Efficient computational approaches for analysis of thin and flexible multibody structures


A large group of real mechanical problems can be modeled and analyzed using the approaches of flexible multibody dynamics. The computational models in the form of differential–algebraic equations can be quite complex, and therefore, it is suitable to develop efficient approaches for the analysis of such models. This paper deals with the development of fast and efficient computational techniques for the analysis of flexible and thin mechanical structures modeled using the absolute nodal coordinate formulation (ANCF). The first original contribution of this paper is the introduction of fast evaluation of nonlinear elastic forces of the ANCF cable element based on the pre-computation of various terms that are constant when evaluated numerically at Gaussian points. The second part of the paper is aimed at the usage of quasi-Newton methods, which led to the reduction in iteration matrix re-evaluations in case of Newmark type of numerical integration methods for equations of motion. Proposed improvements are implemented in an in-house software in MATLAB environment, and their effects are tested on the cable-mass system considered as a flexible multibody system. The numerical results shown in the paper have proven the efficiency of the proposed algorithms. Real behavior of testing mechanical systems was supported by presented experimental results.

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  1. 1.

    Arnold, M., Brüls, O.: Convergence of the generalized-\(\alpha \) scheme for constrained mechanical systems. Multibody Syst. Dyn. 18, 185–202 (2007)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Berzeri, M., Shabana, A.A.: Development of simple models for elastic forces in the absolute nodal coordinate formulation. J. Sound Vib. 235, 539–565 (2000)

    Article  Google Scholar 

  3. 3.

    Bottasso, C.L., Dopico, D., Trainelli, L.: On the optimal scaling of index three DAEs in multibody dynamics. Multibody Syst. Dyn. 19, 3–20 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bulín, R., Hajžman, M., Polach, P.: Nonlinear dynamics of a cable-pulley system using the absolute nodal coordinate formulation. Mech. Res. Commun. 82, 21–28 (2017). https://doi.org/10.1016/j.mechrescom.2017.01.001

    Article  Google Scholar 

  5. 5.

    Bulín, R., Hajžman, M., Polach, P.: Investigation of falling control rods in deformed guiding tubes in nuclear reactors using multibody approaches. In: Proceedings of The 5th Joint International Conference on Multibody System Dynamics. Lisboa, Instituto Superior Técnico (2018)

  6. 6.

    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics (1996)

  7. 7.

    Dmitrochenko, O., Yoo, W.-S., Pogorelov, D.: Helicoseir as shape of a rotating string (II): 3D theory ans simulation using ANCF. Multibody Syst. Dyn. 15, 181–200 (2006)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Epple, A.: Methods for increased computational efficiency of multibody simulations, Ph.D. thesis, School of Aerospace Engineering, Georgia Institute of Technology (2008)

  9. 9.

    García-Vallejo, D., Mayo, J., Escalona, J.L., Domínguez, J.: Efficient evaluation of the elastic forces and the Jacobian in the absolute nodal coordinate formulation. Nonlinear Dyn. 35, 313–329 (2004). https://doi.org/10.1023/B:NODY.0000027747.41604.20

    Article  MATH  Google Scholar 

  10. 10.

    Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using absolute nodal co-ordinate formulation. In: Proceedings of The ECCOMAS Thematic Conference on Multibody Dynamics 2005, Madrid (2005)

  11. 11.

    Gerstmayr, J., Shabana, A.A.: Efficient integration of the elastic forces and thin three-dimensional beam elements in the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109–130 (2006)

    Article  Google Scholar 

  12. 12.

    Gerstmayr, J., Sugiyama, H., Mikkola, A.: Review on the absolute nodal coordinate formulation for large deformation analysis of multibody systems. J. Comput. Nonlinear Dyn. 8, 031016 (2013)

    Article  Google Scholar 

  13. 13.

    Géradin, M., Cardona, A.: Flexible Multibody Dynamics. Wiley, Chichester (2001). ISBN 0-471-48990-5

    Google Scholar 

  14. 14.

    Geradin, M., Idelsohn, S., Hogge, M.: Computational strategies for the solution of large nonlinear problems via quasi-Newton methods. Comput. Struct. 13, 73–81 (1981)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Géradin, M., Rixen, D.J.: Mechanical Vibrations, Theory and Application to Structural Dynamics, 3rd edn. Wiley, Chichester (2015). ISBN 978-1-118-90020-8

    Google Scholar 

  16. 16.

    Gosselin, C., Grenier, M.: On the determination of the force distribution in overconstrained cable-driven parallel mechanisms. Meccanica 46, 3–15 (2011)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Goyal, R., Skelton, R.E.: Tensegrity system dynamics with rigid bars and massive strings. Multibody Syst. Dyn. 46, 203–228 (2019). https://doi.org/10.1007/s11044-019-09666-4

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Guo, X., Zhang, D.G., Li, L., Zhang, L.: Application of the two-loop procedure in multibody dynamics with contact and constraint. J. Sound Vib. 427, 15–27 (2018)

    Article  Google Scholar 

  19. 19.

    Han, J., Kim, J., Kim, S.: An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates. Multibody Syst. Dyn. 47, 293–316 (2019). https://doi.org/10.1007/s11044-019-09690-4

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Hara, K., Watanabe, M.: Development of an efficient calculation procedure for elastic forces in the ANCF beam element by using a constrained formulation. Multibody Syst. Dyn. 43, 369–386 (2018). https://doi.org/10.1007/s11044-017-9594-3

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    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)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hwang, S.W., Bak, J.H., Yoon, J., Park, J.H., Park, J.O.: Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots. J. Mech. Sci. Technol. 30, 5689–5697 (2016)

    Article  Google Scholar 

  23. 23.

    Jeong, S., Yoo, H.H.: Generalized classical Ritz method for modeling geometrically nonlinear flexible multibody systems having a general topology. Int. J. Mech. Sci. 181, 105687 (2020)

    Article  Google Scholar 

  24. 24.

    Koutsovasilis, P., Beitelschmidt, M.: Comparison of model reduction techniques for large mechanical systems. Multibody Syst. Dyn. 20, 111–128 (2008). https://doi.org/10.1007/s11044-008-9116-4

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Krinner, A., Schindler, T., Rixen, D.J.: Time integration of mechanical systems with elastohydrodynamic lubricated joints using Quasi-Newton method and projection formulations. Int. J. Numer. Methods Eng. 110, 523–548 (2017). https://doi.org/10.1002/nme.5365

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Liu, Ch., Tian, Q., Hu, H.: New spatial curved beam and cylindrical shell elements of gradient-deficient Absolute Nodal Coordinate Formulation. Nonlinear Dyn. 70, 1903–1918 (2012). https://doi.org/10.1007/s11071-012-0582-0

    MathSciNet  Article  Google Scholar 

  27. 27.

    Lugrís, U., Naya, M.A., Luaces, A., Cuadrado, J.: Efficient calculation of the inertia terms in floating frame of reference formulations for flexible multibody dynamics. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn. 223, 147–157 (2009). https://doi.org/10.1243/14644193JMBD164

    Article  Google Scholar 

  28. 28.

    Luo, K., Hu, H., Liu, C., Tian, Q.: Model order reduction for dynamic simulation of a flexible multibody system via absolute nodal coordinate formulation. Comput. Methods Appl. Mech. Eng. 324, 573–594 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Negrut, A., Jay, L.O., Khude, N.: A discussion of low-order numerical integration formulas for rigid and flexible multibody dynamics. J. Comput. Nonlinear Dyn. 4, 021008 (2009)

    Article  Google Scholar 

  30. 30.

    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 (DETC2005-85096). J. Comput. Nonlinear Dyn. 2, 73–85 (2007)

    Article  Google Scholar 

  31. 31.

    Nowakowski, C., Fehr, J., Fischer, M., Eberhard, P.: Model order reduction in elastic multibody systems using the floating frame of reference formulation. IFAC Proc. Vol. 45, 40–48 (2012)

    Article  Google Scholar 

  32. 32.

    Marques, F., Flores, P., Lankarani, M.H.: On the frictional contacts in multibody system dynamics. In: Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, pp. 565–576 (2015)

  33. 33.

    Orzechowski, G., Fraczek, J.: Integration of the equations of motion of multibody systems using absolute nodal coordinate formulation. Acta Mech. Autom. 6, 75–83 (2012)

    Google Scholar 

  34. 34.

    Orzechowski, G., Matikainen, M.K., Mikkola, A.M.: Inertia forces and shape integrals in the floating frame of reference formulation. Nonlinear Dyn. 88, 1953–1968 (2017). https://doi.org/10.1007/s11071-017-3355-y

    Article  Google Scholar 

  35. 35.

    Polach, P., Hajžman, M., Václavík, J.: Experimental and computational investigation of a simple fibre-mass system. In: Proceedings of the 19th International Conference Engineering Mechanics 2013, CD-ROM, Svratka (2013)

  36. 36.

    Rong, B., Rui, X., Tao, L., et al.: Theoretical modeling and numerical solution methods for flexible multibody system dynamics. Nonlinear Dyn. 98, 1519–1553 (2019). https://doi.org/10.1007/s11071-019-05191-3

    Article  Google Scholar 

  37. 37.

    Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: finite element method and absolute nodal coordinate formulation. In: Proceeding of IDETC/CIE 2005, pp. 1–9 (2005)

  38. 38.

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

    Google Scholar 

  39. 39.

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

    MathSciNet  Article  Google Scholar 

  40. 40.

    Sugiyama, H., Koyama, H., Yamashita, H.: Gradient deficient curved beam element using the absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 5, 021001-(1-8) (2010). https://doi.org/10.1115/1.4000793

    Article  Google Scholar 

  41. 41.

    Svatoš, P., Šika, Z., Beneš, P., Hajžman, M., Zavřel, J.: Cable driven mechanisms with added piezo active platform. Bull. Appl. Mech. 11, 19–24 (2015)

    Google Scholar 

  42. 42.

    Tang, D., Bao, S., Lv, B., Guo, H., Luo, L., Mao, J.: A derivative-free algorithm for nonlinear equations and its applications in multibody dynamics. J. Algorithms Comput. Technol. 12, 30–42 (2018). https://doi.org/10.1177/1748301817729990

    MathSciNet  Article  Google Scholar 

  43. 43.

    Tian, Q., Chen, L.P., Zhang, Y.Q., Yang, J.: An efficient hybrid method for multibody dynamics simulation based on absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 4, 21009 (2009)

    Article  Google Scholar 

  44. 44.

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

    Article  Google Scholar 

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The work was supported from European Regional Development Fund-Project Research and Development of Intelligent Components of Advanced Technologies for the Pilsen Metropolitan Area (InteCom) (No. CZ.02.1.01/0.0/0.0/17_048/0007267) and by the Czech Science Foundation project number 20-21893S. The first author was further supported by the Motivation system of the University of West Bohemia—Part POSTDOC.

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Bulín, R., Hajžman, M. Efficient computational approaches for analysis of thin and flexible multibody structures. Nonlinear Dyn (2021). https://doi.org/10.1007/s11071-021-06225-5

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  • Multibody dynamics
  • Nonlinear equations of motion
  • Numerical solution
  • ANCF
  • Quasi-Newton method