数字状张拉整体结构的构型设计与力学性能模拟
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摘要
针对大型张拉整体结构的设计问题,选取四棱柱状张拉整体结构和截角正八面体状张拉整体结构作为基本胞元,采用节点连接节点的方式建立球柱组合式数字状张拉整体结构,并使用基于结构刚度矩阵的大变形非线性数值求解方法对其进行力学性能分析.在两类胞元满足各自的自平衡条件和稳定性条件的前提下,组合得到的数字状张拉整体结构亦处于自平衡稳定状态,搭建了实物模型进行验证.以数字8状张拉整体结构为例,模拟研究了结构承受自重等分布载荷和单轴拉压等端部载荷时的静力学响应,以及结构无阻尼振动时的固有频率和模态等动力学性能.结果表明,结构在自重作用下的变形行为受初始预应力、压杆密度和拉索刚度的影响较大,对其进行合理配置方可确保结构具有足够刚度抵抗自重;结构在单轴拉压作用下呈现非线性的载荷-位移曲线,拉伸刚度随变形量的增大而增大,压缩刚度随变形量的增大而减小;结构的固有频率随初始预应力的增大而增大,而模态振型基本不变.研究结果丰富了大型张拉整体结构的外形种类,有望推动此类结构在土木建筑、结构材料等领域的应用.
Due to the novel mechanical properties, tensegrity structures have found various applications in science and engineering, and the design of large-scale tensegrities becomes a vital issue. In this paper, a series of number-shaped tensegrity structures are proposed by assembling the cylindrical and spherical tensegrity elementary cells. Specifically,the quadruplex prismatic tensegrities and the truncated regular octahedral tensegrities are selected as the elementary cells and then connected by using the node-on-node assembly scheme. Furthermore, structural stiffness matrix-based numerical method is employed to simulate the mechanical responses of the assembled tensegrities. Our results show that the obtained number-shaped tensegrities are self-equilibrated and stable when the elementary cells satisfy their selfequilibrium and stability conditions, respectively. A physical sculpture is also constructed using the aluminium alloy bars and nylon strings. Taking the eight-shaped tensegrity structure as an example, the static mechanical responses of the structure subjected to self-weight loading and uniaxial tension/compression are simulated, as well as the structural natural frequencies and modes of its free vibration. The simulations show that the tensegrity could have enough rigidity to bear the self-weight when the structural pre-stress level, the mass density of the compressed bars, and the stiffness of the tensioned strings match well. The load-displacement curves of the tensegrity under uniaxial loading are nonlinear, that is,the tensile stiffness increases with the tensile deformation, while the compressive stiffness decreases with the compressive deformation. The structural natural frequencies are dependent on the pre-stress level, while the vibration modes change little. The present work enriches the shapes of large-scale tensegrities and would promote their applications in civil and material engineering.
Due to the novel mechanical properties, tensegrity structures have found various applications in science and engineering, and the design of large-scale tensegrities becomes a vital issue. In this paper, a series of number-shaped tensegrity structures are proposed by assembling the cylindrical and spherical tensegrity elementary cells. Specifically,the quadruplex prismatic tensegrities and the truncated regular octahedral tensegrities are selected as the elementary cells and then connected by using the node-on-node assembly scheme. Furthermore, structural stiffness matrix-based numerical method is employed to simulate the mechanical responses of the assembled tensegrities. Our results show that the obtained number-shaped tensegrities are self-equilibrated and stable when the elementary cells satisfy their selfequilibrium and stability conditions, respectively. A physical sculpture is also constructed using the aluminium alloy bars and nylon strings. Taking the eight-shaped tensegrity structure as an example, the static mechanical responses of the structure subjected to self-weight loading and uniaxial tension/compression are simulated, as well as the structural natural frequencies and modes of its free vibration. The simulations show that the tensegrity could have enough rigidity to bear the self-weight when the structural pre-stress level, the mass density of the compressed bars, and the stiffness of the tensioned strings match well. The load-displacement curves of the tensegrity under uniaxial loading are nonlinear, that is,the tensile stiffness increases with the tensile deformation, while the compressive stiffness decreases with the compressive deformation. The structural natural frequencies are dependent on the pre-stress level, while the vibration modes change little. The present work enriches the shapes of large-scale tensegrities and would promote their applications in civil and material engineering.
引文
1刘锡良.现代空间结构.天津:天津大学出版社,2003(Liu Xiliang.Modern Space Structure.Tianjin:Tianjin University Press2003(in Chinese))
2 Motro R.Tensegrity:Structural Systems for the Future.London Butterworth-Heinemann,2003
3 Zhang JY,Ohsaki M.Tensegrity Structures:Form,Stability,and Symmetry.Tokyo:Springer,2015
4王博,周演,周昳鸣.面向连续体拓扑优化的多样性设计求解方法.力学学报,2016,48(4):984-993(Wang Bo,Zhou Yan,Zhou Yiming.Multiple designs approach for continuum topology optimization.Chinese Journal of Theoretical and Applied Mechanics2016,48(4):984-993(in Chinese))
5刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展.力学学报,2017,49(3):487-506(Liu Renhuai,Xue Jianghong.Development of nonlinear mechanics for laminated composite plates and shells.Chinese Journal of Theoretical and Applied Mechanics2017,49(3):487-506(in Chinese))
6 Murakami H,Nishimura Y.Static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules International Journal of Solids and Structures,2001,38(50-51)9359-9381
7 Murakami H,Nishimura Y.Infinitesimal mechanism modes of tensegrity modules.Solid Mechanics and Its Applications,2003 106:273-284
8姜涛.球状张拉整体单元的找形问题研究.[硕士论文].杭州浙江大学,2005(Jiang Tao.Form-finding of spherical tensegrity structures.[Master Thesis].Hangzhou:Zhejiang University,2005(in Chinese))
9 Lee S,Lee J.A novel method for topology design of tensegrity structures.Composite Structures,2016,152:11-19
10 Koohestani K.On the analytical form-finding of tensegrities.Composite Structures,2017,166:114-119
11 Feng XD.The optimal initial self-stress design for tensegrity grid structures.Computers&Structures,2017,193:21-30
12 Cai JG,Wang XY,Deng XW,et al.Form-finding method for multimode tensegrity structures using extended force density method by grouping elements.Composite Structures,2018,187:1-9
13张沛,冯健.张拉整体结构的稳定性判定及刚度分析.土木工程学报,2013(10):48-57(Zhang Pei,Feng Jian.Stability criterion and stiffness analysis of tensegrity structures.China Civil Engineering Journal,2013(10):48-57(in Chinese))
14 Zhang LY,Li Y,Cao YP,et al.Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures:A unified analytical solution.Proceedings of the Royal Society A,2012,4683323-3347
15 Zhang LY,Li Y,Cao YP,et al.A unified solution for self-equilibrium and super-stability of rhombic truncated regular polyhedral tensegrities.International Journal of Solids and Structures,2013,50:234-245
16罗阿妮,王龙昆,刘贺平等.张拉整体三棱柱构型和结构稳定性分析.哈尔滨工业大学学报,2016,48(7):82-87(Luo Ani,Wang Longkun,Liu Heping,et al.Analysis of configuration and structura stability of 3-bar tensegrity prism.Journal of Harbin Institute of Technology,2016,48(7):82-87(in Chinese))
17 Liu HP,Zhang JY,Ohsaki M.New 3-bar prismatic tensegrity units.Composite Structures,2018,84:306-313
18 Li Y,Feng XQ,Cao YP,et al.A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures.International Journal of Solids and Structures,2016,47(14):1888-1898
19 Zhang LY,Li Y,Cao YP,et al.Stiffness matrix-based form-finding method of tensegrity structures.Engineering Structures,2014,58(7):36-48
20 Zhang LY,Zhu SX,Li SX,et al.Analytical form-finding of tensegrities using determinant of force-density matrix.Composite Structures,2018,189:87-98
21 Kebiche K,Kazi-Aoual MN,Motro R.Geomerical non-linear analysis of tensegrity systems.Engineering Structures,1999,21:864-876
22 Tran HC,Lee J.Geometric and material nonlinear analysis of tensegrity structures.Acta Mechanica Sinica,2011,27(6):938-949
23 Zhang LY,Li Y,Cao YP,et al.A numerical method for simulating nonlinear mechanical responses of tensegrity structures under large deformations.Journal of Applied Mechanics,2013,80(6):061018
24 Zhang LY,Xu GK.Negative stiffness behaviors emerging in elastic instabilities of prismatic tensegrities under torsional loading.International Journal of Mechanical Sciences,2015,103:189-198
25 Zhang LY,Zhao ZL,Zhang QD,et al.Chirality induced by structural transformation in a tensegrity:Theory and experiment.Journal of Applied Mechanics,2016,83(4):041003
26 Zhang L,Cao Q,Zhang HW.An efficient algorithm for mechanical analysis of bimodular truss and tensegrity structures.International Journal of Mechanical Sciences,2013,70(5):57-68
27 Zhang L,Lu MK,Zhang HW,et al.Geometrically nonlinear elastoplastic analysis of clustered tensegrity based on the co-rotational approach.International Journal of Mechanical Sciences,2015,93:154-165
28 Zhang L,Cao Q,Liu Y,et al.An efficient finite element formulation for nonlinear analysis of clustered tensegrity.Engineering Computations,2016,33(1):252-273
29 Luo H,Bewley TR.Accurate simulation of near-wall turbulence over a compliant tensegrity fabric.Proceedings of SPIE,2005,5757:184-197
30 Li Y,Feng XQ,Cao YP,et al.Constructing tensegrity structures from one-bar elementary cells.Proceedings Mathematical Physical and Engineering Sciences,2010,466:45-61
31 Feng XQ,Li Y,Cao YP,et al.Design methods of rhombic tensegrity structures.Acta Mechanica Sinica,2010,466(2113):559-565
32张幸锵,袁行飞.新型三棱柱张拉整体平板结构研究.建筑结构,2011(3):24-27(Zhang Xinqiang,Yuan Xingfei.Research of a new triangular prism tensegrity plate structure.Building Structure,2011(3):24-27(in Chinese))
33 Liu HP,Geng JS,Luo AN.Tensegrity configuration method for connecting tensegrity units along their axes.Composite Structures.2017,162:341-350
34 Zhang LY,Zhao HP,Feng XQ.Constructing large-scale tensegrity structures with bar-bar connecting using prismatic elementary cells.Archive of Applied Mechanics,2015,85(3):383-394
35 Fraddosio A,Marzano S,Pavone G,et al.Morphology and selfstress design of V-expander tensegrity cells.Composites Part B Engineering,2017,115:102-116
36 Rimoli JJ,Pal RK.Mechanical response of 3-dimensional tensegrity lattices.Composites Part B,2017,115:30-42
37 Zhang LY,Li SX,Zhu SX,et al.Automatically assembled largescale tensegrities by truncated regular polyhedral and prismatic elementary cells.Composite Structures,2018,184:30-40
38章孝顺,章定国,洪嘉振.考虑曲率纵向变形效应的大变形柔性梁刚柔耦合动力学建模与仿真.力学学报,2016,48(3):692-701(Zhang Xiaoshun,Zhang Dingguo,Hong Jiazhen.Rigid-flexible coupling dynamic modeling and simulation with the longitudinal deformation induced curvature effect for a rotating flexible beam under large deformation.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):692-701(in Chinese))
39 Masic M,Skelton RE,Gill PE.Algebraic tensegrity form-finding.International Journal of Solids and Structures,2005,42(16):4833-4858
40 Zhang JY.Structural morphology and stability of tensegrity structures.Kyoto:Kyoto University,2007
41 Ali NBH,Smith IFC.Dynamic behavior and vibration control of a tensegrity structure.International Journal of Solids and Structures,2010,47(9):1285-1296
2 Motro R.Tensegrity:Structural Systems for the Future.London Butterworth-Heinemann,2003
3 Zhang JY,Ohsaki M.Tensegrity Structures:Form,Stability,and Symmetry.Tokyo:Springer,2015
4王博,周演,周昳鸣.面向连续体拓扑优化的多样性设计求解方法.力学学报,2016,48(4):984-993(Wang Bo,Zhou Yan,Zhou Yiming.Multiple designs approach for continuum topology optimization.Chinese Journal of Theoretical and Applied Mechanics2016,48(4):984-993(in Chinese))
5刘人怀,薛江红.复合材料层合板壳非线性力学的研究进展.力学学报,2017,49(3):487-506(Liu Renhuai,Xue Jianghong.Development of nonlinear mechanics for laminated composite plates and shells.Chinese Journal of Theoretical and Applied Mechanics2017,49(3):487-506(in Chinese))
6 Murakami H,Nishimura Y.Static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules International Journal of Solids and Structures,2001,38(50-51)9359-9381
7 Murakami H,Nishimura Y.Infinitesimal mechanism modes of tensegrity modules.Solid Mechanics and Its Applications,2003 106:273-284
8姜涛.球状张拉整体单元的找形问题研究.[硕士论文].杭州浙江大学,2005(Jiang Tao.Form-finding of spherical tensegrity structures.[Master Thesis].Hangzhou:Zhejiang University,2005(in Chinese))
9 Lee S,Lee J.A novel method for topology design of tensegrity structures.Composite Structures,2016,152:11-19
10 Koohestani K.On the analytical form-finding of tensegrities.Composite Structures,2017,166:114-119
11 Feng XD.The optimal initial self-stress design for tensegrity grid structures.Computers&Structures,2017,193:21-30
12 Cai JG,Wang XY,Deng XW,et al.Form-finding method for multimode tensegrity structures using extended force density method by grouping elements.Composite Structures,2018,187:1-9
13张沛,冯健.张拉整体结构的稳定性判定及刚度分析.土木工程学报,2013(10):48-57(Zhang Pei,Feng Jian.Stability criterion and stiffness analysis of tensegrity structures.China Civil Engineering Journal,2013(10):48-57(in Chinese))
14 Zhang LY,Li Y,Cao YP,et al.Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures:A unified analytical solution.Proceedings of the Royal Society A,2012,4683323-3347
15 Zhang LY,Li Y,Cao YP,et al.A unified solution for self-equilibrium and super-stability of rhombic truncated regular polyhedral tensegrities.International Journal of Solids and Structures,2013,50:234-245
16罗阿妮,王龙昆,刘贺平等.张拉整体三棱柱构型和结构稳定性分析.哈尔滨工业大学学报,2016,48(7):82-87(Luo Ani,Wang Longkun,Liu Heping,et al.Analysis of configuration and structura stability of 3-bar tensegrity prism.Journal of Harbin Institute of Technology,2016,48(7):82-87(in Chinese))
17 Liu HP,Zhang JY,Ohsaki M.New 3-bar prismatic tensegrity units.Composite Structures,2018,84:306-313
18 Li Y,Feng XQ,Cao YP,et al.A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures.International Journal of Solids and Structures,2016,47(14):1888-1898
19 Zhang LY,Li Y,Cao YP,et al.Stiffness matrix-based form-finding method of tensegrity structures.Engineering Structures,2014,58(7):36-48
20 Zhang LY,Zhu SX,Li SX,et al.Analytical form-finding of tensegrities using determinant of force-density matrix.Composite Structures,2018,189:87-98
21 Kebiche K,Kazi-Aoual MN,Motro R.Geomerical non-linear analysis of tensegrity systems.Engineering Structures,1999,21:864-876
22 Tran HC,Lee J.Geometric and material nonlinear analysis of tensegrity structures.Acta Mechanica Sinica,2011,27(6):938-949
23 Zhang LY,Li Y,Cao YP,et al.A numerical method for simulating nonlinear mechanical responses of tensegrity structures under large deformations.Journal of Applied Mechanics,2013,80(6):061018
24 Zhang LY,Xu GK.Negative stiffness behaviors emerging in elastic instabilities of prismatic tensegrities under torsional loading.International Journal of Mechanical Sciences,2015,103:189-198
25 Zhang LY,Zhao ZL,Zhang QD,et al.Chirality induced by structural transformation in a tensegrity:Theory and experiment.Journal of Applied Mechanics,2016,83(4):041003
26 Zhang L,Cao Q,Zhang HW.An efficient algorithm for mechanical analysis of bimodular truss and tensegrity structures.International Journal of Mechanical Sciences,2013,70(5):57-68
27 Zhang L,Lu MK,Zhang HW,et al.Geometrically nonlinear elastoplastic analysis of clustered tensegrity based on the co-rotational approach.International Journal of Mechanical Sciences,2015,93:154-165
28 Zhang L,Cao Q,Liu Y,et al.An efficient finite element formulation for nonlinear analysis of clustered tensegrity.Engineering Computations,2016,33(1):252-273
29 Luo H,Bewley TR.Accurate simulation of near-wall turbulence over a compliant tensegrity fabric.Proceedings of SPIE,2005,5757:184-197
30 Li Y,Feng XQ,Cao YP,et al.Constructing tensegrity structures from one-bar elementary cells.Proceedings Mathematical Physical and Engineering Sciences,2010,466:45-61
31 Feng XQ,Li Y,Cao YP,et al.Design methods of rhombic tensegrity structures.Acta Mechanica Sinica,2010,466(2113):559-565
32张幸锵,袁行飞.新型三棱柱张拉整体平板结构研究.建筑结构,2011(3):24-27(Zhang Xinqiang,Yuan Xingfei.Research of a new triangular prism tensegrity plate structure.Building Structure,2011(3):24-27(in Chinese))
33 Liu HP,Geng JS,Luo AN.Tensegrity configuration method for connecting tensegrity units along their axes.Composite Structures.2017,162:341-350
34 Zhang LY,Zhao HP,Feng XQ.Constructing large-scale tensegrity structures with bar-bar connecting using prismatic elementary cells.Archive of Applied Mechanics,2015,85(3):383-394
35 Fraddosio A,Marzano S,Pavone G,et al.Morphology and selfstress design of V-expander tensegrity cells.Composites Part B Engineering,2017,115:102-116
36 Rimoli JJ,Pal RK.Mechanical response of 3-dimensional tensegrity lattices.Composites Part B,2017,115:30-42
37 Zhang LY,Li SX,Zhu SX,et al.Automatically assembled largescale tensegrities by truncated regular polyhedral and prismatic elementary cells.Composite Structures,2018,184:30-40
38章孝顺,章定国,洪嘉振.考虑曲率纵向变形效应的大变形柔性梁刚柔耦合动力学建模与仿真.力学学报,2016,48(3):692-701(Zhang Xiaoshun,Zhang Dingguo,Hong Jiazhen.Rigid-flexible coupling dynamic modeling and simulation with the longitudinal deformation induced curvature effect for a rotating flexible beam under large deformation.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):692-701(in Chinese))
39 Masic M,Skelton RE,Gill PE.Algebraic tensegrity form-finding.International Journal of Solids and Structures,2005,42(16):4833-4858
40 Zhang JY.Structural morphology and stability of tensegrity structures.Kyoto:Kyoto University,2007
41 Ali NBH,Smith IFC.Dynamic behavior and vibration control of a tensegrity structure.International Journal of Solids and Structures,2010,47(9):1285-1296