Control of friction oscillator by Lyapunov redesign based on delayed state feedback

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.

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

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.

Transient Mould Friction Based on the Wavelet Theory

The mould friction is an important parameter reflecting the initial shell and mould lubrication conditions.The research of mould friction is very important to the optimization and developing of continuous casting.The measured mould friction under hydraulic oscillation mode was researched with wavelet analysis to reveal its time-frequency characteristics.Firstly,the mould friction signals under different production conditions were monitored and the mould friction was calculated.Then,mother wavelet function was selected from three wavelet functions which were chosen preliminary according to the characteristics of mould friction and wavelet theory.Through wavelet transformation,mould friction signal was projected onto the wavelet domain,and the time-frequency characteristics of mould friction under different production conditions were obtained and discussed.Mould friction under different production conditions such as different oscillation mode,casting speed fluctuation,increasing and decreasing stage of casting speed and breakout occurrence was reported in detail in the wavelet time-frequency maps.The characteristics of mould friction were reflected well through wavelet transformation which proved that wavelet analysis had a good feasibility for mould friction study and can further reveal the nature of mould friction.