Dynamics of a bistable Miura-origami structure

Origami-inspired structures and materials have shown extraordinary properties and performances originating from the intricate geometries of folding. However, current state of the art studies have mostly focused on static and quasistatic characteristics. This research performs a comprehensive experimental and analytical study on the dynamics of origami folding through investigating a stacked Miura-Ori (SMO) structure with intrinsic bistability. We fabricate and experimentally investigated a bistable SMO prototype with rigid facets and flexible crease lines. Under harmonic base excitation, the SMO exhibits both intrawell and interwell oscillations. Spectrum analyses reveal that the dominant nonlinearities of SMO are quadratic and cubic, which generate rich dynamics including subharmonic and chaotic oscillations. The identified nonlinearities indicate that a third-order polynomial can be employed to approximate the measured force-displacement relationship. Such an approximation is validated via numerical study by qualitatively reproducing the phenomena observed in the experiments. The dynamic characteristics of the bistable SMO resemble those of a Helmholtz-Duffing oscillator (HDO); this suggests the possibility of applying the established tools and insights of HDO to predict origami dynamics. We also show that the bistability of SMO can be programmed within a large design space via tailoring the crease stiffness and initial stress-free configurations. The results of this research offer a wealth of fundamental insights into the dynamics of origami folding, and provide a solid foundation for developing foldable and deployable structures and materials with embedded dynamic functionalities.

Figure 1

Geometry and kinematics of the SMO structure. (a) Geometry of a single SMO structure. The plane crease pattern of the bottom cell A and top cell B are given, with the internal solid and dashed lines indicating the “mountain” and “valley” creases, respectively. (b) A single SMO structure at the bulged-out (${\theta }_{\mathrm{A}}<0$), intermediate (${\theta }_{\mathrm{A}}=0$), and nested-in (${\theta }_{\mathrm{A}}>0$) configurations. (c) Origami-based metamaterials expanded from SMO units, shown in three states.

Figure 2

The normalized potential energy landscape with respect to the folding angle ${\theta }_{\mathrm{A}}$. (a) With stress-free configuration ${\theta }_{\mathrm{A}}^0=\pi /3$, bistability (dot-dashed) shows up when $k_{\mathrm{B}}=\mu k$ is significantly higher than $k_{\mathrm{A}}=k$. (b) With $k_{\mathrm{B}}=10k$, bistability (dot-dashed) shows up when the stress-free configuration ${\theta }_{\mathrm{A}}^0$ is far away from 0. Note that the bistability profile presents obvious asymmetry.

Figure 3

Asymmetric bistability in the SMO structure, with ${\theta }_{\mathrm{A}}^0=-\pi /3$ and $k_{\mathrm{B}}=20k_{\mathrm{A}}=1(\mathrm{N}\mathrm{m})/\mathrm{rad}$. (a) The normalized potential energy with respect to the height $H$. (b) The force-displacement curve. The two stable configurations are denoted by solid dots $H_1$ and $H_2$.

Figure 4

SMO structure prototype. (a) Illustration of the prototyping method. (b) SMO prototype with 3D-printed connectors and screw rods.

Figure 5

Measured and fitted force-displacement curves of the SMO prototype. Insets display the prototype at the nested-in configuration (i), bulged-out configuration (ii), and a middle state (iii).

Figure 6

(a) Schematic illustration of the dynamic experiment setup; insets show the photos of the nested-in and bulged-out configurations of the SMO prototype. (b) Equivalent single degree of freedom model for numerical simulation; inset shows the fitted force-displacement curve.

Figure 7

Experiment results. (a) Displacement transmissibility in terms of the rms value with respect to the excitation frequency $\omega$. (b) From top to bottom, showing the displacement transmissibility of the main frequency components for the intrawell (in), intrawell (out), and interwell responses, respectively. (c) Three types of responses at $\omega =5.8\text{H}z$: from top to bottom, single-periodic intrawell (in), $2T$-subharmonic intrawell (out), and $3T$-subharmonic interwell responses. (d) Three types of responses at $\omega =11\mathrm{Hz}$: From top to bottom, $2T$-subharmonic intrawell (in), $2T$-subharmonic intrawell (out), and interwell chaotic responses. Both the displacement time histories and the FFT spectra are displayed in (c,d), and for reference, the excitation signals are also included.

Figure 8

Numerical simulation results. (a) Displacement transmissibility in terms of the rms value with respect to the excitation frequency $\omega$. (b) From top to bottom, showing the displacement transmissibility of the main frequency components for the intrawell (in), intrawell (out), and interwell responses, respectively.

Figure 9

Similarities between the HDO potential and the SMO potential profiles. (a)–(c) show the potential energy landscapes of the HDO at $\kappa =-0.1,\sigma =0.3$, $\kappa =1,\sigma =0.4$, and $\kappa =1,\sigma =3$, respectively, denoted by solid curves. For reference, the degenerated situations at $\sigma =1$ are also plotted with dotted curves. (d)–(f) show the potential energy landscapes of the SMO structure with a small stress-free angle ${\theta }_{\mathrm{A}}^0=\pi /6$, a nested-in stress-free configuration at ${\theta }_{\mathrm{A}}^0=\pi /3$, and a bulged-out stress-free configuration at ${\theta }_{\mathrm{A}}^0=-\pi /3$, respectively. The SMO structure's potential is calculated based on Eqs. (4) and (5). Comparing the subfigures in the same row could indicate the similarities between the HDO and the SMO potential profiles.

Figure 10

Programmable bistability. (a) Monostable region I and bistable regions II and III on the $\mu -{\theta }_{\mathrm{A}}^0$ plane; in the bistable regions II and III, the contours indicate the potential difference between the two wells ($\mathrm{\Delta }V$). Insets show sketches of the monostable and bistable potential curves. The arrows indicate the increase direction of the distance ($d$), potential depth ($V_{\mathrm{in}}$ and $V_{\mathrm{out}}$), and asymmetry level ($\mathrm{\Delta }V$). (b) Potential energy landscapes corresponding to points $a_1$, $a_2$, and $a_3$ illustrate the effects of the stiffness ratio $\mu$. (c) Potential energy landscapes corresponding to points $a_4$, $a_5$, $a_6$, and $a_7$ illustrate the effects of the stress-free folding angle ${\theta }_{\mathrm{A}}^0$. The changes of potential well positions are denoted by arrows.