental validation of two damping force models for the ANCF. In: Proceedings of ASME International Design Engineering Technical Conferences and Computer Information in Engineering Conference (DETC2007/ MSNDC-34460), Las Vegas, NV, USA (2007)

  8. Maqueda, L.G., Shabana, A.A.: Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams. Multibody Syst. Dyn. 18, 375–396 (2007)

    Article  MATH  Google Scholar 

  9. Omar, M.A., Shabana, A.A.: A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243, 565–576 (2001)

    Article  Google Scholar 

  10. Rankin, C.C., Brogan, F.A.: An element independent corotational procedure for the treatment of large rotations. ASME J. Press. Vessel Technol. 108, 165–174 (1986)

    Article  Google Scholar 

  11. Schwab, A., Gerstmayr, J., Meijaard, J.: Comparison of three-dimensional flexible thin plate elements for multibody dynamic analysis: finite element formulation and absolute nodal coordinate formulation. In: Proceedings of ASME International Design Engineering Technical Conferences and Computer Information in Engineering Conference (DETC2007/ MSNDC-34754), Las Vegas, NV, USA (2007)

  12. Seo, J.H., Jung, I.H., Han, H.S., Park, T.W.: Computation of dynamic stress in flexible multi-body dynamics using absolute nodal coordinate formulation. Key Eng. Mater. 270–273, 1427–1433 (2004)

    Article  Google Scholar 

  13. Shabana, A.A.: Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics. Nonlinear Dyn. 16, 293–306 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  15. Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  16. Shabana, A.A., Maqueda, L.G.: Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements. J. Multibody Syst. Dyn. 20, 239–249 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shabana, A.A., Mikkola, A.M.: Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity. ASME J. Mech. Des. 125, 342–350 (2003)

    Article  Google Scholar 

  18. Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motion—the plane case: part I. ASME J. Appl. Mech. 53, 849–854 (1986)

    Article  MATH  Google Scholar 

  19. Sugiyama, H., Suda, Y.: On the curved beam element using the absolute nodal coordinates. In: Proceedings of the ECCOMAS Thematic Conference in Multibody Dynamics, Milan, Italy (2007)

  20. Yakoub, R.Y., Shabana, A.A.: Three-dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. ASME J. Mech. Des. 123, 614–621 (2001)

    Article  Google Scholar 

  21. Yoo, W.S., Lee, J.H., Park, S.J., Sohn, J.H., Pogorelov, D., Dimitrochenko, O.: Large deflection analysis of a thin plate: computer simulation and experiment. Multibody Syst. Dyn. 11(2), 185–208 (2004)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL, 60607-7022, USA

    Luis G. Maqueda & Ahmed A. Shabana

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  1. Luis G. Maqueda
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  2. Ahmed A. Shabana
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Correspondence to Ahmed A. Shabana.

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Maqueda, L.G., Shabana, A.A. Numerical investigation of the slope discontinuities in large deformation finite element formulations. Nonlinear Dyn 58, 23–37 (2009). https://doi.org/10.1007/s11071-008-9458-8

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  • DOI: https://doi.org/10.1007/s11071-008-9458-8

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