Improved Symmetry Method for the Mobility of Regular Structures Using Graph Products

Mobility analysis plays a key role in form finding and design of novel kinematically indeterminate structures. For large-scale or complex structures, it demands considerable computations and analyses, and, thus, efficient method is of great interest. Because many structures could be viewed as the product of two or three subgraphs, such structures are called regular structures and usually hold certain symmetries. Combining graph theory with group representation theory, this paper proposes an improved symmetry method for the mobility of kinematically indeterminate pin-jointed structures. The concepts of graph products are described and utilized, to simplify the conventional symmetry-extended mobility rule. Based on the definitions of the Cartesian product, the direct product, and the strong Cartesian product, the authors establish the representations of nodes and members for the graph products, respectively. The proposed method focuses on the simple subgraphs, which generate the entire structure, and computes the matrix representations of the nodes and members under each symmetry operation. Therefore, symmetry analysis of the entire structure is transformed into independent evaluations on the subgraphs. Mobility of symmetric structures with a large number of nodes and members is studied, and the static and kinematic indeterminacy of the structures is evaluated using the proposed method.