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

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April 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

Alain Léger ^1, and Elaine Pratt ^1,

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

Abstract
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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.

Keywords: Coulomb friction, unilateral contact., mass-spring systems, non-smooth dynamics, equilibrium states, SD oscillator.

Mathematics Subject Classification: Primary: 70K42, 70K05; Secondary: 37M9.

Citation: Alain Léger, Elaine Pratt. On the equilibria and qualitative dynamics of a forced nonlinear oscillator with contact and friction. Discrete and 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), 409-420. 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), 046218, 5pp. 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), 1-25. doi: 10.1051/m2an/2013092.
[5]	A. Cimetière and A. Léger, Some problems about elastic-plastic post-buckling, Int. J. Solids Structures, 32 (1996), 1519-1533.
[6]	M. Jean, The nonsmooth contact dynamics method, Computer Methods Appl. Mech. Engn, 177 (1999), 235-257. 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), 529-541.
[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), 1-17.
[9]	A. Léger, E. Pratt and Q. J. Cao, A fully nonlinear oscillator with contact and friction, Nonlinear Dynamics, 70 (2012), 511-522. 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), CISM Courses and Lectures, 302, Springer-Verlag, Vienne-New York, 1988, 1-82. doi: 10.1007/978-3-7091-2624-0_1.

show all references

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), 409-420. 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), 046218, 5pp. 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), 1-25. doi: 10.1051/m2an/2013092.
[5]	A. Cimetière and A. Léger, Some problems about elastic-plastic post-buckling, Int. J. Solids Structures, 32 (1996), 1519-1533.
[6]	M. Jean, The nonsmooth contact dynamics method, Computer Methods Appl. Mech. Engn, 177 (1999), 235-257. 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), 529-541.
[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), 1-17.
[9]	A. Léger, E. Pratt and Q. J. Cao, A fully nonlinear oscillator with contact and friction, Nonlinear Dynamics, 70 (2012), 511-522. 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), CISM Courses and Lectures, 302, Springer-Verlag, Vienne-New York, 1988, 1-82. doi: 10.1007/978-3-7091-2624-0_1.

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