# American Institute of Mathematical Sciences

2016, 9(2): 501-527. doi: 10.3934/dcdss.2016009

## On the equilibria and qualitative dynamics of a forced nonlinear oscillator with contact and friction

 1 Laboratoire de Mécanique et d'Acoustique, LMA, CNRS, UPR 7051, Aix-Marseille Univ., Centrale Marseille, F-13402 Marseille Cedex 20, France, France

Received  December 2014 Revised  October 2015 Published  March 2016

After previous works related to the equilibrium states, this paper goes deeper into the study of the effect of coupling between smooth and non-smooth non-linearities on the qualitative behavior of low dimensional dynamical systems. The non-smooth non-linearity is due to non-regularized unilateral contact and Coulomb friction while the smooth one is due to large strains of a simple mass spring system, which lead to a nonlinear restoring force. The main qualitative differences with the case of a linear restoring force are due to the shape of the set of equilibrium states.
Citation: Alain Léger, Elaine Pratt. On the equilibria and qualitative dynamics of a forced nonlinear oscillator with contact and friction. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 501-527. doi: 10.3934/dcdss.2016009
##### References:
 [1] S. Basseville, A. Léger and E. Pratt, Investigation of the equilibrium states and their stability for a simple model with unilateral contact and Coulomb friction,, Arch. Appl. Mech., 73 (2003), 409. doi: 10.1007/s00419-003-0300-y. [2] Q. J. Cao, M. Wiercigroch, E. Pavvlovskaia, C. Grebogi, J. Thompson, An archetypal oscillator for smooth and discontinuous dynamics,, Phys. Review, 74 (2006). doi: 10.1103/PhysRevE.74.046218. [3] Q. J. Cao, A. Léger and Z. X. Li, The equilibrium stability of a smooth to discontinous oscillator with dry friction,, J. of Computational and Nonlinear Dynamics, (2013). [4] A. Charles and P. Ballard, Existence and uniqueness of solution to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points,, Mathematical Modelling and Numerical Analysis, 48 (2014), 1. doi: 10.1051/m2an/2013092. [5] A. Cimetière and A. Léger, Some problems about elastic-plastic post-buckling,, Int. J. Solids Structures, 32 (1996), 1519. [6] M. Jean, The nonsmooth contact dynamics method,, Computer Methods Appl. Mech. Engn, 177 (1999), 235. doi: 10.1016/S0045-7825(98)00383-1. [7] A. Klarbring, Examples of nonuniqueness and nonexistence of solutions to quasistatic contact problems with friction,, Ing. Arch., 60 (1990), 529. [8] A. Léger and E. Pratt, Qualitative analysis of a forced nonsmooth oscillator with contact and friction,, Annals of Solid and Structural Mechanics, 2 (2011), 1. [9] A. Léger, E. Pratt and Q. J. Cao, A fully nonlinear oscillator with contact and friction,, Nonlinear Dynamics, 70 (2012), 511. doi: 10.1007/s11071-012-0471-6. [10] J. J. Moreau, Unilateral contact and dry friction in finite freedom dynamics,, in Nonsmooth Mechanics and Applications (eds. J. J. Moreau and P. D. Panagiotopoulos), (1988), 1. doi: 10.1007/978-3-7091-2624-0_1.

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##### References:
 [1] S. Basseville, A. Léger and E. Pratt, Investigation of the equilibrium states and their stability for a simple model with unilateral contact and Coulomb friction,, Arch. Appl. Mech., 73 (2003), 409. doi: 10.1007/s00419-003-0300-y. [2] Q. J. Cao, M. Wiercigroch, E. Pavvlovskaia, C. Grebogi, J. Thompson, An archetypal oscillator for smooth and discontinuous dynamics,, Phys. Review, 74 (2006). doi: 10.1103/PhysRevE.74.046218. [3] Q. J. Cao, A. Léger and Z. X. Li, The equilibrium stability of a smooth to discontinous oscillator with dry friction,, J. of Computational and Nonlinear Dynamics, (2013). [4] A. Charles and P. Ballard, Existence and uniqueness of solution to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points,, Mathematical Modelling and Numerical Analysis, 48 (2014), 1. doi: 10.1051/m2an/2013092. [5] A. Cimetière and A. Léger, Some problems about elastic-plastic post-buckling,, Int. J. Solids Structures, 32 (1996), 1519. [6] M. Jean, The nonsmooth contact dynamics method,, Computer Methods Appl. Mech. Engn, 177 (1999), 235. doi: 10.1016/S0045-7825(98)00383-1. [7] A. Klarbring, Examples of nonuniqueness and nonexistence of solutions to quasistatic contact problems with friction,, Ing. Arch., 60 (1990), 529. [8] A. Léger and E. Pratt, Qualitative analysis of a forced nonsmooth oscillator with contact and friction,, Annals of Solid and Structural Mechanics, 2 (2011), 1. [9] A. Léger, E. Pratt and Q. J. Cao, A fully nonlinear oscillator with contact and friction,, Nonlinear Dynamics, 70 (2012), 511. doi: 10.1007/s11071-012-0471-6. [10] J. J. Moreau, Unilateral contact and dry friction in finite freedom dynamics,, in Nonsmooth Mechanics and Applications (eds. J. J. Moreau and P. D. Panagiotopoulos), (1988), 1. doi: 10.1007/978-3-7091-2624-0_1.
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