Prestress Stability Research Articles

Generalized prismatic tensegrity derived by dihedral symmetric lines

Classic prismatic tensegrity structures, characterized by dihedral symmetry with one orbit of nodes, are among the simplest and possibly the earliest spatial tensegrity structures invented. This paper introduces a generalized form of the prismatic tensegrity structures by converting a single-loop linkage into truss, in which the lines of joint axes rather than the nodes have dihedral symmetry. Since the vector space formed by the line coordinates of these joints has rank degeneracy one, the generated tensegrity structures are kinematically and statically indeterminate. These tensegrity structures are further proved to be prestress-stable, generally, for the total or partial parameter space of lines based on a second-order analysis of screws, and are called dihedral-line tensegrity structures in this paper. Specifically, this paper focuses on symmetric dihedral-line tensegrity structures, in which the nodes also have dihedral symmetry but in two orbits and members in seven orbits, and are called two-orbit dihedral-line tensegrity structures. It is found that there are at least N struts for the generated tensegrity with DN symmetry. And the classic prismatic tensegrity structures can be recovered from these dihedral-line tensegrity structures by removing certain zero-force members. Symmetric-adapted force density matrices are also provided as well as the relation to that of the classic prismatic tensegrity. Given 4N+6 dimensional parameters inherent to these tensegrity structures, a rich variety of tensegrity structure family is presented.

Equilibrium stresses and rigidity for infinite tensegrities and frameworks

Asymptotic equilibrium stresses are defined for countably infinite tensegrities and generalisations of the Roth-Whiteley characterisation of first-order rigidity are obtained. Generalisations of prestress stability and second order rigidity are given for countably infinite bar-joint frameworks and are shown to give sufficient conditions for continuous rigidity relative to certain prescribed motions. The proofs are based on a new short proof for finite frameworks that prestress stability ensures continuous rigidity.

Stability Conditions for General Tensegrity with Rigid Bodies

This study develops stability conditions for general tensegrity regarding the three stability criteria (e.g., prestress stability, general stability, and super stability) based on an energy approach. The coupling relations of specific nodal degrees of freedom caused by the rigid bodies are described by a set of constraint functions and incorporated into the potential energy function by the Lagrangian multiplier method. Stiffness matrices of general tensegrity systems are derived from the augmented potential energy function. It shows that the stability of general tensegrity can be evaluated through the stiffness matrices expressed in the constrained motion space generated by the constraint functions. Elegant formulations are developed for the evaluation of the three types of stability conditions for general tensegrity systems. Moreover, it turns out that the formulations developed in this study will degenerate into those for the stability analysis of classic tensegrity if no rigid bodies exist in the system, which indicates that the proposed approach is a general framework for the stability analysis of any type of tensegrity systems.

Transverse rigidity is prestress stability

Recently, V. Alexandrov proposed an intriguing sufficient condition for rigidity, which we will call “transverse rigidity”. We show that transverse rigidity is actually equivalent to the known sufficient condition for rigidity called “prestress stability”. Indeed this leads to a novel interpretation of the prestress condition.

Infinitesimal rigidity and prestress stability for frameworks in normed spaces

A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid – i.e. whether every other suitably close framework with the same distance constraints is an isometric copy – is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces.

Graphical Stability Analysis of Maillart’s Roof at Chiasso

The newly-developed method of graphic stability is applied to Robert Maillart’s iconic roof design at Chiasso. The truss has an unusual form that emerged from Maillart’s graphic statics as the shape dual to the symmetrical equilibrium forces. Although graphic statics can lead to highly efficient structural forms, there can be a tendency for diagonal bracing to be minimised or even eliminated, as at Chiasso. Whilst equilibrium is satisfied, there may be a decrease in stability, and it has not previously been possible to quantify this within the traditional graphical approach. Using Chiasso as an example, this paper demonstrates how the prestress stability of this and similar truss structures may now be assessed within the same graphical setting.

On rigid origami III: local rigidity analysis.

In this article, rigid origami is examined from the perspective of rigidity theory. First- and second-order rigidity are defined from local differential analysis of the consistency constraint; while the static rigidity and prestress stability are defined after finding the form of internal force and load. We will show the hierarchical relation among these local rigidities with examples representing different levels. The development of theory here follows the same path as the conventional rigidity theory for bar-joint frameworks, but starts with different high-order rotational constraints. We also bring new interpretation to the internal force and geometric error of constraints associated with energy. Examining the different aspects of the rigidity of origami might give a novel perspective for the development of new folding patterns, or for the design of origami structures where some rigidity is required.

Stability of trusses by graphic statics.

This paper presents a graphical method for determining the linearized stiffness and stability of prestressed trusses consisting of rigid bars connected at pinned joints and which possess kinematic freedoms. Key to the construction are the rectangular areas which combine the reciprocal form and force diagrams in the unified Maxwell–Minkowski diagram. The area of each such rectangle is the product of the bar tension and the bar length, and this corresponds to the rotational stiffness of the bar that arises due to the axial force that it carries. The prestress stability of any kinematic freedom may then be assessed using a weighted sum of these areas. The method is generalized to describe the out-of-plane stability of two-dimensional trusses, and to describe three-dimensional trusses in general. The paper also gives a graphical representation of the ‘product forces’ that were introduced by Pellegrino and Calladine to describe the prestress stability of trusses.

Almost‐Rigidity of Frameworks

AbstractWe extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost‐rigidity. We briefly discuss how the theory may be applied to tensegrities. © 2020 Wiley Periodicals LLC.

Topology-Finding of Tensegrity Structures Considering Global Stability Condition

This study proposes a general framework for topology-finding or topology optimization of tensegrity structures. The existing topology-finding formulation of tensegrity structures based on mixed-integer linear programming (MILP) was improved and transformed into a formulation based on mixed-integer semidefinite programming (MISDP) which considers the global stability condition of tensegrity. We illustrated and analyzed two undesirable phenomena, loss of prestress stability and loss of integrity, caused by the missing global stability condition in previous MILP-based approaches. A branch-and-bound algorithm combined with a primal-dual interior-point algorithm is employed to solve the proposed MISDP model. Some numerical examples illustrated the improvements and effectiveness of the proposed approach. The proposed approach successfully can avoid the two undesirable phenomena and ensure the global stability of the found tensegrity structures. By using different stability conditions in the topology-finding process, the proposed approach can find general stable tensegrity structures and superstable tensegrity structures.

On the design, elastic modeling and experimental characterization of novel tensegrity units

Purpose This study aims to focus on a short review on recent results dealing with the mechanical modelling and experimental characterization of a novel class of tensegrity structures, named class θ = 1 tensegrity prisms. The examined structures exhibit six bars connected by two disjoint sets of strings. Design/methodology/approach First, the self-equilibrium problem of tensegrity θ = 1 prisms is numerically investigated for varying values of two aspect parameters and, next, their prestress stability is studied. The mechanical behavior of the examined structures in the large displacements regime under uniform compression loading is also numerically computed through a path-following procedure. Finally, the predicted constitutive response is validated through experimental tests. Findings The presented results highlight that the examined structures exhibit a large number of infinitesimal mechanisms from the freestanding configuration, and reveal that they exhibit tunable elastic response switching from stiffening to softening. Originality/value This multi-faceted elastic response is in agreement with previous literature results on the elastic response of minimal tensegrity prism, and suggests that such units can be usefully used as non-linear springs in next-generation tensegrity metamaterials.

Prestress Stability of Triangulated Convex Polytopes and Universal Second-Order Rigidity

We prove that universal second-order rigidity implies universal prestress stability and that triangulated convex polytopes in 3-space (with holes appropriately positioned) are prestress stable.

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.

Dihedral ‘star’ tensegrity structures

This paper presents conditions for self-equilibrium and super stability of dihedral ‘star’ tensegrity structures, based on their dihedral symmetry. It is demonstrated that the structures are super stable if and only if they have an odd number of struts, and the struts are as close as possible to each other. Numerical investigations show that their prestress stability is sensitive to the geometry realisation.

Automated discovery and optimization of large irregular tensegrity structures

Tensegrities consist of disjoint struts connected by tensile strings which maintain shape due to pre-stress stability. Because of their rigidity, foldability and deployability, tensegrities are becoming increasingly popular in engineering. Unfortunately few effective analytical methods for discovering tensegrity geometries exist. We introduce an evolutionary algorithm which produces large tensegrity structures, and demonstrate its efficacy and scalability relative to previous methods. A generative representation allows the discovery of underlying structural patterns. These techniques have produced the largest and most complex irregular tensegrities known in the field, paving the way toward novel solutions ranging from space antennas to soft robotics.

Two-dimensional analysis of bar-and-joint assemblies on a sphere: Equilibrium, compatibility and stiffness

This paper discusses a 2D truss formulation with spherical angular variables for bar-and-joint assemblies existing in 3D on the surface of a sphere. Based on the principle of potential energy, coupled problems of equilibrium and compatibility, as well as of stiffness and prestress stability are investigated. Finally, the results are illustrated by numerical examples.

Second-Order Rigidity and Prestress Stability for Tensegrity Frameworks

This paper defines two concepts of rigidity for tensegrity frameworks (frameworks with cables, bars, and struts): prestress stability and second-order rigidity. We demonstrate a hierarchy of rigidity—first-order rigidity implies prestress stability implies second-order rigidity implies rigidity—for any framework. Examples show that none of these implications are reversible, even for bar frameworks. Other examples illustrate how these results can be used to create rigid tensegrity frameworks. This paper also develops a duality for second-order rigidity, leading to a test which combines information on the self stresses and the first-order flexes of a framework to detect second-order rigidity. Using this test, the following conjecture of Roth is proven: a plane tensegrity framework, in which the vertices and bars form a strictly convex polygon with additional cables across the interior, is rigid if and only if it is first-order rigid.

The Stability of Tensegrity Frameworks

For a tensegrity framework with bars, cables (unilateral tension members) and struts (unilateral compression members), this paper presents a sequence of four mathematical concepts for detecting its rigidity. In order from the strongest to the weakest, these concepts are called infinitesimal (or static) rigidity, prestress stability, second-order rigidity and rigidity. Emphasis is placed on pre-stress stability, which lies between infinitesimal rigidity and second-order rigidity.