Multibody System Dynamics

, Volume 14, Issue 2, pp 137–154

Simulation of Unilateral Constrained Systems with Many Bodies

Article

Abstract

Nowadays the theory of multi-body systems including unilateral constraints is quite well established. However, the tendency towards more and more detailed and complex models may not be compensated with increasing computer power. In fact the growing computational effort demands for improved numerical methods in order to solve large systems. In this paper a time-stepping method is proposed for the computation of multi-body systems with many unilateral constraints. Stability and accuracy are discussed with respect to the given discretisation. In order to handle many contacts an iterative algorithm is applied based on a Gauss-Seidel relaxation scheme. A numerical example shows the efficiency of the relaxation scheme in comparison with Lemke's method and an Augmented Lagrangian approach.

Keywords

multi-body system unilateral constraints impacts friction time-stepping 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pfeiffer, F. and Glocker, Ch., Multibody Dynamics with Unilateral Contacts, Wiley Series in Nonlinear Science, Wiley, New York, 1996.Google Scholar
  2. 2.
    Jean, M., ‘The non-smooth contact dynamics method’, Computational Methods in Applied and Mechanical Engineering 177, 1999, 235–257.CrossRefGoogle Scholar
  3. 3.
    Stewart, D. E. and Trinkle, J. C., ‘An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction’, International Journal of Numerical Methods in Engineering 39, 1996, 2673–2691.CrossRefGoogle Scholar
  4. 4.
    Stiegelmeyr, A., ‘A time stepping algorithm for mechanical systems with unilateral contacts’, in Proceedings of DETC'99, ASME, Las Vegas, 1999.Google Scholar
  5. 5.
    Johansson, L., ‘A linear complementarity algorithm for rigid body impact with friction’, European Journal of Mechanics A/Solids 18, 1999, 703–717.CrossRefGoogle Scholar
  6. 6.
    Rockafellar, R. T., ‘Augmented Lagrangians and applications of the proximal point algorithm in convex programming’, Mathematics of Operations Research 1(2), 1976, 97–116.Google Scholar
  7. 7.
    Alart, P. and Curnier, A., ‘A mixed formulation for frictional contact problems prone to Newton like solution methods,’ Computer Methods in Applied Mechanics and Engineering 92, 1991, 353–375.CrossRefGoogle Scholar
  8. 8.
    Leine, R. I. and Glocker, Ch., ‘A set-valued force law for spatial Coulomb–Contensou friction’, in Proceedings of DETC'03, ASME, Chicago, 2003.Google Scholar
  9. 9.
    Glocker, Ch., Dynamik von Starrkörpersystemen mit Reibung und Stöen, Fortschritt-berichte VDI, Reihe 18, Nr. 182, VDI Verlag Düsseldorf, 1995.Google Scholar
  10. 10.
    Glocker, Ch., Set-Valued Force Laws, Dynamics of Non-Smooth Systems, Lecture Notes in Applied Mechanics, Vol. 1, Springer-Verlag, Berlin, 2001.Google Scholar
  11. 11.
    Moreau, J. J., Unilateral Contact and Dry Friction in Finite Freedom Dynamics, Volume 302 of International Centre for Mechanical Sciences, Courses and Lectures. J.J. Moreau P.D. Panagiotopoulos, Springer, Vienna, 1988.Google Scholar
  12. 12.
    Stiegelmeyr, A., Zur numerischen Berechnung strukturvarianter Mehrkörpersysteme, VDI Verlag, Reihe 18, No. 271, Düsseldorf, 2001.Google Scholar
  13. 13.
    Cottle, R. W., Pang, J.-S. and Stone, R. E., The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, San Diego, 1992.Google Scholar
  14. 14.
    Moreau, J. J., ‘Some numerical methods in multibody dynamics: Application to granular materials’, European Journal of Mechanics A/Solids 13(4), Suppl., Special Issue; 2nd European Solid Mechanics Conference Euromech, Genoa, Italy, 1994, pp. 93–114.Google Scholar
  15. 15.
    Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer Verlag, Berlin, 1991.Google Scholar
  16. 16.
    Funk, K., Simulation eindimensionaler Kontinua mit Unstetigkeiten, Fortschritt-berichte VDI, Reihe 18, Nr. 294, VDI Verlag Düsseldorf, 2003.Google Scholar
  17. 17.
    Funk, K. and Pfeiffer, F., ‘A time-stepping algorithm for stiff mechanical systems with unilateral constraints’, in Proceedings of the International Conference on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, pp. 307–314, Thessaloniki, 2002.Google Scholar
  18. 18.
    Arnold, M., Zur Theorie und zur numerischen Lösung von Anfangswertproblemen für differential-algebraische Systeme von höherem Index, Fortschritt-berichte VDI, Reihe 20, Nr. 264, VDI Verlag Düsseldorf, 1998.Google Scholar
  19. 19.
    Rockafellar, R. T., Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1970.Google Scholar
  20. 20.
    Hestenes, M., ‘Multiplier and gradient methods’, Journal of Optical Theory Applications 4, 303–320, 1969.CrossRefGoogle Scholar
  21. 21.
    Powell, B. T., ‘A method for nonlinear constraints in minimization problems’, in Optimization, Ed. R. Fletcher, Academic Press, London, 1969.Google Scholar
  22. 22.
    Dennis, J. E. and Schable, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1983.Google Scholar
  23. 23.
    ATLAS, Automatically Tuned Linear Algebra Software package, http://math-atlas.source-forge.net.

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Martin Förg
    • 1
  • Friedrich Pfeiffer
    • 1
  • Heinz Ulbrich
    • 1
  1. 1.Institute of Applied MechanicsTechnical University of MunichMunichGermany

Personalised recommendations