Geometry and design of origami bellows with tunable response

Origami folded cylinders (origami bellows) have found increasingly sophisticated applications in space flight and medicine. In spite of this interest, a general understanding of the mechanics of an origami folded cylinder has been elusive. With a newly developed set of geometrical tools, we have found an analytic solution for all possible cylindrical rigid-face states of both Miura-ori and triangular tessellations. Although an idealized bellows in both of these families may have two allowed rigid-face configurations over a well-defined region, the corresponding physical device, limited by nonzero material thickness and forced to balance hinge and plate-bending energy, often cannot stably maintain a stowed configuration. We have identified the parameters that control this emergent bistability, and we have demonstrated the ability to design and fabricate bellows with tunable deployability.

Figure 1

Cylindrical Miura-ori corrugation: thick left and right edges connect to one another. The framework used in this paper does not specify whether edges are valley or mountain folded, a property that instead emerges from the solution of its geometry. The red dotted rectangle indicates a unit cell. The lower blue edges are implicitly assumed to mirror ${\omega }_2$ and ${\omega }_3$ about the plane established by ${\omega }_0$ and ${\omega }_1$. A single band of circumferential rigid panels is defined by its lateral size $h$.

Figure 2

Various unit cells (delimited by red dashed rectangles) derived from the Miura-ori folding patterns. A global scaling factor determines the length of the unit cell. At least three circumferential cells are required to make a bellows.

Figure 3

First row: Miura-ori (left) and Kresling (right) bellows models annotated with example unit-cell vectors. Second row: annotated vector illustration of unit cell in the $XY$ plane. Photographs are at approximately the same scale, and these structures correspond to $n=5$ unit cells.

Figure 4

Parameter space for class 1 and class 2 family bellows for a number $n\ge 3$ of unit cells.

Figure 5

Detailed figure of the (red) bistable region of Fig. 4. Grayscale shading indicates the angle difference $\mathrm{\Delta }\psi$ (in radians) between collapsed and deployed solutions, with colored call outs for the contours indicated on the figure's right. The purple curve represents zero separation between the solutions, indicating the solution's degeneracy along the curve $f({\varphi }_1,n)$.

Figure 6

Isotropic bellows construction. Individual strips are patterned with the laser cutter before being attached by the small tabs seen at rightmost edge of (a). Strips are then rolled into a tube (b), which is then collapsed one cell at a time (c) to form the final bellows (d).

Figure 7

Typical mechanical fatigue curve as a bellows (here Class 2–Type A) is repeatedly actuated. The snap-through transition rapidly fatigues the thin polymer bellows, leading to a softening of the device on subsequent (darker) cycles without modifying qualitatively the mechanical response of the device. The top (red) curves are from multiple compressions, and the bottom (blue) curves are data from the corresponding extensions.

Figure 8

Class 1–Type A (Miura-ori) bellows with ${\varphi }_1={68}^{\circ }$. This device collapses smoothly, with only a small step as its transition panels collapse. Force measurement corresponding to the image on the left is indicated as a solid point. The colored gradient trailing the ball indicates previous force or position measurements, and the shadowed line indicates future measurements. See also Movie-S1 in [26].

Figure 9

Class 2–Type A (Kresling) bellows with ${\varphi }_1={68}^{\circ }$. This device is constructed with identical angles ${\varphi }_1$ and ${\varphi }_2$ as Fig. 8, yet it collapses with small steps as each paired panel crumples. It deploys smoothly without external forcing. See also Movie-S2 in [26].

Figure 10

Class 1–Type A (Miura-ori) bellows with ${\varphi }_1={76}^{\circ }$. This device collapses with a single large step at its transition panels, and deploys in a fairly smooth manner. See also Movie-S3 in [26].

Figure 11

Class 2–Type A (Kresling) bellows with ${\varphi }_1={76}^{\circ }$ shows the most dramatic behavior of the lot. Contrary to class 1–type A shown in Fig. 10, this bellows must be pulled apart at each step, and snaps shut as it collapses with increasing rapidity. See also Movie-S4 in [26].

Figure 12

Class 1–Type B bellows with ${\varphi }_1={105}^{\circ }$ and ${\varphi }_2={69}^{\circ }$. This hexagonal bellows is expected to have a single accessible rigid-face configuration. Without a geometrically allowed collapsed state, this bellows buckles under load, breaking its rotational symmetry. See also Movie-S5 in [26].

Figure 13

Class 2–Type B bellows with ${\varphi }_1={105}^{\circ }$ and ${\varphi }_2={69}^{\circ }$. Much like the hexagonal bellows of Fig. 12, without a geometrically allowed collapsed state this bellows crumples, breaking its rotational symmetry. See also Movie-S6 in [26].