2ND-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 se cond-order rigidity. We demonstrate a hierarchy of rigidity-first-orde r rigidity implies prestress stability implies second-order rigidity i mplies rigidity-for any framework. Examples show that none of these im plications are reversible, even for bar frameworks. Other examples ill ustrate how these results can be used to create rigid tensegrity frame works. This paper also develops a duality for second-order rigidity, l eading to a test which combines information on the self stresses and t he first-order flexes of a framework to detect second-order rigidity. Using this test, the following conjecture of Roth is proven: a plane t ensegrity framework, in which the vertices and bars form a strictly co nvex polygon with additional cables across the interior, is rigid if a nd only if it is first-order rigid.