Analytical study of the nonlinear vibration of an electrostatically actuated microbeam using the extended Galerkin method

Document Type : Regular Articles

Authors

1 1-Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt 2-Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, 42351, Saudi Arabia

2 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt

3 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

Abstract

Nano/microelectromechanical systems (N/MEMS) have garnered significant attention in recent decades due to their miniature size, potential for batch fabrication, high reliability, and low power consumption. This study examines the nonlinear vibration behavior of an electrostatically actuated clamped-clamped microbeam, described by a second-order nonlinear ordinary differential equation that accounts for both mid-plane stretching and electrostatic forces. In contrast to earlier analytical approaches, which primarily relied on perturbation techniques or purely numerical solutions, the present work applies the extended Galerkin method (EGM) to obtain higher-order approximate solutions for the system’s nonlinear dynamic response. The key novelty lies in the implementation of EGM for a strongly nonlinear MEMS configuration and its extension to higher-order terms, which enables the accurate characterization of hardening/softening behaviors and resonance shifts. The derived solutions are validated through comparison with numerical simulations based on the Runge-Kutta method and with existing analytical results, showing that EGM delivers more accurate frequency-amplitude predictions while avoiding small-parameter assumptions and reducing computational effort. The findings underscore the potential of EGM as an efficient and precise analytical approach for the design and performance analysis of nonlinear MEMS resonators.

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