﻿ 考虑单元界面关联特性的微定位平台刚度建模<sup>*</sup>
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Stiffness modeling of micro-positioning platform considering unit interface relevance characteristics
GONG Jinliang, ZHENG Yangyang, ZHANG Yanfei
School of Mechanical Engineering, Shandong University of Technology, Zibo 255049, China
Received: 2016-12-28; Accepted: 2017-02-15; Published online: 2017-03-24 16:04
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: 84374294@qq.com
Abstract: A parallel micro-positioning platform with folded beams was designed, which has characteristics of large stroke and low resistance. Stiffness was solved by transfer matrix method. First, transfer matrix of flexible subunit was established. The transitivity characteristics became available by taking advantage of the common node belonging to adjacent elements. Finally, the stiffness matrix between input force and output displacement of flexible mechanism was solved according to the force balance equation and compatibility equation of deformation. A method to solve the stiffness of compliant folded beam and compliant prismatic pair considering full flexibility was put forward. The result of deformation error is less than 20.5% compared with finite element analysis. On this basis, this method ignored the correlation between the subunits because the modular stiffness analysis method regarded each subunit as independent. A modified method considering each subunit dependency was put forward, which reduced the error to less than 10% and made the results better meet the actual engineering needs.
Key words: flexible mechanism     stiffness     transfer matrix     interface relevance characteristics     revision method

1 传递矩阵法

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 图 1 梁单元结点力及外载作用 Fig. 1 Node force and external load of beam unit

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2 大行程并联微定位平台刚度 2.1 机构描述

 图 2 大行程并联微定位平台 Fig. 2 Large stroke parallel micro-positioning platform
2.2 子单元刚度

2.2.1 弹性移动副

 图 3 弹性移动副单元划分 Fig. 3 Compliant prismatic pair unit division

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2.2.2 弹性折叠梁

 图 4 弹性折叠梁单元划分 Fig. 4 Compliant folded beam unit division

 图 5 弹性折叠梁单元A划分 Fig. 5 Compliant folded beam unit A division

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2.3 大行程并联微定位平台刚度求解

2.3.1 杠杆放大机构

 图 6 柔性杠杆放大机构单元划分 Fig. 6 Flexible lever magnifying mechanism unit division

F7p′为平行四杆机构对其产生的作用力。结点7为杠杆放大机构、平行四杆机构以及折叠梁的结合处，因此在下文的求解过程中，将结点7处的力与位移的命名与其单元相联系，如F7h表示在单元h上结点7的力。

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 图 7 单元f受力分析 Fig. 7 Force analysis of unit f

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2.3.2 平行四杆机构

 图 8 柔性平行四杆机构单元划分 Fig. 8 Flexible parallel four-bar mechanism unit division

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2.3.3 计算整体刚度

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2条支链对动平台作用力及运动示意图如图 9所示，由动平台力平衡得

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 图 9 动平台作用力及运动示意图 Fig. 9 Moving platform force and motion diagram

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θd值接近零，因此sin θdθd；cos θd≈1。式(37)可简化为

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3 算例分析 3.1 传递矩阵法与有限元法结果对比

 图 10 大行程并联微定位平台参数模型 Fig. 10 Large stroke parallel micro-positioning platform parameter model

 图 11 大行程并联微定位平台有限元模型 Fig. 11 Finite element model of large stroke parallel micro-positioning platform

 fwy/N 0 50 100 150 200 ux有限元法/μm -2.030 63 1.556 45 5.143 53 8.730 61 12.317 70 ux修正前/μm -2.094 30 1.308 69 4.711 68 8.114 66 9.816 16 ux修正后/μm -2.013 07 1.410 88 4.834 82 8.258 77 11.682 70 ux修正前误差/% 3.135 15.918 8.396 7.055 20.309 ux修正后误差/% 0.865 9.353 6.002 5.404 5.155 uy有限元法/μm 7.184 39 6.170 25 5.156 11 4.141 96 3.127 82 uy修正前/μm 6.805 97 5.758 83 4.711 68 4.188 10 3.140 65 uy修正后/μm 6.847 90 5.841 36 4.834 82 3.828 29 2.821 75 uy修正前误差/% 5.267 6.668 8.619 1.114 0.410 uy修正后误差/% 4.684 5.330 6.231 7.573 9.785

3.2 考虑界面关联特性的修正方法

 图 12 单元a、b结构图 Fig. 12 Unit a, b structure chart

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4 结论

1) 本文设计了一种压电陶瓷驱动的大行程并联微定位平台，采用含折叠梁的杠杆放大机构，提高其位移输出倍率，减少传动链的阻力，实现大行程位移输出。

2) 利用传递矩阵法求解并联微定位平台，将平台按结构划分单元，分别求解各单元传递矩阵，根据各相邻单元公共结点实现其传递性；再由变形协调方程、力平衡方程求解整体刚度；并求解了弹性移动副及弹性折叠梁考虑全柔性的刚度公式。

3) 将本文求解结果与有限元分析结果对比，误差在20.5%以内，分析其误差产生的原因并提出了考虑各子单元界面关联特性的修正方法，修正后误差降低到10%以内，证明了修正方法的有效性和传递矩阵的准确性。

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#### 文章信息

GONG Jinliang, ZHENG Yangyang, ZHANG Yanfei

Stiffness modeling of micro-positioning platform considering unit interface relevance characteristics

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(1): 10-17
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0972