Skip to main content
SpringerLink
Log in

A New Flexible Multibody Beam Element Based on the Absolute Nodal Coordinate Formulation Using the Global Shape Function and the Analytical Mode Shape Function

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Several techniques for the reduced dimensionality of finite elementformulations were considered as component mode reduction methods in themiddle sixties. These techniques are widely used in flexiblemultibody simulations for solving small deformation problems. Theabsolute nodal coordinate formulation for solving large rotation anddeformation problems has been established as a full finite elementmethod instead of using similar kinds of reduction techniques. In thispaper, a reduced order absolute nodal coordinate formulation is newlyestablished by introducing the global beam shape function and theanalytical deformation modes as a full finite element. This formulationleads to a constant and symmetric mass matrix as the conventionalabsolute nodal coordinate formulation, and makes it possible to reducethe number of elements and system coordinates of the beam structurewhich undergoes large rotations and large deformations. Numericalexamples show that the excellent agreements between thepresent formulation and the conventional absolute nodal coordinateformulation using a large number of elements are examined. These results demonstratethat the present formulation has high accuracy in the sense that thepresent solutions are similar to the conventional ones with fewersystem coordinates, and high efficiency in computation.

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. Agrawal, O. P. and Shabana, A. A., ‘Dynamic analysis of multibody systems using component modes’, Computers & Structures 21(6), 1985, 1303–1312.

    Google Scholar 

  2. Banerjee, A. K. and Dicknes, J. M., ‘Dynamics of an arbitrary body in large rotation and translation’, AIAA Journal of Guidance, Control, and Dynamics 13(2), 1990, 221–227.

    Google Scholar 

  3. Banerjee, A. K. and Kane, T. R., ‘Dynamics of a plate in large overall motion’, ASME Journal of Applied Mechanics 56, 1989, 887–892.

    Google Scholar 

  4. Belytschko, T. and Glaum, L.W., ‘Application of higher order corotational stretch theories to nonlinear finite element analysis’, Computers & Structures 10, 1979, 175–182.

    Google Scholar 

  5. Belytschko, T. and Hsieh, B. J., ‘Non-linear transient finite element analysis with convected coordinates’, International Journal for Numerical Methods in Engineering 7, 1973, 255–271.

    Google Scholar 

  6. Berzeri, M., Campanelli, M., and Shabana, A. A., ‘Definition of the elastic forces in the finite-element absolute nodal coordinate formulation and the floating frame of reference formulation’, Multibody System Dynamics 5, 2001, 21–54.

    Google Scholar 

  7. Berzeri, M. and Shabana, A. A., ‘A finite element study of the geometric centrifugal stiffening effect’, Technical Report #MBS99-5-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, 1999.

    Google Scholar 

  8. Berzeri, M. and Shabana, A. A., ‘Development of simple models for the elastic forces in the absolute nodal coordinate formulation’, Journal of Sound and Vibration 235(4), 2000, 539–565.

    Google Scholar 

  9. Campanelli, M., Berzeri, M., and Shabana, A. A., ‘Performance of the incremental and non-incremental finite element formulations in flexible multibody problems’, ASME Journal of Mechanical Design 122, 2000, 498–507.

    Google Scholar 

  10. Cardona, A. and Geradin, M., ‘Modeling of superelements in mechanism analysis’, International Journal for Numerical Methods in Engineering 32(8), 1991, 1565–1593.

    Google Scholar 

  11. Escalona, J. L., Hussien, H. A., and Shabana, A. A., ‘Application of the absolute nodal coordinate formulation to multibody system dynamics’, Journal of Sound and Vibration 214(5), 1998, 833–851.

    Google Scholar 

  12. Flanagan, D. P. and Taylor, L. M., ‘An accurate numerical algorithm for stress integration with finite rotations’, Computer Methods in Applied Mechanics and Engineering 62, 1987, 305–320.

    Google Scholar 

  13. Geradin, M., Cardona, A., Doan, D. B., and Duysens, J., ‘Finite element modeling concepts in multibody dynamics’, in Computer Aided Analysis of Rigid and Flexible Mechanical Systems, M. F. O. S. Pereira and J. Ambrosio (eds.), Kluwer, Dordrecht, 1994, pp. 233–284.

    Google Scholar 

  14. Guyan, R. J., ‘Reduction of stiffness and mass matrices’, AIAA Journal 3(2), 1965, 380.

    Google Scholar 

  15. Hughes, T. J. R. and Winget, J., ‘Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis’, International Journal for Numerical Methods in Engineering 15(12), 1980, 1862–1867.

    Google Scholar 

  16. Hurty, W. C., ‘Dynamic analysis of structural systems using component modes’, AIAA Journal 3(4), 1965, 678–685.

    Google Scholar 

  17. Iwai, R. and Kobayashi, N., ‘A new type component mode synthesis beam element based on the absolute nodal coordinate formulation’, in Proceedings of ASME 2003 Design Engineering Technical Conferences, Illinois, Paper No. DETC2003/VIB-48353, 2003.

  18. Kane, T. R. and Levinson, D. A., ‘Formulation of equations of motion for complex spacecraft’, AIAA Journal of Guidance and Control 3(2), 1980, 99–112.

    Google Scholar 

  19. Kane, T. R., Ryan, R. R., and Banerjee, A. K., ‘Dynamics of a cantilever beam attached to a moving base’, AIAA Journal of Guidance, Control, and Dynamics 10(2), 1987, 139–151.

    Google Scholar 

  20. Kim, S. S. and Haug, E. J., ‘Selection of deformation modes for flexible multibody dynamics’, Journal of Mechanics, Structures, and Machines 18(4), 1990, 565–586.

    Google Scholar 

  21. Kobayashi, N., ‘Modeling methodology of beam structure for vibration suppression control’, Proceedings of JSME 890(61), 1989, 86–87 [in Japanese].

    Google Scholar 

  22. Kobayashi, N., Sugiyama, H., and Watanabe, M., ‘Dynamics of flexible beam using a component mode synthesis based formulation’, in Proceedings of ASME 2001 Design Engineering Technical Conferences, Pittsburgh, Paper No. DETC2001/VIB-21351, 2001.

  23. MacNeal, R. H., ‘A hybrid method of component mode synthesis’, Computers & Structures 1, 1971, 581–601.

    Google Scholar 

  24. Mayo, J. and Dominguez, J., ‘A finite element geometrically nonlinear dynamic formulation of flexible multibody systems using a new displacements representation’, ASME Journal of Vibration and Acoustics 119(4), 1997, 573–581.

    Google Scholar 

  25. Mayo, J., Dominguez, J., and Shabana, A. A., ‘Geometrically nonlinear formulations of beams in flexible multibody dynamics’, ASME Journal of Vibration and Acoustics 117, 1995, 501–509.

    Google Scholar 

  26. Mikkola, A. M. and Shabana, A. A., ‘A large deformation plate element for multibody applications’, Technical Report #MBS00-6-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, IL, 2000.

    Google Scholar 

  27. Omar, M. A. and Shabana, A. A., ‘Development of a shear deformable element using the absolute nodal coordinate formulation’, Technical Report #MBS00-3-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, IL, 2000.

    Google Scholar 

  28. Rankin, C. C. and Brogan, F. A., ‘An element independent corotational procedure for the treatment of large rotations’, ASME Journal of Pressure Vessel Technology 108, 1986, 165–174.

    Google Scholar 

  29. Ryan, R. R., ‘Flexibility modeling methods in multibody dynamics’, in Proceedings AAS/AIAA Astrodynamics Specialist Conference, Paper AAS, 1987, pp. 87–431.

  30. Shabana, A. A., ‘An absolute nodal coordinate formulation for the large rotation and deformation analysis of flexible bodies’, Technical Report #MBS96-1-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, 1996.

    Google Scholar 

  31. Shabana, A. A., ‘Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics’, Nonlinear Dynamics 16, 1998, 293–306.

    Google Scholar 

  32. Shabana, A. A., Dynamics of Multibody Systems, second edition, Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  33. Shabana, A. A., ‘Flexible multibody dynamics: Review of past and recent development’, Multibody System Dynamics 1(2), 1997, 189–222.

    Google Scholar 

  34. Shabana, A. A., ‘Resonance conditions and deformable body co-ordinate systems’, Journal of Sound and Vibration 192(1), 1996, 389–398.

    Google Scholar 

  35. Shabana, A. A. and Christensen, A. P., ‘Three-dimensional absolute nodal coordinate formulation: Plate problem’, International Journal for Numerical Methods in Engineering 40, 1997, 2775–2790.

    Google Scholar 

  36. Shabana, A. A., Hussien, H. A., and Escalona, J. L., ‘Application of the absolute nodal coordinate formulation to large rotation and large deformation problems’, ASME Journal of Mechanical Design 120(2), 1998, 188–195.

    Google Scholar 

  37. Shabana, A. A. and Schwertassek, R., ‘Equivalence of the floating frame of reference approach and finite element formulations’, International Journal of Non-Linear Mechanics 33(3), 1998, 417–432.

    Google Scholar 

  38. Shabana, A. A. and Yakoub, R. Y., ‘Three dimensional absolute nodal coordinate formulation for beam elements: Theory’, ASME Journal of Mechanical Design 123, 2001, 606–613.

    Google Scholar 

  39. Simo, J. C., ‘A three-dimensional finite-strain rod model. Part II: Computational aspects’, Journal of Computer Method in Applied Mechanics and Engineering 58, 1986, 79–116.

    Google Scholar 

  40. Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – The plane case: Parts I and II’, ASME Journal of Applied Mechanics 53, 1986, 849–863.

    Google Scholar 

  41. Simo, J. C. and Vu-Quoc, L., ‘The role of non-linear theories in transient dynamic analysis of flexible structures’, Journal of Sound and Vibration 119(3), 1987, 487–508.

    Google Scholar 

  42. Sunada, W. H. and Dubowsky, S., ‘On the dynamic analysis and behavior of industrial robotic manipulators with elastic members’, ASME Journal of Mechanisms, Transmission, and Automation in Design 105(1), 1983, 42–51.

    Google Scholar 

  43. Sunada, W. H. and Dubowsky, S., ‘The application of finite element methods to dynamic analysis of flexible spatial and co-planar linkage systems’, ASME Journal of Mechanical Design 103, 1981, 643–651.

    Google Scholar 

  44. Takahashi, Y. and Shimizu, N., ‘Study on elastic forces of the absolute nodal coordinate formulation for deformable beams’, in Proceedings of the Second Symposium on Multibody Dynamics and Vibration, ASME Conferences, Paper No. DETC99/VIB-8203, 1999.

  45. Wu, S. C. and Haug, E. J., ‘Geometric non-linear substructuring for dynamics of flexible mechanical systems’, International Journal for Numerical Methods in Engineering 26, 1988, 2211–2226.

    Google Scholar 

  46. Yakoub, R. Y. and Shabana, A. A., ‘Three dimensional absolute nodal coordinate formulation for beam elements: Implementation and application’, ASME Journal of Mechanical Design 123, 2001, 614–621.

    Google Scholar 

  47. Yakoub, R. Y. and Shabana, A. A., ‘Use of cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems’, Nonlinear Dynamics 20, 1999, 267–282.

    Google Scholar 

  48. Yamada, K. and Kanoh, H., ‘Modeling of flexible systems’, Simulation 8(2), 1989, 24–33 [in Japanese].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Integrated Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan

    Riki Iwai

  2. Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa, 229-8558, Japan

    Nobuyuki Kobayashi

Authors
  1. Riki Iwai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nobuyuki Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

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 Dynamics 34, 207–232 (2003). https://doi.org/10.1023/B:NODY.0000014560.78333.76

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:NODY.0000014560.78333.76

Search

Navigation