﻿ 多外力柔性微动机构输出位移求解方法<sup>*</sup>
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Method for output displacement solving of compliant micro-motion mechanism with multi-input forces
GONG Jinliang, JIA Guopeng, ZHANG Yanfei
School of Mechanical Engineering, Shandong University of Technology, Zibo 255049, China
Received: 2017-04-11; Accepted: 2017-05-26; Published online: 2017-06-30 10:00
Foundation item: National Natural Science Foundation of China (61303006); the Research Award Fund for Outstanding Young Scholars of Shandong Province (BS2012ZZ009)
Corresponding author. ZHANG Yanfei. E-mail:88659258@qq.com
Abstract: For the compliant micro-motion mechanism with one input force, compliance describes the relation between output displacement and input force, and is an important performance index for the dynamic performance and positioning precision. For the one with many input forces, the relation equation between output displacement and input forces has the same role with the compliance. For obtaining this equation, the method which combined the compliance matrix method and the motion raw of rigid body was proposed. Firstly, the whole structure is divided to elements, and the relation equation between displacement and force of element end is established. Secondly, the superposition or coordinate relation equation about displacements or forces of different element ends was solved according to their structural relation. Finally, the equation of relation between the output displacement and the input forces was worked out by synthesizing all the solved equations. The output displacement of a micro-motion gripper was worked out by this method and contrasted with the one from finite element analysis method. The results show that this method has the enough precision and a good adaptability for micro-motion mechanism performance analysis and optimization. The theory suggestion about dimension optimization was obtained by analyzing the equation by using MATLAB software.
Key words: compliant micro-motion mechanism     output displacement     compliance matrix     micro-motion gripper     finite element analysis

1 多外力下输出位移求解原理 1.1 柔性单元末端力与末端位移的关系

 (1)
 图 1 矩形杆单元 Fig. 1 Square bar element

 (2)

 图 2 柔性铰链单元 Fig. 2 Compliant hinge element
1.2 刚体单元上力与位移的关系

 (3)

 (4)

 (5)

1.3 整体结构上力与位移的关系

 (6)
 图 3 三连杆串联支链 Fig. 3 Serial chain of three bars

2 微动夹持器的输出位移求解 2.1 微动夹持器结构模型

 图 4 微动夹持器结构模型 Fig. 4 Structural model of micro-motion gripper
2.2 参数模型和力学模型

 图 5 微动夹持器参数模型 Fig. 5 Parameterized model of micro-motion gripper

 图 6 微动夹持器力学模型 Fig. 6 Mechanical model of micro-motion gripper
2.3 位移求解

2.3.1 单元3的位移求解

 (7)
 图 7 单元3结构 Fig. 7 Structure of Element 3

 (8)
 (9)
 (10)

 (11)

2.3.2 单元6末端的位移求解

 (12)
 图 8 单元6结构 Fig. 8 Structure of Element 6

 (13)

 (14)
 (15)

 (16)

2.3.3 单元7的位移求解

 (17)
 图 9 单元8和单元9的串联支链结构 Fig. 9 Structure of series chain of Element 8 and 9

 (18)
 图 10 单元7结构 Fig. 10 Structure of Element 7

 (19)
 (20)

 (21)

3 有限元方法验证

 图 11 微动夹持器有限元网格划分模型 Fig. 11 Finite element meshing model of micro-motion gripper

F2选取0~20 N内的5组值，F1选取10~50 N内的5组值，两者相互组合得到25组输入数对，对其进行理论计算与有限元分析，对比结果如表 1所示。

 外力 位移uIy 位移uIx F2/N F1/N 理论值/μm 有限元分析值/μm 相对误差/% 理论值/μm 有限元分析值/μm 相对误差/% 0 10 -1.31 -1.42 7.7 -0.122 -0.132 7.6 20 -2.62 -2.84 7.7 -0.245 -0.265 7.5 30 -3.93 -4.27 7.9 -0.367 -0.397 7.6 40 -5.24 -5.69 7.9 -0.490 -0.530 7.5 50 -6.56 -7.12 7.8 -0.612 -0.663 7.7 5 10 0.020 9 0.047 8 56 0.003 75 0.007 45 49 20 -1.29 -1.37 5.8 -0.119 -0.125 4.8 30 -2.60 -2.80 7.1 -0.241 -0.258 6.5 40 -3.91 -4.22 7.3 -0.364 -0.390 6.6 50 -5.22 -5.65 7.6 -0.486 -0.523 7.1 10 10 1.34 1.46 8.2 0.132 0.142 7.0 20 0.041 7 0.095 6 56 0.007 51 0.014 9 49 30 -1.27 -1.33 4.5 -0.115 -0.118 2.5 40 -2.58 -2.75 6.2 -0.237 -0.250 5.2 50 -3.89 -4.17 6.7 -0.360 -0.382 5.7 15 10 2.68 2.92 8.2 0.256 0.281 8.8 20 1.37 1.50 8.6 0.134 0.146 8.2 30 0.062 6 0.143 56 -0.011 3 -0.022 3 49 40 -1.25 -1.28 2.3 -0.111 -0.110 1.0 50 -2.56 -2.70 5.2 -0.234 -0.242 3.3 20 10 4.02 4.38 8.2 0.382 0.418 8.6 20 2.71 2.97 8.7 0.260 0.284 8.4 30 1.40 1.53 8.4 0.141 0.154 8.4 40 0.083 5 0.191 56 0.015 0.029 8 49 50 -1.23 -1.24 0.8 -0.107 -0.103 3.9

 图 12 uIx的理论值与有限元分析值 Fig. 12 Theoretical analysis and finit element analysis results of uIx
 图 13 uIy的理论值与有限元分析值 Fig. 13 Theoretical analysis and finit element analysis results of uIy

4 基于解析式的参数分析

 图 14 参数α与柔度系数cIx的关系曲线 Fig. 14 Relation curves of parameter α and compliance coefficient cIx
 图 15 参数α与柔度系数cIy的关系曲线 Fig. 15 Relation curves of parameter α and compliance coefficient cIy

5 结论

1) 本文结合柔度矩阵法和刚体受力与移动规律得到了求解多外力作用下柔性机构输出位移解析解的方法，从而满足了实际情况中多外力作用情况的工程需要。该方法具有普遍适用性，能为一般柔性机构的性能分析和进一步的参数优化提供方法支持。

2) 运用该方法求解了两外力下的微动夹持器的输出位移，并与有限元方法的计算结果进行了对比。结果表明该方法具有可靠的精度及规范的求解过程。给出了机构参数α与柔度的解析表达式，根据关系曲线对结构参数α的选择做了定性分析，表明了该方法在参数优化方面的实际应用价值。

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#### 文章信息

GONG Jinliang, JIA Guopeng, ZHANG Yanfei

Method for output displacement solving of compliant micro-motion mechanism with multi-input forces

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(3): 429-436
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0219