Multiple impacts and Painlevé paradox

Bernard Brogliato
This talk will focus on the modelling of multiple impact phenomena between rigid bodies (or systems of rigid bodies), and on singularities due to Coulomb's friction with a focus on the Painlevé paradox.

The problem of multiple impacts is still an open issue in the realm of impact mechanics. We will try to introduce the basics of multiple impact modeling, in particular the fundamental reasons why multiple impacts are not a direct extension of single impacts, in the sense that the dissipation of the energy at the contact points is no longer sufficient to get realistic models: one has to incorporate also the energy dispersion that is mainly due to the wave propagation in the system. The classical example of chains of balls will serve as an illustration, which is worth studying in view of a better understanding of "simple" granular systems. This is also an opportunity to realize that the classical "textbooks" solution concerning the Newton's cradle, is quite far from the problem's real complexity. The case of the rocking block, a system that is widely used in the field of Earthquake Engineering for the prediction of building motion under earthquake excitations, will also be tackled.

Singularities are ubiquitous in Solid Mechanics, especially when Coulomb's friction is present. The so-called Painlevé paradoxes appear when unilateral contact is coupled with Coulomb's model of dry friction. We will review in detail the dynamics of the classical example of a rod sliding on a rough plane, that gives rise to a very and unexpectedly rich dynamical behaviour: tangential velocity jumps (impacts without collisions) due to inconsistencies (non-existence of the normal contact force), and singular ODEs. General theories fail to encompass these particular features and a tailored analysis is necessary. The numerical integration issue with implicit Euler methods will be also presented.

Lecture slides

References:
  • Z. Zhao, C. Liu, B. Brogliato, 2009 ``Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations'', Proceedings of the Royal Society A, Mathematical, Physical and Engineering Sciences, vol.465, no 2107, pp. 2267-2292, July.
  • C. Liu, Z. Zhao, B. Brogliato, 2009 ``Frictionless multiple impacts in multibody systems: Part II. Numerical algorithm and simulation results'', Proceedings of the Royal Society A, Mathematical, Physical and Engineering Sciences, vol.465, no 2101, pp.1-23, January.
  • C. Liu, Z. Zhao, B. Brogliato, 2008 ``Frictionless multiple impacts in multibody systems: Part I. Theoretical framework'', Proceedings of the Royal Society A, Mathematical, Physical and Engineering Sciences, vol.464, no 2100, pp.3193-3211, December.
  • C. Liu, Z. Zhao, B. Brogliato, 2008 ``Energy dissipation and dispersion effects in a granular media'', Physical Review E, vol.78, no 3, 031307, September.
  • B. Brogliato, H. Zhang, C. Liu, 2012 "Analysis of a generalized kinematic impact law for multibody-multicontact systems, with application to the planar rocking block and chains of balls", Multibody System Dynamics, DOI: 10.1007/s11044-012-9301-3.
  • F. Génot, B. Brogliato, 1999 "New results on Painlevé paradoxes", European Journal of Mechanics A/Solids, vol.18, pp.653-677.

  • The research reports corresponding to these papers are available here and here .