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When to Hold and When to Fold: Studies on the topology of origami and linkages

Linkages and mechanisms are pervasive in physics and engineering as models for a variety of structures and systems, from jamming to biomechanics. With the increase in physical realizations of discrete shape-changing materials, such as metamaterials, programmable materials, and self-actuating structures, an increased understanding of mechanisms and how they can be designed is crucial. At a basic level, linkages or mechanisms can be understood to be rigid bars connected at pivots around which they can rotate freely. We will have a particular focus on origami-like materials, an extension to linkages with the added constraint of faces. Self-actuated versions typ- ically start flat and when exposed to an external stimulus - such as a temperature change or magnetic field - spontaneously fold. Since these structures fold all at once, and the number of folding patterns accessible to a given origami are exponential, they are prone to folding to a configuration other than the desired one. Other work has suggested methods for avoiding this misfolding, but it assumes ideal, rigid origami. Here, we expand on these models to account for the elasticity of real structures and introduce methods for accounting for Gaussian curvature in them. We also explore how to find and set an upper bound on minimal forcing sets, or the minimum set of folds required to force an origami, and present a graph theory algorithm for finding them in arbitrary origami. Taken altogether, these origami studies give insight into how the physical properties of origami influence folding and a new set of tools for avoiding misfolding. Next, we turn back to a more fundamental study of linkages and present a new method for finding the manifold of their critical points. We then demonstrate a design protocol that utilizes this manifold to create linkages with tun- able motions, before turning to several example structures, including the four-bar linkage and the Kane-Lubensky chain.
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