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A numerical study of the dynamic properties of Miura folded metamaterials

  • Dapeng Zhang
  • Bin Ji
  • Xiang ZhouEmail author
Original Paper
  • 84 Downloads

Abstract

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.

Keywords

Miura origami structure Harmonic excitation Impact Frequency response Wave propagation 

1 Introduction

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 [1]. For example, Zhou et al. [2] 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. [3] 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. [4] 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. [5] 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 [6] 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

Three types of origami models are investigated in this paper. Figure 1a shows the basic model that consists of a single Miura unit cell with two rigid plates attached to both ends, where the edges of the Miura unit cell can rotate and slide freely with respect to the rigid plates. Each facet of the Miura unit cell is a parallelogram with an interior angle of 60 degrees and equal side lengths of 100√2 mm. The initial height of the Miura unit cell is 200 mm. For simplicity, the facets of the Miura unit cell are modelled as rigid plates and the creases as elastic hinges with prescribed torsional stiffness. Specifically, we assume that the torsional stiffness per unit length of the four creases inside a single Miura unit cell (the red lines) have the same value of ka.
Fig. 1

The origami models. a The single Miura unit cell model. b The model consisting of two directly connected unit cells. c The multi-DOF model consisting of ten unit cells separated by rigid separators

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

To investigate the dynamic responses of the Miura-based origami metamaterials, three types of excitations applied in the vertical direction are investigated, namely harmonic force, harmonic displacement and impact force. In the case of the harmonic force excitation, the bottom plate is constrained and a harmonic force is applied to the top plate. The load F is defined as
$$F = F_{0} \times \sin \left( {t \times \omega } \right),$$
(1)
where F0 and ω denote the amplitude and frequency of the excitation force. In the case of the harmonic displacement excitation, the top plate is left unconstrained and a harmonic displacement is applied to the bottom plate. The displacement X satisfies
$$X = X_{0} \times \sin \left( {t \times \omega } \right),$$
(2)
where X0 and ω are the amplitude and frequency of the excitation displacement. For both the harmonic force and displacement cases, a wide spectrum of excitation frequencies ranging from 0.01 to 100 Hz is examined. For each excitation frequency, the simulation time lasts for 20 excitation periods. Finally, in the case of the impact load, the bottom plate is fixed and an impact force is applied to the top plate. The load is given by
$$F = F_{0} \times {\text{step}}\left( {t, 0, 0, 0.1, 1} \right) + F_{0} \times {\text{step}}\left( {t, 0.1, 0, 0.2, - 1} \right).$$
(3)

In ADAMS, Eq. (3) represents a single triangular wave impulse that lasts for 0.2 s with a magnitude of F0.

3 Results

In this section, the simulation results of the above-mentioned origami models with various crease parameters under different types of excitations are discussed. In the case of the harmonic force excitation, the main quantity of interest is the magnitude of the response force Fmrf measured at the fixed end, which is defined by
$$F_{\text{mrf}} = (F_{\hbox{max} } - F_{\hbox{min} } ),$$
(4)
where Fmax and Fmin are the maximum and minimum values of the response force. In the case of the displacement excitation, due to the high nonlinearity of the responses of the models, we employ the root mean square (RMS) value of the displacement to characterize the average vibrational energy. The displacement transmissibility Trms of the models is then defined by dividing the RMS value of the free end (i.e. the top plate) by the RMS value of the excitation (i.e. the bottom plate) as
$$T_{\text{rms}} = \frac{{X_{\text{rms}} }}{{Y_{\text{rms}} }} = \frac{{\sqrt {\left( {x_{1}^{2} + x_{2}^{2} + \cdots + x_{N}^{2} } \right)/N} }}{{\sqrt {\left( {y_{1}^{2} + y_{2}^{2} + \cdots + y_{N}^{2} } \right)/N} }},$$
(5)
where xi and yi\((i = 1,2, \ldots ,N)\) denote the displacements of the top and bottom plates at different time points, respectively, and N is the total number of displacement data collected over the simulation time. In the impact load case, to quantify the deformation of the origami structure, we define the strain of a single Miura unit cell as
$$\varepsilon = \Delta L/L,$$
(6)
where ΔL denotes the change in height, which is positive when reduction in height occurs, and L denotes the initial height of the unit cell. The deformation inside the entire structure is then plotted as a strain contour plot of all unit cells, where different strain levels are indicated by different colours.

3.1 Vibration responses under harmonic force excitations

First, the frequency responses of the single-unit-cell models with various crease stiffness ranging from 0.05 to 100 N mm/deg/mm under harmonic force excitations is analysed. The simulation results are shown in Fig. 2. According to the figure, the behaviour of the response force can be classified into three stages according to excitation frequency. In the first stage, the amplitude of the response force and that of the exciting force are nearly identical when the excitation frequency is low. In the second stage, the amplitude of the response force is much larger than that of the excitation force because of resonance. In the third stage where the excitation frequency continues to increase, the amplitude of response force decreases. It is shown that by changing the crease stiffness, different resonant frequencies can be obtained.
Fig. 2

The response forces of single-unit-cell models under harmonic force excitations. a The frequency response of the response force. b The time history of response force when ka = 0.1 and ω = 0.01 Hz. c The time history of response force when ka = 0.1 and ω = 2 Hz. d The time history of response force when ka = 0.1 and ω = 100 Hz

The simulation results of the two-unit-cell models under harmonic force excitations are shown in Fig. 3, where ka = kc = 0.1 N mm/deg/mm and kb varies from 0.05 to 100 N mm/deg/mm. The frequency responses of these models are quite similar to those of the single-unit-cell models except that the resonant frequencies for the two models are slightly different. This result is expectable because both models have single DOF of folding motion. Again, by changing the crease stiffness, different resonant frequencies can be obtained.
Fig. 3

The response forces of double-unit-cell models under harmonic force excitations. a The frequency response of the response force. b The time history of response force when kb = 0.1 and ω = 0.01 Hz. c The time history of response force when kb = 0.1 and ω = 1 Hz. d The time history of response force when kb = 0.1 and ω = 100 Hz

3.2 Vibration responses under harmonic displacement excitations

Figure 4 shows the displacement transmissibility results of the single-unit-cell models with various crease stiffness ranging from 0.05 to 100 N mm/deg/mm under harmonic displacement excitations. The overall trend of the displacement transmissibility is quite similar to that of the response force under harmonic force excitation. The displacement transmissibility is close to 1 at low excitation frequencies. When resonance occurs, there is a sudden increase in the magnitude of the displacement transmissibility. As the excitation frequency further increases, the displacement transmissibility drops to a level lower than 1. It is shown that the higher the crease stiffness, the higher is the resonant frequency. By changing the crease stiffness, a good low-frequency vibration isolation can be achieved.
Fig. 4

The displacement transmissibilities of single-unit-cell models under harmonic displacement excitations. a The frequency response of displacement transmissibility. b The time history of displacement transmissibility when ka = 0.1 and ω = 0.01 Hz. c The time history of displacement transmissibility when ka = 0.1 and ω = 2 Hz. d The time history of displacement transmissibility when ka = 0.1 and ω = 100 Hz

The displacement transmissibility results of the two-unit-cell models under the harmonic displacement excitation are shown in Fig. 5. The model parameters are the same as those for Fig. 3. Again, the results of the two-unit-cell models are quite similar to those of the single-unit-cell model. It can been seen that the resonant frequency increases with the increase in the crease stiffness, whereas the resonant displacement transmissibility does not change.
Fig. 5

The displacement transmissibilities of double-unit-cell models under harmonic displacement excitations. a The frequency response of displacement transmissibility. b The time history of displacement transmissibility when kb = 0.1 and ω = 0.01 Hz. c The time history of displacement transmissibility when kb = 0.1 and ω = 1 Hz. d The time history of displacement transmissibility when ka = 0.1 and ω = 100 Hz

Figure 6 shows the displacement transmissibility of the multi-DOF model with ten identical unit cells separated by separators. It is shown that the frequency response of displacement transmissibility has multiple resonant peaks. With the increase in the excitation frequency, the resonances at higher frequencies become out of phase and the resonant peaks become lower. It is also noted that as the excitation frequency increases, the response time starts lagging behind the excitation and the delay time increases.
Fig. 6

The displacement transmissibility of the multi-DOF model with ten unit cells under harmonic displacement excitations. a The frequency response of displacement transmissibility. b The time history of displacement transmissibility at the first peak. c The time history of displacement transmissibility at the second peak. d The time history of displacement transmissibility at the third peak

3.3 Impact responses

The impact studies are all carried out based on the multi-DOF model with ten unit cells. Figure 7 shows the impact response of the model with ka of 0.1 N mm/deg/mm. First, it is noted that the impact load generates two types of mechanical waves propagating through the model: one is the impact wave with large amplitude and slow propagation speed, which is indicated by the yellow (the compression wave) and dark blue (the tensile wave) patterns, and the other is the oscillatory wave with a much smaller oscillation amplitude and fast changing rate (see the light-blue grating in Fig. 7a). When the response reaches a steady state as shown in Fig. 7b, the amplitude of the strains tends to be more uniform and the peak strain rate is much smaller than that of the initial stage. It is also noted that during the initial stage, the largest strain magnitude occurs in the top unit cell which is subjected to the direct impact, and the strain level decays quickly from the second unit cells onwards. This is clear indication that the stacked Miura unit cell model has a good capability to impede the propagation of the impact through the model. It is interesting to find that except the top unit cell, the impact wave has a tensile forefront despite the compressive nature of the impact load. This is due to the fact that the Miura unit cell is a single-DOF structure and hence when the impact load causes compressive deformation of the top unit cell, the upper side of the unit cell moves downwards while the lower side of the unit cell moves upwards, which applies a tensile load to the rest unit cells.
Fig. 7

The impact response of the multi-DOF model with ten unit cells where F0 = 0.1, ka = 0.1. a Strain contour plot versus time during the initial 100 s. b Strain contour versus time at steady state. c Strain–time curves in layers 1 to 10

To explore the influence of the crease stiffness on the impact response of the multi-DOF model, we simulated two models with ka equal to 1 and 10 N mm/deg/mm, respectively. The simulation results are shown in Figs. 8 and 9, respectively. According to the results, the main differences among the models with different crease stiffness are in the maximum strain magnitude and the propagation speed of the impact wave. Specifically, with the increase in the crease stiffness, the propagation of the impact wave becomes faster whereas the maximum strain magnitude becomes smaller.
Fig. 8

The impact response of the multi-DOF model with ten unit cells where F0 = 0.1, ka = 1. a Strain contour plot versus time during the initial 20 s. b Strain contour versus time at steady state. c Strain–time curves in layers 1–10

Fig. 9

The impact response of the multi-DOF model with ten unit cells where F0 = 0.1, ka = 10. a Strain contour plot versus time during the initial 10 s. b Strain contour versus time at steady state. c Strain–time curves in layers 1–10

In the above analysis, all the creases are treated as idealized ones with no damping in the models. In real structures, however, damping always exists. To investigate the influence of damping on the impact response of the origami metamaterials, we simulated three models in which we introduced different damping coefficient c ranging from 0.001 to 0.1 to the creases. The simulation results are shown in Fig. 10. It is shown that the propagation speed of the impact waves remains the same for models with various damping coefficients. For models with higher damping coefficient, the magnitude of the impact wave decays faster, indicating that damping can effectively dissipate the impact energy and thus provide a better impact mitigation capacity.
Fig. 10

Strain contour plot versus time of the multi-DOF models with ten unit cells with damping: a c = 0, b c = 0.001 and c c = 0.01

4 Conclusion

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.

Notes

Acknowledgement

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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Copyright information

© Shanghai Jiao Tong University 2019

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Shanghai Key Laboratory of Spacecraft MechanismShanghaiChina
  3. 3.Aerospace System Engineering ShanghaiShanghaiChina

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