基于拓扑优化的平面2-DOF柔顺并联机构构型设计
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摘要
目的运用拓扑优化技术设计一种平面2-DOF(Degrees-of-Freedom)柔顺并联机构,使其具有微米级别的微运动特性。方法依据平面2-DOF并联原型机构的受力情况进行运动特性分析,基于同构封闭矢量映射原理构建平面2-DOF柔顺并联机构的微分运动Jacobian矩阵,表达多自由度的柔顺并联机构从输入到输出间各关节的运动关系。定义柔度为优化目标函数,微分运动Jacobian矩阵为运动条件,材料体积分数为约束条件,构建平面2-DOF柔顺并联机构材料属性的有理近似模型,并运用移动渐进线法优化求解。结果优化后的平面2-DOF柔顺并联机构构型在x方向的位移为-0.0089mm,在y方向的位移为0.0053 mm,同时,其沿x方向和y方向的微位移理论值为-0.0031 mm和0.0067 mm。结论与并联原型机构的运动特性相比,平面2-DOF柔顺并联机构优化后的微运动特性具有一致性,均为微米级。
The paper aims to design a planar 2-DOF compliant parallel mechanism with the topology optimization technology to make it have kinetic characteristics at the micron level. The kinetic characteristic was analyzed according to the loading situation of planar 2-DOF parallel prototype mechanism. The kinetic differential Jacobian matrix of planar2-DOF compliant parallel mechanism was built based on isomorphic closed vector mapping principle to show the kinematic relations of the joints from input to output in multiple DOF compliant parallel mechanism. In this paper, the optimization model of rational approximation of material properties for planar 2-DOF compliant parallel mechanism was built, which defined the compliance as the optimization objective function, the differential kinetic Jacobian matrix as kinetic condition and the material volume ratio as constraint condition. The Moving Asymptotes Method was used to solve the optimization problem. For the configuration of planar 2-DOF compliant parallel mechanism after optimization, its displacement in x and y direction was-0.0089 mm and 0.0053 mm; however, the theoretical displacement in x and y direction was-0.0031 mm and 0.0067 mm. Results showed that the differential kinetic characteristics of planar 2-DOF compliant parallel mechanism optimal configuration is consistent with its parallel prototype mechanism, and the positional accuracy is achieved at the micron level.
The paper aims to design a planar 2-DOF compliant parallel mechanism with the topology optimization technology to make it have kinetic characteristics at the micron level. The kinetic characteristic was analyzed according to the loading situation of planar 2-DOF parallel prototype mechanism. The kinetic differential Jacobian matrix of planar2-DOF compliant parallel mechanism was built based on isomorphic closed vector mapping principle to show the kinematic relations of the joints from input to output in multiple DOF compliant parallel mechanism. In this paper, the optimization model of rational approximation of material properties for planar 2-DOF compliant parallel mechanism was built, which defined the compliance as the optimization objective function, the differential kinetic Jacobian matrix as kinetic condition and the material volume ratio as constraint condition. The Moving Asymptotes Method was used to solve the optimization problem. For the configuration of planar 2-DOF compliant parallel mechanism after optimization, its displacement in x and y direction was-0.0089 mm and 0.0053 mm; however, the theoretical displacement in x and y direction was-0.0031 mm and 0.0067 mm. Results showed that the differential kinetic characteristics of planar 2-DOF compliant parallel mechanism optimal configuration is consistent with its parallel prototype mechanism, and the positional accuracy is achieved at the micron level.
引文
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[3]余跃庆,徐齐平.柔顺机构PR伪刚体动力学建模与特性分析[J].农业机械学报,2013,44(3):225-229.YU Yue-qing,XU Qi-ping.Dynamic Modeling and Characteristic Analysis of Compliant Mechanisms Based on PR Pseudo-rigid-body Model[J].Transactions of the Chinese Society for Agricultural Machinery,2013,44(3):225-229.
[4]李仕华,句彦儒,马琦翔,等.新型3-RRRRR并联微动平台的位置及误差分析[J].中国机械工程,2014,25(23):3142-3146.LI Shi-hua,JU Yan-ru,MA Qi-xiang,et al.Analysis of Position and Error for A Novel 3-RRRRR Parallel Micromanipulator[J].China Mechanical Engineering,2014,25(23):3142-3146.
[5]朱大昌,宋马军,李雅琼.平面两自由度全柔顺微运动并联机构的拓扑优化设计[J].机械设计,2015,32(10):15-18.ZHU Da-chang,SONG Ma-jun,LI Ya-qiong.Topological Optimization Design of Full Complaint Parallel Micro-motion Mechanism under Plane Two Degree-offreedom[J].Journal of Machine Design,2015,32(10):15-18.
[6]焦洪宇,周奇才,李文军,等.基于变密度法的周期性拓扑优化[J].机械工程学报,2013,49(13):132-138.JIAO Hong-yu,ZHOU Qi-cai,LI Wen-jun,et al.Periodic Topology Optimization Using Variable Density Method[J].Journal of Mechanical Engineering,2013,49(13):132-138.
[7]MARTINEZ J M.A Note on the Theoretical Convergence Properties of the SIMP Method[J].Struct Multidisc Optim,2005,29:319-323.
[8]张晖,刘书田,张雄.考虑自重载荷作用的连续体结构拓扑优化[J].力学学报,2009,41(1):98-103.ZHANG Hui,LIU Shu-tian,ZHANG Xiong.Topology Optimization of Continuum Structures Subjected to Self-weight Loads[J].Chinese Journal of Theoretical and Applied Mechanics,2006,27(6):1209-1216.
[9]SVANBERG K.The Method of Moving Asymp-totes:A New Method for Structural Optimization[J].International Journal for Numerical Methods in Engineering,1987,24(2):359-373.
[10]张云清,罗震,陈立平,等.基于移动渐近线方法的结构多刚度拓扑优化设计[J].航空学报,2006,27(6):1209-1215.ZHANG Yun-qing,LUO Zhen,CHEN Li-ping,et al.Topological Optimization Design for Multiple Stiffness Structures Using Method of Moving Asymptotes[J].Acta Aeronautica ET Astronautica Sinica,2006,27(6):1209-1215.
[11]SIGMUND O.A 99 Line Topology Optimization Code Written in Matlab[J].Lecture Notes in Computer Science,2007,4573(21):120-127.
[12]BENDSOE M P.Optimization of Structural Topology,Shape and Material[M].Berlin,Heidelberg,New York:Shape&Material,1995.
[13]HER I,CHANG J C.A Linear Scheme for the Displacement Analysis of Micropositioning Stages with Flexure Hinges[J].Transactions of the ASME,Journal of Mechanical Design,1994,116(3):770-776.
[14]SIGMUND O.Design of Mulitphysics Actuators Using Topology Optimization-PartⅠTwo-meterial Structures[J].Comput Methods Appl Mech Engrg,2001,190:6577-6664.
[15]SIGMUND O.Design of Mulitphysics Actuators Using Topology Optimization-PartⅡOne-material Structures[J].Compute Methods Appl Mech Engrg,2001,490:6605-6627.