A numerical study of the dynamic properties of Miura folded metamaterials
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In this paper, we investigated the dynamic properties of origami structures composed of Miura unit cells with rigid facets and elastic hinges under three types of excitations: harmonic force, harmonic displacement and impact. Under the simple harmonic force or displacement excitations, different crease stiffness affects the vibration responses of Miura folded metamaterials. The single degree-of-freedom (DOF) models have one resonant peak, after which the vibration amplitude at the response end is lower than that of the excitation end. Increasing the crease stiffness can increase the resonant frequency. The multi-DOF model exhibits multiple resonant peaks under harmonic excitations, where the lowest resonant frequency has the highest peak. Increasing the number of layers can reduce the resonant frequency. When the multi-DOF model is subjected to impact load, the magnitude of the impact wave decays quickly after the impact and finally reaches a steady state with a low average strain magnitude. When the crease stiffness is increased, the propagation of the impact wave becomes faster, whereas the maximum strain magnitude becomes smaller. Introducing different damping coefficient to the crease has no influence on the propagation speed of the impact waves, but can accelerate the decay in the magnitude of the impact wave.
KeywordsMiura origami structure Harmonic excitation Impact Frequency response Wave propagation
Most research on folded metamaterials has been focused on Miura origami (Miura-ori) metamaterials and their variation forms, and mainly studied their quasi-static mechanical properties . For example, Zhou et al.  studied the quasi-static mechanical properties of Miura-ori folded cores for sandwich structures and analysed their bending and shear properties using finite element analysis. Schenk et al.  studied the Poisson’s ratios of stacked Miura-based cellular metamaterials based on theoretical analysis and developed the manufacturing method for prototyping. Liu et al.  studied quasi-static deformation characteristics of folded plates made of Elvaloy by comparing the experimental analysis with the simulation results. The results showed that folded plates made from this material have a high reuse rate.
The research on the dynamical properties of folded metamaterials is relatively rare. In the existing work on the dynamic responses, the folded structure is usually simplified as a spring-mass equivalent model or a multi-link equivalent model. Most of the system characteristics are neglected. For example, Fang et al.  studied the self-stability characteristic and dynamic response of a Miura folded metamaterial under the simple harmonic displacement excitation. The main frequency of the structure was analysed based on the Fourier transformation. It was noted that the dynamic responses of the folded metamaterial can be designed by adjusting the stiffness of the creases. Sadeghi and Li  designed a folded metamaterial with quasi-zero stiffness characteristic by means of sealing compressed air in the unit cells. A spring-mass equivalent model was developed to analyse the vibration properties of the structure. It was found that the structure could provide effective low-frequency vibration isolation. Yasuda et al. [7, 8] developed the equivalent multi-link and spring-mass models to study the impact responses of the Tachi-Miura polyhedron (TMP) tubular folded metamaterial. In their study, the TMP structure was divided into individual TMP modules by massless rigid separators, and the strain wave propagation characteristics inside the structure under the impulse load were studied. It was revealed that the impact load acting on one end propagates through the structure in the form of small-amplitude soliton wave, indicating that the metamaterial has good impact resistance performance. In this paper, we investigated the dynamic properties of single degree-of-freedom (DOF) and multiple-DOF Miura-based origami structures by means of virtual vibration and impact tests using the dynamics simulation software ADAMS (MSC, USA).
2 Origami modelling method
2.1 Origami models
The second model contains two directly connected unit cells, as shown in Fig. 1b. The torsional stiffnesses of the four creases of the first and second units cells (the red and blue lines, respectively) equal ka and kb, respectively, and the torsional stiffness of the two inter-unit creases (the yellow lines) is denoted as kc.
Either a single Miura unit cell or multiple directly connected unit cells, e.g. the second model, has a single degree of freedom (DOF) of folding motion, which tends to be rigid under dynamic loads. In the third model, we consider a multi-layered structure consisting of ten multiple stacked Miura unit cells separated by rigid separators, as shown in Fig. 1c. The massless separation surfaces allow each constitutive Miura unit cell to deform independently, hence making the entire structure have multiple DOFs and be more flexible under dynamic loads. The creases of all unit cells inside the model are assigned with the same torsional stiffness ka. Unless otherwise specified, all creases in the models are assumed to be idealized ones with zero damping.
2.2 Load and boundary conditions
In ADAMS, Eq. (3) represents a single triangular wave impulse that lasts for 0.2 s with a magnitude of F0.
3.1 Vibration responses under harmonic force excitations
3.2 Vibration responses under harmonic displacement excitations
3.3 Impact responses
In this paper, we presented a preliminary numerical study on the dynamic responses of origami structures built from standard Miura unit cells consisting of parallelogram rigid facets and elastic hinges. Three types of origami models, i.e. singe Miura unit cell, two directly connected unit cells and ten stacked unit cells with rigid separators, and three types of excitations, i.e. harmonic end force, harmonic end displacement and impact load, are considered. The main findings are as follows. First, under the simple harmonic force or displacement excitation, the single DOF models have one resonant peak. Before the resonant peak, the vibration amplitudes at the response end and the excitation end are almost the same whereas after the resonant peak, the vibration amplitude at the response end is lower than that of the excitation end. By changing the crease stiffness, the resonant frequency of the single-DOF models can be altered. The higher the crease stiffness, the higher is the resonant frequency. Second, the multi-DOF model exhibits multiple resonant peaks under harmonic excitations, where the lowest resonant frequency has the highest peak. Increasing the number of layers can reduce the resonant frequency. Third, when the multi-DOF model is subjected to impact load, the magnitude of the impact wave decays quickly after the impact and finally reaches a steady state with a low average strain magnitude. When the crease stiffness is increased, the propagation of the impact wave becomes faster, whereas the maximum strain magnitude becomes smaller. Introducing different damping coefficients to the crease has no influence on the propagation speed of the impact waves but can accelerate the decay in the magnitude of the impact wave.
Because of the scale-independent nature of Miura origami structures, the models studied in this paper can be easily extended to bulky metamaterials with a large number of unit cells. Such metamaterials have great potential applications in aerospace structures. For example, such metamaterials can be used as the core material for lightweight sandwich beams and panels for vibration filtering and impact mitigation. As a preliminary study on the dynamics properties of Miura-based metamaterials, there are several limitations of the present work. First, our work considered the influences of crease stiffness and damping coefficient. The influences of different geometrical parameters of the unit cell have not been studied. Second, the impact responses of the multi-DOF structures subjected to multiple impact loads have not been investigated yet. Third, the facets were modelled as rigid plates. However, it is more realistic to model the facets as elastic plates. These limitations will be considered in our future work.
The financial supports from the National Science Foundation of China (No. 11602147) and the Shanghai Aerospace Science and Technology Innovation Fund (SAST) are gratefully acknowledged.
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