Mathematical approach for rapid determination of pull-in displacement in MEMS devices

Introduction Microelectromechanical systems (MEMS) are pivotal in diverse fields such as telecommunications, healthcare, and aerospace. A critical challenge in MEMS devices is accurately determining the pull-in displacement and voltage, which significantly impacts device performance. Existing methods, including the variational iteration method and homotopy perturbation method, often fall short in providing precise estimations of these parameters. Methods This study introduces a novel mathematical approach that combines physical insights into the pull-in phenomenon with variational theory. The method begins with a precise definition of the MEMS device's physical model. By uniquely applying the variational principle and incorporating a custom-designed functional, a set of equations is derived. These equations are transformed into an iterative algorithm for calculating pull-in displacement, with nonlinear terms addressed through approximation and perturbation techniques tailored to the MEMS system’s characteristics. Results Validation using specific examples demonstrates the method's accuracy in determining pull-in displacement and voltage. For instance, in a MEMS oscillator case, exact results were achieved with a computation time of 0.015 s. Compared to traditional methods, this approach yields exact values rather than approximations, showcasing superior precision and efficiency. Discussion The proposed method offers significant advantages, including enhanced accuracy, reduced computational time, and minimized error accumulation by solving algebraic equations instead of iterating differential equations. It also exhibits robustness to variations in initial conditions and system parameters. Limitations include the need for modifying the pull-in criterion when variational formulation is unattainable and the exclusion of environmental factors like temperature and pressure fluctuations. Future research should focus on refining MEMS models to incorporate these factors and integrating the approach with techniques such as Galerkin technology. Conclusion This research advances the mathematical understanding of MEMS device behavior and holds substantial potential for the design and optimization of MEMS devices across various applications, further driving the progression of MEMS technology.