西安建筑科技大学学报(自然科学版)

为解决给定张拉整体结构的可行力密度找形问题,本文提出一种基于二次奇异值分解的优化设计方法.该方法需事先根据结构的几何对称性对构件进行合理分组,并在此基础上重组平衡矩阵,通过奇异值分解法求解线性齐次方程组以获得满足要求的可行初始力密度.同时引入了切线刚度矩阵特征值的计算方法来检验张拉整体结构的稳定性.最后,基于空间十二杆张拉整体结构不同分组的计算结果,利用木棒和弹力绳模拟拉压单元制作模型,验证了结构形态的合理性及本文算法的有效性.

In order to solve the form-finding problem of feasible force density for a given tensegrity structure,an optimal design method based on quadratic singular value decomposition is presented. In this method,the component groups should be reasonably divided according to the geometric symmetry of the structure,and then the balance matrix should be reorganized. The singular value decomposition method is then used to solve the linear homogeneous system of equations in seeking the feasible force density that meets the requirements. The method of calculating the eigenvalue of the tangent stiffness matrix is also introduced to check the stability of the tensegrity structure. Finally,the wooden bar and elastic rope are used to simulate the tensegrity components,and the model is assembled based on the calculation results of different groups of the 12-strut spatial tensegrity structure,verifying the rationality of the structure configuration and the effectiveness of the proposed algorithm.

引言
1 找形算法
2 稳定性判断
3 算例
4 张拉整体结构模型制作
5 结论

图1 新型张拉整体结构模型<br/>Fig.1 The new tensegrity structure model

图1 新型张拉整体结构模型
Fig.1 The new tensegrity structure model

表1 节点的坐标<br/>Tab.1 The nodal coordinates of the structure

表1 节点的坐标
Tab.1 The nodal coordinates of the structure

表2 张拉整体结构初始力密度(h=5)<br/>Tab.2 Initial force density of the tensegrity structure(h=5)

表2 张拉整体结构初始力密度(h=5)
Tab.2 Initial force density of the tensegrity structure(h=5)

表3 12杆张拉整体结构切线刚度矩阵特征值(h=5)<br/>Tab.3 Eigenvalues of the tangent stiffness matrix of the 12-strut tensegrity structure(h=5)

表3 12杆张拉整体结构切线刚度矩阵特征值(h=5)
Tab.3 Eigenvalues of the tangent stiffness matrix of the 12-strut tensegrity structure(h=5)

图2 空间12杆张拉整体结构单元分组情况(h=5)<br/>Fig.2 Elemental grouping of the 12-strut spatial tensegrity structure(h=5)

图2 空间12杆张拉整体结构单元分组情况(h=5)
Fig.2 Elemental grouping of the 12-strut spatial tensegrity structure(h=5)

图3 目标函数的迭代过程(h=12)<br/>Fig.3 Convergence process of the target function(h=12)

图3 目标函数的迭代过程(h=12)
Fig.3 Convergence process of the target function(h=12)

表4 张拉整体结构初始力密度(h=12)<br/>Tab.4 Initial force density of the tensegrity structure(h=12)

表4 张拉整体结构初始力密度(h=12)
Tab.4 Initial force density of the tensegrity structure(h=12)

表5 空间12杆张拉整体结构切线刚度矩阵特征值(h=12)<br/>Tab.5 Eigenvalues of the tangent stiffness matrix of the 12-struts tensegrity structure in space(h=12)

表5 空间12杆张拉整体结构切线刚度矩阵特征值(h=12)
Tab.5 Eigenvalues of the tangent stiffness matrix of the 12-struts tensegrity structure in space(h=12)

图4 空间12杆张拉整体结构分组情况(h=12)<br/>Fig.4 Elemental grouping of the 12-strut spatial tensegrity structure(h=12)

图4 空间12杆张拉整体结构分组情况(h=12)
Fig.4 Elemental grouping of the 12-strut spatial tensegrity structure(h=12)

图5 空间12杆张拉整体结构(h=36)<br/>Fig.5 The 12-strut spatial tensegrity structure(h=36)

图5 空间12杆张拉整体结构(h=36)
Fig.5 The 12-strut spatial tensegrity structure(h=36)

图6 目标函数的迭代过程(h=36)<br/>Fig.6 Convergence process of the target function(h=36)

图6 目标函数的迭代过程(h=36)
Fig.6 Convergence process of the target function(h=36)

表6 张拉整体结构初始力密度(h=36)<br/>Tab.6 Initial force density of the tensegrity structure(h=36)

表6 张拉整体结构初始力密度(h=36)
Tab.6 Initial force density of the tensegrity structure(h=36)

表7 空间12杆张拉整体结构切线刚度矩阵特征值(h=36)<br/>Tab.7 Eigenvalues of the tangent stiffness matrix of the 12-strut spatial tensegrity structure(h=36)

表7 空间12杆张拉整体结构切线刚度矩阵特征值(h=36)
Tab.7 Eigenvalues of the tangent stiffness matrix of the 12-strut spatial tensegrity structure(h=36)

图7 材料准备
Fig.7 Material preparation

图8 空间十二杆张拉整体结构实体模型<br/>Fig.8 Entity model of the 12-strut spatialtensegrity structure

图8 空间十二杆张拉整体结构实体模型
Fig.8 Entity model of the 12-strut spatialtensegrity structure

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备注

引言

1 找形算法

2 稳定性判断

3 算例

4 张拉整体结构模型制作

5 结论

期刊信息