Tensegrity Systems Research Articles

Stratégies de mise en précontrainte dans les systèmes de tenségrité et contrôle

ABSTRACT Tensegrity systems are innovative structures, whose stiffness is the result of internal effects occurring at the assembling stage. The chosen method for establishing the initial self-stress depends on design choices and also on the structure scale. The self-stress target is conditioned by the project conditions, and specifically by static and dynamic loads which will be applied on the structure. For the present case, choices that have been made for tensegrity systems design imply a tension control in active elements, with a dynamic procedure. Then a very good knowledge of constitutive laws for the components is essential. At the beginning of the process, a first self-stress state is established by a geometrical control of active element shortening. Then, a self-stress implementation method, based on an identified state, is submitted. Then, once a unitary influence basis has been chosen, the convergence toward the target state is realised by active element shortening. By this mean, final state is ...

STABILITY CONJECTURE IN THE THEORY OF TENSEGRITY STRUCTURES

The general problem of the stability of tensegrity structures comprising struts and cables is formulated. It is conjectured that any tensegrity system with totally tensioned cables is stable independently of its topology, geometry and specific magnitudes of member forces.

Effect of a Cable Rupture on Tensegrity Systems

A numerical investigation of the effect of a sudden rupture of a cable component in a Tensegrity assemblage is performed using nonlinear dynamic time history analysis to determine element axial forces and nodal displacements responses of the system. Details of the modeling of the Tensegrity elements as well as the numerical scheme used to integrate the equations of motion are discussed. The investigation is carried on a continuous struts Tensegrity system. The system is constituted by an assemblage of five expanded-octahedron basic modules. The simulation of the sudden rupture of a single cable is realized by replacing this element by its tension, applied as an external force to the extremities of the struts to which this element is attached. This force is then removed instantaneously provoking a load imbalance in the system. This imbalance acts as a triggering mechanism sending an impulsive shock to the damaged structure resulting in a damped vibration motion and large magnitude element axial forces.

Evolution of Natural Frequencies in Tensegrity Systems: A Case Study

In this study, a dynamic analysis of self-stress levels of tensegrity systems is proposed. A direct relation between the first natural frequency of these systems and the levels of self-stress in the elements is established. It constitutes a first step toward the study of the adaptability of tensegrity systems to external loads; this adaptability could be realized by adjusting the levels of self-stress of one or several cable elements. The study is carried out using continuous struts tensegrity systems made of linear associations of simplex type modules. When subjected to static loading, tensegrity systems show an evolution of their frequencies with respect to the tension levels in the elements. The frequencies are computed assuming the structure oscillates linearly about its static equilibrium position. This evolution can be used, to associate the level of self-stress to a desired resonant frequency of the system. The harmonic dynamic loading of these systems, confirms their non linear characters and show a particular evolution of the tension-frequency relation.

Can melatonin regulate the expression of prohormone convertase 1 and 2 genes via monomeric and dimeric forms of RZR/ROR nuclear receptor, and can melatonin influence the processes of embryogenesis or carcinogenesis by disturbing the proportion of cAMP and cGMP concentrations? Theoretic model of controlled apoptosis

The presented model of controlled apoptosis has been based on the assumption that correct information exchange between an organism as a whole, and each of its cells is conditioned by mutual proportions of cAMP and cGMP concentrations (C cAMP, C cGMP), according to the formula C cAMP× C cGMP= ‘a’ (constant). The regulation of balance of these ‘second messengers’ in a cell and an extracellular space would depend on the mutual proportions of concentrations of Melatonin and monomers of Melanin. These indoloderived compounds could be the activators of the transcription factors i.e. RZR and NFkappa-B, regulating the expression of Prohormone Convertase (PC) gen and Nitric Oxide Synthase (NOS) gen, respectively. Additionally, maternal Melatonin and Nitric Oxide (NO), being able to pass through trophoblast or placenta freely, would play decisive role in the synchronization of embryogenesis and intrauterine development of the fetus. In case of an embryo or a fetus, the result of C cAMPand C cGMPmultiplication, different from the proper constant ‘a’-value, would mean occurrence of disorders in the structure and functioning of the cellular tensegrity system and, in consequence, disturbances in the intercellular information exchange. It would lead to deviation in cellular metabolism, oriented cell movement, uncontrolled apoptosis, and as a consequence, would lead to the development of fetal defects. In case of a child or an adult, a sudden occurrence and prolongation of such disturbances in C cAMP–C cGMPproportions would induce a process of apoptosis of normal cells and an initiation of a cancerogenesis. On the other hand, the recovery of equilibrium in the information exchange system would initiate apoptosis of neoplastic cells, and simultaneously, proliferation of connective tissue cells. According to the presented hypothesis, a decrease in C cAMPand destabilization of the C cAMP–C cGMPbalance in an embryo or a fetus would result from relatively excessive amounts of maternal Melatonin (monomers) in fetal circulation, while a decrease of C cAMPand destabilization of the C cAMP–C cGMPbalance in a child or an adult would be a consequence of relatively insufficient amounts of Melatonin (dimers) in an organism. It seems possible, that determination of both C cAMPand C cGMPwould enable an early detection of high risk of developmental defects occurrence in an embryo or a fetus and neoplastic processes in a child or an adult. This method might also be considerably useful in monitoring a safe substitutional hormonotherapy. Copyright 2001 Harcourt Publishers Ltd INTRODUCTION

Nonlinear elastoplastic analysis of tensegrity systems

A procedure for the nonlinear elastoplastic analysis of tensegrity systems under static loads is presented. This procedure considers both geometric and material nonlinearities, using an updated Lagrangian formulation and a modified Newton–Raphson iterative scheme with incremental loading. The applicability of this procedure is demonstrated through the study of the elastoplastic behavior of a five module tensegrity beam. The effect of the yielding of some cable elements on the initial defined relational geometry once a tensegrity system is unloaded is outlined.

Multiparametered Formfinding Method: Application to Tensegrity Systems

A method allowing a multiparametered formfinding for prestressed and selfstressed reticulated systems with tensile and compressive members is presented. Known methods, based on geometric analysis and dynamic (dynamic relaxation) considerations have been developed for these systems but they allow generally the evolution of only one parameter. But, in case of shape finding of non-regular new forms or when the sought-after form is subject to a set of geometrical constraints, it becomes obligatory to elaborate a multiparametered form-finding process. The proposed numerical method, which is described in this paper, exploits the force density method, already used for form finding of pure tensile structures. However, equilibrium matrix of pure tensile structures as cable nets systems, admits always an inverse, which might be false when tensile and compressive members coexist in the system. In this paper, different processes allowing to define prestressed (or selfstressed) equilibrium geometry are described. Except for the relational structure which is considered as known at the beginning of the process, two sets of form-finding parameters can be identified for this method: prestress (or selfstress) coefficients of members and coordinates or redundant nodes. The proposed method does not yield a unique geometry but it is very convenient for a multiparametered formfinding, and has produced very interesting results, especially for Tensegrity Systems. Application of this method of multiparametered formfinding to Tensegrity Systems, provides the designer with an efficient way to achieve interesting new selfstressed geometries, such as the generation of double-layer grids by agglomeration of Tensegrity modules.

Geometrical non-linear analysis of tensegrity systems

A calculation method for structures with large deformations and displacements, based on works by Bathe [7–9], is developed so as to determine the tangent stiffness matrix and the internal stress vector. The formulation is established for a `bar' element. Application of this method for tensegrity systems allowed the study of behaviour for a simple self-stressed system, the four-strut tensegrity system, in case of `traction, compression, flexion and torsion' loading. Except for compression, the structure is rigidified when the loads increase. Influence of the self-stress level on this behaviour is also evaluated. Secondly a structure, generated by assembly of several four-strut tensegrity systems, has been calculated only under traction, concentrated and distributed. The behaviour under uniformly distributed load can be related to the `isolate' cell behaviour, even if some difficulties appear, mainly because of boundary conditions and possible choices for common cables between two adjacent cells.

Active Control of Tensegrity Systems

Tensegrity systems have been known for almost half a century. Their mechanical and geometrical characteristics make them suitable to terrestrial and extraterrestrial applications. This paper describes a numerical scheme of active control of tensegrity systems in extraterrestrial applications. An algorithm based on instantaneous optimal control has been developed and applied to an assembly of elementary tensegrity cells. The whole structural system comprises four cells forming a linear structure with 27 cables and 12 struts. Active control is realized in the first simulation by means of three actuators and in the second by means of six actuators located on two of the four cells. The results show that vibration amplitudes of nodes are reduced by such an active control scheme.

Definitions and Feasibility Studies of Tensegrity Systems

In this paper, the developments concerning tensegrity are briefly reviewed. The concept of tensegrity is systematically redefined. The concept is then expanded to cable-strut systems. The feasibility of tensegrity (cable-strut) systems is discussed, including some tools for further research. Finally, work concerning further research is considered.

Cable-strut systems: part I — tensegrity

The concept of cable-strut is extended from that of tensegrity. The broad interpretation of cable-strut systems includes tensegrity systems, RP (Reciprocal Prism) and CP (Crystal-cell Pyramid) system, etc. Its narrow interpretation excludes tensegrity systems. Thus this paper is divided into two parts. Part I gives concept, properties and feasibility studies of tensegrity structures to put forward cable-strut systems. Part II presents the theory and novel concepts concerning the application of cable-strut systems, and concludes that cable-strut systems are revolutionary in space structures. In this part, the essential idea of tensegrity is analyzed and the concept of tensegrity is systematically redefined. Tensegrity grids can be classified into two types of configurations: non-contiguous strut and contiguous strut. Their properties are presented and compared. The properties of the latter are also compared with those of RP grids. The low efficiency of tensegrity grids is analyzed. The feasibility studies, concerning applicable tensegrity forms and their application scale, are also introduced in this paper.

Cable-strut systems: Part II — Cable-strut

In this part, the basic concept of cable-strut systems is introduced. Cable-strut systems, including reciprocal prism (RP) and cell pyramidal (CP) grids invented by the author, as successful attempts to introduce cables into simplexes to form grids, are revolutions in space structures. Their properties are presented in this paper. Cable-strut systems possess self-stressed equilibrium, avoiding reliance on a bulky anchorage system, which is a most important advantage in construction over conventional flexible structures. It has improved greatly the structural properties of tensegrity systems, which also possess self-stressed equilibrium. Its planar form becomes the lightest self-stressed equilibrium space bar systems. The additional advantages over conventional space bar systems are that its joint design can be simplified, and its grid depth and grid length can be adjusted easily to sustain large bar forces and to lower bar forces further. Moreover, its stiffness can be increased by introducing bars to replace connecting cables to form double-layer and triple-layer forms. Super-span domical and cylindrical forms of RP system have also proved to be feasible and economical. The advantage of cable-strut systems in architecture is that the systems are clear in sense of sight, which makes them very attractive.

Mobile Tensegrity Systems

Popularized by R. B. Fuller, and simultaneously by G.D. Emmerich, Tensegrity Systems appeared for the first time in the sculptural work of K. Snelson at the end of the forties. Understanding of these structural systems is not a simple task. Many publications have been devoted to their study under plastic, mathematical, mechanical, geometrical and even symbolic aspects. They belong to a specific class of space called structures, which is mainly characterized by the fact that tension stresses are equilibrated by elements which have no rigidity in compression. Some basic concepts are described in the following paragraphs in order to explain how these space structures can be used in mobile architecture.

Structural Morphology Of Tensegrity Systems

Tensegrity systems are structures in which morphological and mechanical aspects are closely related by selfstress requirements. In the first part, this paper describes the birth of the idea which led to these systems. Snelson's work seems to be the main contribution for their initial design, even if Fuller created the word tensegrity and contributed with other searchers, like Emmerich to the popularization of these systems. The second part is devoted to form-finding methods which are necessary to reach a morphology compatible with mechanical requirements. References for main computation methods are given. Among them numerical methods like force density method could constitute an useful design tool.

Tensegrity Systems: The State of the Art

Tensegrity systems are space structures of a special nature and their industrial use has not yet been fully developed. The article describes the state of the art. The systems examined are geometrically and mechanically complex. Solutions have been found for some of the problems and others require development. A large set of references identifies the main contributions of researchers encouraging the designing of tensegrity systems open up new prospects for construction and application.

The Correspondence Between Convex Polydedra and Tensegrity Systems: A Classification System

Since tensegrity systems are often topologically and geometrically complex it is important to have a simple and effective method to describe and classify them in an orderly manner. This paper sets forth a system of relating any convex polyhedron to a tensegrity by defining a one-to-one correspondence between the elements of the polyhedron and the elements of the tensegrity. This system, which was first introduced by the author in a 1978 booklet while working in Buckminster Fuller's Philadelphia office, is now extended to include agglomerations of convex polyhedra — both regular and irregular space-filling arrays. In both the original system and its extension, the principle of geometric duality is clearly demonstrated. More importantly for engineers, the method will allow those familiar with space structures an easy and convenient way to relate any space frame or truss to a corresponding tensegrity structure.

Tensegrity Systems and Geodesic Domes

The name Buckminster Fuller is associated with the development of geodesic domes and with tensegrity systems. The principle of the latter is that rigidity is obtained by selfstressing alone. The shapes of domes constructed using tensegrity systems are not generally known a priori; these domes belong to the class of systems with indeterminate shape, such as cable systems. A form-finding method using the dynamic relaxation technique with attenuation of kinetic energy is presented here. The results given by this general method are compared with known solutions in certain cases and make it possible to quantify the deviations between geometries of tensegrity systems and polyhedral geometries. An example is given to illustrate the possibilities of the combination of the principle of tensegity systems with that of geodesic domes. It is a double-layer, double-curvature system whose shape respects the selfstressing criterion. Industrial construction of the system can be envisaged thanks to the design of an assembly node which permits variations in curve radius.

A correspondence between scene analysis and motions of frameworks

For over 100 years there has been a known correspondence between plane pictures of spherical polyhedra and static stresses in bar and joint frameworks with planar graphs. We define a new map from any plane picture of a general scene to a corresponding plate and bar framework in the plane. This map, and its return map, give a simple isomorphism between the space of scenes over the picture and the space of instantaneous motions of the framework. When two simple equivalence relations on the pictures and on the frameworks are introduced, the maps define an isomorphism of the equivalence clases which permits the full translation of questions, techniques and theorems between the two fields of study. A direct geometric interpretation of this correspondence is described and an extension to systems with occlusion and tensegrity (systems of inequalities) is presented. The analogous patterns in higher dimensions are described, yielding a correspondence between motions of bar and body frameworks in 3-space and 6-dimensional scenes over 5-dimensional pictures.

Steam engine governors

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.