Adaptive force density method for form-finding problem of tensegrity structures

A numerical method is presented for form-finding of tensegrity structures. Eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates. The equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation. A unique and non-degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates. A simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non-degenerate structure. Several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method.     

基于动力松弛法的松弛索杆体系找形分析

索杆张力结构施工成形分析需要解决一个松弛态索杆系统的平衡形态求解问题,该问题可归结为一个给定构件原长的受荷索杆机构系统的找形问题。文中利用动力松弛法进行该类松弛索杆体系的找形分析。由于不建立刚度矩阵,避免了体系几何不稳定性所引起的刚度矩阵奇异问题。该方法采用悬链线索单元模型,可以考虑索大垂度效应。文中还推导了反映索原长和内力之间关系的悬链线索单元协调方程。通过一个正交松弛索网算例分析,考察了该找形方法的计算精度和收敛性。最后还模拟了一个索穹顶的施工成形过程,表明了该找形方法用于索杆张力结构施工成形分析的有效性。     

A novel method of determining the sole configuration of tensegrity structures

A novel analysis method is presented for form-finding of tensegrity structures. The spectral decomposition of the force density matrix and the singular value decomposition of the equilibrium matrix are performed iteratively to find the feasible sets of nodal coordinates and force densities. An algorithm of determining the sole configuration of free-form tensegrities is provided by specifying an independent set of nodal coordinates, which indicates the geometrical and mechanical properties of the structures can be at least partly controlled by the proposed method. Several illustrative examples are presented to demonstrate the efficiency and robustness in finding self-equilibrium configurations of tensegrity structures.     

考虑支承的大跨度索膜结构整体分析与设计

基于修正的拉格朗日法,建立非线性有限元平衡方程,介绍了空间梁单元非线性刚度矩阵。针对此类结构在形态分析及设计中的难点问题,提出利用定力单元、定形单元和微调单元建立数值模型。采用AutoCAD二次开发工具ObjectARX编制有限元软件,能够进行考虑支承的大跨度索膜结构整体分析与设计。以芜湖体育场为工程背景,进行了形态分析和长期荷载组合下的反应。计算结果表明,本文所提方法和有限元软件显著地提高了设计工作的效率,并能准确地获得满足设计要求的结构初始形态及受力分析数据,便于实际工程的设计应用。     

Numerical form-finding of tensegrity structures

A novel and versatile numerical form-finding procedure that requires only a minimal knowledge of the structure is presented. The procedure only needs the type of each member, i.e. either compression or tension, and the connectivity of the nodes to be known. Both equilibrium geometry and force densities are iteratively calculated. A condition of a maximal rank of the force density matrix and minimal member length, were included in the form-finding procedure to guide the search of a state of self-stress with minimal elastic potential energy. It is indeed able to calculate novel configurations, with no assumptions on cable lengths or cable-to-strut ratios. Moreover, the proposed approach compares favourably with all the leading techniques in the field. This is clearly exemplified through a series of examples.     

Form-finding of tensegrity structures via genetic algorithm

We propose an efficient method for the form-finding of tensegrity structures. The force densities of each tensegrity are obtained by the minimisation of a particular objective function, leading to a semi-positive definite force density matrix (a super-stable tensegrity) with a required rank deficiency. A genetic algorithm is used as a global search technique for the minimisation. The geometry of a tensegrity is subsequently formed based on those eigenvectors of the force density matrix corresponding to zero eigenvalues. Furthermore, two other methods are introduced to convert the asymmetrical geometry obtained from the main algorithm into its symmetrical counterparts. This transformation in geometry is performed by finding a suitable linear combination of the mentioned eigenvectors. Examples from well-known tensegrities including prismatic, truncated tetrahedron, expandable octahedron and truncated icosahedron tensegrities are studied using the present method, and the results obtained are compared with those documented in the literature to verify the efficiency of the present method.     

力密度找形分析方法及计算机实现

本文通过对力密度法的分析研究，概括了力密度法的基本原理和找形分析的过程，用矩阵形式给出了力密度的基本列式。通过对各个矩阵基本元素分析，找到了力密度矩阵的组装规律和一维变带宽存储的规则及实现算法和过程，并给出了详细的力密度矩阵的装配过程。通过基于C 语言的系统框架的程序，实现了该基于一维变带宽存储的力密度找形分析方法。最后，通过几个数值算例表明了，力密度法找形理论的先进性，及本文提出的算法过程实现的可行性和正确性。由于本文的算法是基于效率比较高的一维变带宽压缩存储格式，因此，可以实现大型索膜结构的找形分析。     

A computational framework for the form-finding and design of tensegrity structures

A new computational framework is proposed for the form-finding and design of tensegrity structures with or without super-stability. The form-finding of tensegrities is formulated as two unconstrained minimisation problems where their objective functions are defined based on eigenvalues of a modified force density matrix. The Nelder–Mead simplex method is then used to solve the minimisation problems. Furthermore, another efficient method is suggested for the interactive form-finding and design of tensegrities with geometrical and force constraints. Examples of the form-finding of tensegrities are presented and the results obtained are compared and contrasted with those analytical results documented in the literature, to verify the accuracy and efficiency of the developed methods.     

Self-stress design of tensegrity grid structures with exostresses

A numerical method is presented for initial self-stress design of tensegrity grid structures with exostresses, which is defined as a linear combination of the coefficients of independent self-stress modes. A discussion on proper division of the number of member groups for the purpose of existence of a single integral feasible self-stress mode has been explicitly given. Dummy elements to transform the tensegrity grid structure with statically indeterminate supports into self-stressed pin-jointed system without supports are employed. The unilateral properties of the stresses in cables and struts are taken into account. Evaluation of the stability for the structure is also considered. Several numerical examples are presented to demonstrate the efficiency and robustness in searching initial single integral feasible self-stress mode for tensegrity grid structures.     

含可动机构的杆系结构非线性力法分析

推导了基于矩阵奇异值分解的小变形条件下杆系结构的力法分析（LFM）过程，并采用杆件内力修正的方法，提出了考虑大位移的几何非线性力法分析（NFM）方法思想和步骤。该方法有效地解决了结构中同时存在刚体位移、弹性位移的结构分析问题，适合计算机编程，结构的强度计算、机构分析和形态分析，讨论了力法与位移法的优缺点，给出的算例表明所提出的方法是正确的、有效的。     

A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures

We propose a Monte Carlo form-finding method that employs a stochastic procedure to determine equilibrium configurations of a tensegrity structure. This method does not involve complicated matrix operations or symmetry analysis, works for arbitrary initial configurations, and can handle large scale regular or irregular tensegrity structures with or without material/geometrical constraints.     

基于构件层次的铰接杆系结构几何稳定性讨论

考察构件刚度和构件撤除对杆件系统几何稳定性的影响.从常规结构稳定理论的角度审视铰接杆件系统几何稳定性问题.基于结构稳定的能量准则和刚度矩阵的构成分析,重新考察了Maxwell准则和平衡矩阵准则的充分必要性.解释了构件零刚度和构件撤除对体系几何稳定性影响的一致性.利用自应力矩阵的特性,提出并证明了一种快速识别杆系结构中“必需杆”的方法.一种多根构件撤除后体系几何稳定性的判别准则进而被发展.该判别准则的数值效率体现在仅利用原结构平衡矩阵一次分解后的信息,杆件撤除后体系平衡矩阵的秩可通过两小规模矩阵秩之间的关系来表示.     

Dynamic Analysis of Large Space Structures Using Transfer Matrices and Joint Coupling Matrices

Abstract

Linear dynamic analysis of lattice structures using transfer matrices and joint coupling matrices is presented. A lattice structure is defined as a network of one-dimensional members that are connected by joints. Two examples are considered to illustrate how transfer matrices and joint coupling matrices may be used to compute natural frequencies of vibration. These two examples indicate that the transfer matrix and joint coupling matrix analysis is numerically accurate over a wide range of frequencies and becomes increasingly efficient, compared to the finite element method, as the frequency increases. Some suggestions for further improvements in computational efficiency and some comments about applicability to numerical analysis of wave propagation problems are given.     

Form-finding of a new kind of tensegrity tori using overlapping modules

In the past decades, the form-finding of tensegrity structures of regular geometric shapes, such as cylindrical tensegrities, polyhedral tensegrities, spherical tensegrities and so on, has been systematically studied. However, seldom studies on the form-finding of tensegrity tori have been reported. Considering the potential applications of the tensegrity tori in a number of fields, including architecture, sculpture, and other relevant fields, this paper carries out an exploration on a new kind of tensegrity tori. The topology of the new kind of tensegrity tori is based on the well-known cylindrical tensegrities and overlapping between every two adjacent tensegrity modules is allowed. Incorporating the singular value decomposition of equilibrium matrix with a force-finding algorithm, a general procedure for determining the feasible configurations for the new kind of tensegrity tori is proposed. Parametric analyses on several typical forms of the tensegrity tori are conducted and the feasible ranges of the design parameters and applicability of the feasible configurations are discussed.     

Stability conditions of prestressed pin-jointed structures

Pin-jointed structures are first classified to trusses, tensile structures, and tensegrity structures in view of their respective stability properties. A sufficient condition for stability of an equilibrium state is derived for tensegrity structures. The condition is based on the bilinear forms of the linear and geometrical stiffness matrices considering the flexibility of members. The stability is defined by the positive definiteness of the tangent stiffness matrix, whereas the definition of prestress-stability is based on the geometrical stiffness matrix and the infinitesimal mechanisms. Numerical examples verify that the so-called super-stability condition might not be satisfied by a stable tensegrity structure, and that a prestress-stable structure can be unstable if the prestresses are moderately large.     

Algebraic tensegrity form-finding

This paper concerns the form-finding problem for general and symmetric tensegrity structures with shape constraints. A number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form-finding problem. The concept of a tensegrity invariance (similarity) transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations. This result provides the basis for a new tensegrity form-finding tool. The generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives. State-of-the-art numerical algorithms are applied to solve several example problems. Examples are given for tensegrity plates, shell-class symmetric tensegrity structures and structures generated by applying similarity transformation.     

杆件去除与内力变更时空间网架结构的简捷计算法

本文采用矩阵位移法,引入初内力的概念,提出并建立了初内力准方程式,再运用性叠加原理来简化分析计算杆件去除与内力变更时的空间时的空间网架结构。这种简捷计算法的所讨论的工况下只要一次性的建立、分解和应用网架结构的总刚度矩阵,无须重复多次建立和分解总刚度矩阵,从而使会计工作量成倍地甚至几十倍地大幅度减少。特别是当网架结构多根杆、依次先后断杆、预应力及分期分批施加预应力时,采用本文方法可更方便地求得各工况     

Finite element based form-finding algorithm for tensegrity structures

This paper presents a novel form-finding algorithm for tensegrity structures that is based on the finite element method. The required data for the form-finding is the topology of the structure, undeformed bar lengths, total cable length, prestress of cables and stiffness of bars. The form-finding is done by modifying the single cable lengths such that the total cable length is preserved and the potential energy of the system is minimized. Two- and three-dimensional examples are presented that demonstrate the excellent performance of the proposed algorithm.     

随机振动问题的广义坐标合成法

工程中的随机振动分析多采用完全二次型组合法(CQC)及其改进算法,如虚拟激励法(PEM)和谐波激励法(HEM)。广义坐标合成法(GCS)提出了一种计算随机响应的新思路,其基本原理是将物理空间的计算转移到自由度较小的振型空间进行,从而缩减了计算量和计算规模。对于较大规模结构体系的随机振动问题,GCS方法计算响应协方差矩阵的计算量只相当于PEM的2r/sT,其中s、r和T分别为激励功率谱矩阵的秩、振型总数和离散频率点数。此外,对于给定激励时程的问题,由于GCS方法直接求解广义坐标运动方程,因而可以方便得出响应时程。通过对几种方法的详细对比,说明对于大多数只需要求解响应方差的随机振动问题,GCS是最优的计算方法。PEM只有在s相似文献   

Symmetric prismatic tensegrity structures. Part II: Symmetry-adapted formulations

This paper presents analytical formulations for the symmetry-adapted equilibrium, force density and geometrical stiffness matrices for prismatic tensegrity structures with dihedral symmetries. An analytical expression for the infinitesimal mechanisms is also derived.