Casimir-Induced Pull-In Instability and Nonlinear Dynamics in Fractal RF MEMS: Mathematical Modeling and Analysis

Abstract

Abstract

Radio-frequency microelectromechanical systems (RF MEMS) technology has developed rapidly in recent years and has been widely applied in communication systems, sensing devices, and other advanced electronic platforms. Motivated by the significant role of nanoscale effects, this paper investigates the influence of the Casimir force on pull-in instability in RF MEMS structures. A theoretical framework is first established by incorporating Casimir interactions into the classical MEMS formulation, leading to a Casimir RF MEMS model. To further capture multiscale geometric features, the model is extended into fractal space, resulting in a fractal Casimir RF MEMS formulation. Within this unified framework, the pull-in dynamics are systematically analyzed and compared with the conventional model to show the impact of nanoscale physics and fractal geometry. The results demonstrate that the Casimir force substantially alters the instability behavior by lowering the critical pull-in actuation threshold and modifying the stability characteristics, while the fractal structure further affects the evolution of the pull-in process. In addition, He’s frequency formulation is applied to obtain numerical periodic solutions of the governing equations, enabling an effective characterization of nonlinear oscillatory responses. Overall, this study provides a mathematical framework for Casimir- and fractal-induced effects in RF MEMS and offers new insights into the design and optimization of next-generation micro- and nano-electromechanical systems.