Zero modes of origami-inspired trusses

The subject of this paper is to use methods from linear algebra to study the zero modes, or folding modes, of periodic structures or mechanisms. First, we introduce a first-order analytical method to find the zero modes of one cell in these patterns. By adding multipliers, this gives the trend throughout the pattern. We apply this method to a case study of two planar mechanisms: a simple essentially 1D lattice from literature, and a slightly more complicated 2D case. 3D prints of this second case matched the analytically predicted behavior, validating this method. With these insights, we turned to the fully 3D case of the Yoshimura origami pattern and related patterns, by considering the trusses formed by the pattern. For this case, we primarily applied numerical methods. This allowed easy visualization of its modes and modification of the pattern. The Yoshimura pattern constructed from n-sided polygons was found to have 2(n - 3) zero modes (in addition to the rigid-body transformations), regardless of the number of layers, with both methods. For n = 4, each of the two modes were entirely localized to either end of the structure. n [greater than] 4 had modes localized towards either end, but not completely so. Modes remained symmetric, occurring on both ends, despite various modifications made to the n = 4 pattern. The twist angle had the largest impact on these modes at critical angles. Twisting opposite of the Yoshimura pattern gave the Kresling pattern, which had more than two bulk modes. Zero twist gave a cylindrical pattern with one linear and one constant bulk mode. In between, modes were somewhat localized, approaching the full localization of the Yoshimura case when approaching the Yoshimura twist angle.