The objective of this work is to create an analytical framework to study the problem of instability and squeezed film damping in bi-axial micro-scanners under electromagnetic actuation considering Casimir effects and size dependence, simultaneously. Also, the concept of Eulerian angles has also been used to achieve this idea. The novelty of this work is analytical solution of squeeze film damping considering changes of pressure distribution for micro-mirror in the horizontal and vertical rotations. Moreover, the modified couple stress theory is employed to assess the effect of the small-scale on the dynamic instability of a biaxial micro-scanner, and the governing equations are derived based on the concept of Eulerian angles. This study addresses the nonlinear effect of air squeeze film damping on the stability of the micro-scanner according to its geometry. Then, the influence of magnetic field, Casimir force, and size are examined, followed by the generation of phase portrait and plot of parametric analysis. Based on the obtained results, the Casimir force accounts for the instability threshold of the system. Moreover, the phase portrait diagrams indicates that small size and air squeeze film damping improves the rotational stability of the micro-scanner. AC voltage is applied to induce electromagnetic actuation, and the graphs of nonlinear frequency response are plotted versus the micromirror rotation amplitudes considering various effects of magnetic field actuation and air squeeze film damping. Furthermore, increasing the magnetic field diminishes the instability threshold, and augmentation of the air squeeze film damping enhances the system stability and decreases the maximum rotation amplitude of the micro-scanner. The obtained results are validated against the empirical/theoretical results in the literature.

这项工作的目的是创建一个分析框架，以同时考虑卡西米尔效应和尺寸依赖性，研究电磁驱动下双轴微型扫描仪的不稳定性和挤压膜阻尼问题。同样，欧拉角的概念也已经用于实现该想法。这项工作的新颖性是考虑到微镜在水平和垂直旋转中压力分布变化的挤压膜阻尼分析解决方案。此外，采用改进的耦合应力理论来评估小尺度对双轴微扫描仪动态不稳定性的影响，并根据欧拉角的概念推导了控制方程。这项研究根据其几何形状，研究了压缩膜阻尼对微型扫描仪稳定性的非线性影响。然后，检查磁场，卡西米尔（Casimir）力和大小的影响，然后生成相图和参数分析图。基于获得的结果，卡西米尔力解释了系统的不稳定性阈值。此外，相图表明，小尺寸和空气挤压膜阻尼可改善微型扫描仪的旋转稳定性。施加交流电压以感应电磁激励，并考虑了磁场激励和空气挤压膜阻尼的各种影响，绘制了非线性频率响应曲线与微镜旋转幅度的关系图。此外，增大磁场会减小不稳定性阈值，而增大空气挤压膜阻尼会增强系统稳定性并减小微扫描仪的最大旋转幅度。相对于文献中的经验/理论结果验证了所获得的结果。