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Electrostatic actuators have major role in many MEMS devices, e.g. sensors, actuators. The amount of applied voltage to an electrostatic micro-actuator has a direct impact on the amplitude of deflection throughout the cantilever. This research aims to study the effect of the electrode length and the applied voltage on the amplitude of deflection of the micro-cantilever. Also, the resonant frequency for the cantilever was computed for full length and compared with simulation results. Finite element method, ANSYS was used as a simulation tool.

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. INDEX TERMS Boundary control, distributed parameter system (DPS), global sliding mode control (GSMC), method of lines (MOL), partial differential equation (PDE), Timoshenko beam.