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...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.展开更多
The stability and boundedness of mechanical system have been one of important research topics. In this paper ultimate boundedness of a dry friction oscillator, belonging to nonsmooth mechanical system, is investigated...The stability and boundedness of mechanical system have been one of important research topics. In this paper ultimate boundedness of a dry friction oscillator, belonging to nonsmooth mechanical system, is investigated by proposing a controller design method. Firstly a sufficient condition of the stability for the nominal system with delayed state feedback is derived by constructing a Lyapunov-Krasovskii function. The delayed feedback gain matrix is calculated by applying linear matrix inequality method. Secondly on the basis of the delayed state feedback, a continuous function is designed by Lyapunov redesign to ensure that the solutions of the friction oscillator system are ultimately bounded under the overall control. Moreover, the ultimate bound can be adjusted in practice by choosing appropriate parameter. Accordingly friction-induced vibration or instability can be controlled effectively. Numerical results show that the pro- posed method is valid.展开更多
基金supported by the National Natural Science Foundation of China(Grant 11372082)the National Basic Research Program of China(Grant 2015CB057405)
文摘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.
基金supported by the National Natural Science Foundation of China (10672007)
文摘The stability and boundedness of mechanical system have been one of important research topics. In this paper ultimate boundedness of a dry friction oscillator, belonging to nonsmooth mechanical system, is investigated by proposing a controller design method. Firstly a sufficient condition of the stability for the nominal system with delayed state feedback is derived by constructing a Lyapunov-Krasovskii function. The delayed feedback gain matrix is calculated by applying linear matrix inequality method. Secondly on the basis of the delayed state feedback, a continuous function is designed by Lyapunov redesign to ensure that the solutions of the friction oscillator system are ultimately bounded under the overall control. Moreover, the ultimate bound can be adjusted in practice by choosing appropriate parameter. Accordingly friction-induced vibration or instability can be controlled effectively. Numerical results show that the pro- posed method is valid.
