Distributed microscopic actuation analysis of deformable plate membrane mirrors

To further reduce the areal density of optical mirrors used in space telescopes and other space-borne optical structures, the concept of flexible membrane deformable mirror has been proposed. Because of their high flexibility, poor stiffness and low damping properties, environmental excitations such as orbital maneuver, path changing, and non-uniform heating may induce unexpected vibrations and thus reduce working performance. Therefore, active vibration control is essential for these membrane mirrors. In this paper, two different mirror models, i.e., the plate membrane model and pure membrane model, are studied respectively. In order to investigate the modal vibration characteristics of the mirror, a piezoelectric layer is fully laminated on its non-reflective side to serve as actuators. Dynamic equations of the mirror laminated with piezoelectric actuators are presented first. Then, the actuator induced modal control force is defined. When the actuator area shrinks to infinitesimal, the expressions of microscopic local modal control force and its two components are obtained to predict the spatial microscopic actuation behavior of the mirror. Different membrane pretension forces are also applied to reveal the tension effects on the actuation of the mirror. Analyses indicate that the spatial distribution of modal micro-control forces is exactly the same with the sensing signals distribution of the mirror, which provides crucial guidelines for optimal actuator placement of membrane deformable mirrors.