Second-Order Rigidity and Prestress Stability for Tensegrity Frameworks

Abstract

Abstract. 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 rigiditymfor 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.

Tensegrity frameworks: Static analysis review

“…. ., p T n ) ∈ R nd and is called a placement [83], embedding [14], realization [61] or configuration [19].…”

“…Then, the rigidity matrix R(p) [19], compatibility matrix (B) [62] or geometrical matrix (Π) [83], which are equivalent concepts, can be defined. Such a matrix has a row for each edge and a column for each vertex and dimension, so it is an e by nd matrix.…”

“…Given two different configurations p, q ∈ R nd for the same abstract framework G, denoted G(p) and G(q), they are said to be congruent [61,19] if one is the result of a rigid motion from the other. This means that both realizations have the same topology and distances between vertices, but their position in the d-space is different.…”

“…In the literature it is possible to find both, a mathematical definition, [19], or a more intuitive physical derivation [32]. Its non-diagonal elements are the stress, with changed sign, between any two adjacent nodes, and the elements of the main diagonal are the sum of the corresponding row (column) as shown in eq.…”

“…Connelly and Whiteley [19] adapted two concepts for the rigidity and stability of tensegrity frameworks: Second order rigidity, which is an extension to tensegrity frameworks of the second-order rigidity concept for bar frameworks, previously introduced by Connelly [13], and pre-stress stability, which is based on the energy methodology introduced by Connelly [14]. For the first time a general hierarchy for some rigidity and stability definitions was presented by those authors, and the conditions for some of the direct and inverse implications were established.…”