Mobility of symmetric deployable structures subjected to external loads

• Mobility of deployable structures under external loads is established using symmetry.
• Symmetry of mechanisms and loads can be identified from irreducible representations.
• A mobile structure has fully symmetric mechanism non-orthogonal to external loads.
• A deployable structure can be rigid if symmetry of loads is higher than that of mechanisms.

Deployable structures can exhibit remarkable and continuous geometric transformations, however, they are likely to be rigid under certain external loads. This study adopts group theory to evaluate the mobility of symmetric deployable structures under external loads. Mobility analysis is expressed as determining the orthogonality of internal mechanism modes and external loads. Based on the symmetry groups, both the mechanism modes and external loads are associated with specific symmetry subspaces. Thus, it can be evaluated whether the external loads stiffen all the internal mechanism modes. Illustrative examples on pin-jointed structures and over-constrained mechanisms are given to verify the proposed method. It turns out that the product of the internal mechanisms and external loads is equivalent to that of the mechanisms and loads in the symmetry subspaces associated with different irreducible representations. A deployable structure will be immobile under external loads if symmetry order of the loads is higher than that of the mechanisms. In addition, the structure will be immobile, if the internal mechanisms and the external loads are equisymmetric and orthogonal to each other. The conclusions agree with published results, and need much fewer computations. The proposed method is efficient and applicable to most symmetric deployable structures.