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Stiffness analysis of a flexible lever magnifying mechanism based on transfer matrix method
ZHENG Yangyang, GONG Jinliang, ZHANG Yanfei
School of Mechanical Engineering, Shandong University of Technology, Zibo 255049, China
Received: 2016-03-29; Accepted: 2016-06-03; Published online: 2016-06-08 16:54
Foundation item: National Natural Science Foundation of China (61303006); Foundation for Outstanding Young Scientist in Shandong Province (BS2012ZZ009); Young Teachers Development Support Plan of Shandong University of Technology (2013-02)
Corresponding author. GONG Jinliang, E-mail: 84374294@qq.com
Abstract: Stiffness is an important performance index for the dynamic performance and positioning precision of compliant micromanipulator. Concept of transfer matrix in engineering was applied to the stiffness analysis here. First, according to its structure characteristics, the compliant micromanipulator was modularized and each unit was treated as flexible body. Taking axial, shear and bending deformation into consideration, we solved transfer matrix of the subunit. Then each unit was assembled through the transfer matrix. Finally, relational model between input force and output displacement of compliant micromanipulator was established according to the force balance. The research result indicates that because multi-dimensional real deformation of each unit was taken into consideration, high-precision result was guaranteed. At the same time, the deformation compatibility equations between flexible and rigid units did not need to be solved during the analysis, and conversion of compliant micromanipulator global coordinate system was avoided. The analysis and computation time was also reduced. A kind of flexible lever magnifying mechanism stiffness model was established with this method. The error is less than 6.4% compared with the result of finite element analysis. The accuracy of analysis is improved effectively, and important theoretical basis is provided for parameter design.
Key words: compliant micromanipulator     transfer matrix     flexible lever magnifying mechanism     stiffness     finite element

1 传递矩阵法 1.1 基本原理

 图 1 梁单元结点力与位移 Fig. 1 Beam element node force and displacement

 (1)

 (2)

 (3)

L为梁的长度；E为弹性模量; G为剪切模量；u为应力分布不均匀系数，当截面为矩形时取u=1.2；A (x) 为截面剪切面积函数；I (x) 为惯性积函数。

1.2 传递矩阵坐标变换

 图 2 梁单元旋转变换 Fig. 2 Rotation transform of beam element
 (4)
 (5)

 (6)

 (7)

 (8)
2 柔性杠杆放大机构刚度求解 2.1 机构描述

 图 3 柔性杠杆放大机构示意图 Fig. 3 Schematic diagram of flexible lever magnifying mechanism

2.2 子单元刚度建模

 图 4 柔性杠杆放大机构单元划分 Fig. 4 Element partition of flexible lever magnifying mechanism
 图 5 柔性杠杆放大机构参数模型 Fig. 5 Parameter model of flexible lever magnifying mechanism

2.2.1 弹性移动副刚度

 图 6 弹性移动副 Fig. 6 Flexible prismatic pair

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2.2.2 柔性梁传递矩阵

 图 7 柔性梁单元 Fig. 7 Flexible beam element
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T3=λ(90°)T Tl(l2, h1)λ(90°)

T5=λ(180°)TTl(l4, h2)λ(180°)

T6=λ(180°)TTl(l5, h2)λ(180°)

T8=λ(-90°)TTl(l6, h3)λ(-90°)

2.2.3 柔性铰链传递矩阵

 图 8 柔性铰链 Fig. 8 Flexible hinge

 (13)

 (14)

Tr中的元素

T2=λ(90°)TTr(R1, t2)λ(90°)

T4=λ(90°)TTr(R2, t3)λ(90°)

T7=λ(-90°)TTr(R4, t5)λ(-90°)

T9=λ(-90°)TTr(R5, t6)λ(-90°)

2.3 力平衡建立

 图 9 子单元2、3、4、5受力分析 Fig. 9 Force analysis of subunit 2, 3, 4, 5

 (15)

 (16)

 (17)

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e点位移连续得

 (19)

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 (21)

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 (23)

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3 算例分析

 图 10 柔性杠杆放大机构ANSYS网格划分 Fig. 10 ANSYS mesh generation of flexible lever magnifying mechanism

 l5/mm 刚度/(MN·m-1) 误差/% 有限元法 传递矩阵法 50.5 10.931 9 11.618 2 6.278 28 60.5 11.326 9 12.034 7 6.248 46 70.5 12.008 9 12.768 9 6.329 08 80.5 12.903 9 13.273 1 6.348 12

 图 11 有限元法与传递矩阵法关系曲线 Fig. 11 Relation curves of finite element method and transfer matrix method

 图 12 刚度k与l5的关系曲线 Fig. 12 Relation curve of stiffness k and l5

4 结论

1) 传递矩阵法将柔性微动机构划分为若干子单元，考虑各个子单元的轴向力、剪切力、弯矩变形。根据子单元传递矩阵的传递关系及力平衡求解整体机构刚度。避免了刚柔单元的位移协调方程及将各单元转换到整体坐标系，减少了计算工作量，且具有递推性，利于编程。

2) 以柔性杠杆放大机构为例，将有限元法分析结果与传递矩阵法分析结果相对比，误差在6.4%以内，且刚度的变化规律是一致的，证明了此方法的精确性，与实际情况相符。因此可通过传递矩阵法求解柔性杠杆放大机构刚度的最小值，使其在相同的输入力的情况下，输出位置最大。对柔性杠杆放大机构参数设计提供了理论依据。同时，此方法同样适用于其他柔性微动机构的刚度及运动学分析。

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#### 文章信息

ZHENG Yangyang, GONG Jinliang, ZHANG Yanfei

Stiffness analysis of a flexible lever magnifying mechanism based on transfer matrix method

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(4): 849-856
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0245