工程科学与技术   2017, Vol. 49 Issue (1): 103-108

Precise Stiffness Matrix Modeling of Irregular Complex Component Based on Multiple Linear Regression and According Test
ZHANG Yanfei, LI Chunxia, GONG Jinliang
School of Mechanical Eng., Shandong Univ. of Technol., Zibo 255049, China
Abstract: 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%.
Key words: irregular component    stiffness matrix    multiple linear regression    synthetic test

1 刚度矩阵多元线性回归分析

 $\mathit{\boldsymbol{F}} = \mathit{\boldsymbol{KX}}$ (1)

 ${\mathit{\boldsymbol{K}}_i} = \left[ {\begin{array}{*{20}{c}} {{k_{i1}}}&{{k_{i2}} \ldots }&{{k_{i6}}} \end{array}} \right],{\rm{ }}i = 1,2, \ldots ,6。$

K矩阵第1行方程为例，其方程为：

 ${F_x} = {k_{11}}{\Delta _x} + {k_{12}}{\Delta _y} + {k_{13}}{\Delta _z} + {k_{14}}{\Delta _{\theta x}} + {k_{15}}{\Delta _{\theta y}} + {k_{16}}{\Delta _{\theta z}}$ (2)

 $\left\{ \begin{array}{l} {F_{1x}} = {k_{11}}{\Delta _{1x}} + {k_{12}}{\Delta _{1y}} + {k_{13}}{\Delta _{1z}} + {k_{14}}{\Delta _{1\theta x}} + \\ \;\;\;\;\;\;\;\;{k_{15}}{\Delta _{1\theta y}} + {k_{16}}{\Delta _{1\theta z}},\\ {F_{2x}} = {k_{11}}{\Delta _{2x}} + {k_{12}}{\Delta _{2y}} + {k_{13}}{\Delta _{2z}} + {k_{14}}{\Delta _{2\theta x}} + \\ \;\;\;\;\;\;\;\;{k_{15}}{\Delta _{2\theta y}} + {k_{16}}{\Delta _{2\theta z}},\\ \cdots \\ {F_{nx}} = {k_{11}}{\Delta _{nx}} + {k_{12}}{\Delta _{ny}} + {k_{13}}{\Delta _{nz}} + {k_{14}}{\Delta _{n\theta x}} + \\ \;\;\;\;\;\;\;\;{k_{15}}{\Delta _{n\theta y}} + {k_{16}}{\Delta _{n\theta z}} \end{array} \right.$ (2)

 $\mathit{\boldsymbol{Y}} = \Delta \mathit{\boldsymbol{XK}}_1^{\rm{T}}$ (3)
 $\mathit{\boldsymbol{Y}} = {\left[ {\begin{array}{*{20}{c}} {{F_{1x}}}&{{F_{2x}}}& \cdots &{{F_{nx}}} \end{array}} \right]^{\rm{T}}}$ (4)
 $\Delta \mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{\Delta _{1x}}}&{{\Delta _{1y}}}&{{\Delta _{1z}}}&{{\Delta _{1\theta x}}}&{{\Delta _{1\theta y}}}&{{\Delta _{1\theta z}}}\\ {{\Delta _{2x}}}&{{\Delta _{2y}}}&{{\Delta _{2z}}}&{{\Delta _{2\theta x}}}&{{\Delta _{2\theta y}}}&{{\Delta _{2\theta z}}}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {{\Delta _{nx}}}&{{\Delta _{ny}}}&{{\Delta _{nz}}}&{{\Delta _{n\theta x}}}&{{\Delta _{n\theta y}}}&{{\Delta _{n\theta z}}} \end{array}} \right]$ (5)

 图1 刚度矩阵多元线性回归分析流程图 Fig. 1 Flow chart of rigid matrix analysis based on multiple linear regression method

 $\begin{array}{l} M(\mathop {{k_{11}}}\limits^ \wedge ,\mathop {{k_{12}}}\limits^ \wedge , \cdots ,\mathop {{k_{16}}}\limits^ \wedge ) = \sum\limits_{i = 1}^n {\left( {{{({y_i} - \mathop {{k_{11}}}\limits^ \wedge {x_{i1}} - \mathop {{k_{12}}}\limits^ \wedge {x_{i2}} - \ldots - \mathop {{k_{1n}}}\limits^ \wedge {x_{in}})}^2}} \right)} = \\ \mathop {\min }\limits_{{k_{11}},{k_{12}}, \cdots ,{k_{16}}} 1\sum\limits_{i = 1}^n {{{({y_i} - {k_{11}}{x_{i1}} - {k_{12}}{x_{i2}} - \ldots - {k_{1n}}{x_{in}})}^2}} \end{array}$ (6)
2 刚度矩阵多元线性回归检验

 ${t_\alpha } = \frac{{{K_{1j}}}}{{\sqrt {{c_{jj}}} \sqrt {\frac{{{Q_{{\rm{res}}}}}}{{n - m - 1}}} }}$ (7)

 $F = \frac{{{Q_{{\rm{res}}}}/m}}{{{Q_{{\rm{res}}}}/\left( {n - m - 1} \right)}}$ (8)

 ${R^2} = \frac{{{Q_{{\rm{res}}}}}}{{{Q_{\rm{T}}}}} = \frac{{{Q_{\rm{T}}} - {Q_{{\rm{res}}}}}}{{{Q_{\rm{T}}}}}$ (9)

3 数值算例

 1. 动平台；2. 上连杆；3. 滑动槽；4. 下连杆；5. U型轴。 图2 3-URS并联机构 Fig. 2 3-URS parallel mechanism

3-URS并联机构中球铰、动平台、底座支架等均为规则组件，可以采用传统方法建立刚度模型。而上下连杆、滑动槽、U型轴等构件结构复杂，均为不规则形状。

 图3 U型轴不规则构件 Fig. 3 U-shape axis irregular component

 $\begin{array}{l} \mathit{\boldsymbol{K}} = \\ \left[ {\begin{array}{*{20}{c}} {5.338{\rm{ }}78 \times {{10}^8}}&{7.237{\rm{ }}12 \times {{10}^4}}&{ - 1.222{\rm{ }}14 \times {{10}^5}}&{7.927{\rm{ }}35 \times {{10}^2}}&{2.187{\rm{ }}72 \times {{10}^2}}&{2.182{\rm{ }}18 \times {{10}^5}}\\ { - 1.288{\rm{ }}96 \times {{10}^5}}&{4.382{\rm{ }}29 \times {{10}^7}}&{ - 1.099{\rm{ }}33 \times {{10}^5}}&{1.274{\rm{ }}74 \times {{10}^3}}&{4.704{\rm{ }}96 \times {{10}^2}}&{8.741{\rm{ }}35 \times {{10}^4}}\\ {1.635{\rm{ }}09 \times {{10}^5}}&{ - 2.815{\rm{ }}65 \times {{10}^4}}&{1.739{\rm{ }}04 \times {{10}^8}}&{ - 1.474{\rm{ }}37 \times {{10}^6}}&{ - 4.801{\rm{ }}19 \times {{10}^5}}&{5.169{\rm{ }}35 \times {{10}^2}}\\ { - 2.963{\rm{ }}54 \times {{10}^5}}&{9.118{\rm{ }}17 \times {{10}^2}}&{ - 2.681{\rm{ }}78 \times {{10}^6}}&{3.004{\rm{ }}08 \times {{10}^4}}&{8.196{\rm{ }}44 \times {{10}^3}}&{ - 7.069{\rm{ }}42}\\ { - 7.773{\rm{ }}49 \times {{10}^2}}&{1.329{\rm{ }}65 \times {{10}^3}}&{ - 6.414{\rm{ }}28 \times {{10}^5}}&{5.228{\rm{ }}89 \times {{10}^3}}&{1.292{\rm{ }}13 \times {{10}^4}}&{ - 2.152{\rm{ }}21 \times {{10}^{ - 1}}}\\ {3.627{\rm{ }}36 \times {{10}^5}}&{1.706{\rm{ }}88 \times {{10}^5}}&{ - 2.008{\rm{ }}89 \times {{10}^3}}&{1.801{\rm{ }}09 \times 10}&{5.523{\rm{ }}76}&{3.954{\rm{ }}19 \times {{10}^3}} \end{array}} \right]。\end{array}$

 $\begin{array}{l} \mathit{\boldsymbol{T}} = \\ \left[ {\begin{array}{*{20}{c}} {551.777{\rm{ }}6}&{0.891{\rm{ }}4}&{ - 0.421{\rm{ }}9}&{0.3231}&{0.2730}&{663.173}\\ { - 3.888{\rm{ }}1}&{1{\rm{ }}575.420{\rm{ }}4}&{ - 1.107{\rm{ }}8}&{1.516{\rm{ }}5}&{1.713{\rm{ }}8}&{775.382{\rm{ }}1}\\ {1.045{\rm{ }}3}&{ - 0.214{\rm{ }}5}&{371.457{\rm{ }}1}&{ - 371.759{\rm{ }}9}&{ - 370.682{\rm{ }}4}&{0.971{\rm{ }}8}\\ { - 0.110{\rm{ }}2}&{0.406{\rm{ }}9}&{ - 335.113}&{443.663{\rm{ }}2}&{370.649{\rm{ }}9}&{ - 0.778{\rm{ }}4}\\ { - 0.597{\rm{ }}5}&{1.208{\rm{ }}0}&{ - 164.726{\rm{ }}0}&{158.519{\rm{ }}3}&{1{\rm{ }}199.428{\rm{ }}2}&{ - 0.048{\rm{ }}5}\\ {240.816{\rm{ }}9}&{135.048{\rm{ }}1}&{ - 0.445{\rm{ }}5}&{0.471{\rm{ }}5}&{0.442{\rm{ }}8}&{771.913{\rm{ }}0} \end{array}} \right]。\end{array}$

 $\begin{array}{l} \mathit{\boldsymbol{K}} = \\ \left[ {\begin{array}{*{20}{c}} {5.309{\rm{ }}59 \times {{10}^8}}&0&0&0&0&{2.168{\rm{ }}17 \times {{10}^5}}\\ { - 5.801{\rm{ }}015 \times {{10}^5}}&{4.442{\rm{ }}67 \times {{10}^7}}&0&0&{4.100{\rm{ }}05 \times {{10}^2}}&{8.680{\rm{ }}89 \times {{10}^4}}\\ 0&0&{1.732{\rm{ }}34 \times {{10}^8}}&{ - 1.468{\rm{ }}79 \times {{10}^6}}&{ - 4.785{\rm{ }}68 \times {{10}^5}}&0\\ 0&0&{ - 2.652{\rm{ }}76 \times {{10}^6}}&{2.979{\rm{ }}80 \times {{10}^4}}&{8.126{\rm{ }}20 \times {{10}^3}}&0\\ 0&0&{ - 6.359{\rm{ }}86 \times {{10}^5}}&{8.183{\rm{ }}10 \times {{10}^3}}&{1.290{\rm{ }}71 \times {{10}^4}}&0\\ {3.622{\rm{ }}74 \times {{10}^5}}&{1.716{\rm{ }}40 \times {{10}^5}}&0&0&0&{3.954{\rm{ }}39 \times {{10}^3}} \end{array}} \right]。\end{array}$

4 结论

1) 基于多元线性回归理论提出求解不规则构件刚度矩阵的统一解法。首先采用ANSYS分析得到力和位移值的多组数对，利用多元线性回归分析求解回归系数，然后应用t检验对方程形式进行简化，并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，从而解决了传统方法无法求解不规则构件刚度矩阵的难题。

2) 应用该方法求解了U型轴复杂构件的刚度矩阵，在相同外力作用下，对构件变形量的理论计算值和ANSYS分析值进行了对比，最大误差仅为0.11%，验证了方法的准确性。

3) 该方法为后续对整机做以刚度为目标的优化设计提供了重要依据。可将复杂构件的刚度矩阵作为常量与规则构件的刚度矩阵统一建模得到整机的刚度矩阵解析模型，解决了多数刚度优化设计中需要对不规则构件进行简化处理的问题，能够有效提高分析精度。

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