二自由度快速控制反射镜系统固有频率优化设计

在二自由度快速控制反射镜系统设计中，为提高系统的控制带宽，应尽量降低工作方向上的低阶固有频率，提高非工作方向上的高阶固有频率。该课题以某深切口柔性铰链快反镜系统作为研究对象，首先对系统前四阶固有频率的振型运动方向进行了分析，并针对传统刚度计算方法不适用于第三阶振型方向的问题，重新推导了第三阶振型方向上的刚度计算公式；其次，利用能量法和卡式第二定理对深切口柔性铰链上的工作刚度进行了推导，并进行了非线性拟合化简，得出的简化计算公式计算结果与有限元仿真结果误差不超过8.9%，证明了推导的铰链工作刚度理论公式的准确性；然后，将第三阶振型方向刚度计算公式和柔性铰链刚度计算公式代入固有频率计算公式，并进行有限元验证，结果表明理论公式计算结果与有限元仿真结果误差不超过1.7%，证明了新的三阶振型方向上的刚度计算公式的准确性。最后利用遗传算法，对系统前四阶固有频率进行了多目标优化设计，到达设计要求，所求出的优化结构较初始结构有明显优化，工作方向刚度减小19.04%，非工作方向刚度提高297.83%和77.09%。此外还对其进行了有限元仿真验证，结果证明一、二阶固有频率减小8.08%、5.40%，三、四阶固有频率提高了112.59%、16.80%。证明优化结构较初始结构有较大提高，能有效提高系统控制带宽。

Optimal design of natural frequency of two-degree-of-freedom fast steering mirror system

In the design of a two-degree-of-freedom fast steering mirror system, in order to increase the control bandwidth of the system, the low-order natural frequency in the working direction should be reduced as much as possible, and the high-order natural frequency in the non-working direction should be increased. This subject used a deep-cut flexure hinge fast-reflection mirror system as the research object. First, the vibration mode movement direction of the first four-order natural frequency of the system was analyzed, and considering that the traditional stiffness calculation method was not suitable for the third-order mode direction problem, the stiffness calculation formula in the third-order mode shape direction was re-derived; secondly, the working stiffness of the deep-cut flexure hinge was deduced by the energy method and the second card theorem, and the nonlinear fitting was simplified. The error between the simplified calculation formula calculation result and the finite element simulation result did not exceed 8.9%, which proves the accuracy of the derived hinge working stiffness theoretical formula; then, the third-order mode shape direction stiffness calculation formula and the flexible hinge stiffness calculation formula were substituted into natural frequency calculation formula and finite element verification. The results showed that the error between the theoretical formula calculation result and the finite element simulation result did not exceed 1.7%, which proves the accuracy of the new third-order mode shape direction stiffness calculation formula. Finally, using genetic algorithm, multi-objective optimization design was carried out on the first four-order natural frequency of the system, and the design requirements were reached. The optimized structure obtained was significantly optimized compared with the initial structure, the stiffness in the working direction was reduced by 19.04%, and the stiffness in the non-working direction was increased by 297.83% and 77.09%. In addition, it has been verified by finite element simulation, and the results showed that the first and second order fundamental frequencies were reduced by 8.08% and 5.40%. The third and fourth-order fundamental frequencies have increased by 112.59% and 16.80%. It proves that the optimized structure is greater than the initial structure, which can effectively increase the system control bandwidth.