基于线性回归与检验的不规则构件精确刚度矩阵建模
Precise Stiffness Matrix Modeling of Irregular Complex Component Based on Multiple Linear Regression and According Test
作者：张彦斐(山东理工大学 机械工程学院, 山东 淄博 255049)；李春霞(山东理工大学 机械工程学院, 山东 淄博 255049)；宫金良(山东理工大学 机械工程学院, 山东 淄博 255049)
Author：ZHANG Yanfei(School of Mechanical Eng., Shandong Univ. of Technol., Zibo 255049, China)；LI Chunxia(School of Mechanical Eng., Shandong Univ. of Technol., Zibo 255049, China)；GONG Jinliang(School of Mechanical Eng., Shandong Univ. of Technol., Zibo 255049, China)
收稿日期：2016-09-04 年卷（期）页码：2017,49(1):103-108
期刊名称：工程科学与技术
Journal Name：Advanced Engineering Sciences
关键字：不规则构件;刚度矩阵;多元线性回归;综合检验
Key words：irregular component;stiffness matrix;multiple linear regression;synthetic test
基金项目：国家自然科学基金资助项目（61303006）；山东省优秀中青年科学家科研奖励基金资助项目（BS2012ZZ009）；山东理工大学青年教师发展支持计划资助项目（2014-03）
中文摘要
传统刚度分析方法只能得到规则结构的刚度解析表达式，而针对不规则构件，通常采用将复杂构件简化为规则件的办法求解，由于过度简化得到的刚度矩阵精确度下降。为了便于进行面向刚度的机械系统参数优化设计，必须解决不规则构件的规范化刚度矩阵求解问题。以3-URS（U-虎克铰、R-转动副、S-球铰）并联机构中U型轴复杂构件为例，提出一种基于多元线性回归原理求解复杂构件刚度矩阵的方法。将刚度矩阵的每个元素看作回归系数，采用ANSYS分析得到力和位移值的多组数对，基于多元线性回归理论求解回归系数，并将其组成6×6的刚度矩阵。由于该刚度矩阵主要应用于后续的刚度矩阵叠加，为降低计算难度，提高计算效率，在保证一定精度的前提下，对回归方程中的每一个回归系数作t检验，剔除不显著变量，对方程形式进行简化。对简化后的方程进行F检验和拟合优度检验，验证方程的回归效果。应用该方法求解了U型轴复杂构件的刚度矩阵，通过实验给出了24组样本数据，依据多元线性回归分析得到6×6刚度矩阵，进行t检验后部分矩阵元素变为零，在对其进行F检验和拟合优度检验后得到机构的刚度矩阵。在相同外力作用下，对构件变形量应用该理论计算得到的刚度矩阵的理论计算值和ANSYS分析值进行了对比，结果表明最大误差仅为0.11%，验证了方法的准确性。
英文摘要
The traditional stiffness analysis method can only get the analytical stiffness expression of the regular structure.But for the irregular components,it should be simplified as many parts of certain regular component.The accuracy of stiffness matrix is reduced due to over-simplification.In order to optimize the mechanical system parameters based on stiffness,it is necessary to solve the problem of normalized stiffness matrix of irregular components.Taking the U-shape axis in 3-URS (U-hooke joint,R-revolute joint,S-spherical joint) parallel mechanism as an example,the method for solving stiffness matrix of the complex component was proposed based on multiple linear regression theory.Each element of the stiffness matrix was considered as a regression coefficient.Multiple sets of force and displacement pairs were obtained by adopting ANSYS software and then used to solve the regression coefficients based on multiple linear regression analysis.The regression coefficients were arranged in a form of 6×6 stiffness matrix and it was mainly used for superposition.In order to reduce the computational difficulty and improve the computational efficiency,t-test was applied on every regression coefficients and those non-significant variables would be eliminated to simplify the equation form with a certain precision.Then F-test and goodness of fit test were both performed on the simplified results to verify the regression effect.Finally,the method was applied to solve the stiffness matrix of U-shaped axis complex element.Twenty four sets of sample data were obtained by the experiment method.The 6×6 stiffness matrix of regression coefficients were computed based on multiple linear regression theory.The partial matrix elements became zero after the t-test.Finally,the stiffness matrix of the mechanism was obtained after F-test and goodness of fit test.The accuracy of the method was verified by comparing the deformation values computed by ANSYS analysis and calculated by the stiffness matrix under the same external forces.The results showed that the maximum error is only 0.11%.
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