The equilibrium stability for a smooth and discontinuous oscillator with dry friction  

作　　者：Zhi-Xin Li Qing-Jie Cao Alain Léger 

机构地区：[1]School of Astronautics, Centre for Nonlinear Dynamics Research, Harbin Institute of Technology, Harbin 150001 China [2]Laboratoire de Mecanique et d' Acoustique, CNRS, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France

出　　处：《Acta Mechanica Sinica》2016年第2期309-319,共11页力学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant 11372082);the National Basic Research Program of China(Grant 2015CB057405)

摘　　要：In this paper,we investigate the equilibrium stability of a Filippov-type system having multiple stick regions based upon a smooth and discontinuous（SD） oscillator with dry friction.The sets of equilibrium states of the system are analyzed together with Coulomb friction conditions in both（ f_n,f_s） and（x,˙x） planes.In the stability analysis,Lyapunov functions are constructed to derive the instability for the equilibrium set of the hyperbolic type and La Salle＇s invariance principle is employed to obtain the stability of the nonhyperbolic type.Analysis demonstrates the existence of a thick stable manifold and the interior stability of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property,and also shows that the unstable manifolds of the hyperbolic-type are that of the endpoints with their saddle property.Numerical calculations show a good agreement with the theoretical analysis and an excellent efficien y of the approach for equilibrium states in this particular Filippov system.Furthermore,the equilibrium bifurcations are presented to demonstrate the transition between the smooth and the discontinuous regimes.In this paper,we investigate the equilibrium stability of a Filippov-type system having multiple stick regions based upon a smooth and discontinuous（SD） oscillator with dry friction.The sets of equilibrium states of the system are analyzed together with Coulomb friction conditions in both（ f_n,f_s） and（x,˙x） planes.In the stability analysis,Lyapunov functions are constructed to derive the instability for the equilibrium set of the hyperbolic type and La Salle＇s invariance principle is employed to obtain the stability of the nonhyperbolic type.Analysis demonstrates the existence of a thick stable manifold and the interior stability of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property,and also shows that the unstable manifolds of the hyperbolic-type are that of the endpoints with their saddle property.Numerical calculations show a good agreement with the theoretical analysis and an excellent efficien y of the approach for equilibrium states in this particular Filippov system.Furthermore,the equilibrium bifurcations are presented to demonstrate the transition between the smooth and the discontinuous regimes.

关 键 词：SD oscillator Equilibrium set Dry friction Coulomb’s cone 

分 类 号：O313.5[理学—一般力学与力学基础]