Multibody System Dynamics

, Volume 32, Issue 4, pp 535–549 | Cite as

Efficient dynamics analysis of large-deformation flexible beams by using the absolute nodal coordinate transfer matrix method

  • Bao Rong


This paper describes an efficiently computational method for dynamics simulating of planar multibody system with flexible beams undergoing large deformations by combining the absolute nodal coordinate formulation (ANCF) and the transfer matrix method (TMM). Firstly, according to ANCF and the geometrically nonlinear theory, the two-dimensional shear-deformable beam element and its dynamic equations of motion are presented. Then according to TMM and the dynamic equations of beam elements, by defining the new state vectors and deducing the new transfer equations of elements, an efficient computational method named as ANC–TMM, is presented for dynamics of planar multibody system with flexible beams undergoing large deformations. Compared with the ordinary dynamics methods, the proposed method combines the strengths of TMM and ANCF. It does not need the global dynamic equations of system and has the low order of the system matrix and high computational efficiency. This method can be extended to solve the nonlinear dynamics problems of general flexible multibody systems undergoing large deformations. Formulations as well as a numerical example of a flexible four-bar linkage mechanism are given to validate the method.


Multibody system Geometric nonlinearity Computational method Shear-deformable beam element Large rotation 



The research was supported by the Natural Science Foundation of China Government (Grant No. 10902051) and the Program for New Century Excellent Talents in University (Grant No. NCET-10-0075). We owe special thanks to Professor Jorge A.C. Ambrósio (Instituto Superior Técnico, Lisbon, Portugal) and Professor Werner Schiehlen (University of Stuttgart, Germany) for their heuristic suggestions.


  1. 1.
    Schiehlen, W.: Research trends in multibody system dynamics. Multibody Syst. Dyn. 18(1), 3–13 (2007) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Wasfy, T.M., Noor, A.K.: Computational strategies for flexible multibody system. Appl. Mech. Rev. 56(6), 553–613 (2003) CrossRefGoogle Scholar
  3. 3.
    Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, New York (2005) CrossRefMATHGoogle Scholar
  4. 4.
    Berzeri, M., Campanelli, M., Shabana, A.A.: Definition of the elastic forces in the finite-element absolute nodal coordinate formulation and the floating frame of reference formulation. Multibody Syst. Dyn. 5(1), 21–54 (2001) CrossRefMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Matikainen, M.: Development of beam and plate finite elements based on the absolute nodal coordinate formulation. Ph.D. dissertation, Lappeenranta University of Technology, Finland (2009) Google Scholar
  7. 7.
    Tian, Q., Zhang, Y.Q., Chen, L.P., Qin, G.: Advances in the absolute nodal coordinate method for the flexible multibody dynamics. Adv. Mech. 40(2), 189–202 (2010) Google Scholar
  8. 8.
    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) CrossRefMATHGoogle Scholar
  9. 9.
    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, pp. 21–24 (2005) Google Scholar
  10. 10.
    Liu, C., Tian, Q., Hu, H.Y.: Efficient computational method for dynamics of flexible multibody systems based on absolute nodal coordinate. Theor. Appl. Mech. Lett. 42(6), 1197–1204 (2010) Google Scholar
  11. 11.
    Yakoub, R.Y., Shabana, A.A.: Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems. Nonlinear Dyn. 20(3), 267–282 (1999) CrossRefMATHGoogle Scholar
  12. 12.
    Chijie, L.: Dynamic and thermal analyses of flexible structures in orbit. Ph.D. dissertation, University of Connecticut, Storrs (2006) Google Scholar
  13. 13.
    Pestel, E.C., Leckie, F.A.: Matrix Method in Elastomechanics. McGraw-Hill, New York (1963) Google Scholar
  14. 14.
    Kumar, A.S., Sankar, T.S.: A new transfer matrix method for response analysis of large dynamic systems. Comput. Struct. 23(4), 545–552 (1986) CrossRefMATHGoogle Scholar
  15. 15.
    Dokanish, M.A.: A new approach for plate vibration: combination of transfer matrix and finite element technique. J. Mech. Des. 94(2), 526–530 (1972) Google Scholar
  16. 16.
    Loewy, R.G., Bhntani, N.: Combined finite element-transfer matrix method. J. Sound Vib. 226(5), 1048–1052 (1999) CrossRefGoogle Scholar
  17. 17.
    Xue, H.Y.: A combined finite element-stiffness equation transfer method for steady state vibration response analysis of structures. J. Sound Vib. 265(4), 783–793 (2003) CrossRefGoogle Scholar
  18. 18.
    Rong, B., Rui, X.T., Wang, G.P., et al.: Modified finite element transfer matrix method for eigenvalue problem of flexible structures. J. Appl. Mech. 78(2), 021016 (2011) CrossRefGoogle Scholar
  19. 19.
    Rong, B., Rui, X.T., Wang, G.P.: New method for dynamics modeling and analysis on flexible plate undergoing large overall motion. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 224(K1), 33–44 (2010) Google Scholar
  20. 20.
    Rong, B., Rui, X.T., Wang, G.P., et al.: New efficient method for dynamics modeling and simulation of flexible multibody systems moving in plane. Multibody Syst. Dyn. 24(2), 181–200 (2010) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Horner, G.C.: The Riccati transfer matrix method. Ph.D. dissertation, University of Virginia, USA (1975) Google Scholar
  22. 22.
    Wang, G.P., Rong, B., Tao, L., Rui, X.T.: Riccati discrete time transfer matrix method for dynamic modeling and simulation of an underwater towed system. J. Appl. Mech. 79(4), 041014 (2012) CrossRefGoogle Scholar
  23. 23.
    Zhai, W.M.: Two simple fast integration methods for large-scale dynamic problems in engineering. Int. J. Numer. Methods Eng. 39(24), 4199–4214 (1996) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Nanchang Military AcademyNanchangP.R. China

Personalised recommendations