Static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules

Static and dynamic properties of a pair of dual spherical tensegrity modules invented by Buckminster Fuller are investigated. They are regular truncated icosahedral and dodecahedral tensegrity modules. The computation of the Maxwell number and the use of Calladine's relation reveal that regular truncated icosahedral and dodecahedral tensegrity modules possess 55 infinitesimal mechanism modes. A reduced equilibrium matrix is presented for the initial shape finding to economically impose the existence of a pre-stress mode. Both the initial shape and the corresponding pre-stress mode are analytically obtained by using graphs of the icosahedral group and the reduced equilibrium matrix. For both icosahedral and dodecahedral modules the maximum values of the cable tension is always less than the absolute value of bar compression. In order to classify a large number of infinitesimal mechanism modes, modal analyses are conducted. Infinitesimal mechanism modes have the stiffness due to pre-stress and are associated with lowest natural frequencies. Their natural frequencies increase proportionally to the square root of the amplitude of pre-stress. It is found that there are only 15 distinct natural frequencies associated with the infinitesimal mechanism modes.