On the symmetries of the origami waterbomb pattern: kinematics and mechanical investigations

Abstract

Origamis are becoming the inspiration of new adaptive structures applied for several purposes. One of the challenges of the design of the origami inspired structures is to deal with the large number of variables and degrees of freedom (DoFs) associated with such complex structures. Closed tessellations have a reduced number of DoF when compared to the opened ones. Besides, the coupling due to the closure of the tessellation promotes some periodicity along the structure. Symmetric behaviors allow the description of the structure from a unit cell behavior, establishing reduced-order models. This paper investigates the origami waterbomb pattern, exploring the unit cell behavior and its symmetries. Initially, kinematics analysis based on an equivalent mechanism approach establishes a reduced-order model associated with symmetry hypotheses. Afterward, mechanical analysis is investigated using a nonlinear finite element analysis through bar-and-hinge formulation. A comparison between both formulations is performed showing the range of validity of the reduced-order model description. The general conclusions are applied to a cylindrical tessellation under symmetric actuation showing the capability of the reduced-order model for the origami description. Results show that the rigid foldability hypothesis is the essential point for the equivalence between the two descriptions.

Appendix

This appendix presents details about the origami formulation.

1.1 Equivalent mechanism analysis

The waterbomb unit cell is a closed-loop mechanism and, for the formulation used in this paper, it is assumed that the first linkage $\left(i=1\right)$ is associated with the crease $OB$, being numbered counterclockwise. Therefore, the last linkage $\left(i=6\right)$ is related to the crease $OA$ (see Fig. 2). The frame definition is summarized as:

• 1. The first frame $\left(i=1\right)$ is defined as the crease $OB$.

• 2. Frames are disposed following a counterclockwise sequence, following the vertex order $B,\,C,\,D,\,E,\,F$ and $A$, starting from $\left(i=1\right)$ at vertex $B$ and ending at $\left(i=6\right)$ at vertex $A$.

• 3. The ${z}_{i}$ axis of each $i$ frame is aligned with the crease, with the origin at $O$ (see Fig. 2).

• 4. The ${y}_{i}$ axis of each frame is coplanar with the origami face delimited by the joints $i$ and $i-1$, and the ${y}_{1}$ frame is coplanar with the origami face delimited by frames $1$ and $6$.

• 5. The ${x}_{i}$ axis of each $i$ frame is the normal of the face delimited by joints $i$ and $i-1$, and the ${x}_{1}$ frame is the normal to the face delimited by frames $1$ and $6$.

• 6. The waterbomb defines an inner region and an outer region, where the inner region is contained within the waterbomb edges $AB,\,BC,\,CD,\,DE,\,EF$ and $FA$. Each ${x}_{i}$ axis points outwards the inner region.

Each ${z}_{i}$ axis is defined such that every ${\theta }_{i}$ angle belongs to the range $\left[0,\pi \right]$. With this consideration, ${z}_{i}$ axis associated with valley folds (creases $OA,\,OC,\,OD$ and $OF$) are positioned along the crease, pointing from ${O}_{i}$ to the correspondent vertex ($A,\,C,\,D$ or $F$), while ${z}_{i}$ axis associated with mountain folds ($OB$ and $OE$) are positioned along the crease, pointing to the opposite direction of the correspondent vertex ($B$ or $E$). The values of the D–H parameters for a generic waterbomb cell are given in Table 3.

Table 3 D–H parameters and its correspondence to each vertex of the unit cell

The waterbomb pattern has a characteristic that all joints intercept at a common point (point $O$ in Fig. 2), resulting in ${a}_{i}={R}_{i}=0\,(i=1,\ldots 6)$. In addition, ${\alpha }_{i}$ is fixed for each pair of consecutive joints, being associated with the angle λ that defines the shape of the waterbomb cell wherein, for a squared waterbomb cell, $\lambda =\pi/4$.

1.2 Finite element analysis

The behavior of origami structures is described assuming quasi-static equilibrium, where the shape change is due to a succession of equilibrium configurations. It is assumed that the total potential energy $\mathrm{\Phi }$ is the sum of the strain energy stored in bars, ${U}_{bar}$, the strain energy stored in folding (torsional springs on the creases) and bending (torsional springs as virtual folds), ${U}_{spr}$, the work done by external loads, ${V}_{ext}$,

$$\mathrm{\Phi }={U}_{bar}+{U}_{spr}-{V}_{ext}$$

(8)

By considering quasi-static equilibrium, the ith bar element is represented by

$$\begin{aligned}&{\boldsymbol{T}}_{bar}^{i}={A}_{i}{L}_{i}{S}_{x}\frac{\partial {E}_{x}}{\partial {\boldsymbol{u}}_{i}}\\ &{\boldsymbol{K}}_{bar}^{i}={A}_{i}{L}_{i}\left[{S}_{x}\frac{{\partial }^{2}{E}_{x}}{\partial {\boldsymbol{u}}_{i}^{2}}+C\frac{\partial {E}_{x}}{\partial {\boldsymbol{u}}_{i}}{\left(\frac{\partial {E}_{x}}{\partial {\boldsymbol{u}}_{i}}\right)}^{T}\right]\end{aligned}$$

(9)

where ${S}_{x}$ is the second Piola–Kirchhoff (P–K) tensor, $C$ is a tangent modulus, ${A}_{i}$ is the transversal section area of the bar element and ${L}_{i}$ is the length of the bar element. The degree of freedom that describes the torsional spring (its rotation) is given by the dihedral angle between the panels and can be obtained straight from the displacements and the original coordinates of the vertices. Besides, the torsional spring has its behavior assumed as linear elastic. Thus, for the jth torsional spring,

$$\begin{aligned}&{\boldsymbol{T}}_{spr}^{j}={L}_{j}\frac{\partial \mathrm{\Psi }}{\partial \theta }\frac{\partial \theta }{\partial {\boldsymbol{u}}_{j}}{L}_{j}{M}_{RES}\frac{\partial \theta }{\partial {\boldsymbol{u}}_{j}}\\ &{\boldsymbol{K}}_{spr}^{j}={L}_{j}\left[{M}_{RES}\frac{{\partial }^{2}\theta }{\partial {\boldsymbol{u}}_{j}^{2}}+k\frac{\partial \theta }{\partial {\boldsymbol{u}}_{j}}{\left(\frac{\partial \theta }{\partial {\boldsymbol{u}}_{j}}\right)}^{T}\right]\end{aligned}$$

(10)

where ${M}_{RES}$ is the resisting moment per unit length, k is the rotational stiffness modulus per unit length, $\mathrm{\Psi }$ is the stored energy function and $\theta$ is the dihedral angle. The linear formulation of the moment ${M}_{RES}$ does not detect local penetration of origami panels and, to avoid that, additional kinematic constraints are considered. Based on that, the moment per unit length is given by

$${M}_{RES}=\left\{\begin{array}{l}k\left({\theta }_{1}-{\theta }_{0}\right)+\left(\frac{2k{\theta }_{1}}{\pi }\right)\mathrm{tan}\left(\frac{\pi \left(\theta -{\theta }_{1}\right)}{2{\theta }_{1}}\right)\\ k\left(\theta -{\theta }_{0}\right)\\ k\left({\theta }_{2}-{\theta }_{0}\right)+\left[\frac{2k\left(2\pi -{\theta }_{2}\right)}{\pi }\right]\mathrm{tan}\left(\frac{\pi \left(\theta -{\theta }_{2}\right)}{4\pi -2{\theta }_{2}}\right)\end{array}\,\begin{array}{c}\begin{array}{c},\,\theta \in \,\left]0,{\theta }_{1}\right[\\ \,\end{array}\\ ,\,\theta \in \left[{\theta }_{1},{\theta }_{2}\right]\\ \begin{array}{c}\,\\ ,\theta \in \,\left]{\theta }_{2},2\pi \right[\end{array}\end{array}\right..$$

(11)

The original Merlin Code assumes that the load is applied with respect to the undeformed configuration, keeping its initial characteristic during all time steps. Here, this input load follows the deformed configuration and therefore, follows the node movement. This approach allows a proper description of the origami that does not present any incorrect extra stretching. A workflow for the modified FEA is presented in Fig. 21, illustrated by a single cell. The XYZ coordinates of each node of the origami is used as input, being reshaped as a combination of nodes and panels, and the creases are properly identified and stored as bars in a trussed-like structure. Additionally, the boundary conditions and the actuation are defined as inputs, being either force or displacement type. This set of inputs are fed to the solver that, using an iteration method and with the formulation previously presented, converge the solution through a quasi-static analysis of the unbalanced system, until it reaches the equilibrium.

Fig. 21

Overview of simulation framework. The initial configuration is inserted in a combination of Nodes and Bars, and the external forces and constraints are carefully considered. At each iteration, a convergence is performed on each node displacement using MGDCM, and, after converging, a new input is generated with the revaluation of the forces based on the new Nodes (Nodes considering the displacement)