Method for output displacement solving of compliant micro-motion mechanism with multi-input forces
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摘要:
对于单个外力作用下的微动机构，柔度描述了外力与其输出位移之间的关系，是影响微动机构动态性能和精度的重要性能指标。对于多个外力作用下的情况，与之相对应的是微动机构输出位移与其多外力间的关系式。为求解该关系式，采用了柔度矩阵和刚体运动规律相结合的方法。首先，对整体进行单元划分，并求解结构中各柔性单元的端点位移与端点力之间的关系式；然后，根据各单元间的结构关系求解各不同单元端点力或端点位移间的叠加和协调关系式；最后，综合所有求得的关系式整理出输出位移与外力之间的关系式。运用该方法求解了一种微动夹持器的工作端位移，并与有限元分析软件的计算结果进行了对比。研究结果表明：该方法具有普遍适用性以及可靠的精度，能够用于一般微动机构的性能分析与优化。应用MATLAB软件对求得的关系式进行分析，获得了结构参数优化所需的理论依据。
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.
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图 12 uIx的理论值与有限元分析值
Figure 12. Theoretical analysis and finit element analysis results of uIx
图 13 uIy的理论值与有限元分析值
Figure 13. Theoretical analysis and finit element analysis results of uIy
图 14 参数α与柔度系数cIx的关系曲线
Figure 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
Figure 15. Relation curves of parameter α and compliance coefficient cIy
表 1 位移uIx和uIy的理论值与有限元分析值结果对比
Table 1. Comparison of displacement uIx and uIy results between theoretical analysis and finit element analysis
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 
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