The maximum dissipation principle in rigid-body dynamics with inelastic impacts
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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.
KeywordsImpulse (physics) Coulomb friction Collisions (physics) Rigid body dynamics Contact dynamics Impact dynamics Measure differential inclusions Complementarity problems
The authors would like to acknowledge the support through the Cluster of Excellence Engineering of Advanced Materials (EAM).
- 3.Bonnefon O, Daviet G (2011) Quartic formulation of coulomb 3D frictional contact. Technical report RT-0400. INRIAGoogle Scholar
- 4.Diebel J (2006) representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix 58:1–35Google Scholar
- 5.Erleben K (2004) Stable, robust, and versatile multibody dynamics animation. PhD thesis. University of CopenhagenGoogle Scholar
- 13.Mirtich B (1996) Impulse-based dynamic simulation of rigid body systems. PhD thesis. University of CaliforniaGoogle Scholar
- 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–ffGoogle Scholar
- 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 1–82Google Scholar
- 21.Preclik T (2014) Models and algorithms for ultrascale simulations of non-smooth granular dynamics. PhD thesis. Friedrich-Alexander-Universität Erlangen-NürnbergGoogle Scholar
- 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 153–162Google Scholar
- 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 169–181Google Scholar