ABSTRACT

In this article, we derive a mechanical distributed parameter model for the annular sector plate segments of the Extremely Large Telescope’s deformable mirror M4. Additionally, we modally analyse the derived model via analytical and numerical approaches. The deformable mirror M4 is used to reject wavefront disturbances and enhance the optical imaging quality. We present a control oriented annular sector Kirchhoff–Love plate model featuring an elastic boundary condition and its modal analysis for one of the six identical M4 segments. Subsequently, we show that the well-known method of separation of variables is incompatible with the modal analysis of the presented distributed parameter model in cylindrical coordinates. Moreover, we successfully modally analyse the model using a finite difference approximation and a realistic construction of an M4 segment via a finite element approximation to compare the results. The modal analyses provide consistent results and therefore, both models underlying the analyses are consistent.

摘要

在本文中，我们推导出了超大望远镜可变形反射镜 M4 的环形扇形板段的机械分布参数模型。此外，我们通过分析和数值方法对派生模型进行模态分析。可变形反射镜M4用于抑制波前干扰并提高光学成像质量。我们提出了一个面向控制的环形扇形 Kirchhoff-Love 板模型，该模型具有弹性边界条件及其对六个相同 M4 段之一的模态分析。随后，我们证明了众所周知的变量分离方法与所提出的柱坐标分布参数模型的模态分析不兼容。而且，我们使用有限差分近似成功地对模型进行了模态分析，并通过有限元近似对 M4 段的实际构造进行了模态分析，以比较结果。模态分析提供一致的结果，因此，作为分析基础的两个模型是一致的。