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图1为磁流变抛光的初始面形,其由10、70、150、300 Hz四个频率成分组成正弦信号,各频率对应的振幅分别为0.8、1.2、1.5、0.5 mm,其表达式为:
图2为磁流变抛光的去除函数,其主要的几何参数有长度(参考尺度为20 mm)、宽度(参考尺度为12 mm)和束径(最小外接圆的直径,参考尺度为15 mm)。
图3为确定性加工中常用的ρ-θ螺旋线抛光轨迹。图4为初始面形图,将初始正弦面形减去驻留时间与去除函数按照公式(3)进行的二维卷积得到ρ-θ螺旋线抛光后的面形如图5所示。可以发现按照ρ-θ螺旋轨迹抛光后的面形存在中心过抛的现象。
根据公式(2)在频域对其误差频率的变化进行分析,计算得到螺旋轨迹抛光前后的面形沿x方向和y方向的一维PSD曲线,计算结果如图6所示。
Figure 6. PSD analysis of the (a) x direction and (b) y direction before and after the spiral polishing
图6为螺距为10 mm、进给距离为5 mm时对正弦面形进行加工前后的面形PSD曲线,可以发现沿着该轨迹抛光可以对4 mm处的尖峰误差进行有效收敛,但同时也引入了很多比较碎的误差频率。ρ-θ螺旋线加工轨迹,可以消除圆形工件中的非对称性误差;极坐标加工的方式使得加工设备对X轴和Y轴的行程要求有所降低,加工时不需要换向,但同时也存在工件转动不平稳、中心转速过快而造成中心过抛等缺点。
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图7为磁流变抛光加工中常用的光栅轨迹,由于轨迹有抛光头的进给距离和光栅的栅距,材料去除量是不均匀分布的。图8为初始面形,图9为光栅加工后的面形。
图10为在抛光头的进给距离为5 mm、光栅的栅距为10 mm时抛光前后的功率谱密度曲线,可以发现等间距光栅线会在频谱上存在一个尖峰误差,而且尖峰误差对应的频率是轨迹线行距对应的频率。由对功率谱密度函数的描述可知,合格的光学元件表面频谱连续且光滑,尖峰误差的出现对光学系统有不良影响。如果以LLNL对中频误差的划分方式来看,磁流变抛光的栅距为5 mm,实际加工过程中栅距的范围是0.12 mm< x <33.3 mm,那么出现对应的尖峰误差频率恰好出现在中频误差范围之内。所以光栅轨迹带来的尖峰误差是中频恶化的一个关键因素。
Figure 10. PSD analysis in the (a) x direction and (b) y direction before and after the grating line polishing
光栅线轨迹加工时工件做周期往复运动,抛光头沿着工件的直径做进给运动,可应用于所有平面的加工。对于光栅线轨迹,抛光模仅有X轴和Y轴运动,要求加工设备行程要覆盖整个光学表面,而且加工时抛光头要反复换向,增加了加工时间。
因此,可以看出,规则抛光轨迹都是离散间隔分布的,轨迹无法遍历每个误差点,导致卷积过程与实际去除的不一致,会带来对称的迭代误差。
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根据公式(4)将螺旋矩阵轨迹加上一个实际位置量10%的白噪声随机扰动后对面形进行加工,再分别与传统螺旋线轨迹加工、光栅轨迹加工进行对比,分析其优化效果。变距螺旋矩阵轨迹加工同一面行前后结果如图11所示。对其进行两方向的功率谱密度分析可以分别得到x、y方向的PSD,如图12所示。
Figure 12. PSD analysis of the width screw matrix trajectory (a) x direction (b) y direction PSD analysis
如图12所示,面形在不规则的螺旋矩阵轨迹抛光后的功率谱密度曲线比螺旋线和光栅线的功率谱密度曲线整体下移,在10−4 mm−1处和10−3 mm−1处的频率误差均能够得到抑制。具体情况如表1所示。
Direction Key
frequencyPSD before
optimizedAfter polishing Improvement Raster Spiral New method PSD Converging efficiency PSD Converging efficiency PSD Converging efficiency x 9.43E-05 9735 1361 86% 1278 87% 1125 88% 2.00% y 9.43E-05 8292 8167 2% 7843 5% 0.00001893 100% 96.54% x 1.29E-03 8888 1152 87% 1235 86% 394.6 96% 8.99% y 1.29E-03 96.44 1.34E-05 100% 0.00001064 100% 1.12 99% −1.16% Table 1. Comparison before and after PSD optimized
可见,对于两个关键频率,x、y两个方向共计四个关键点位PSD数据,相对于传统抛光轨迹,文中所提方法收敛率较传统光栅轨迹和螺旋线轨迹的平均水平分别提高了2.00%、96.54%、8.99%、−1.16%。总体来说,对于观测点位收敛效率比传统方法提高率不太明显,但是通过随机优化,其绝对收敛效率已经很高。另外,在观测点外总体PSD的平滑程度已经得到了改善。观测点位处的峰值在抛光后已经出现了频率漂移的现象,说明文中的随机优化方法改善了原有频率点的中频误差。第四个点位较传统方法略有下降,因为在这一频率观测点位关键频率被平滑,不至于产生峰值,其PSD绝对值虽然增加,但是平滑程度有明显改善。总体来讲,通过不规则的螺旋矩阵轨迹加工面形可以降低面形表面的中频误差,较传统方法绝对收敛率提高了26.59%。
Trajectory optimization method of variable pitch spiral matrix for intermediate frequency error
doi: 10.3788/IRLA20210443
- Received Date: 2021-07-02
- Rev Recd Date: 2021-09-30
- Publish Date: 2022-04-07
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Key words:
- magnetorheological polishing /
- intermediate frequency error /
- power spectral density /
- spiral matrix trajectory /
- trajectory optimization
Abstract: Magnetorheological polishing can efficiently remove low-frequency errors on the surface of optical elements, but it also introduces intermediate-frequency errors. The existence of intermediate-frequency errors has a serious impact on the performance of the optical system, which must be effectively controlled. The commonly used grating trajectories and spiral trajectories were studied. It was found that the regular polishing trajectory caused the convolution process to be inconsistent with the actual removal process, which would introduce symmetrical iterative errors. The iterative error is an important factor in the deterioration of the intermediate frequency error. Based on the research of grating trajectory and spiral trajectory, a trajectory optimization method for variable pitch spiral matrix trajectory was proposed to reduce the intermediate frequency error of optical components. The trajectory retained the advantages of simplicity and ease of the grating trajectory and the spiral trajectory by disrupting the pitch of the spiral matrix trajectory, and also changed the randomness between the trajectory lines. Through the power spectrum analysis of the surface shape before and after processing, it was verified that the trajectory can effectively reduce the intermediate frequency error of the surface, and the intermediate frequency convergence efficiency of the grating line and the spiral line is comprehensively improved by 26.59%.