Advertisement

Multibody System Dynamics

, Volume 29, Issue 3, pp 255–269 | Cite as

Three- and four-noded planar elements using absolute nodal coordinate formulation

  • Alexander Olshevskiy
  • Oleg Dmitrochenko
  • Changwan Kim
Article

Abstract

This paper investigates two new types of planar finite elements containing three and four nodes. These elements are the reduced forms of the spatial plate elements employing the absolute nodal coordinate approach. Elements of the first type use translations of nodes and global slopes as nodal coordinates and have 18 and 24 degrees of freedom. The slopes facilitate the prevention of the shear locking effect in bending problems. Furthermore, the slopes accurately describe the deformed shape of the elements. Triangular and quadrilateral elements of the second type use translational degrees of freedom only and, therefore, can be utilized successfully in problems without bending. These simple elements with 6 and 8 degrees of freedom are identical to the elements used in conventional formulation of the finite element method from the kinematical point of view. Similarly to the famous problem called “flying spaghetti” which is used often as a benchmark for beam elements, a kind of “flying lasagna” is simulated for the planar elements. Numerical results of simulations are presented.

Keywords

Finite elements Absolute nodal coordinate formulation Flexible multibody system dynamics Large displacements 

Notes

Acknowledgements

This paper was supported by Konkuk University in 2012.

References

  1. 1.
    Song, J.O., Haug, E.J.: Dynamic analysis of planar flexible mechanisms. Comput. Methods Appl. Mech. Eng. 24, 359–381 (1980) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Shabana, A.A., Wehage, R.A.: Coordinate reduction technique for transient analysis of special substructures with large angular rotations. J. Struct. Mech. 11(3), 401–431 (1983) CrossRefGoogle 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) MATHCrossRefGoogle Scholar
  4. 4.
    Rankin, C.C., Brogan, F.A.: An element independent corotational procedure for the treatment of large rotations. J. Press. Vessel Technol. 108, 165–174 (1986) CrossRefGoogle Scholar
  5. 5.
    Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem, Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985) MATHCrossRefGoogle Scholar
  6. 6.
    Simo, J.C., Vu-Quoc, L.: A three-dimensional finite strain rod model, Part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986) MATHCrossRefGoogle Scholar
  7. 7.
    Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1, 189–222 (1997) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1(3), 339–348 (1997) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hughes, T.J.R.: The Finite Element Method. Prentice-Hall, Englewood Cliffs (1987) MATHGoogle Scholar
  10. 10.
    Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123(4), 606–613 (2001) CrossRefGoogle Scholar
  11. 11.
    Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dyn. 45(1–2), 109–130 (2006) MATHCrossRefGoogle Scholar
  12. 12.
    Von Dombrowski, S.: Analysis of large flexible body deformation in multibody systems using absolute coordinates. Multibody Syst. Dyn. 8(4), 409–432 (2002) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. J. Multibody Syst. Dyn. 20, 359–384 (2008) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: classical finite element formulation and absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 5(1), 011010 (2010) (10 pages) CrossRefGoogle Scholar
  15. 15.
    Nachbagauer, K., Gruber, P.G., Vetyukov, Yu., Gerstmayr, J.: A spatial thin beam finite element based on the absolute nodal coordinate formulation without singularities. In: Proc. of the ASME 2011 Int. Design Eng. Techn. Conf. & Computers and Information in Eng. Conf. IDETC/CIE 201. Washington, DC, USA, August, pp. 28–31 (2011) Google Scholar
  16. 16.
    Mikkola, A.M., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn. 9(3), 283–309 (2003) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Dmitrochenko, O.N., Pogorelov, D.Yu.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10, 17–43 (2003) MATHCrossRefGoogle Scholar
  18. 18.
    Omar, M.A., Shabana, A.A.: A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243(3), 565–576 (2001) CrossRefGoogle Scholar
  19. 19.
    Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34(1), 53–74 (2003) MATHCrossRefGoogle Scholar
  20. 20.
    Garcia-Vallejo, D., Mikkola, A.M., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50(1–2), 249–264 (2007) MATHCrossRefGoogle Scholar
  21. 21.
    Nachbagauer, K., Pechstein, A., Irschik, H., Gerstmayr, J.: New locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation. J. Multibody Syst. Dyn. 26(3), 245–263 (2011) MATHCrossRefGoogle Scholar
  22. 22.
    Kubler, L., Eberhard, P., Geisler, J.: Flexible multibody systems with large deformations and nonlinear structural damping using absolute nodal coordinates. Nonlinear Dyn. 34(1–2), 31–52 (2003) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gerstmayr, J., Schoberl, J.: A 3D finite element method for flexible multibody systems. Multibody Syst. Dyn. 15(4), 309–324 (2006) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Olshevskiy, A., Dmitrochenko, O.: Three-dimensional solid elements employing slopes in the absolute nodal coordinate formulation. In: Proc. of the 24th Nordic Seminar on Computational Mechanics, Helsinki, 2011.11.3–4, pp. 162–165 (2011) Google Scholar
  25. 25.
    Dmitrochenko, O., Mikkola, A.: A formal procedure and invariants of a transition from conventional finite elements to the absolute nodal coordinate formulation. Multibody Syst. Dyn. 22(4), 323–339 (2009) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Dufva, K., et al.: Nonlinear dynamics of three-dimensional belt drives using the finite-element method. Nonlinear Dyn. 48(4), 449–466 (2007) MATHCrossRefGoogle Scholar
  27. 27.
    Weed, D., Maqueda, L., Brown, M., Hussein, B., Shabana, A.: A new nonlinear multibody/finite element formulation for knee joint ligaments. Nonlinear Dyn. 60(4), 357–367 (2010) MATHCrossRefGoogle Scholar
  28. 28.
    Gantoi, F., Brown, M., Shabana, A.: ANCF finite element/multibody system formulation of the ligament/bone insertion site constraints. J. Comput. Nonlinear Dyn. 5(3), (2010), 9 pages Google Scholar
  29. 29.
    Abdullah, M.A., Michitsuji, Y., Nagai, M., Miyajima, N.: Analysis of contact force variation between contact wire and pantograph based on multibody dynamics. J. Mech. Syst. Transp. Logist. 3(3), 552–567 (2010) CrossRefGoogle Scholar
  30. 30.
    Wan, H., Dong, H., Ren, Y.: Study of strain energy in deformed insect wings dynamic. In: Behavior of Materials. Conference Proceedings of the Society for Experimental Mechanics Series, vol. 1, pp. 323–328. Springer, New York (2011) Google Scholar
  31. 31.
    Hrennikoff, A.: Solution of problems of elasticity by the frame-work method. ASME J. Appl. Mech. 8, A619–A715 (1941) MathSciNetGoogle Scholar
  32. 32.
    Melosh, R.J.: Structural analysis of solids. ASCE Struct. J. 4, 205–223 (1963) Google Scholar
  33. 33.
    Dmitrochenko, O., Mikkola, A.: Digital nomenclature code (dnc) of finite element kinematics and its modification (dncm) for absolute nodal coordinates. In: Proc. of 1st Int. Joint Conf. for Multibody System Dynamics (IMSD-2010), Lappeenranta, 25–27.05.2010 (2010) Google Scholar
  34. 34.
    Dufva, K., Shabana, A.: Analysis of thin plate structures using the absolute nodal coordinate formulation. J. Multibody Dyn. 219, 345–355 (2005). Proceedings of the Institution of Mechanical Engineers, Part K Google Scholar
  35. 35.
    Specht, B.: Modified shape functions for the three node plate bending element passing the patch test. Int. J. Numer. Methods Eng. 26, 705–715 (1988) MATHCrossRefGoogle Scholar
  36. 36.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Solid Mechanics, vol. 2. Butterworth, London (2000) MATHGoogle Scholar
  37. 37.
    Gere, J.M., Timoshenko, S.P.: Mechanics of Materials, 4th edn. PWS, Boston (1997), 912 pages Google Scholar
  38. 38.
    Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions—the plane case: Parts I and II. J. Appl. Mech. 53, 849–863 (1986) MATHCrossRefGoogle Scholar
  39. 39.
    Pogorelov, D.: Some developments in computational techniques in modeling advanced mechanical systems. In: van Campen, D.H. (ed.) Proc. of IUTAM Symp. on Interaction between Dynamics and Control in Advanced Mech. Systems, pp. 313–320. Kluwer Academic, Dordrecht (1997) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Alexander Olshevskiy
    • 1
  • Oleg Dmitrochenko
    • 2
  • Changwan Kim
    • 1
  1. 1.School of Mechanical EngineeringKonkuk UniversitySeoulKorea
  2. 2.Department of Mechanical EngineeringLappeenranta University of TechnologyLappeenrantaFinland

Personalised recommendations