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.     

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 studies on the collapse behavior of tensegrity systems considering cable rupture and strut collapse with snap-through

Researches on the collapse behavior of a 3×3×0.7 m tensegrity grid have been conducted with the aim of examining the accuracy of the proposed numerical procedure for investigating the localization or propagation of collapse in these systems. The experimental program consists of tests on the constituent elements and collapse test on the whole system. In the current study, two types of collapse due to sudden rupture of a cable element and buckling of a strut were examined in the studied tensegrity model under load control. It was found that the most important factors that influence the collapse behavior of the tensegrity model are the imperfection amplitude, damping factors and residual stresses of the buckled struts. Based on the obtained results, the finite element model were adjusted, compared and validated with the experimental results until reliable and robust numerical model were achieved.     

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.     

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.     

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.     

Duality between plane trusses and grillages

In the paper, projective plane duality, that is, a point-to-line, line-to-point, incidence-to-incidence correspondence between plane trusses and grillages of simple connection is treated. By means of linear algebra it is proved that the rank of the equilibrium matrix of plane trusses and grillages does not change under projective transformations and polarities: consequently the number of infinitesimal inextensional mechanisms and the number of independent states of self-stress are preserved under these transformations. The results obtained are also applied to structures with unilateral constraints, and by using several examples it is shown that plane tensegrity trusses have projective dual counterparts among grillages which can be physically modelled with popsicle sticks by weaving.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Buckminster Fuller's “Tensegrity” structures and Clerk Maxwell's rules for the construction of stiff frames

Maxwell has shown that b bars assembled into a frame having j joints would, in general, be simply stiff if b = 3j?6. Some of Buckminster Fuller's “Tensegrity” structures have fewer bars than are necessary to satisfy Maxwell's rule, and yet are not “mechanisms” as one might expect, but are actually stiff structures. Maxwell anticipates special cases of this sort, and states that their stiffness will “be of a low order”. In fact, the conditions under which Maxwell's exceptional cases occur also permit at least one state of “self-stress” in the frame.Linear algebra enables us to find the number of “incipient” modes of low-order stiffness of the frame in terms of the numbers of bars, joints and independent states of self-stress. Self-stress in the frame has the effect of imparting first-order stiffness to the frame, and it seems from experiments that a single state of self-stress can stiffen a large number of modes. It is this factor which Fuller exploits to make satisfactory structures.     

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.     

Detection of finite mechanisms in symmetric structures

Using group representation theory, it is possible to block-diagonalise the equilibrium matrix of a symmetric structure. This analysis can identify the symmetry properties of any states of self-stress or mechanisms present in the structure. This paper will show that in some cases, this linear analysis, combined with symmetry arguments, can show that particular mechanisms of a symmetric structure must be finite.     

Snapping instability in prismatic tensegrities under torsion

Tensegrities are a class of lightweight and reticulated structures consisting of stressed strings and bars. It is shown that each prismatic tensegrity can have two self-equilibrated and stable states, leading to a snapping instability behavior under an applied torque. The predicted mechanism is experimentally validated, and can be used in areas such as advanced sensors and actuators, energy storage/adsorption equipments, and folding/unfolding devices.     

Stability of an elastic cytoskeletal tensegrity model

An elastic cytoskeletal tensegrity structure composed by six inextensible elastic struts and 24 elastic cables is considered. The model is studied, adopting delay convention for stability. Critical conditions for simple and compound instabilities are defined. Post-critical behavior is also described. Equilibrium states with buckling of the struts are also considered. It is revealed that critical Euler buckling load of the struts is a necessary but not a sufficient condition for the existence of bifurcated equilibrium states, caused by buckling of the struts.     

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.