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.     

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.     

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.     

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。     

数字状张拉整体结构的构型设计与力学性能模拟

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.      

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。     

Advanced form-finding for cable-strut structures

A numerical method is presented for form-finding of cable-strut structures. The topology and the types of members are the only information that is required in this form-finding process. Dummy members are used to transform the cable-strut structure with supports into self-stressed system without supports. The requirement on rank deficiencies of the force density and equilibrium matrices for the purpose of obtaining a non-degenerate d-dimensional self-stressed structure has been explicitly discussed. 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 which satisfy the minimum required rank deficiencies of the force density and equilibrium matrices, respectively. Based on numerical examples it is found that the proposed method is very efficient, robust and versatile in searching self-equilibrium configurations of cable-strut 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。     

GEOMETRIC DRAWING METHOD FOR FORM-FINDING AND FORCE-FINDING FOR TRIANGULAR PRISM OF TENSEGRITY

力学和几何学是密不可分的,本文给出了一种几何作图法进行端面平行时张拉整体三棱柱的找形方法和作图确定自平衡内力大小的找力方法,并推导出找形后张拉整体三棱柱自平衡内力的力密度计算公式,通过算例验证了几何作图法找形与找力的正确性。几何作图法找形和找力方法操作简单,直观可控,可以在CAD软件中实现,甚至可以纸上手绘实现;自平衡力密度公式简单,表达直观,是力密度法的一种几何实现。     

An operative algebraic formulation for the unilaterally-constrained mechanical problem of smart tensegrities

Kinematics and statics of tensegrities are addressed by means of a novel algebraic formulation. The inequality constraints, associated to cable-type unilateral structural members, are explicitly enforced in the equilibrium and compatibility problems. Fundamental tensegrity properties (rigidity, pre-stressability, and stability) are focused by a novel structural perspective and algebraic criteria for their assessment are established. Some classical results are generalized to the case of tensegrity models involving both deformable and non-deformable structural members. An operative algorithm for the analysis of the large-displacement elastic tensegrity response is proposed, not limited by special requirements in terms of structural symmetries or member connectivity, and therefore resulting a general design tool. Exemplary applications highlight the effectiveness of the proposed approach for designing tensegrity structures endowed with smart global behavior related to the optimal tuning of structural stiffness.     

索网天线找形分析的参变量变分有限元法

针对柔性空间索网天线的非线性特点,建立了基于参变量变分描述索网拉压非线性和共旋列式描述几何非线性的有限元控制方程,应用Lernke与改进牛顿法进行求解.通过对索网预张力平衡计算,证明改进牛顿法比Newton-Raphson法具有更强的收敛能力.进一步将力密度法迭代原理与有限元法结合应用于索网天线的非线性找形分析中,获得了理想的索网构型.本文的索网找形方法可广泛应用于空间索网天线结构的设计.     

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。     

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.     

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.     

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.     

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.     

The load distribution among three legs on the wall: model predictions for cockroaches

In a former study on terrestrial locomotion of cockroaches in the sagittal plane, it was hypothesised that the ground reaction force distribution among three legs synchronously in contact with a substrate is predominantly explained by joint torque minimisation within all three legs. We verified this hypothesis with a simple mechanical model in two dimensions, consisting of one body and three mass-less legs. Hereto, we calculated force distributions resulting from different optimisation criteria for varying slope angles of the substrate. We compared these distributions to each other and the few experimental findings available. We found that, for any slope angle, the force distribution rather seems to be derived from the fundamental “table” solution, i.e. equalised vertical and vanishing horizontal components (equivalent to pure force minimisation at zero slope), than from pure torque minimisation. For cockroaches, the “table” solution is likely to be modified by torque minimisation within the leading and the trailing leg. We demonstrate that the minimisation of leg forces and of interaction forces is fully equivalent. Moreover, our model predicts the force distribution for arbitrary slope angles. Based on our model calculations, we speculate that in terrestrial locomotion, some animals may rely on spring-mass model dynamics whatever slope angle to be overcome. This might only become evident when focusing movement analyses strictly on a gravity rather than on a substrate-based coordinate system.     

张拉膜结构力密度法混合找形分析

论述了张拉膜结构力密度法混合找形的基本理论;所谓力密度法混合找形,即部分单元力密度控制,部分单元弹性控制;力密度控制采用线性求解,弹性控制采用非线性求解,通过迭代计算混合找形求出各结点的坐标值。据此编制了相应的计算软件;对工程实例进行了验算,结果表明,本文给出的计算结果与德国著名软件easy的计算结果相吻合。     

Benchmark case studies in structural design optimization using the force method

A structural optimization algorithm is developed for truss and beam structures under stress–displacement or frequency constraints. The algorithm combines the mathematical programming based on the Sequential Quadratic Programming (SQP) technique and the finite element technique based on the Integrated Force Method. A new approach based on the single value decomposition technique has been developed to derive the compatibility matrix required in the force method. Benchmark case studies illustrate the procedure and allow the results obtained to be compared with those reported in the literature. It is shown that the computational effort required by the force method is significantly lower than that of the displacement method and in some cases such as structural optimization problems with multiple frequency constraints, the analysis procedure (force or displacement method) significantly affects the final optimum design and the structural optimization based on the force method may result in a lighter design.     

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.