The maximum dissipation principle in rigid-body dynamics with inelastic impacts

Abstract

Formulating a consistent theory for rigid-body dynamics with impacts is an intricate problem. Twenty years ago Stewart published the first consistent theory with purely inelastic impacts and an impulsive friction model analogous to Coulomb friction. In this paper we demonstrate that the consistent impact model can exhibit multiple solutions with a varying degree of dissipation even in the single-contact case. Replacing the impulsive friction model based on Coulomb friction by a model based on the maximum dissipation principle resolves the non-uniqueness in the single-contact impact problem. The paper constructs the alternative impact model and presents integral equations describing rigid-body dynamics with a non-impulsive and non-compliant contact model and an associated purely inelastic impact model maximizing dissipation. An analytic solution is derived for the single-contact impact problem. The models are then embedded into a time-stepping scheme. The macroscopic behaviour is compared to Coulomb friction in a large-scale granular flow problem.

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References

  1. 1.

    Anitescu M, Potra F (1997) Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn 14(3):231–247

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bannerman MN, Sargant R, Lue L (2011) DynamO: a free O(N) general event-driven molecular dynamics simulator. J Comput Chem 32(15):3329–3338

    Article  Google Scholar 

  3. 3.

    Bonnefon O, Daviet G (2011) Quartic formulation of coulomb 3D frictional contact. Technical report RT-0400. INRIA

  4. 4.

    Diebel J (2006) representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix 58:1–35

    Google Scholar 

  5. 5.

    Erleben K (2004) Stable, robust, and versatile multibody dynamics animation. PhD thesis. University of Copenhagen

  6. 6.

    Gavrea BI, Anitescu M, Potra FA (2008) Convergence of a class of semi-implicit timestepping schemes for nonsmooth rigid multibody dynamics. SIAM J Optim 19(2):969–1001

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Hassanpour A et al (2011) Analysis of particle motion in a paddle mixer using discrete element method (DEM). Powder Technol 206(1):189–194

    Article  Google Scholar 

  8. 8.

    Jayasundara C et al (2011) CFD-DEM modelling of particle flow in IsaMills–Comparison between simulations and PEPT measurements. Miner Eng 24(3):181–187

    Article  Google Scholar 

  9. 9.

    Jean M (1999) The non-smooth contact dynamics method. Comput Methods Appl Mech Eng 177(3):235–257

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Jia Y-B (2013) Three-dimensional impact: energybased modeling of tangential compliance. Int J Robot Res 32(1):56–83

    Article  Google Scholar 

  11. 11.

    Jia Y-B, Wang F (2016) Analysis and computation of two body impact in three dimensions. J Comput Nonlinear Dyn 12:041012

    Article  Google Scholar 

  12. 12.

    Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52(3):225–237

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Mirtich B (1996) Impulse-based dynamic simulation of rigid body systems. PhD thesis. University of California

  14. 14.

    Mirtich B, Canny J (1995) Impulse-based simulation of rigid bodies. In: Proceedings of the 1995 symposium on interactive 3D graphics. ACM, pp 181–ff

  15. 15.

    Mishra B, Rajamani R (1992) The discrete element method for the simulation of ball mills. Appl Math Model 16(11):598–604

    Article  MATH  Google Scholar 

  16. 16.

    Mitarai N, Nakanishi H (2012) Granular flow: dry and wet. Eur Phys J Spec Topic 204(1):5–17

    Article  Google Scholar 

  17. 17.

    Moreau JJ (1988) Unilateral contact and dry friction in finite freedom dynamics. In: Moreau JJ, Panagiotopoulos PD (eds) Nonsmooth mechanics and applications. Springer, New York, pp 1–82

  18. 18.

    Nuseirat AA-F, Stavroulakis G (2000) A complementarity problem formulation of the frictional grasping problem. Comput Methods Appl Mech Eng 190:941–952

    Article  MATH  Google Scholar 

  19. 19.

    Painlevé P (1895) Sur les lois du frottement de glissement. C R Acad Sci Paris 121:112–115

    MATH  Google Scholar 

  20. 20.

    Popa C, Preclik T, Rüde U (2015) Regularized solution of LCP problems with application to rigid body dynamics. Numer Algorithms 69(1):145–156

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Preclik T (2014) Models and algorithms for ultrascale simulations of non-smooth granular dynamics. PhD thesis. Friedrich-Alexander-Universität Erlangen-Nürnberg

  22. 22.

    Preclik T, Rüde U (2015) Ultrascale simulations of non-smooth granular dynamics. Comput Part Mech 2(2):173–196

    Article  Google Scholar 

  23. 23.

    Sauer J, Schömer E (1998) A constraint-based approach to rigid body dynamics for virtual reality applications. In: Proceedings of the ACM symposium on virtual reality software and technology, pp 153–162

  24. 24.

    Shen Y, Stronge W (2011) Painlevé paradox during oblique impact with friction. Eur J Mech A/Solids 30(4):457–467

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Stewart DE (1998) Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem. Arch Ration Mech Anal 145(3):215–260

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Stewart DE (2011) Dynamics with inequalities: impacts and hard constraints. SIAM, Philadelphia

    Google Scholar 

  27. 27.

    Stewart D (2000) Rigid-body dynamics with friction and impact. SIAM Rev 42(1):3–39

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Stewart D, Trinkle J (1996) An implicit timestepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. Int J Numer Methods Eng 39(15):2673–2691

    Article  MATH  Google Scholar 

  29. 29.

    Stronge WJ (1990 ) Rigid body collisions with friction. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences, vol. 431, no. 1881. The Royal Society, pp 169–181

  30. 30.

    Stronge WJ (2004) Impact mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  31. 31.

    Tasora A, Anitescu M (2010) A convex complementarity approach for simulating large granular flows. J Comput Nonlinear Dyn 5(3):1–10

    Article  MATH  Google Scholar 

  32. 32.

    Tasora A, Anitescu M (2011) A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput Methods Appl Mech Eng 200(5):439–453

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the support through the Cluster of Excellence Engineering of Advanced Materials (EAM).

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Correspondence to Tobias Preclik.

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Preclik, T., Eibl, S. & Rüde, U. The maximum dissipation principle in rigid-body dynamics with inelastic impacts. Comput Mech 62, 81–96 (2018). https://doi.org/10.1007/s00466-017-1486-0

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Keywords

  • Impulse (physics)
  • Coulomb friction
  • Collisions (physics)
  • Rigid body dynamics
  • Contact dynamics
  • Impact dynamics
  • Measure differential inclusions
  • Complementarity problems