基于力密度矩阵和平衡矩阵的张拉整体结构找形方法
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摘要
提出了一种基于力密度矩阵和平衡矩阵的张拉整体结构找形新方法,结构的几何拓扑关系和单元类型是本找形方法所需的唯一条件.为了找到合适的节点坐标和力密度,将结构体系的力密度矩阵和平衡矩阵分别采用谱分解和奇异值分解进行循环迭代,直至满足对应矩阵最小秩缺失的必要条件.通过几个典型算例证明了利用本方法寻找张拉整体结构自平衡状态的效率和鲁棒性,可为寻找新型、非对称并且复杂的张拉整体结构提供借鉴.
A novel method is presented for form-finding of tensegrity structures based on the force density matrix and equilibrium matrix. The topology and the types of members are the only information required in this form-finding process. To find the feasible sets of nodal coordinates and force densities,the spectral decomposition of force density matrix and the singular value decomposition of the equilibrium matrix were iteratively performed until the necessary minimum rank deficiency conditions satisfied. Several illustrative examples are presented to demonstrate the efficiency and robustness in finding self-equilibrium configurations of tensegrity structures,and a theoretical guide is provided to find new,asymmetric and complex tensegrity structures.
A novel method is presented for form-finding of tensegrity structures based on the force density matrix and equilibrium matrix. The topology and the types of members are the only information required in this form-finding process. To find the feasible sets of nodal coordinates and force densities,the spectral decomposition of force density matrix and the singular value decomposition of the equilibrium matrix were iteratively performed until the necessary minimum rank deficiency conditions satisfied. Several illustrative examples are presented to demonstrate the efficiency and robustness in finding self-equilibrium configurations of tensegrity structures,and a theoretical guide is provided to find new,asymmetric and complex tensegrity structures.
引文
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[2]INGBER D E,TENSEGRITY I.Cell structure and hierarchical systems biology[J].Journal of cell science,2003,116:1157-1173.
[3]LAZOPOULOSK A.Stability of an elastic tensegrity structure[J].Acta mechanica,2005,179(1-2):1-10.
[4]PIRENTIS A P,LAZOPOULOS K A.On the singularities of a constrained(incompressible-like)tensegrity-cytoskeleton model under equitriaxial loading[J].International Journal of Solids and Structures,2010,47(6):759-767.
[5]SCHEKH J.The force density method for form finding and computation of general networks[J].Computer Methods in Applied Mechanics and Engineering,1974,3(2):115-134.
[6]MOTRO R,NAJARI S,JOUANNA P.Static and dynamic analysis of tensegrity systems[C]∥Proceedings of the international symposium on shell and spatial structures,computational aspects.Springer,1986:270-279.
[7]ZHANG J Y,OHSAKI M.Adaptive force density method for form-finding problem of tensegrity structures[J].International Journal of Solids and Structures,2006,43(18-19):5658-5673.
[8]ESTRADA G,BUNGARTZ H,MOHRDIECK C.Numerical form-finding of tensegrity structures[J].International Journal of Solids and Structures,2006,43(22-23):6855-6868.
[9]TRAN H C,LEE J.Advanced form-finding of tensegrity structures[J].Computers&Structures,2010,88(3–4):237-246.
[10]PAGITZ M,JUR J M M.Tensegrity frameworks:static analysis review[J].Mechanism and Machine Theory,2008,43(7):859-881.
[11]卢成江,吴知丰,赵洪斌.非线性有限元法分析张拉整体结构初始形态[J].沈阳建筑大学学报(自然科学版),2007,23(6):900-904.
[12]XU X,LUO Y.Form-finding of nonregular tensegrities using a gentic algorithm[J].Mechanics Research Communications,2010,37(1):85-91.
[13]TRAN H C,LEE J.Form-finding of tensegrity structures with multiple states of self-stress[J].Acta Mechanica,2011,222(1-2):131-147.
[14]LEE S,LEE J.Form-finding of tensegrity structures with arbitrary strut and cable members[J].International Journal of Mechanical Science,2014,85(8):55-62.
[15]ZHANG L Y,LI Y,CAO Y P,et al.Stiffness matrix based form-finding method of tensegrity structures[J].Engineering Structures,2014,58(7):36-48.
[16]KOOHESTANI K,GUEST S.A new approach to the analytical and numerical form-finding of tensegrity structures[J].International Journal of Solids and Structures,2013,50(19):2995-3007.
[17]TIBERT A G,PELLEGRINO S.Review of formfinding methods for tensegrity structures[J].International Journal of Space Structure,2003,18(4):209-223.
[18]MASIC M,SKELTON R,GILL P.Algebraic tensegrity form-finding[J].International Journal of Solids and Structures,2005,42(16-17):4833-4858.
[19]MOTRO R.Tensegrity:Structural Systems for the Future[M].London:Kogan Page,2003.
[20]陈志华,王小盾,刘锡良.张拉整体结构的力密度法找形分析[J].建筑结构学报,1999,20(5):29-35.
[21]PELLEGRINO S,CALLADINE C R.Matrix analysis of statically and kinematically indeterminate frameworks[J].International Journal of Solids and Structures,1986,22(4):409-428.
[22]MEYER C D.Matrix analysis and applied linear algebra[M].Philadelphia:SIAM,2000.
[23]CONNELLY R.Rigidity and energy[J].Inventiones Mathematicae,1982,66(1):11-33.
[24]CONNELLY R.TERRELL M.Globally rigid symmetric tensegrities[J].Structural Topology,1995,21:59-78.
[25]CONNELLY R.Generic global rigidity[J].Discrete&Computational Geometry,2005,33(4):549-563.
[26]OHSAKI M,ZHANG J Y.Stability conditions of prestressed pin-jointed structures[J].International Journal of Nonlinear Mechanics,2006,41(10):1109-1117.
[27]PELLEGRINOS.Structural computations with the singular value decomposition of the equilibrium matrix[J].International Journal of Space Structure,1993,30(21):3025-3035.
[2]INGBER D E,TENSEGRITY I.Cell structure and hierarchical systems biology[J].Journal of cell science,2003,116:1157-1173.
[3]LAZOPOULOSK A.Stability of an elastic tensegrity structure[J].Acta mechanica,2005,179(1-2):1-10.
[4]PIRENTIS A P,LAZOPOULOS K A.On the singularities of a constrained(incompressible-like)tensegrity-cytoskeleton model under equitriaxial loading[J].International Journal of Solids and Structures,2010,47(6):759-767.
[5]SCHEKH J.The force density method for form finding and computation of general networks[J].Computer Methods in Applied Mechanics and Engineering,1974,3(2):115-134.
[6]MOTRO R,NAJARI S,JOUANNA P.Static and dynamic analysis of tensegrity systems[C]∥Proceedings of the international symposium on shell and spatial structures,computational aspects.Springer,1986:270-279.
[7]ZHANG J Y,OHSAKI M.Adaptive force density method for form-finding problem of tensegrity structures[J].International Journal of Solids and Structures,2006,43(18-19):5658-5673.
[8]ESTRADA G,BUNGARTZ H,MOHRDIECK C.Numerical form-finding of tensegrity structures[J].International Journal of Solids and Structures,2006,43(22-23):6855-6868.
[9]TRAN H C,LEE J.Advanced form-finding of tensegrity structures[J].Computers&Structures,2010,88(3–4):237-246.
[10]PAGITZ M,JUR J M M.Tensegrity frameworks:static analysis review[J].Mechanism and Machine Theory,2008,43(7):859-881.
[11]卢成江,吴知丰,赵洪斌.非线性有限元法分析张拉整体结构初始形态[J].沈阳建筑大学学报(自然科学版),2007,23(6):900-904.
[12]XU X,LUO Y.Form-finding of nonregular tensegrities using a gentic algorithm[J].Mechanics Research Communications,2010,37(1):85-91.
[13]TRAN H C,LEE J.Form-finding of tensegrity structures with multiple states of self-stress[J].Acta Mechanica,2011,222(1-2):131-147.
[14]LEE S,LEE J.Form-finding of tensegrity structures with arbitrary strut and cable members[J].International Journal of Mechanical Science,2014,85(8):55-62.
[15]ZHANG L Y,LI Y,CAO Y P,et al.Stiffness matrix based form-finding method of tensegrity structures[J].Engineering Structures,2014,58(7):36-48.
[16]KOOHESTANI K,GUEST S.A new approach to the analytical and numerical form-finding of tensegrity structures[J].International Journal of Solids and Structures,2013,50(19):2995-3007.
[17]TIBERT A G,PELLEGRINO S.Review of formfinding methods for tensegrity structures[J].International Journal of Space Structure,2003,18(4):209-223.
[18]MASIC M,SKELTON R,GILL P.Algebraic tensegrity form-finding[J].International Journal of Solids and Structures,2005,42(16-17):4833-4858.
[19]MOTRO R.Tensegrity:Structural Systems for the Future[M].London:Kogan Page,2003.
[20]陈志华,王小盾,刘锡良.张拉整体结构的力密度法找形分析[J].建筑结构学报,1999,20(5):29-35.
[21]PELLEGRINO S,CALLADINE C R.Matrix analysis of statically and kinematically indeterminate frameworks[J].International Journal of Solids and Structures,1986,22(4):409-428.
[22]MEYER C D.Matrix analysis and applied linear algebra[M].Philadelphia:SIAM,2000.
[23]CONNELLY R.Rigidity and energy[J].Inventiones Mathematicae,1982,66(1):11-33.
[24]CONNELLY R.TERRELL M.Globally rigid symmetric tensegrities[J].Structural Topology,1995,21:59-78.
[25]CONNELLY R.Generic global rigidity[J].Discrete&Computational Geometry,2005,33(4):549-563.
[26]OHSAKI M,ZHANG J Y.Stability conditions of prestressed pin-jointed structures[J].International Journal of Nonlinear Mechanics,2006,41(10):1109-1117.
[27]PELLEGRINOS.Structural computations with the singular value decomposition of the equilibrium matrix[J].International Journal of Space Structure,1993,30(21):3025-3035.