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This paper uses He's Homotopy Perturbation Method (HPM) to analyze the nonlinear free vibrational behavior of clamped-clamped and clamped-free microbeams considering the effects of rotary inertia and shear deformation. Galerkin's projection method is used to reduce the governing nonlinear partial differential equation. to a nonlinear ordinary differential equation. HPM is used to find analytic expressions for nonlinear natural frequencies of the pre-stretched microbeam. A parametric study investigated the effects of design parameters such as applied axial loads and slenderness ratio. The effect of rotary inertia and shear deformation on the nonlinear natural frequency was investigated. For verification, a numerical approach was implemented to solve the nonlinear equation. of vibration. A comparison between analytical and numerical results shows that HPM can predict system nonlinear vibrational behavior significantly more accurately than previously used methods in the literature.

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This paper applies the homotopy perturbation method to the simulation of the static response of nano-switches to electrostatic actuation and intermolecular surface forces. The model accounts for the electric force nonlinearity of the excitation and for the fringing field effect. Using a mode approximation in the Galerkin projection method, the nonlinear boundary value differential equation describing the statical behavior of nano-switch is reduced to a nonlinear algebraic equation which is solved using the homotopy perturbation method. The number of included terms in the perturbation expansion for achieving a reasonable response has been investigated. Three cases have been specifically studied. These cases correspond to when the effective external force is the electrostatic force, the combined electrostatic and Casimir force and the combined electrostatic and van der Waals force. In all three cases the pull-in characteristics has been investigated thoroughly. Results have been compared with numerical results and also analytical results available in the literature. It was found that HPM modifies the overestimation of N/MEMS instability limits reported in the literature and can be used as an effective and accurate design tool in the analysis of N/MEMS.

The objective of this paper is to present a supervised multi-level fuzzy controller to control the deflection of an electrostatically actuated microplate within and beyond its pull-in range. The mode shapes of the microplate are derived using Extended Kantorovich Method (EKM) which are shown to be in great agreement with finite element results. Using open loop simulations, it is shown that the first mode shape is effectively the dominant one. Then by utilizing a single mode approximation along with employing the Lagrange equation, the dynamic behavior of the microplate is described in modal space by an ordinary differential equation. By static and dynamic simulations, dependence of the plate deflection on the applied voltage is identified linguistically. Then based on the linguistic description of the system, a fuzzy controller is designed to stabilize the microplate at desired deflections. To improve the performance specifications of the closedloop system, another fuzzy controller at a higher level is proposed to adjust the parameters of the main controller in real time. To guarantee the stability of the closed-loop system, a non-fuzzy supervisory unit is attached to the control architecture. The simulations results reveal that by using the presented single level and supervised adaptive controllers, the control objective is met effectively with good performance specifications. It is also observed that adding a second level and a supervisory unit to the main controller can reduce the overshoot and the settling time for within and beyond pull-in stabilization of electrostatically actuated microplates in following the step commands. Excellent performance of the system in the presence of the proposed controller is further demonstrated using multiple step and also sinusoidal commands. The qualitative knowledge resulting from this research can be generalized and used for development of efficient controllers for N/MEMS actuators and electrostatically actuated nano/micro positioning systems.

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