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.      

Geometric and material nonlinear analysis of tensegrity structures

A numerical method is presented for the large deflection in elastic analysis of tensegrity structures including both geometric and material nonlinearities.The geometric nonlinearity is considered based on both total Lagrangian and updated Lagrangian formulations,while the material nonlinearity is treated through elastoplastic stress-strain relationship.The nonlinear equilibrium equations are solved using an incremental-iterative scheme in conjunction with the modified Newton-Raphson method.A computer program is developed to predict the mechanical responses of tensegrity systems under tensile,compressive and flexural loadings.Numerical results obtained are compared with those reported in the literature to demonstrate the accuracy and efficiency of the proposed program.The flexural behavior of the double layer quadruplex tensegrity grid is sufficiently good for lightweight large-span structural applications.On the other hand,its bending strength capacity is not sensitive to the self-stress level.     

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.     

Structural morphology of tensegrity systems

The coupling between form and forces, their structural morphology, is a key point for tensegrity systems. In the first part of this paper we describe the design process of the simplest tensegrity system which was achieved by Kenneth Snelson. Some other simple cells are presented and tensypolyhedra are defined as tensegrity systems which meet polyhedra geometry in a stable equilibrium state. A numerical model giving access to more complex systems, in terms of number of components and geometrical properties, is then evoked. The third part is devoted to linear assemblies of annular cells which can be folded. Some experimental models of the tensegrity ring which is the basic component of this “hollow rope” have been realized and are examined.     

Investigation of clustered actuation in tensegrity structures

As tensegrity research is moving away from static structures toward active structures it is becoming critical that new actuation strategies and comprehensive active structures theories are developed to fully exploit the properties of tensegrity structures. In this paper a new general tensegrity paradigm is presented that incorporates a concept referred to as clustered actuation. Clustered actuation exploits the existence of cable elements in a tensegrity structure by allowing cables to be run over frictionless pulleys or through frictionless loops at the nodes. This actuation strategy is a scalable solution that can be utilized for active structures that incorporate many active elements and can reduce the number of actuators necessary for complex shape changes. Clustered actuation also has secondary benefits, specifically reducing the force requirements of actuators in dynamic structures, reducing the number of pre-stress modes to potentially one global mode and relieving element size limitations that occur with embedded actuation. Newly formulated clustered equilibrium equations are developed using energy methods and are shown to be a generalization of the classic tensegrity governing equations. Pre-stress analysis, mechanism analysis and stability of clustered structures are discussed. Lastly, examples compare the mechanics of a clustered structure to an equivalent classic structure and the utility of clustering is highlighted by allowing for actuation throughout a class 1 (no bar-to-bar connections) tensegrity while not embedding the actuators into the structure.     

On a connectivity matrix formula for tensegrity prism plates

This paper considers tensegrity structures constructed from repetition of simple fundamental units. The tensegrity prism is chosen as a fundamental unit, which allows us to build plates, columns, towers, and their variations. The connectivity matrix plays a central role in analysis and design of tensegrity structures. This paper provides a systematic way to construct connectivity matrices for tensegrity structures constructed from repetition of tensegrity prisms. The number of units and node locations (shape) can be chosen arbitrarily. As an application of the connectivity matrix, a minimal-mass design subjected to force equilibrium (force balance) and yield and buckling stress constraints is shown.     

Bracing rhombic structure by one-dimensional tensegrities

Certain behaviors of some material are characterized by the periodic bar and joint elements. We present a kinematic, geometric and graph theoretic connected model that describes the stability, rigidity property of these rhombic tiling materials in two dimensions. Cables, struts or rods are placed as bracing elements between opposite pairs of diagonal joints, to prevent the rhombic tiling from the rotation of the bars around their common joint. We characterize the rigidity of the finite parts of the rhombic bracing structure in the plane. The results of this paper are based on the theorem of the rigidity of one-dimensional tensegrity framework from Recski and Shai. We have applied our results to describe some auxetic type structures that were mentioned earlier in the scientific literature. We also introduce the model as a possible candidate for the mechanical information processing system in a repetitive bar and joint structure.     

Analysis of clustered tensegrity structures using a modified dynamic relaxation algorithm

Tensegrities are spatial, reticulated and lightweight structures that are increasingly investigated as structural solutions for active and deployable structures. Tensegrity systems are composed only of axially loaded elements and this provides opportunities for actuation and deployment through changing element lengths. In cable-based actuation strategies, the deficiency of having to control too many cable elements can be overcome by connecting several cables. However, clustering active cables significantly changes the mechanics of classical tensegrity structures. Challenges emerge for structural analysis, control and actuation. In this paper, a modified dynamic relaxation (DR) algorithm is presented for static analysis and form-finding. The method is extended to accommodate clustered tensegrity structures. The applicability of the modified DR to this type of structure is demonstrated. Furthermore, the performance of the proposed method is compared with that of a transient stiffness method. Results obtained from two numerical examples show that the values predicted by the DR method are in a good agreement with those generated by the transient stiffness method. Finally it is shown that the DR method scales up to larger structures more efficiently.     

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.     

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.     

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.     

A direct approach to design of geometry and forces of tensegrity systems

In the process of designing a tensegrity system, some constraints are usually introduced for geometry and/or forces to ensure uniqueness of the solution, because the tensegrity systems are underdetermined in most cases. In this paper, a new approach is presented to enable designers to specify independent sets of axial forces and nodal coordinates consecutively, under the equilibrium conditions and the given constraints, to satisfy the distinctly different requirements of architects and structural engineers. The proposed method can be used very efficiently for practical applications because only linear algebraic equations are to be solved, and no equation of kinematics or material property is needed. Some numerical examples are given to show not only efficiency of the proposed method but also its ability of searching new configurations.     

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.     

张拉整体结构的动力学等效建模与实验验证

为了实现张拉整体结构高效动力学计算, 并考虑其大范围运动中柔性杆局部动态屈曲, 提出了一种受压细长杆动力学降阶模型, 采用五节点弹/扭簧集中质量离散模型等效连续杆的静力学和动力学特性. 首先, 通过静力学等效分析推导了弹簧拉压刚度和扭簧弯曲刚度表达式, 可准确预测杆件受压屈曲和近似预测其后屈曲行为. 第二, 通过动能等效分析推导了集中质量表达式, 可准确预测杆在线速度场下的运动. 第三, 通过弯曲振动固有模态等效分析确定弯曲刚度和节点质量的分布参数, 合适的分布参数取值组合可将降阶模型前两阶固有频率相对误差均降低至1%以内. 第四, 在全局坐标系下建立张拉整体结构瞬态动力学方程, 并利用静力凝聚法实现方程高效迭代求解. 最后, 分别对球形张拉整体结构准静态压缩、模态分析和碰撞动力学进行仿真和实验对比分析, 证明了提出的动力学降阶模型可有效预测张拉整体结构的静力学行为、固有振动特性及瞬态动力学响应, 并分析了结构参数变化对其力学特性的影响规律. 本文提出的动力学等效建模与计算方法, 可望用于软着陆行星探测器、大型可展开空间结构及点阵材料等复杂张拉整体系统的动力学分析与控制.      

Wave analysis and control of double cascade-connected damped mass-spring systems

This paper presents wave analysis and control for double cascade-connected damped mass-spring systems, whose mass is connected beyond the adjacent masses. The system is motivated by a cantilevered tensegrity beam supporting tensile and compressive forces. The wave solution is derived from a recurrent formula, and the properties of the propagation constants are precisely investigated. Elimination of reflected waves provides the impedance matching controller. We show that the impedance matching controller can be constructed from a similarity transformation of the characteristic impedance matrix by a matrix composed of the propagation constants. A numerical example of vibration control of a tensegrity beam is shown.     

Equilibrium conditions of a tensegrity structure

This paper characterizes the necessary and sufficient conditions for tensegrity equilibria. Static models of tensegrity structures are reduced to linear algebra problems, after first characterizing the problem in a vector space where direction cosines are not needed. This is possible by describing the components of all member vectors. While our approach enlarges (by a factor of 3) the vector space required to describe the problem, the advantage of enlarging the vector space makes the mathematical structure of the problem amenable to linear algebra treatment. Using the linear algebraic techniques, many variables are eliminated from the final existence equations.     

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.     

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.     

Harmonic model of graphene based on a tight binding interatomic potential

Like in many other materials, the presence of topological defects in graphene has been demonstrated to modify its behavior, thus enhancing features aimed at several technological applications, more specifically, its electronic and transport properties. In particular, pristine defect-free graphene has been shown to be of limited use for semiconductor-based electronics, whereas the presence of individual or cluster defect rings along grain boundaries hinders electron transport and introduce a transport gap, unveiling the possibility of novel electronic device applications based on the structural engineering of graphene-based materials. In this work, we present an atomic bondwise force-constant model from the tight binding potential by Xu et al. (1992), that accounts for the electron-mechanical coupling effects in graphene. First we verify that this computational scheme is capable of accurately predicting the defect energies and core structures of dislocation dipoles based on the theory of discrete dislocations of Ariza and Ortiz (2005). In order to demonstrate our ability to characterize the effect of patterned distributions of structural defects on the electronic structure of graphene, we present the electronic band structures and density of states curves of several defective graphene sheets.     

A numerical algorithm for tensegrity dynamics with non-minimal coordinates

This paper provides a numerical correction algorithm for implementation of the dynamics of tensegrity systems described by non-minimal coordinates. This correction algorithm corrects any numerical error that would violate the fixed-length bar constraints. A recursive form of the correction algorithm is proposed, and simulation results support the validity of the proposed scheme.