Robust Global Boundary Vibration Control of Uncertain Timoshenko Beam With Exogenous Disturbances

In this paper, the dynamics solution problem and the boundary control problem for the Timoshenko beam under uncertainties and exogenous disturbances are addressed. The dynamics of the Timoshenko beam are represented by one non-homogenous hyperbolic partial differential equation (PDE), one homogenous hyperbolic PDE, and three ordinary differential equations (ODEs). The authors suggest a method of lines (MOL) for obtaining the dynamics of the Timoshenko beam in the form of the ODE formula instead of the PDE formula. A global sliding mode boundary control (GSMBC) is designed for vibration reduction of the Timoshenko beam influenced by uncertainties, distributed disturbance, displacement boundary disturbance, and rotation boundary disturbance. Chattering phenomena are avoided by using exponential reaching law reinforced by a relay function. Along the time and position axis: a) the boundary displacements and rotation of the Timoshenko beam are converged to equilibrium; b) the distributed vibrations on both displacements and rotation axis of the Timoshenko are attenuated; c) influence of the exogenous disturbances and system parameters uncertainties are compensated. By using the Lyapunov direct approach, exponential convergence and robustness of the closed-loop system are guaranteed. Finally, simulations are carried out to show that the proposed GSMBC-based MOL scheme is effective for vanishing the vibrations of the Timoshenko beam under uncertainties and external disturbances.