Mathematical modeling of an electrostatic MEMS with tilted elastomeric micro-pillars

Abstract

The aim of this study is to develop a comprehensive mathematical model for an electrostatic MEMS (Micro-Electro-Mechanical Systems) with gap-filling tilted micro-pillars. Elastomeric pillars are used due to their high dielectric constant, resulting in a higher equivalent permittivity of the gap medium. This leads to increased sensitivity and decreased required voltage. Although this system has gained popularity in recent years, a thorough understanding and accurate predictions of its behavior require a detailed mathematical model. Regrettably, the existing literature does not offer adequate information. The present model incorporates three coupled set of nonlinear differential equations that govern the longitudinal and transverse vibrations of the PDMS (Polydimethylsiloxane) pillars, as well as the transverse vibrations of the moving electrode. It demonstrates how the device's behavior is affected by varying tilt angles and numbers of micro-pillars. The viscoelastic properties of the PDMS pillars are also taken into account, following the Kelvin-Voigt model with non-linear strains. Numerical methods are used to solve the governing equations, enabling investigation of static deformation and vibrational responses under different DC (Direct Current) and DC+AC (Alternating Current) actuations. The nonlinear equations proposed in this study provide a valuable tool for accurately predicting the behavior of capacitive MEMS devices that use microstructured gaps with tilted micro-pillar arrays.