Comptes Rendus
no. 3
The legacy of Jean-Jacques Moreau in mechanics
Dynamics of a particle with friction and delay
Manuel D.P. Monteiro Marques1 ; Raoul Dzonou2
1 Faculdade de Ciências, D.Mat. et C.M.A.F.-C.I.O., Universidade de Lisboa, Portugal
2 1 bis, rue de Craonne, 64000 Pau, France
Comptes Rendus. Mécanique, Volume 346 (2018) no. 3, pp. 237-246.

We are interested in the motion of a simple mechanical system having a finite number of degrees of freedom subjected to a unilateral constraint with dry friction and delay effects (with maximal duration τ>0). At the contact point, we characterize the friction by a Coulomb law associated with a friction cone. Starting from a formulation of the problem that was given by Jean-Jacques Moreau in the form of a second-order differential inclusion in the sense of measures, we consider a sweeping process algorithm that converges towards a solution to the dynamical contact problem. The mathematical machinery as well as the general plan of the existence proof may seem much too heavy in order to treat just this simple case, but they have proved useful in more complex settings.

Nous nous intéressons au mouvement d'un système mécanique ayant un nombre fini de degrés de liberté soumis à une contrainte unilatérale avec frottement sec et des forces qui peuvent dépendre de l'histoire du mouvement avec facteur de retard τ. Au contact, nous caractérisons le frottement par une loi de Coulomb associée à un cône de frottement, en suivant la formulation du problème proposée par Jean-Jacques Moreau sous la forme d'une inclusion différentielle du second ordre au sens des mesures (la réaction et l'accéleration pouvant être des mesures). Un algorithme de type « sweeping process » permet de montrer l'existence d'une suite qui converge vers une solution du problème de contact dynamique. L'outillage mathématique ainsi que la démarche de la preuve semblent trop lourds pour traiter ce problème plutôt simple, mais les deux peuvent être utiles dans des cadres plus complexes.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2017.12.008
Keywords: Contact with friction and delay, Numerical methods, Sweeping process algorithm
Mot clés : Contact avec frottement et retard, Méthodes numériques, Sweeping process

Manuel D.P. Monteiro Marques 1; Raoul Dzonou 2

1 Faculdade de Ciências, D.Mat. et C.M.A.F.-C.I.O., Universidade de Lisboa, Portugal
2 1 bis, rue de Craonne, 64000 Pau, France 
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Manuel D.P. Monteiro Marques; Raoul Dzonou. Dynamics of a particle with friction and delay. Comptes Rendus. Mécanique, Volume 346 (2018) no. 3, pp. 237-246. doi : 10.1016/j.crme.2017.12.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.12.008/

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