Skip to main content
SpringerLink
Log in

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

An Erratum to this article was published on 06 February 2009

Abstract

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hughes, T.J.R., Simo, J.C.: Computational Inelasticity. Springer, New York (1998)

    MATH  Google Scholar 

  2. Cortés, F., Elejabarrieta, M.J.: Finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models. Int. J. Numer. Mech. Eng. 69, 2173–2195 (2007)

    Article  Google Scholar 

  3. Nutting, P.G.: A new general law of deformation. J. Franklin Inst. 191, 679–685 (1921)

    Article  Google Scholar 

  4. Gemant, A.: A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 7, 311–317 (1936)

    Article  Google Scholar 

  5. Leibniz, G.: Leibnizsche Mathematische Schriften. Georg Olm Verlag, Hildesheim (1962)

    Google Scholar 

  6. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  7. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3), 201–210 (1983)

    Article  MATH  Google Scholar 

  8. Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behaviors. J. Rheol. 30(1), 133–155 (1986)

    Article  MATH  Google Scholar 

  9. Enelund, M., Mähler, L., Runesson, B., Josefson, B.M.: Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws. J. Solids Struct. 36, 2417–2442 (1999)

    Article  MATH  Google Scholar 

  10. Padovan, J.: Computational algorithms for FE formulations involving fractional operators. Comput. Mech. 2, 271–287 (1987)

    Article  MATH  Google Scholar 

  11. Schmidt, A., Gaul, L.: Finite element formulation of viscoelastic constitutive equations using fractional time derivatives. Nonlinear Dyn. 29, 37–55 (2002)

    Article  MATH  Google Scholar 

  12. Galucio, A.C., Deü, J.F., Ohayon, R.: Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech. 33, 282–291 (2004)

    Article  MATH  Google Scholar 

  13. Deü, J.F., Galucio, A.C., Ohayon, R.: Dynamic responses of flexible-link mechanisms with passive active damping treatment. Comput. Struct. 86, 258–265 (2008)

    Article  Google Scholar 

  14. Shabana, A.A.: Flexible multi-body dynamics review of past and recent developments. Multibody Syst. Dyn. 1, 189–222 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Shabana, A.A.: An absolute nodal coordinates formulation for the large rotation and deformation analysis of flexible bodies. Technical Report. No. MBS96-1-UIC, University of Illinois at Chicago (1996)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

  19. Yoo, W.S., Dmitrochenko, O., Yu, D.: Review of finite elements using absolute nodal coordinates for large-deformation problems and matching physical experiments. In: Proceedings of the ASME DETC Conference, California, Long Beach, DETC2005-84720

  20. JongHwi, S., SeokWon, K., IlHo, J., TaeWon, P., JinYong, M., YoungGuk, K., JangBom, C.: Dynamic analysis of a pantograph–catenary system using absolute nodal coordinates. Veh. Syst. Dyn. 44(8), 615–630 (2006)

    Article  Google Scholar 

  21. Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 5, 109–130 (2006)

    Article  Google Scholar 

  22. 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  MathSciNet  Google Scholar 

  23. 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 

  24. García-Vallejo, D., Mayo, J., Escalona, J.L., Domínguez, J.: Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements. Multibody Syst. Dyn. 20, 1–28 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. García-Vallejo, D., Valverde, J., Domínguez, J.: An internal damping model for the absolute nodal coordinate formulation. Nonlinear Dyn. 42, 347–369 (2005)

    Article  Google Scholar 

  26. Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, New York (2008)

    MATH  Google Scholar 

  27. Zhang, W., Shimizu, N.: FE formulation for the viscoelastic body modeled by fractional constitutive law. Acta Mech. Sin. 17(4), 354–365 (2001)

    Article  Google Scholar 

  28. Omar, A., Shabana, A.A.: A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243(3), 565–576 (2001)

    Article  Google Scholar 

  29. García-Vallejo, D., Mayo, J., Escalona, J.L.: Efficient evaluation of the elastic forces and the Jacobian in the absolute nodal coordinate formulation. Nonlinear Dyn. 35, 313–329 (2004)

    Article  MATH  Google Scholar 

  30. Gerstmayr, J., Shabana, A.A.: Efficient integration of the elastic forces and thin three-dimensional beam elements in the absolute nodal coordinate formulation. In: ECCOMAS Thematic Conference, Madrid, Spain, 21–24 June 2005

  31. Newmark, N.M.: A method of computation for structural dynamics. ASCE J. Eng. Mech. Div. 67–94 (1959)

  32. Fung, T.C.: Complex-time-step Newmark methods with controllable numerical dissipation. Int. J. Numer. Methods Eng. 41, 65–93 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 20, 287–306 (2008)

    Article  MathSciNet  Google Scholar 

  34. Géradin, M., Rixen, D.: Mechanical Vibrations: Theory and Applications to Structural Dynamics. Wiley, New York (1997)

    Google Scholar 

  35. Negrut, D., Rampalli, R., Ottarsson, G., Sajdak, A.: On an implementation of the Hilber–Hughes–Taylor method in the context of index 3 differential-algebraic equations of multibody dynamics. ASME J. Comput. Nonlinear Dyn. 2(1), 73–85 (2007)

    Article  Google Scholar 

  36. Hussein, B., Negrut, D., Shabana, A.A.: Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations. Nonlinear Dyn. 54, 283–296 (2008)

    Article  Google Scholar 

  37. 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 

  38. Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34, 53–74 (2003)

    Article  MATH  Google Scholar 

  39. García-Vallejo, D., Mikkola, A.M., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50, 249–264 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Computer-Aided Design, School of Mechanical Science & Engineering, Huazhong University of Science & Technology, Wuhan, Hubei, 430074, China

    Yunqing Zhang, Qiang Tian & Liping Chen

  2. Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, 79409, USA

    Jingzhou (James) Yang

Authors
  1. Yunqing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jingzhou (James) Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liping Chen.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s11044-008-9142-2

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Y., Tian, Q., Chen, L. et al. Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods. Multibody Syst Dyn 21, 281–303 (2009). https://doi.org/10.1007/s11044-008-9139-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-008-9139-x

Keywords

Search

Navigation