Analysis of Electrostatically Actuated Narrow Microcantilevers Using Rayleigh-Ritz Energy Technique

Abstract

The operation of electrostatically actuated MEMS devices involves an inherent phenomenon termed as the pull-in instability which reduces their useful travel range. Accurate prediction of pull-in parameters (pull-in displacement and pull-in voltage) is hence vital in the design of such microdevices. In this article, we present a complete nondimensional formulation and implementation of an efficient numerical scheme based on the Rayleigh-Ritz energy technique to determine the static pull-in parameters of an electrostatically actuated narrow microcantilever beam. Deflection of the beam is approximated by an admissible polynomial function that satisfies its mechanical boundary conditions. The principle of stationary potential energy and the condition of instability are then applied to obtain a highly nonlinear algebraic equation which is solved using the Fibonacci minimization algorithm. An optimum number of Gauss quadrature points is used to integrate the nonlinear electrostatic terms. Two mathematical models accounting for the fringing field capacitance are examined in turn. A comparison is made between the normalized values of pull-in parameters obtained by considering the two aforementioned fringing field models. Two examples of electrostatically actuated narrow microcantilevers are then solved using the proposed scheme for the validation purpose. For these examples, a comparison is made between the values of pull-in parameters obtained using the proposed scheme and those previously published in the literature. An excellent agreement between the two establishes the utility of the proposed scheme.