Self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities using symmetry-adapted force-density matrix method

Tensegrities consisting of axially loaded strings and bars have various technologically important applications, for which symmetric configurations are preferred as prototypes. In this study, we investigate the self-equilibrated states of rhombic truncated regular tetrahedral and cubic tensegrities by combining group representation theory and force-density matrix method. The force-density matrices of these tensegrities are analytically block-diagonalized using the irreducible representation matrices of tetrahedral and cubic groups. By making the symmetry-adapted force-density matrix satisfy the necessary number of nullity and positive semi-definiteness, we derive the analytical expressions for self-equilibrium and super-stability of rhombic truncated regular tetrahedral and cubic tensegrities. This work helps to design highly symmetric tensegrities for developing biomechanical models, mechanical metamaterials, and flexible robotics.