Control oriented modelling and modal analysis of the deformable mirror M4 of the extremely large telescope

Abstract

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.