Analytical form-finding of tensegrities using determinant of force-density matrix

Tensegrities have found great importance and numerous applications in many civil, aerospace and biological systems, and form-finding analysis is a vital step to obtain their self-equilibrated configurations before applying external loads. In this paper, we present a concise and general analytical scheme for tensegrity form-finding analysis. Additions and multiplications are employed as major computational operations, that can guarantee the solving process computationally efficient. Based on the characteristic polynomial of the symbolic force-density matrix, the two (three) lower-order coefficients that are necessary for the form-finding of planar (three-dimensional) tensegrities are expressed by a unified compact equation using the matrix determinants. The force-densities of tensegrity elements satisfying the established equation can determine the self-equilibrated state of tensegrity. A large number of representative planar and three-dimensional examples are analyzed to verify the validity and efficiency of our analytical form-finding method. The predictions of our scheme are in broad agreement with the results obtained by many other methods. This study produces continuously variable force-densities of self-equilibrated tensegrities, and helps to design their unusual mechanical properties for scientific and engineering applications.