Electrostatic micromechanical actuators have numerous applications in science and technology. In many applications, they are operated in a narrow frequency range close to resonance and at a drive voltage of low variation. Recently, new applications, such as microelectromechanical systems (MEMS) microspeakers (µSpeakers), have emerged that require operation over a wide frequency and dynamic range. Simulating the dynamic performance under such circumstances is still highly cumbersome. State-of-the-art finite element analysis struggles with pull-in instability and does not deliver the necessary information about unstable equilibrium states accordingly. Convincing lumped-parameter models amenable to direct physical interpretation are missing. This inhibits the indispensable in-depth analysis of the dynamic stability of such systems. In this paper, we take a major step towards mending the situation. By combining the finite element method (FEM) with an arc-length solver, we obtain the full bifurcation diagram for electrostatic actuators based on prismatic Euler-Bernoulli beams. A subsequent modal analysis then shows that within very narrow error margins, it is exclusively the lowest Euler-Bernoulli eigenmode that dominates the beam physics over the entire relevant drive voltage range. An experiment directly recording the deflection profile of a MEMS microbeam is performed and confirms the numerical findings with astonishing precision. This enables modeling the system using a single spatial degree of freedom.

静电微机械执行器在科学技术领域有着广泛的应用。在许多应用中，它们在接近谐振的窄频率范围和低变化的驱动电压下运行。最近，出现了需要在较宽频率和动态范围内运行的微机电系统 (MEMS) 微型扬声器 (μSpeakers) 等新应用。在这种情况下模拟动态性能仍然非常麻烦。最先进的有限元分析与拉入不稳定性作斗争，并且不能相应地提供有关不稳定平衡状态的必要信息。缺乏令人信服的可直接物理解释的集总参数模型。这阻碍了对此类系统的动态稳定性进行必不可少的深入分析。在本文中，我们朝着改善这种情况迈出了重要一步。通过将有限元法（FEM）与弧长求解器相结合，我们获得了基于棱柱欧拉-伯努利梁的静电驱动器的全分岔图。随后的模态分析表明，在非常窄的误差范围内，在整个相关驱动电压范围内，完全是最低的欧拉-伯努利本征模在梁物理中占主导地位。进行了直接记录 MEMS 微梁偏转轮廓的实验，并以惊人的精度证实了数值结果。这使得能够使用单个空间自由度对系统进行建模。