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A Nonlinear Finite Element Framework for Flexible Multibody Dynamics: Rotationless Formulation and Energy-Momentum Conserving Discretization

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 12)

A uniform framework for rigid body dynamics and nonlinear structural dynamics is presented. The advocated approach is based on a rotationless formu lation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holo-nomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional mechanical systems.

Keywords

Multibody System Multibody Dynamics Holonomic Constraint Flexible Multibody System Spherical Pendulum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    .Bauchau OA, Bottasso CL (1999) On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems. Comput Meth Appl Mech Eng 169:61–79MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    .Bauchau OA, Choi JY, Bottasso CL (2002) On the modeling of shells in multi-body dynamics. Mult Syst Dyn 8:459–489MATHCrossRefGoogle Scholar
  3. 3.
    .Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New YorkMATHGoogle Scholar
  4. 4.
    .Betsch P (2005) The discrete null space method for the energy consistent inte gration of constrained mechanical systems. Part I: Holonomic constraints. Com put Methods Appl Mech Eng 194(50–52):5159–5190MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Betsch P, Hesch C (2007) Energy-momentum conserving schemes for frictionless dynamic contact problems. Part I: NTS method. In: Wriggers P, Nackenhorst U (eds) IUTAM Symposium on Computational Methods in Contact Mechanics. Volume 3 of IUTAM Bookseries, pp 77–96. SpringerGoogle Scholar
  6. 6.
    .Betsch P, Leyendecker S (2006) The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: Multibody dynamics. Int J Numer Meth Eng 67(4):499–552MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Betsch P, S¨anger N (2007) On the use of geometrically exact shells in a con serving framework for flexible multibody dynamics. In preparation.Google Scholar
  8. 8.
    .Betsch P, Steinmann P (2001) Conservation properties of a time FE method. Part II: Time-stepping schemes for nonlinear elastodynamics. Int J Numer Meth Eng 50:1931–1955MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    .Betsch P, Steinmann P (2001) Constrained integration of rigid body dynamics. Comput Methods Appl Mech Eng 191:467–488MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    .Betsch P, Steinmann P (2002) Conservation properties of a time FE method. Part III: Mechanical systems with holonomic constraints. Int J Numer Meth Eng 53:2271–2304MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    .Betsch P, Steinmann P (2002) A DAE approach to flexible multibody dynamics. Mult Syst Dyn 8:367–391MATHMathSciNetGoogle Scholar
  12. 12.
    .Betsch P, Steinmann P (2002) Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int J Numer Meth Eng 54:1775– 1788MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    .Betsch P, Uhlar S (2007) Energy-momentum conserving integration of multi-body dynamics. Mult Syst Dyn 17(4):243–289MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    .Bottasso CL, Borri M, Trainelli L (2001) Integration of elastic multibody sys tems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications. Comput Methods Appl Mech Eng 190:3701–3733CrossRefMathSciNetGoogle Scholar
  15. 15.
    .Bottasso CL, Borri M, Trainelli L (2002) Geometric invariance. Comput Mech 29(2):163–169MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    .Brank B, Korelc J, Ibrahimbegović A (2003) Dynamics and time-stepping schemes for elastic shells undergoing finite rotations. Comput & Struct 81(12): 1193–1210CrossRefGoogle Scholar
  17. 17.
    .Büchter N, Ramm E (1992) Shell theory versus degeneration — A comparison in large rotation finite element analysis. Int J Numer Methods Eng 34:39–59MATHCrossRefGoogle Scholar
  18. 18.
    .Crisfield MA (1991) Non-linear finite element analysis of solids and structures. Volume 1: Essentials. Wiley, New YorkGoogle Scholar
  19. 19.
    .Garíia de Jalón J (2007) Twenty-five years of natural coordinates. Mult Syst Dyn 18(1):15–33MATHCrossRefGoogle Scholar
  20. 20.
    .Géradin M, Cardona A (2001) Flexible multibody dynamics: A finite element approach. Wiley, New YorkGoogle Scholar
  21. 21.
    .Goicolea JM, Garcia Orden JC (2000) Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy-momentum schemes. Comput Meth Appl Mech Eng 188:789–804MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    .Gonzalez O (1999) Mechanical systems subject to holonomic constraints: Differential-algebraic formulations and conservative integration. Physica D 132: 165–174MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    .Gonzalez O, Simo JC (1996) On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Comput Methods Appl Mech Eng 134:197–222MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    .Göttlicher B, Schweizerhof K (2005) Analysis of flexible structures with occa sionally rigid parts under transient loading. Comput & Struct 83:2035–2051CrossRefGoogle Scholar
  25. 25.
    .Ibrahimbegović A, Mamouri S, Taylor RL, Chen AJ (2000) Finite element method in dynamics of flexible multibody systems: Modeling of holonomic constraints and energy conserving integration schemes. Multy Syst Dyn 4(2– 3): 195–223MATHCrossRefGoogle Scholar
  26. 26.
    .Jelenić G, Crisfield MA (2001) Dynamic analysis of 3D beams with joints in presence of large rotations. Comput Methods Appl Mech Engrg 190:4195–4230MATHCrossRefGoogle Scholar
  27. 27.
    .Kuhl D, Ramm E (1999) Generalized energy-momentum method for non-linear adaptive shell dynamics. Comput Meth Appl Mech Eng 178:343–366MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    .Kunkel P, Mehrmann V (2006) Differential-algebraic equations. European Mathematical Society, ZurichMATHGoogle Scholar
  29. 29.
    .Leyendecker S, Betsch P, Steinmann P (2008) The discrete null space method for the energy consistent integration of constrained mechanical systems. Part III: Flexible multibody dynamics. Mult Syst Dyn 19(1–2):45–72MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    .Muñoz J, Jelenić G, Crisfield MA (2003) Master-slave approach for the mod elling of joints with dependent degrees of freedom in flexible mechanisms. Com mun Numer Meth Eng 19:689–702MATHCrossRefGoogle Scholar
  31. 31.
    .Puso MA (2002) An energy and momentum conserving method for rigid-flexible body dynamics. Int J Numer Meth Eng 53:1393–1414MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    .Romero I (2004) The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput Mech 34:121–133MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    .Romero I, Armero F (2002) Numerical integration of the stiff dynamics of geometrically exact shells: An energy-dissipative momentum-conserving scheme. Int J Numer Meth Eng 54:1043–1086MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    .Romero I, Armero F (2002) An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. Int J Numer Meth Eng 54:1683–1716MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    .Rosenberg RM (1977) Analytical dynamics of discrete systems. Plenum, New YorkMATHGoogle Scholar
  36. 36.
    .Sansour C, Wagner W, Wriggers P, Sansour J (2002) An energy-momentum integration scheme and enhanced strain finite elements for the non-linear dy namics of shells. Non-linear Mech 37:951–966MATHCrossRefGoogle Scholar
  37. 37.
    .Simo JC, Rifai MS, Fox DD (1992) On a stress resultant geometrically exact shell model. Part VI: Conserving algorithms for non-linear dynamics. Int J Numer Meth Eng 34:117–164MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    .Simo J, Tarnow N (1992) The discretes energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z Angew Math Phys (ZAMP) 43:757– 792MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    .Simo JC, Tarnow N (1994) A new energy and momentum conserving algorithm for the nonlinear dynamics of shells. Int J Num Meth Eng 37:2527–2549MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    .Taylor RL (2001) Finite element analysis of rigid-flexible systems. In: Ambŕosio JAC, Kleiber M (eds) Computational aspects of nonlinear structural systems with large rigid body motion. Volume 179 of NATO Science Series: Computer & Systems Sciences, pp 63–84. IOS Press, AmsterdamGoogle Scholar
  41. 41.
    .Uhlar S, Betsch P (2007) On the rotationless formulation of multibody dy namics and its conserving numerical integration. In: Bottasso CL, Masarati P, Trainelli L (eds) Proceedings of the ECCOMAS Thematic Conference on Multi-body Dynamics, Politecnico di Milano, MilanoGoogle Scholar

Copyright information

© Springer Science + Business Media B.V 2009

Authors and Affiliations

  1. 1.Chair of Computational MechanicsUniversity of SiegenSiegenGermany

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