为满足4 m望远镜底座结构高刚度和轻量化的需求，研究了考虑结构变形和模态约束下的拓扑优化问题。首先，基于连续体结构拓扑优化的思想，以单元虚拟密度为设计变量，以位移变形和一阶频率及质量为优化约束，应变能最小为优化目标，建立了底座设计的拓扑优化数学模型，并详细推导目标函数及约束条件的灵敏度；然后，对底座结构应用拓扑优化设计，并以所得的理想概念构型为基础，进行底座结构的详细设计；最后，采用有限元法对优化模型进行静刚度和动刚度的分析与校核。设计结果表明，底座质量从27.66 t减至22.15 t；最大变形量由0.0377 mm减小为0.014 mm；一阶频率从217.1 Hz提高至247.45 Hz；在减小质量的同时，有效提高了底座结构的静刚度和动刚度，验证了拓扑优化方法的有效性。该方法将对4 m望远镜跟踪架的其他部件设计提供帮助。

To meet the requirement for high stiffness and lightweight, topology optimization method with constraints of static displacements and first resonance frequency was studied in 4 m telescope mount base design. Firstly, mathematical model of topology optimization was built subjected to minimum compliance, with pseudo density of the finite elements as design variables, static displacements, first resonance frequency and total mass as optimization constraints. Procedure of calculating sensitivities of optimization objective was presented in detail. Furthermore, topology optimization method was applied in mount base design, based on which shape and size design were carried out. Finally, the static stiffness and dynamic behavior of the optimized structure was analyzed and checked using the finite element method. The achieved results show that total mass of mount base reduces from 27.66 t to 22.15 t, while the maximum displacement decreases from 0.037 7 mm to 0.014 mm and first resonance frequency increases from 217.1 Hz to 247.45 Hz, ie, improving static and dynamic performances with lots of mass cut, which validate the presented topology optimization method. This topology optimization method will provide efficient help to other components′ design of 4m telescope mount.