Skip to main content
SpringerLink
Log in

Reduction of system matrices of planar beam in ANCF by component mode synthesis method

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

A method of reducing the system matrices of a planar flexible beam described by an absolute nodal coordinate formulation (ANCF) is presented. In this method, we focus that the bending stiffness matrix expressed by adopting a continuum mechanics approach to the ANCF beam element is constant when the axial strain is not very large. This feature allows to apply the Craig–Bampton method to the equation of motion that is composed of the independent coordinates when the constraint forces are eliminated. Four numerical examples that compare the proposed method and the conventional ANCF are demonstrated to verify the performance and accuracy of the proposed method. From these examples, it is verified that the proposed method can describe the large deformation effects such as dynamic stiffening due to the centrifugal force, as well as the conventional ANCF does. The use of this method also reduces the computing time, while maintaining an acceptable degree of accuracy for the expression characteristics of the conventional ANCF when the modal truncation number is adequately employed. This reduction in CPU time particularly pronounced in the case of a large element number and small modal truncation number; the reduction can be verified not only in the case of small deformation but also in the case of a fair bit large deformation.

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

Abbreviations

A :

cross-sectional area of beam

B :

velocity transformation matrix

C :

constraint equation

e :

nodal coordinate vector

E :

Young’s modulus of beam

I :

second moment of area of beam

K :

stiffness matrix

l :

beam length

M :

mass matrix

m c :

number of constraint equation

n b :

number of degrees of freedom of the boundary region in a component

n c :

number of normal modes

n e :

number of elements in a component

n i :

number of degrees of freedom of interior region in a component with n e elements

n t :

number of truncation modes

n r :

number of degrees of freedom of the reduced component

q :

generalized coordinate vector

x :

local coordinate of element in undeformed configuration

ρ :

density of beam element

ξ :

modal coordinate vector of normal modes

\(\varOmega _{m}^{2}\) :

eigen value of mth normal mode

subscript b,i :

boundary and interior regions of beam element

subscript dp, in :

related to dependent and independent coordinates

subscript l :

related to axial direction of beam element

subscript m :

the mode number

subscript t :

related to bending of beam element

References

  1. Rupp, C.C., Laue, J.H.: Shuttle/tethered satellite system. J. Aeronaut. Sci. 26(1), 1–17 (1978)

    Google Scholar 

  2. Funase, R., Sugita, M., Mori, O., Tsuda, Y., Kawaguchi, J.: Modeling of spinning solar sail by multi particle model and its application to attitude control system. In: Proceedings of Design Engineering Technical Conference, DETC2009-87821, ASME, pp. 1–12 (2009)

    Google Scholar 

  3. Müftü, S.: Mechanics of a thin, tensioned shell, wrapped helically around a turn-bar. J. Fluids Struct. 23(5), 767–785 (2007)

    Article  Google Scholar 

  4. Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Shabana, A.A.: An absolute nodal coordinate formulation for the large rotation and deformation analysis of flexible bodies. Technical report MBS96-1-UIC, University of Illinois at Chicago, pp. 1–26 (1996)

  6. Escalona, J.L., Hussien, H.A., Shabana, A.A.: Application of the absolute nodal coordinate formulation to multibody system dynamics. J. Sound Vib. 214(5), 833–851 (1998)

    Article  Google Scholar 

  7. Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn., pp. 314–343. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  8. Berzeri, M., Shabana, A.A.: Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation. J. Sound Vib. 235(4), 539–565 (2000)

    Article  Google Scholar 

  9. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123(4), 606–613 (2001)

    Article  Google Scholar 

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dmitrochenko, O.N., Pogorelov, D.Y.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10(1), 17–43 (2003)

    Article  MATH  Google Scholar 

  12. Dmitrochenko, O.N., Mikkola, A.M.: Two simple triangular plate elements based on the absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 3(4), 0410121–0410128 (2008)

    Article  Google Scholar 

  13. Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678–685 (1965)

    Article  Google Scholar 

  14. Craig, R.R. Jr., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968)

    Article  MATH  Google Scholar 

  15. MacNeal, R.H.: A hybrid method of component mode synthesis. Comput. Struct. 1, 581–601 (1971)

    Article  Google Scholar 

  16. Géradin, M., Cardona, A.: Flexible Multibody Dynamics—A Finite Element Approach, pp. 185–202. Wiley, Chichester (2001)

    Google Scholar 

  17. Gerstmayr, J., Ambrósio, J.A.C.: Component mode synthesis with constant mass and stiffness matrices applied to flexible multibody systems. Int. J. Numer. Methods Eng. 7(11), 1518–1546 (2008)

    Article  Google Scholar 

  18. 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(1–2), 207–232 (2003)

    Article  MATH  Google Scholar 

  19. Guyan, R.J.: Reduction of stiffness and mass matrices. AIAA J. 3(2), 380 (1965)

    Article  Google Scholar 

  20. Inman, D.J.: Engineering Vibration, pp. 329–337. Prentice Hall, New York (1994)

    Google Scholar 

  21. Berzeri, M., Shabana, A.A.: Study of the centrifugal stiffening effect using the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 7(4), 357–387 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1, Fuchinobe, Chuo-ku, Sagamihara, Kanagawa, 2525258, Japan

    Nobuyuki Kobayashi, Tsubasa Wago & Yoshiki Sugawara

Authors
  1. Nobuyuki Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsubasa Wago
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoshiki Sugawara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nobuyuki Kobayashi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kobayashi, N., Wago, T. & Sugawara, Y. Reduction of system matrices of planar beam in ANCF by component mode synthesis method. Multibody Syst Dyn 26, 265–281 (2011). https://doi.org/10.1007/s11044-011-9259-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-011-9259-6

Keywords

Search

Navigation