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Multibody System Dynamics

, Volume 8, Issue 4, pp 409–432 | Cite as

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

  • Stefan von Dombrowski
Article

Abstract

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.

This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.

Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.

Euler–Bernoulli beam large deformation large rotation multibody dynamics flexible bodies absolute nodal coordinate formulation 

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References

  1. 1.
    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
  2. 2.
    Shabana, A.A. and Yakoub, R.Y., 'An isoparametric three dimensional beam element using the absolute nodal coordinate formulation', Technical Report MBS97-1-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, 2000.Google Scholar
  3. 3.
    Sugiyama, H., Mikkola, A.M. and Shabana, A.A., 'A nonlinear finite element solution for cable problems using absolute nodal coordinate formulation', Technical ReportMBS02-1-UIC, University of Illinois at Chicago, Department of Mechanical Engineering, 2002.Google Scholar
  4. 4.
    Shabana, A.A., 'Definition of the slopes and the finite element absolute nodal coordinate formulation', Multibody System Dynamics 1(3), 1997, 339-348.Google Scholar
  5. 5.
    Pan, W. and Haug, E.J., 'Dynamic simulation of general flexible multibody systems', Mechanics of Structures & Machines 27(2), 1999, 217-251.Google Scholar
  6. 6.
    Christensen, A.P. and Shabana, A.A., 'Exact modeling of the spatial rigid body inertia using the finite element method', Department of Mechanical Engineering, University of Illinois at Chicago, 1997.Google Scholar
  7. 7.
    Escalona, J.L., Hussien, H.A. and Shabana, A.A., 'Application of the absolute nodal coordinate formulation to multibody system dynamics', Technical Report MBS97-1-UIC, Department of Mechanical Engineering, University of Illinois at Chicago, 1997.Google Scholar
  8. 8.
    Schwertassek, R. and Wallrapp, O., Dynamik flexibler Mehrkörpersysteme, Vieweg Verlag, Braunschweig, 1999.Google Scholar
  9. 9.
    Volterra, E., 'The equations of motion for curved elastic bars deduced by the use of the 'method of internal constraints', Ingenieur-Archiv XXIII, 1955, 402-409.Google Scholar
  10. 10.
    Volterra, E., 'The equations of motion for curved and twisted elastic bars deduced by the use of the 'method of internal constraints', Ingenieur-Archiv XXIV, 1956, 392-400.Google Scholar
  11. 11.
    Hodges, D.H., 'Nonlinear equations for dynamics of pretwisted beams undergoing small strains and large rotations', TP 2470, NASA, 1985.Google Scholar
  12. 12.
    Hodges, D.H. and Ormiston, R.A., 'On the nonlinear deformation geometry of Euler Bernoulli beams', TP 1566, NASA, 1980.Google Scholar
  13. 13.
    Houbolt, J.C. and Brooks, G.W., 'Differential equations of motion for combined flapwise bending', TN 3905, NASA, 1957.Google Scholar
  14. 14.
    Hiller, M., Krupp, T. and Schwertassek, R., 'Quantitativer Vergleich verschiedener Parametrisierungen von Drehbewegungen', Zeitschrift für angewandte Mathematik und Mechanik 73(4-5), 1993, T98-T100.Google Scholar
  15. 15.
    Shabana, A.A., 'Flexible multibody dynamics: Review of past and recent developments', Multibody System Dynamics 1(2), 1997, 189-222.Google Scholar
  16. 16.
    Shabana, A.A. and Christensen, A.P., 'Three dimensional absolute nodal co-ordinate formulation: Plate problem', International Journal for Numerical Methods in Engineering 40(15), 1997, 2775-2790.Google Scholar
  17. 17.
    Fischer, U. and Stephan, W., Prinzipien and Methoden der Dynamik, VEB Fachbuchverlag, Leipzig, 1972.Google Scholar
  18. 18.
    Budó, A., Theoretische Mechanik, 2nd edn., Hochschulbucher für Physik, Vol. 25, VEB Deutscher Verlag der Wissenschaften, 1963.Google Scholar
  19. 19.
    Shabana, A.A., 'Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics', Nonlinear Dynamics 16, 1998, 293-306.Google Scholar
  20. 20.
    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
  21. 21.
    Bremer, H. and Pfeiffer, F., Elastische Mehrkörpersysteme, Studienbücher Mechanik, Teubner, Stuttgart, 1992.Google Scholar
  22. 22.
    Nowak, U. and Weimann, L., 'A family of Newton codes for systems of highly nonlinear equations', Technical Report TR 91-10, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1991.Google Scholar
  23. 23.
    Hairer, E. and Wanner, G., 'Stiff differential equations solved by Radau methods', Journal of Computers and Applied Mathematics 111(1), 1999, 93-111.Google Scholar
  24. 24.
    Hairer, E. and Wanner, G., Solving Ordinary Differential Equations. II. Stiff and Differential Algebraic Problems, 2nd edn., Springer Series in Computers and Mathematics, Vol. 14, Springer-Verlag, Berlin, 1996.Google Scholar
  25. 25.
    Shi, P., McPhee, J. and Heppler, G.R., 'A deformation field for Euler-Bernoulli beams with applications to flexible multibody dynamics', Multibody System Dynamics 5, 2001, 79-104.Google Scholar
  26. 26.
    Valembrois, R.E., Fisette, P. and Samin, J.C., 'Comparison of various techniques for modelling flexible beams in multibody dynamics', Nonlinear Dynamics 12, 1997, 367-397.Google Scholar
  27. 27.
    Géradin, 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.S. Pereira and J.A.C. Ambrósio (eds), Kluwer Academic Publishers, Dordrecht, 1994, 233-284.Google Scholar
  28. 28.
    Simo, J.C., 'A finite strain beam formulation. The three-dimensional dynamic problem, Part I', Computer Methods in Applied Mechanics and Engineering 49, 1985, 55-70.Google Scholar
  29. 29.
    Simo, J.C. and Vu-Quoc, L., 'On the dynamics of flexible beams under large overall motions-The plane case', Journal of Applied Mechanics 53, 1986, 849-863.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Stefan von Dombrowski
    • 1
  1. 1.Institute of Robotics and Mechatronics, German Aerospace Center (DLR)WeßlingGermany

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