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.

中文翻译：

---

具有倾斜弹性体微柱的静电 MEMS 的数学建模

本研究的目的是为具有间隙填充倾斜微柱的静电 MEMS（微机电系统）开发一个综合数学模型。由于其高介电常数而使用弹性柱，从而导致间隙介质具有更高的等效介电常数。这会提高灵敏度并降低所需电压。尽管该系统近年来越来越受欢迎，但对其行为的透彻理解和准确预测需要详细的数学模型。遗憾的是，现有文献没有提供足够的信息。本模型包含三个耦合的非线性微分方程组，用于控制 PDMS（聚二甲基硅氧烷）柱的纵向和横向振动以及移动电极的横向振动。它演示了不同的倾斜角度和微柱数量如何影响设备的行为。遵循具有非线性应变的 Kelvin-Voigt 模型，还考虑了 PDMS 柱的粘弹性特性。采用数值方法求解控制方程，从而能够研究不同 DC（直流）和 DC+AC（交流）驱动下的静态变形和振动响应。本研究中提出的非线性方程为准确预测使用带有倾斜微柱阵列的微结构间隙的电容式 MEMS 器件的行为提供了宝贵的工具。