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.     

Experimental and numerical study on the self-stress design of tensegrity systems

Tensegrity systems as kinematically and statically indeterminate pin-jointed systems are characterized by mechanisms and self-stress states. Unlike the other reticulated systems, in tensegrity systems, unilateral behavior of cables causes some problems in determining the basis of compatible self-stress states. At the present study, self-stress design of tensegrity systems is presented. Experimental study on two 3×3×0.7 m tensegrity grids was conducted to verify the accuracy and validity of the numerical method. Using supporting constraints, an effective method for the implementation of self-stress states in a much reduced number of stages is proposed and calibrated. Considering the results of the present study, the self-stress design of these systems can be improved to obtain specific desired behavior.     

Dynamic behavior and vibration control of a tensegrity structure

Tensegrities are lightweight space reticulated structures composed of cables and struts. Stability is provided by the self-stress state between tensioned and compressed elements. Tensegrity systems have in general low structural damping, leading to challenges with respect to dynamic loading. This paper describes dynamic behavior and vibration control of a full-scale active tensegrity structure. Laboratory testing and numerical simulations confirmed that control of the self-stress influences the dynamic behavior. A multi-objective vibration control strategy is proposed. Vibration control is carried out by modifying the self-stress level of the structure through small movement of active struts in order to shift the natural frequencies away from excitation. The PGSL stochastic search algorithm successfully identifies good control commands enabling reduction of structural response to acceptable levels at minimum control cost.     

Damage assessment of tensegrity structures using piezo transducers

This paper presents the application of surface-bonded piezo-transducers for damage assessment of tensegrity structures through dynamic strain measurement and electro-mechanical impedance (EMI) technique. The two techniques are first applied on a single module tensegrity structure, 1 m×1 m in size and their damage diagnosis results compared. A single piezoelectric-ceramic (PZT) patch bonded on a strut measures the dynamic strain during an impact excitation of the structure. Damage is identified from the changes in global frequencies of the structure obtained from the PZT patch’s response. This is compared with the damage identified using the EMI technique, which is a signature based technique and operates at frequencies of the order of kHz. The dynamic strain approach, which requires commonly available hardware, is found to exhibit satisfactory performance vis-à-vis the EMI technique for damage assessment of tensegrity structures. The damage diagnosis exercise is then extended to a tensegrity grid structure, 2 m×2 m size, fabricated using galvanized iron (GI) pipes and mild steel wire ropes. The damage is localized using changes in natural frequencies observed experimentally using the dynamic strain approach and the corresponding mode shapes of the undamaged structure derived numerically. The dynamic strain approach is found to be very expedient, displays competitive performance and is at the same time cost effective for damage assessment of tensegrity structures.     

Prismatic tensegrity structures with additional cables: Integral symmetric states of self-stress and cable-controlled reconfiguration procedure

A simple energy method is put forward to determine the prestress distribution for symmetric tensegrity structures with multiple states of self-stress and a class of prismatic tensegrities with additional cables are introduced to show the accuracy of presented method. For the purpose of modifying structural shape as well as mechanical properties, a cable-controlled reconfiguration procedure is subsequently proposed for these structures. By defining the length adjustments as the control parameters, the reconfiguration procedure is regarded as a quasi-static process, consisting of a sequence of equilibrium configurations with varying control parameters. Then the nonlinear iterative algorithm based on the tangent stiffness of the structure is presented to simulate and follow this reconfiguration process. By way of example, a particular class of reconfigurations coined symmetrical reconfigurations are investigated carefully and the key features as well as the potential applications are given.     

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.     

Prestress stability of pin-jointed assemblies using ant colony systems

Prestress stability is the key of whether a pin-jointed assembly could be transformed into a tensegrity structure. This study developed an optimization model to investigate the prestress stability of pin-jointed assemblies. The continuous optimization problem was converted into a modified traveling salesman problem (TSP), and the ant colony system (ACS) was used to search for feasible solutions. Coefficients for the independent states of self-stress were taken as different cities in the network. Several typical examples were tested. It could be concluded that the proposed technique is efficient, and applicable to both planar and three-dimensional complex pin-jointed assemblies.     

Symmetric prismatic tensegrity structures: Part I. Configuration and stability

This paper presents a simple and efficient method to determine the self-equilibrated configurations of prismatic tensegrity structures, nodes and members of which have dihedral symmetry. It is demonstrated that stability of this class of structures is not only directly related to the connectivity of members, but is also sensitive to their geometry (height/radius ratio), and is also dependent on the level of self-stress and stiffness of members. A catalogue of the structures with relatively small number of members is presented based on the stability investigations.     

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.     

Steady-state dynamics of a 3D tensegrity structure: Simulations and experiments

This paper considers a modeling and analysis approach for the investigation of the linear and nonlinear steady-state dynamics of a base excited 3D tensegrity module carrying a top mass. The tensegrity module contains three compressive members, which may buckle and six cables (tendons). First, a dynamic model of the system is derived using Lagrange’s equation with constraints. The buckling modeling of the compressive members is based on the assumed-mode method with a single mode discretization. The tendons are modeled as piecewise linear springs, which can only take tensile forces. This research focusses on the dynamic stability of the tensegrity structure by defining the geometrical and material properties in such a way that the system is just below the static stability boundary. Static and linear dynamic analysis is performed. In the nonlinear steady-state analysis, frequency-amplitude plots, power spectral density plots, bifurcation point continuation diagrams, and Poincaré maps are presented. A tensegrity structure is designed and manufactured and an experimental set-up is realized in order to validate the model by comparing experimentally and numerically obtained responses. In the validation stage, the numerical results are based on an amplifier-shaker-tensegrity structure model. It can be concluded that the numerical results match partly quantitatively and partly qualitatively with the experimentally obtained responses.     

Selection of prestress for optimal dynamic/control performance of tensegrity structures

This paper concerns prestress optimization of a tensegrity structure for its optimal LQR performance. A linearized dynamic model of the structure is derived in which the force-density variables that parameterize the prestress of the structure appear linearly. A feasible region for these parameters is defined in terms of the extreme directions of the prestress cone. A numerical method for computing this basis for a structure prestress cone is proposed. The problem is solved using a gradient method that provides a monotonic decrease of the objective function inside the feasible region. A numerical example of a cantilevered planar tensegrity beam is shown.     

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.     

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.     

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.     

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.     

Design methods of rhombic tensegrity structures

As a special type of novel flexible structures, tensegrity holds promise for many potential applications in such fields as materials science, biomechanics, civil and aerospace engineering. Rhombic systems are an important class of tensegrity structures, in which each bar constitutes the longest diagonal of a rhombus of four strings. In this paper, we address the design methods of rhombic structures based on the idea that many tensegrity structures can be constructed by assembling one-bar elementary cells. By analyzing the properties of rhombic cells, we first develop two novel schemes, namely, direct enumeration scheme and cell-substitution scheme. In addition, a facile and efficient method is presented to integrate several rhombic systems into a larger tensegrity structure. To illustrate the applications of these methods, some novel rhombic tensegrity structures are constructed.     

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.     

On the elastica solution of a T3 tensegrity structure

Stability studies of a T3 tensegrity structure are performed. This structure is composed of three slender struts interconnected by six nonlinear elastic tendons and is prestressed. The struts are governed by linear constitutive laws and are allowed to buckle. Since tensegrity is used for modeling structures with quite large deformations, for example the cytoskeleton, and bifurcation theory—valid for small solutions of the nonlinear equations—does not directly apply, a general procedure for studying the stability behavior of the particular tensegrity model based upon the elastica theory is presented. The reference placement is defined by the prestress, and the equilibrium placements are defined by the applied force and moment.     

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.