﻿ 新型单轴柔性铰链拓扑结构设计与柔度分析<sup>*</sup>
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Topological structure design and compliance analysis of a new single-axis flexure hinge
QIU Lifang, CHEN Haixiang, WU Youwei
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
Received: 2017-06-07; Accepted: 2017-10-16; Published online: 2017-11-28 17:09
Foundation item: National Natural Science Foundation of China (51475037)
Corresponding author. QIU Lifang, E-mail:qlf@ustb.edu.cn
Abstract: Based on three-dimensional continuum topology optimization theory, aimed at maximizing compliance ratio, solid isotropic material with penalization model was used to establish the topology optimization model of a single-axis flexure hinge. With the help of OptiStruct, this paper designed a kind of single-axis flexure hinge with a new three-dimensional topological structure. Secondly, combining Castigliano's second theorem and the method of energy for the compliance of flexure hinge in theory, it deduced the compliance matrix of the new flexure hinge. 16 groups' analysis in theory and finite element simulation analysis showed the correctness of the theoretical formula because the relative error of analysis and FEA was within 6.35%. Finally, it compared the difference of compliance between the new flexure hinge and circular flexure hinge with the same cut profile. The results show that the new flexure hinge has much better performance in compliance. Compared with the circular flexure hinge, its compliance can be improved by 300%. Based on the three-dimensional continuum topology optimization method, this paper presented a new thought for the design of single-axis flexure hinge.
Key words: circular flexure hinge     three-dimensional continuum topology optimization     variable density method     compliance ratio     compliance analysis

1 基于变密度法设计柔性铰链 1.1 拓扑优化模型

 图 1 拓扑优化区域与工况示意图 Fig. 1 Schematic of topology optimization area and working conditions

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 图 2 拓扑优化模型的正视图和中心漂移 Fig. 2 Front view and center drift of topology optimization model

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1.2 拓扑优化过程和结果

HyperWorks软件中的OptiStruct模块是基于变密度材料插值法进行拓扑优化迭代计算的, 借助该软件将1.1节建立的拓扑优化模型实现求解, 经过38轮迭代计算, 得到满足各限制条件的拓扑优化结果, 终止迭代计算, 完成单轴柔性铰链的拓扑优化设计。输出拓扑优化结果的密度阈值设置为0.3, 即单元的相对密度xe大于0.3的单元将被保留, 小于0.3的单元将被去除, 得到拓扑优化过程和结果如图 3所示。

 图 3 拓扑优化过程和结果 Fig. 3 Process and results of topology optimization

 图 4 新型柔性铰链三维图 Fig. 4 3D drawing of new flexure hinge
 图 5 新型柔性铰链俯视图、正视图及参数 Fig. 5 Top and front view of new flexure hinge with parameters
2 柔度理论分析

 图 6 柔性铰链等效弹簧刚度示意图 Fig. 6 Schematic of flexure hinge's equivalent spring stiffness
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 图 7 第i个弯曲片段简化悬臂梁示意图 Fig. 7 Schematic of the ith bending segment's simplified cantilever beam

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Cx-Fxi柔度项表达式:

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Cθy-Myi柔度项表达式:

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Ciz-Fz柔度项表达式:

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3 有限元仿真分析验证

 图 8 新型柔性铰链有限元仿真分析 Fig. 8 FEA of new flexure hinge

 R/mm t/mm Cθy-My Cz-Fz Cx-Fx 理论值/(10-4rad·(N·mm)-1) 仿真值/(10-4rad·(N·mm)-1) 相对误差/% 理论值/(10-2N-1·mm) 仿真值/ (10-2N-1· mm) 相对误差/% 理论值/ (10-6N-1· mm) 仿真值/ (10-6N-1· mm) 相对误差/% 4 0.2 64.442 66.383 3.01 10.420 10.770 3.36 37.143 38.302 3.12 4 0.3 23.106 23.298 0.83 3.775 3.805 0.79 28.384 28.538 0.54 4 0.4 11.298 11.005 2.59 1.858 1.810 2.58 23.641 22.985 2.77 4 0.5 6.202 6.123 1.27 1.026 1.014 1.17 19.700 19.352 1.77 6 0.2 77.907 78.843 1.20 28.263 28.695 1.53 42.968 43.792 1.92 6 0.3 28.290 27.365 3.27 10.316 10.023 2.84 33.563 32.419 3.41 6 0.4 13.458 12.810 4.81 4.927 4.724 4.12 27.114 26.029 4.00 6 0.5 7.401 7.073 4.43 2.720 2.626 3.46 22.915 21.882 4.51 8 0.2 86.393 88.427 2.35 55.480 57.161 3.03 46.336 47.892 3.36 8 0.3 31.454 30.408 3.33 20.309 19.784 2.59 36.874 35.342 4.15 8 0.4 14.781 14.133 4.38 9.571 9.259 3.26 29.199 28.345 2.92 8 0.5 8.288 7.761 6.36 5.282 5.012 5.11 25.165 23.827 5.32 10 0.2 95.256 96.204 1.00 95.177 97.148 2.07 50.632 51.210 1.14 10 0.3 34.478 32.827 4.79 34.696 33.372 3.82 38.963 37.730 3.16 10 0.4 16.066 15.174 5.55 16.209 15.534 4.16 31.940 30.258 5.27 10 0.5 8.849 8.299 6.22 8.997 8.557 4.89 26.450 25.449 3.78

Cθy-My柔度项相对误差的最大值为6.36%, 最小值为0.83%, 平均值为3.46%;Cz-Fz柔度项相对误差最大值为5.11%, 最小值0.79%, 平均值为3.05%;Cx-Fx柔度项相对误差最大值为5.32%, 最小值为0.54%, 平均值为3.20%。理论分析和仿真分析数值的相对误差均小于6.36%, 在误差允许范围内, 验证了柔度矩阵理论公式的正确性。

4 与圆弧型柔性铰链的对比

 图 9 2种柔性铰链Cθy-My和Cz-Fz柔度项对比 Fig. 9 Comparison of compliance item Cθy-My and Cz-Fz between two flexure hinges
 图 10 2种柔性铰链柔度比对比 Fig. 10 Comparison of compliance ratio χ between two flexure hinges

5 结论

1) 本文基于三维连续体拓扑优化的变密度法, 建立了单轴柔性铰链的拓扑优化模型, 设计出一种具有全新三维拓扑结构的单轴柔性铰链。

2) 借助等效弹簧模型描述新型柔性铰链的三维拓扑结构, 推导出其柔度矩阵, 设计16组实例进行有限元仿真分析, 并与理论分析值进行对比, 结果表明有限元仿真分析值和理论计算值的相对误差在6.36%以内, 验证了理论分析的正确性。

3) 对比具有相同参数的新型柔性铰链与圆弧型柔性铰链的柔度和柔度比, 结果表明新型柔性铰链比圆弧型柔性铰链柔度提升300%, 柔度比是其1.5倍。

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#### 文章信息

QIU Lifang, CHEN Haixiang, WU Youwei

Topological structure design and compliance analysis of a new single-axis flexure hinge

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(6): 1133-1140
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0388