Engineering structures with adaptive geometry require an optimized integration of actuation, sensing, and packaging. Origami structures, by definition, can “shape-shift” between multiple geometric configurations that are predefined by a pattern of folds. However, most origami models assume ‘rigid origami’, meaning that the facets between fold lines cannot deform. Although a useful, simplifying assumption, in reality no origami pattern is completely rigid. Furthermore, facet compliance could broaden the design space of origami devices. Together, these motivate our investigation of the transition from rigid to compliant origami for a given fold pattern. We approximate the fold pattern as a truss and incorporate relative stiffness magnitudes associated with bar elongation (10^8), folding (10^2), and facet bending (10^1 -10^12).
Following Schenk et al’s formalism (PNAS 2012) to relate nodal forces to the nodal displacement, we determine the eigenvalues and eigenmodes of the stiffness matrix, which correspond to the natural modes of the structure. Due to the significant difference in stiffness values, we are able to separate the eigenmodes associated only with folding from those with bar elongation and/or facet bending. As expected, rigid origami modes appear as “fold-only” modes when facet stiffness is much greater than fold stiffness. However, as facet stiffness approaches fold stiffness, these mode shapes adopt a combination of fold and facet deformations. By identifying these fold/facet bending modes we can determine how to redistribute bar, fold or facet stiffness in the surrounding structure in order to mitigate or amplify their effect. These results have important implications for not only origami topology design, but also for optimal placement of local material properties.