Mathematical modelling for large-aperture mirror systems: enabling optical-mechanical performance analysis and support structure optimization

Abstract

Large-aperture mirror systems (LAMS) are widely used in fields such as electro-optical countermeasures, spaceborne remote sensing, and laser communication. Accurate modeling and analysis of LAMS are crucial for its design and optimization. To transition from one-off simulation models to a generalized theoretical mathematical model, this paper presents a theoretical modeling method that combines the Reciprocal Theorem of Work and the Perturbation Method. First, based on the Kirchhoff plate theory, a variable-coefficient deflection governing equations for LAMS is derived. Then, using the principles of the Perturbation Method, an analytical solution for the governing equations within the framework of the Reciprocal Theorem of Work is obtained. Additionally, the proposed governing equations are validated for convergence, and it is found that the solution converges when m ≥ 3. By comparing with FEM results, the accuracy of the proposed model is confirmed, with the root mean square error (RMSE) below 9.918 × 10^-4 and R² exceeding 0.875. Furthermore, the influence of higher-order perturbation terms and computational efficiency are analyzed. The results show that the first-order perturbation solution satisfies the accuracy requirements, and the proposed method offers significant computational efficiency advantages over FEM. Finally, for a specific LAMS model, the proposed governing equations are applied to optimize its support scheme. The results indicate that, for 9 support points, the optimal radius of the inner supports is 0.267a, while the outer supports have an optimal radius of 0.775a. For 18 support points, the optimal radius of the inner supports is 0.343a, with the outer supports having an optimal radius of 0.811a. Overall, the proposed method establishes a theoretical foundation for designing and optimizing the support structures of LAMS, significantly facilitating engineers in optimizing its optomechanical performance.