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Multibody System Dynamics

, Volume 19, Issue 1–2, pp 45–72 | Cite as

The discrete null space method for the energy-consistent integration of constrained mechanical systems. Part III: Flexible multibody dynamics

  • Sigrid Leyendecker
  • Peter Betsch
  • Paul Steinmann
Article

Abstract

In the present work, the unified framework for the computational treatment of rigid bodies and nonlinear beams developed by Betsch and Steinmann (Multibody Syst. Dyn. 8, 367–391, 2002) is extended to the realm of nonlinear shells. In particular, a specific constrained formulation of shells is proposed which leads to the semi-discrete equations of motion characterized by a set of differential-algebraic equations (DAEs). The DAEs provide a uniform description for rigid bodies, semi-discrete beams and shells and, consequently, flexible multibody systems. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. The present approach thus circumvents the use of rotational variables throughout the whole time discretization, facilitating the design of energy–momentum methods for flexible multibody dynamics. After the discretization has been completed a size-reduction of the discrete system is performed by eliminating the constraint forces. Numerical examples dealing with a spatial slider-crank mechanism and with intersecting shells illustrate the performance of the proposed method.

Keywords

Conserving time integration Constrained mechanical systems Flexible multibody dynamics Nonlinear structural dynamics Differential-algebraic equations 

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References

  1. 1.
    Angeles, J., Lee, S.: The modeling of holonomic mechanical systems using a natural orthogonal complement. Trans. Can. Soc. Mech. Eng. 13(4), 81–89 (1989) Google Scholar
  2. 2.
    Antmann, S.S.: Nonlinear Problems in Elasticity. Springer, Berlin (1995) Google Scholar
  3. 3.
    Bauchau, O.A., Choi, J.-Y.: On the modeling of shells in multibody dynamics. Multibody Dyn. Syst. 459–489 (2002) Google Scholar
  4. 4.
    Belytschko, T., Liu, W.K., Moran, B.: Nonlinear Finite Elements for Continua and Structures. Wiley, New York (2000) zbMATHGoogle Scholar
  5. 5.
    Betsch, P.: A unified approach to the energy consistent numerical integration of nonholonomic mechanical systems and flexible multibody dynamics. GAMM Mitt. 27(1), 66–87 (2004) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Betsch, P.: The discrete null space method for the energy consistent integration of constrained mechanical systems. Part I: Holonomic constraints. Comput. Methods Appl. Mech. Eng. 194(50–52), 5159–5190 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Betsch, P., Leyendecker, S.: The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: Multibody dynamics. Int. J. Numer. Methods Eng. 67(4), 499–552 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Betsch, P., Steinmann, P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191, 467–488 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Betsch, P., Steinmann, P.: Conserving properties of a time FE method. Part III: Mechanical systems with holonomic constraints. Int. J. Numer. Methods Eng. 53, 2271–2304 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Betsch, P., Steinmann, P.: A DAE approach to flexible multibody dynamics. Multibody Syst. Dyn. 8, 367–391 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Betsch, P., Steinmann, P.: Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int. J. Numer. Methods Eng. 54, 1775–1788 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bottasso, C.L., Borri, M., Trainelli, L.: Integration of elastic multibody systems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications. Comput. Methods Appl. Mech. Eng. 190, 3701–3733 (2001) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Brank, B., Korelc, J., Ibrahimbegović, A.: Dynamics and time-stepping schemes for elastic shells undergoing finite rotations. Comput. Struct. 81, 1193–1210 (2003) CrossRefGoogle Scholar
  14. 14.
    Büchter, N., Ramm, E.: Shell theory versus degeneration—a comparison in large rotation finite element analysis. Int. J. Numer. Methods Eng. 34, 39–59 (1992) zbMATHCrossRefGoogle Scholar
  15. 15.
    Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures. Vol. I: Essentials. Wiley, New York (1991) Google Scholar
  16. 16.
    Crisfield, M.A., Jelenić, G.: Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc. Roy. Soc. Lond. A 455, 1125–1147 (1999) zbMATHCrossRefGoogle Scholar
  17. 17.
    Géradin, M., Cardona, A.: Flexible Multibody Dynamics. Wiley, New York (2001) Google Scholar
  18. 18.
    Goicolea, J.M., Orden, J.C.: Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy–momentum schemes. Comput. Methods Appl. Mech. Eng. 188, 789–804 (2000) zbMATHCrossRefGoogle Scholar
  19. 19.
    Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gonzalez, O.: Mechanical systems subject to holonomic constraints: differential-algebraic formulations and conservative integration. Phys. D 132, 165–174 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Göttlicher, B.: Effiziente Finite-Element-Modellierung gekoppeler starrer und flexibler Strukturbereiche bei transienten Einwirkungen. PhD thesis, Universität Karlsruhe (2002) Google Scholar
  22. 22.
    Hughes, T.J.R.: The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover, New York (2000) Google Scholar
  23. 23.
    Hughes, T.J.R., Liu, W.K.: Nonlinear finite element analysis of shells. Part I: Three-dimensional shells. Comput. Methods Appl. Mech. Eng. 26, 331–362 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Ibrahimbegović, A., Mamouri, S.: Finite rotations in dynamics of beams and implicit time-stepping schemes. Int. J. Numer. Methods Eng. 41, 781–814 (1998) CrossRefzbMATHGoogle Scholar
  25. 25.
    Ibrahimbegović, A., Mamouri, S., Taylor, R.L., Chen, A.J.: Finite element method in dynamics of flexible multibody systems: modeling of holonomic constraints and energy conserving integration schemes. Multibody Syst. Dyn. 4, 195–223 (2000) CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Jelenić, G., Crisfield, M.A.: Interpolation of rotational variables in non-linear dynamics of 3d beams. Int. J. Numer. Methods. Eng. 43, 1193–1222 (1998) CrossRefzbMATHGoogle Scholar
  27. 27.
    Jelenić, G., Crisfield, M.A.: Dynamic analysis of 3D beams with joints in presence of large rotations. Comput. Methods Appl. Mech. Eng. 190, 4195–4230 (2001) CrossRefzbMATHGoogle Scholar
  28. 28.
    Kuhl, D., Ramm, E.: Generalized energy–momentum method for non-linear adaptive shell dynamics. Comput. Methods Appl. Mech. Eng. 178, 343–366 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Leyendecker, S., Betsch, P., Steinmann, P.: Energy-conserving integration of constrained Hamiltonian systems—a comparison of approaches. Comput. Mech. 33, 174–185 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Leyendecker, S., Betsch, P., Steinmann, P.: Objective energy–momentum conserving integration for the constrained dynamics of geometrically exact beams. Comput. Methods Appl. Mech. Eng. 195, 2313–2333 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems. Texts in Applied Mathematics, vol. 17. Springer, Berlin (1994) zbMATHGoogle Scholar
  32. 32.
    Munoz, J., Jelenić, G., Crisfield, M.A.: Master-slave approach for the modeling of joints with dependent degrees of freedom in flexible mechanisms. Commun. Numer. Methods Eng. 19, 689–702 (2003) zbMATHCrossRefGoogle Scholar
  33. 33.
    Puso, M.A.: An energy and momentum conserving method for rigid-flexible body dynamics. Int. J. Numer. Methods Eng. 53, 1393–1414 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Romero, I., Armero, F.: Numerical integration of the stiff dynamics of geometrically exact shells: an energy-dissipative momentum-conserving scheme. Int. J. Numer. Methods Eng. 54, 1043–1086 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Romero, I., Armero, F.: An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum scheme in dynamics. Int. J. Numer. Methods Eng. 54, 1683–1716 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Sansour, J., Wagner, W., Wriggers, P.: An energy–momentum integration scheme and enhanced strain finite elements for the non-linear dynamics of shells. Nonlinear Mech. 37, 951–966 (2002) zbMATHCrossRefGoogle Scholar
  37. 37.
    Simo, J.C.: On a stress resultant geometrically exact shell model. Part VII: Shell intersections with 5/6-DOF finite element formulations. Comput. Methods Appl. Mech. Eng. 108, 319–339 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Simo, J.C., Rifai, M.S., Fox, D.D.: On a stress resultant geometrically exact shell model. Part VI: Conserving algorithms for non-linear dynamics. Int. J. Numer. Methods Eng. 34, 117–164 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Simo, J.C., Tarnow, N.: A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int. J. Numer. Methods Eng. 37, 2527–2549 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Taylor, R.L.: Finite element analysis of rigid-flexible systems. In: Ambrósio, J.A.C., Kleiber, M. (eds.) Computational Aspects of Nonlinear Structural Systems with Large Rigid Body Motion, vol. 179, pp. 62–84. IOS, Amsterdam (2001) Google Scholar
  41. 41.
    Warburton, G.B.: The Dynamical Behaviour of Structures. Pergamon, Elmsford (1976) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Sigrid Leyendecker
    • 1
  • Peter Betsch
    • 2
  • Paul Steinmann
    • 1
  1. 1.Chair of Applied Mechanics, Department of Mechanical EngineeringUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Chair of Computational Mechanics, Department of Mechanical EngineeringUniversity of SiegenSiegenGermany

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