For addressing the limitations of traditional elastic metamaterials in opening wide bandgaps below 100 Hz, a tunable pneumatic metamaterial plate with airbag local resonators is proposed. Utilizing the characteristics of airbags, such as small volume, large load-bearing capacity, easy stiffness adjustment, and the ability to provide multi-directional restoring forces, a structured low-stiffness local resonator with a certain load-bearing capacity is designed. By varying the gauge pressure of the airbag, the bandgap can be moved toward lower frequencies, thereby achieving a broad low-frequency vibration suppression capability for various wave propagations. The low-frequency vibration bandgap characteristics of the tunable pneumatic metamaterial are analyzed and verified by applying the finite element method. The results illustrate that this tunable pneumatic metamaterial can attenuate bending waves in the range of 22–121 Hz by adjusting the air pressure. Moreover, increasing the gauge pressure will not only shift the complete bandgap toward lower frequencies but also significantly expand the bandwidth of the complete bandgap. For instance, increasing the gauge pressure from 0 to 50 kPa reduces the opening frequency of the complete bandgap from 36 to 22 Hz while enhancing the relative bandwidth from 0.52 to 0.85. Extending from this, a parametric study was conducted to examine the impact of the structural parameters of airbag-type resonant units on bandgap evolution, summarizing the general principles for achieving wide low-frequency bandgaps. Finally, the bandgap characteristics of the tunable pneumatic metamaterial are confirmed through the frequency response function of a finite periodic structure.

Metamaterials, characterized as artificial periodic composite structures, exhibit exceptional physical properties that can modify the propagation characteristics of electromagnetic, sound, and elastic waves in media.1–3 Elastic metamaterials are a subclass of metamaterials having negative equivalent parameters within the bandgap range. They possess the ability to control and dissipate elastic waves, a feat achieved through the mutual coupling interaction between the elastic waves and the metamaterial structures.4–9 A local resonance structure can be constituted by distributively mounting the local resonance units on or embedding them in a substructure. This formation induces a local resonant bandgap, which in turn suppresses vibration transmission within the bandgap range of the substrate structure. Consequently, this method efficaciously addresses the goals of vibration damping and noise reduction.10–15 

In the realm of mitigating radiated noise, particular emphasis is placed on the control of low-frequency vibrations below 200 Hz within the structure and equipment of ships.16 Local resonance-type elastic metamaterials hold promising potential for achieving low-frequency broadband vibration reduction in ship structures. Nonetheless, conventional elastic metamaterials fall short in making adaptive adjustments to suit the diverse and fluctuating operational conditions and environments encountered in practice.17 They are impeded by a range of issues, including a narrow bandgap range, limited variety, and challenges in opening low-frequency bandgaps below 100 Hz. Therefore, opening low-frequency bandgaps for various wave propagation types and actively modulating the location of these low-frequency bandgaps becomes a key challenge. Overcoming this challenge is essential for broadening the scope of metamaterial applications in the suppression of vibrations within ship structures.

The advent of smart materials in microstructures has been a notable development in recent years, allowing for the active control of metamaterials’ effective properties via external fields, such as electric,18 magnetic,19–21 thermal,22–25 and pressure fields.26,27 Compared to smart materials, such as piezoelectric materials28–33 and shape memory alloys,34–39 hyperelastic materials26,40–53 are prominently characterized by their ability to withstand large deformations and provide restoring forces in multiple directions simultaneously.54–58 By designing air chambers inside the material and utilizing pneumatic adjustments, the natural frequency of local resonant unit structures can be significantly reduced and controlled. Therefore, pneumatically driven elastic metamaterials based on hyperelastic materials (hereinafter referred to as pneumatic metamaterials) hold substantial potential for multi-directional vibration suppression and active control of low-frequency bandgaps. On a related note, Ning et al. engineered a tunable acoustic metamaterial to mitigate vibrations and noise in the ultra-low frequency range, utilizing annular airbags as the intermediary structure between the framework and counterweight. They successfully manipulated the dynamic response of Acoustic Metamaterials (AMMs) by adjusting the air pressure and temperature within the airbags.59 In a similar vein of innovation, Liu et al., drawing inspiration from the operational mechanism of pneumatic soft robots, conceived a novel acoustic switch. This structure, composed of a row of cylindrical airbags, enables the opening and closing of the phononic crystal bandgap by controlling the deformation of airbags through air pressure adjustments.60 Addressing the inefficiencies and sluggish real-time performance of traditionally mechanically loaded tunable negative stiffness mechanical metamaterials, Tan et al. put forth a pneumatically driven negative stiffness mechanical metamaterial with real-time tunable characteristics. This design can achieve negative stiffness properties with a higher energy absorption efficiency.61 Aiming at the difficulty of realizing broadband vibration damping of large mechanical equipment by traditional power vibration absorbers, Yin et al. proposed a new type of airbag dynamic vibration absorber. This absorber, capable of independently adjusting stiffness and damping by modulating the internal pressure of the airbag and the aperture of the throttle holes between the connecting pipelines, yielded favorable damping effects across a broader frequency range.62 Chen designed a kind of pneumatically actuated metamaterials with periodically arranged and interconnected holes. By adjusting the shape of the holes using pneumatic means, it was possible to dynamically modify the stiffness, Poisson’s ratio, and the critical condition necessary for pattern transformation.26 Narang proposed a soft metamaterial inspired by the laminar jamming phenomenon, which implied that a laminate of compliant strips would become strongly coupled via friction when subjected to a pressure gradient and resulted in the changes of its mechanical properties. In their research, an acrylic frame enclosing copy paper sheets and a vacuum tube were fabricated, and its mechanical properties could be easily controlled by regulating its inner pressure by inflation or deflation.49 Yuan developed a programmable 2D honeycomb structure using a pneumatic actuator and studied how the actuator’s local deformation affects the mechanical properties of the structure.46 Barnwell designed a phononic medium consisting of a two-dimensional periodic array of incompressible nonlinear elastic annular cylinders embedded in a homogeneous elastic host. The tunable bandgap of the periodic elastic composite structure is achieved by controlling the nonlinear elastic prestress.47 

The aforementioned research has proven that pneumatic actuation is an effective way to change the structure of metamaterials with specific purposes. Unlike traditional mechanical loading methods that require complex equipment, pneumatic actuation offers a convenient and easy approach that integrates with other structures, thereby expanding the scope of applications of tunable metamaterials. In the field of vibration control, research on certain damping and isolation devices driven by pneumatic methods, such as airbag isolators and airbag dynamic absorbers, has matured and has been successfully applied in engineering practice. However, research on the construction of local resonant pneumatic metamaterials (PMMs) using airbags is still relatively rare. The mechanisms of low-frequency bandgaps in pneumatic metamaterials and the regulation patterns of structural parameters and air pressure on low-frequency bandgaps still require further study. In addition, the engineering application of pneumatic metamaterials faces many challenges, including the complexity of designing and manufacturing micro-airbags, as well as the stability and reliability of the design of inflation and deflation systems.

In this article, a pneumatic metamaterial based on airbag local resonators is designed, which exhibits a maximum bending wave bandgap when unpressurized and attains a bandgap shift toward lower frequencies upon pressurization. The structure of this article is organized as follows: first, the mechanism of bandgap generation is discussed, based on an ideal infinite periodic structure. Subsequently, the influences of surface pressure and structural parameters on the low-frequency bandgap are scrutinized. Finally, the low-frequency bandgap characteristics of the finite periodic structure are affirmed through numerical simulation. The bending waves ranging from 22 to 121 Hz can be attenuated within the adjustment range of air pressure. This provides a theoretical basis and effective methods for the application of pneumatic metamaterials in low-frequency vibration damping and noise reduction.

The pneumatic metamaterial proposed in this paper consists of airbag local resonators arranged periodically on a thin plate, as shown in Fig. 1(a). The airbag local resonator consists of three parts: the plate, the airbag, and the weight. Figures 1(b) and 1(c) represent the dimensions of the unit structure with variables; the lattice constant of the periodic structure is a; the thickness of the thin plate is h; the height and outer diameter of the cylindrical airbag are h1 and r1, respectively; the height and outer diameter of the tungsten weight are h2 and r2, respectively; and the wall thickness of the airbag is t. The airbag has characteristics such as small volume, large load-bearing capacity, and easily adjustable stiffness, enabling the tuning of the natural frequency of the local resonator by adjusting the air pressure, thus achieving active control of the bandgap of the metamaterial.

FIG. 1.

Schematic diagram of the PMM plate and its units. (a) Finite-sized PMM structure consisting of 10 × 10 unit cells. (b) and (c) Geometrical configuration of the unit structure. It consists of three parts: plate, airbag, and weight. The airbag is a closed hollow structure filled with air. By changing the air pressure inside the airbag, the natural frequency of the local resonator can be changed so as to adjust the bandgap characteristics of the PMMs.

FIG. 1.

Schematic diagram of the PMM plate and its units. (a) Finite-sized PMM structure consisting of 10 × 10 unit cells. (b) and (c) Geometrical configuration of the unit structure. It consists of three parts: plate, airbag, and weight. The airbag is a closed hollow structure filled with air. By changing the air pressure inside the airbag, the natural frequency of the local resonator can be changed so as to adjust the bandgap characteristics of the PMMs.

Close modal

This article employs the finite element method for the calculation of band structure. Introducing Bloch wave vectors under periodic boundary conditions, allowing them along the irreducible Brillouin zone boundary of the unit structure, and solving the corresponding eigenvalues, the complete band structure diagram can be obtained. Details about the calculation method can be found in Refs. 63–65.

The change in relative pressure (gauge pressure) inside and outside the airbag causes variations in the volume and stiffness characteristics of the bag, thereby affecting the propagation of elastic waves in PMMs. Therefore, the analysis of bandgap characteristics of PMMs can be divided into the following two steps: (1) structural deformation and stiffness changes caused by pressure variations; (2) eigenvalue solutions when Bloch wave vectors sweep along the deformed unit cell’s irreducible Brillouin zone boundary. The numerical calculations in this paper are all based on the finite element software COMSOL. This section focuses on studying the impact of gauge pressure on the static characteristics of airbag local resonators.

Assuming an initial air pressure of 0.1 MPa inside and outside the airbag, the inflation expansion process of the airbag is simulated by applying a gradually increasing pressure to the inner surface of the airbag. The geometric structural parameters of PMMs are shown in Table I. The material of the thin plate is aluminum, and the material of the weight is tungsten, with specific parameters shown in Table II. The airbag material is neoprene, and its response can be captured using the Ogden model. The form of the Ogden strain energy potential is59 
U=i=1N2μiαi2(λ̄1αi+λ̄2αi+λ̄3αi3)+i=1N1Di(Jel1)2i,
(1)
where λ̄i is the deviatoric principal stretches and λ̄i=J1/3λi, in which λi are the principal stretches. The initial shear modulus and bulk modulus for the Ogden form are given by μ0=i=1Nμi and K0 = 2/D1. Here, N = 3, and the material parameters are listed in Table III.
TABLE I.

Geometrical parameters of local resonators (a for lattice constant; b for plate thickness; h1 for airbag height; h2 for the outer diameter of the weight; r1 and r2 for the outer diameters of the airbag and the weight, respectively; and t for airbag thickness. In this article, the outer diameters of the airbag and the weight are kept equal, i.e., r1 = r2).

a (mm)h (mm)h1 (mm)h2 (mm)r1(r2) (mm)t (mm)
100 2.5 24 40 
a (mm)h (mm)h1 (mm)h2 (mm)r1(r2) (mm)t (mm)
100 2.5 24 40 
TABLE II.

Local resonator material parameters. The material of the plate is aluminum, and the material of the weight is tungsten.

Materialρ (kg/m3)E (GPa)ν
Al/plate 2700 71 0.33 
W/mass 19 100 354.1 0.35 
Materialρ (kg/m3)E (GPa)ν
Al/plate 2700 71 0.33 
W/mass 19 100 354.1 0.35 
TABLE III.

Ogden parameters of the neoprene airbag.59 

iαiμi (MPa)Di
1.3 0.4095 0.2367 
5.0 0.003 
−2.0 0.01 
iαiμi (MPa)Di
1.3 0.4095 0.2367 
5.0 0.003 
−2.0 0.01 

In order to quantitatively describe the change of airbag volume with gauge pressure during the simulated inflation process, the data were dimensionless according to the initial state of the airbag. Figure 2 shows the relationship between the dimensionless volume change and dimensionless gauge pressure of the airbag, where the dimensionless parameters are defined as the ratio of the volume (or pressure) change to the initial state. It can be observed that an increase in gauge pressure causes deformation of the airbag, and the total volume increases with the increase in gauge pressure. For example, when the dimensionless gauge pressure increases from 0.25 to 0.5, the dimensionless volume change increases from 0.22 to 0.68. As the airbag expands, the spatial position of the tungsten weight also changes until a new equilibrium state is reached. In the following, we will utilize the large deformation capability of the airbag to achieve a reduction in the stiffness of the local resonator.

FIG. 2.

Relationships of the dimensionless gauge pressure (Δp/p0) and dimensionless change of cavity volume (ΔV/V0). The three marked points in the figure represent the relative deformation of the airbag at dimensionless gauge pressures of 0, 0.25, and 0.5, as indicated by the black arrows.

FIG. 2.

Relationships of the dimensionless gauge pressure (Δp/p0) and dimensionless change of cavity volume (ΔV/V0). The three marked points in the figure represent the relative deformation of the airbag at dimensionless gauge pressures of 0, 0.25, and 0.5, as indicated by the black arrows.

Close modal

The change in gauge pressure inside the airbag also causes a change in the stiffness characteristics of the local resonator. In this section, the interaction between air and the airbag during the vertical compression process is simulated by establishing the relationship between the internal pressure and volume of the airbag, and then, the variation law of the vertical stiffness of the local resonator with gauge pressure is studied.

Assuming a constant mass of air is filled in the sealed airbag, and treating the air as an ideal gas under adiabatic compression, according to Boyle’s law, there exists the following relationship between pressure and volume:
pp0=V0Vn,
(2)
where p0 and V0 are the initial internal pressure and volume of the airbag before compression, respectively, while p and V are the internal pressure and volume of the deformed airbag during the compression process, respectively, and n is the heat capacity ratio. According to Gauss’ law, the volume V of the deformed airbag can be calculated through the integration over the inner surface area of the airbag, with the expression as follows:66 
V=Ω1dΩ=Ωx00dΩ=Ωznzds,
(3)
where Ω denotes the bounded closed region enclosed by the sealed airbag, ∂Ω denotes the inner surface of the airbag, x and z are the coordinate components of the deformed configuration of the airbag, and nz is the z-component of the outward normal vector on the inner surface. Therefore, the internal pressure of the airbag can be calculated based on the volume of the deformed airbag. The interaction between the air and the airbag during vertical compression can be characterized by applying the gauge pressure Δp as a load to the inner surface of the airbag,
Δp=0.1×[(V0/V)1/41].
(4)
Following f = k · x, the equivalent vertical stiffness of the airbag is measured through the applied concentrated force f. x is the measured small displacement. Figure 3 shows the force–displacement relationship during vertical compression at different gauge pressures. In the uninflated state, the airbag stiffness shows a gradual softening characteristic, i.e., the stiffness decreases with airbag deformation. For example, at the initial stage of compression, the stiffness is about 48 N/mm, and at the end of the compression process, the stiffness is about 18 N/mm. In the inflated state, the vertical compression force and airbag displacement have a linear relationship, and as the inflation pressure increases, the stiffness of the airbag gradually decreases. When the gauge pressure increases from 25 to 50 kPa, the stiffness decreases from 44 to 38 N/mm. In the following, we will discuss the reason why pressurization leads to a reduction in airbag stiffness.
FIG. 3.

Displacement–force relationships of airbag at different gauge pressures. Airbag stiffness consists of two parts: rubber stiffness and compressed air stiffness. (a) Displacement–compression force relationships of airbag. (b) Displacement–support force relationships of rubber. (c) Displacement–support force relationships of compressed air. The red arrow shows the direction of the load.

FIG. 3.

Displacement–force relationships of airbag at different gauge pressures. Airbag stiffness consists of two parts: rubber stiffness and compressed air stiffness. (a) Displacement–compression force relationships of airbag. (b) Displacement–support force relationships of rubber. (c) Displacement–support force relationships of compressed air. The red arrow shows the direction of the load.

Close modal
The stiffness K of the airbag is mainly divided into the stiffness KA of compressed air and the stiffness KR of the rubber, with the general expression as follows:62,67,68
K=KA+KR,
(5)
KA=pdSedz+n(p+pa)VSe2,
(6)
KR(ωt)=kR(ωt)+jωtcR(ωt),
(7)
where p is the gauge pressure, Se is the effective area of the airbag, z is the height of the resonator, n is the heat capacity ratio of the ideal gas, pa is the atmospheric pressure, V is the volume of gas inside the airbag, and ωt is the excitation frequency.

In the study of large size high-pressure airbags, the stiffness of the rubber can be neglected, and the stiffness of the airbag is approximately equal to the stiffness of compressed air. From Eq. (6), it can be seen that the total stiffness increases with the increase in gauge pressure. When the size of the airbag and the gauge pressure are smaller, the situation will be different. Figures 3(b) and 3(c) show the support force–displacement relationship curves provided by rubber and compressed air, respectively. It can be seen that when the size of the airbag and the gauge pressure are smaller, the effect of bladder stiffness is more pronounced. During the pressurization process, the stiffness of the compressed air increases, while the stiffness of the rubber decreases. This is due to the expansion deformation of the airbag (as shown in Fig. 2), leading to a reduction in the supporting force on the tungsten weight. Due to the more pronounced effect of rubber stiffness, the overall stiffness actually decreases with the increase in gauge pressure. In the following, we will use this phenomenon to significantly regulate the bandgap characteristics of PMMs.

This section calculates the band structure of PMMs when not inflated (Δp = 0) and explains the mechanism of bandgap formation from the perspective of the interaction between plate waves and airbag-local resonant units.

The geometry and material parameters of PMMs are the same as in Sec. III A, and the calculated band structure is shown in Fig. 4. The three band branches originating at point Γ in the band diagram represent three basic plate wave modes, namely longitudinal wave (S0 mode), horizontal shear wave (SH0 mode), and bending wave (A0 mode). The band branches marked as S1, SH1, and A1 in the diagram represent the first-order octave modes of the basic plate wave modes.69–72  Figures 5(a)5(d) display the various orders of local resonance modes along with their displacement vector fields, corresponding to the translational and rotational vibrations of the tungsten weight, while the aluminum plate remains almost stationary. Figures 5(e) and 5(f) represent the Bloch mode shapes for the lower boundaries of band branches S1 (or SH1) and A1, respectively. They correspond to in-plane and out-of-plane vibrations of the plate, while the tungsten weight remains almost stationary. Due to the symmetry of the unit structure, modes A, D, and F are all repeated mode frequencies. In the following, we will discuss the role of the aforementioned modes in the formation of bandgaps.

FIG. 4.

Band structures of PMMs in the uninflated state. The inset map shows the first Brillouin zone (square area) and irreducible Brillouin zone (triangle ΓXM). A0, S0, and SH0 correspond to the energy-band branches of the three fundamental plate-wave modes, and A1, S1, SH1 are the corresponding first-order octave modes. A, B, C, and D is at Γ, and E and F is at T, which correspond to the boundary frequencies of the bandgap for different kinds of wave propagation. Frequencies A–E correspond to the in-plane bandgap, and C–F correspond to out-plane bandgap, and their overlap region (C–E) is the complete bandgap.

FIG. 4.

Band structures of PMMs in the uninflated state. The inset map shows the first Brillouin zone (square area) and irreducible Brillouin zone (triangle ΓXM). A0, S0, and SH0 correspond to the energy-band branches of the three fundamental plate-wave modes, and A1, S1, SH1 are the corresponding first-order octave modes. A, B, C, and D is at Γ, and E and F is at T, which correspond to the boundary frequencies of the bandgap for different kinds of wave propagation. Frequencies A–E correspond to the in-plane bandgap, and C–F correspond to out-plane bandgap, and their overlap region (C–E) is the complete bandgap.

Close modal
FIG. 5.

Bloch mode shapes of the selected points in Fig. 4. A and D correspond to the translational modes of the weight, B and D correspond to the rotational modes of the weight, and E and F correspond to the translational modes of the plate. Because of the symmetry of unit cell, there are repeated frequency modes (at A, D, and F).

FIG. 5.

Bloch mode shapes of the selected points in Fig. 4. A and D correspond to the translational modes of the weight, B and D correspond to the rotational modes of the weight, and E and F correspond to the translational modes of the plate. Because of the symmetry of unit cell, there are repeated frequency modes (at A, D, and F).

Close modal

According to the wave theory in plates, the polarization directions of longitudinal waves, horizontal shear waves, and bending waves are x, y, and z, respectively. The repeated mode A (16.4 Hz) corresponds to the longitudinal and transverse vibration modes of tungsten weight within the plane of the plate, and they have strong coupling effects with the in-plane longitudinal and horizontal shear waves with the same polarization directions, respectively. When their energy bands meet, energy level repulsion occurs and they truncate each other, thereby opening up longitudinal and horizontal shear wave bandgaps above the flat band A. The cutoff frequencies of the bandgaps are determined by the lower boundary frequencies of the S1 and SH1 band branches. Figure 5(e) only shows the Bloch mode shapes at the cutoff frequency of the longitudinal wave bandgap. From the figure, it can be seen that the plate is vibrating in the x-direction, indicating that longitudinal waves can propagate smoothly through the plate. It should be noted that, in the low-frequency range studied, the energy band branches corresponding to longitudinal and horizontal shear waves approximately coincide. Therefore, the low-frequency bandgaps of these two types of in-plane waves nearly coincide. This article refers to the low-frequency bandgaps of longitudinal and horizontal shear waves as in-plane bandgaps, with the relative bandwidth defined as the ratio of bandgap width to central frequency. Therefore, the in-plane bandgap is from 16.4 to 61.4 Hz (A–E), and its relative bandwidth is 1.16. Mode B (31.3 Hz) corresponds to the torsional vibration of tungsten weight within the plane of the plate, with the vibration components along the x and y axes within the plane canceling each other out. This implies that no reactive force is applied to the plate structure, and thus, this vibration mode makes no contribution to the formation of the bandgap.

For plate structures, bending vibration is the most common vibration mode; therefore, controlling bending vibration in plates is of great significance for vibration damping. Local resonance mode C (36.8 Hz) corresponds to the vertical vibration mode of the tungsten weight, which can form a strong coupling with the bending waves propagating in the plate, thus forming a bending wave bandgap above the flat band C. Similar to the cases of longitudinal and horizontal shear waves, the cutoff frequency of the bending wave bandgap is determined by the lower boundary frequency of the A1 band branch, with its corresponding Bloch mode shapes (i.e., mode F) shown in Fig. 5(f). Therefore, the out-of-plane bandgap (C–F) ranges from 36.8 to 121.8 Hz, with a relative bandwidth of 1.07. The overlapping area of the out-of-plane and in-plane bandgaps constitutes the complete bandgap, within which no form of wave can propagate in the plate. The complete bandgap (C–E) ranges from 36.8 to 61.4 Hz, with a relative bandwidth of 0.50.

In summary, PMMs have a local resonance bandgap in the low frequency range. The creation of the bandgap is due to the coupling of local resonance modes with the propagating wave modes in the plate. When the polarization direction of a propagating wave mode in the plate is the same as that of a local resonance mode, the corresponding local resonance mode will be excited, thus producing a strong coupling effect between the two. The frequency range formed by the local resonance mode and the lower boundary of the first-order octave mode of the plate wave is where the low-frequency bandgap is located.

As analyzed earlier, the lower boundary frequency of the bandgap is determined by the frequency of the locally resonant modes. In each order of locally resonant modes, the equivalent mass is mainly provided by the tungsten weight, while the equivalent stiffness is mainly provided by the airbags. Thus, the position of the bandgap can be adjusted by changing the physical parameters of the corresponding components. This section focuses on the regulation law of the gauge pressure on the low-frequency bandgaps of PMMs, and the impact of structural parameters on bandgap characteristics will be discussed in Sec. IV C.

Figure 6 shows the band structures of two types of PMMs (Δp = 25 kPa; Δp = 50 kPa) when inflated, with the geometric and material parameters consistent with those in Sec. III A. The results indicate that both types of PMMs exhibit an in-plane bandgap (A–E), an out-of-plane bandgap (C–F), and a complete bandgap (C–E), but the positions of these three types of bandgaps differ. This is because increasing the gauge pressure changes the frequency of the airbag’s locally resonant mode and therefore changes the position of the bandgap. For instance, when the gauge pressure increases from 25 to 50 kPa, the vertical vibration mode of the airbag decreases from 28.7 to 22.5 Hz, thus opening a low-frequency bending wave bandgap at 22.5 Hz.

FIG. 6.

Band structures of PMMs in the inflated state. (a) Δp = 25 kPa. (b) Δp = 50 kPa. Implications of symbols in this figure are the same as those in Fig. 4. The mode shape of each mode is the same as Fig. 5.

FIG. 6.

Band structures of PMMs in the inflated state. (a) Δp = 25 kPa. (b) Δp = 50 kPa. Implications of symbols in this figure are the same as those in Fig. 4. The mode shape of each mode is the same as Fig. 5.

Close modal

The trends of the upper and lower boundary frequencies and the total width of the bandgap with the change of gauge pressure are shown in Fig. 7, where the width of the bandgap is represented by the area enclosed by the curves. In representing the bandgap width, the flat bands corresponding to the locally resonant modes B and D are not shown in the figure because they produce only very narrow passbands. From Fig. 7, it can be seen that as the gauge pressure increases, the modes C and F associated with the boundary of the bending wave bandgap move to lower frequencies, and hence, the bending wave bandgap moves to lower frequencies. This is consistent with the relationship between the vertical stiffness of the airbag local resonator and the gauge pressure in Sec. III. It can also be seen that the bandwidth of the complete bandgap increases as the gauge pressure increases; for example, the relative bandwidth of the complete bandgap increases from 0.52 to 0.85 as the gauge pressure increases from 0 to 50 kPa.

FIG. 7.

Evolution of mode frequencies and bandgaps in the band structure vs gauge pressure in the airbag. C and F represent the upper and lower boundary frequencies of the out-of-plane bandgap, and A and E represent the upper and lower boundary frequencies of the in-plane bandgap. The width of the bandgap is represented by the area enclosed by the curve, and the overlapping part of the out-of-plane bandgap and the in-plane bandgap is the complete bandgap.

FIG. 7.

Evolution of mode frequencies and bandgaps in the band structure vs gauge pressure in the airbag. C and F represent the upper and lower boundary frequencies of the out-of-plane bandgap, and A and E represent the upper and lower boundary frequencies of the in-plane bandgap. The width of the bandgap is represented by the area enclosed by the curve, and the overlapping part of the out-of-plane bandgap and the in-plane bandgap is the complete bandgap.

Close modal

This is due to the different trends of each locally resonant mode with increasing gauge pressure. When the gauge pressure increases, the opening frequency of the complete bandgap (mode C) shifts to low frequencies while the cutoff frequency (mode E) shifts to high frequencies. Therefore, as the gauge pressure increases, the complete bandgap shifts to lower frequencies while its bandwidth increases.

This section studies the effect of airbag structural parameters, such as airbag height h1, airbag diameter r1, airbag thickness t, and weight height h2, on the low-frequency bandgap range. The default baseline values for the geometric parameters are given in Table I. The effect of each of the variables involved in the above structural parameters on the bandgap characteristics is investigated in the inflated state (Δp = 25 kPa), and the results are shown in Fig. 8. The symbols in the figure represent the boundary frequencies of the bandgap, and the area enclosed by the curve represents the bandwidth, the specific meaning of which is consistent with the description in Fig. 7.

FIG. 8.

Evolution of mode frequencies and bandgaps in the band structure vs geometric parameters. (a) Effect of airbag height h1 on the bandgap. (b) Effect of weight height h2 on the bandgap. (c) Effect of airbag outer diameter r1(r2) on the bandgap. (d) Effect of airbag thickness t on the bandgap. In all cases, the gauge pressure is 30 kPa and the outer diameters of the airbag and the weight are always equal, i.e., r1 = r2. Therefore, when changing the outer diameter of the airbag, the height of the ballast, h1, is adjusted accordingly, with the aim of keeping the mass of the weight constant. The default baseline values for the geometric parameters are given in Table I. Implications of symbols in this figure are the same as those in Fig. 7.

FIG. 8.

Evolution of mode frequencies and bandgaps in the band structure vs geometric parameters. (a) Effect of airbag height h1 on the bandgap. (b) Effect of weight height h2 on the bandgap. (c) Effect of airbag outer diameter r1(r2) on the bandgap. (d) Effect of airbag thickness t on the bandgap. In all cases, the gauge pressure is 30 kPa and the outer diameters of the airbag and the weight are always equal, i.e., r1 = r2. Therefore, when changing the outer diameter of the airbag, the height of the ballast, h1, is adjusted accordingly, with the aim of keeping the mass of the weight constant. The default baseline values for the geometric parameters are given in Table I. Implications of symbols in this figure are the same as those in Fig. 7.

Close modal

Comparing the results shown in Fig. 8, it can be seen that for airbags, in order to reduce the opening frequency (Mode C) of the bending wave bandgap, airbags with smaller diameters and thicknesses but larger heights can be selected. It should be noted that for a bending wave bandgap, a lower bandgap opening frequency implies a relatively narrow bandwidth. For the tungsten weight, increasing its height is equivalent to increasing the equivalent mass of the resonance unit, which also shifts the mode C to lower frequencies while increasing the bending wave bandwidth. This is due to the different sensitivities of the modal frequencies to changes in the equivalent mass. As the height of the weight increases, mode C moves to lower frequencies, while mode F (bending wave cutoff frequency) does not change significantly.

Therefore, to obtain a wider bandgap in the low-frequency range, the best method is to increase the equivalent mass of the resonant unit, such as increasing the height of the weight or choosing a material with a higher density. Next is to reduce the equivalent stiffness of the resonant unit, such as reducing the thickness and diameter of the airbag, or increasing its height. However, the latter method will cause a reduction in the width of the bandgap.

To validate the low-frequency bandgap characteristics of PMMs, the finite element method was used to calculate the frequency response functions of two types of finite PMMs (Δp = 0 kPa; Δp = 50 kPa) composed of 10 × 10 units. The materials and structural dimensions of the plate and local resonator are consistent with the example in Sec. III. The finite element model is shown in Fig. 9, where a simple harmonic excitation is applied at point P1 vertically to the plate surface, and the z-direction acceleration at point P2 is captured for calculating the frequency response function of the structure. Assuming that the excitation and steady-state response of the system are represented by Feiωt and Aeiωt, respectively, the amplitude of the frequency response function of the structure is given by the following equation:
H=20lgAF(dB),
(8)
where A represents the complex amplitude at the vibration pickup point and F represents the complex force amplitude at the excitation point.
FIG. 9.

Finite-sized PMM structure consisting of 10 × 10 unit cells. P1 and P2 denote the applied positions of load and the measure position of output, respectively.

FIG. 9.

Finite-sized PMM structure consisting of 10 × 10 unit cells. P1 and P2 denote the applied positions of load and the measure position of output, respectively.

Close modal

Figure 10(a) presents the frequency response function of the finite PMMs at Δp = 0, showing a significant attenuation within the 36–121 Hz range, indicating a vibration bandgap. A resonant peak appears near 50 Hz, which is consistent with the narrow passband in the bandgap range shown in Fig. 4. It is noteworthy that by altering the geometry, such as increasing the thickness and diameter of the airbag or reducing its height, the resonance peak within the passband range can be shifted to higher frequencies. This part generally involves multi-parameter optimization work, which we will explore in future research. The resonance peaks at 36 Hz (P1) and 50 Hz (P2) correspond to resonance flat bands C and D, respectively, in the energy band structure of Fig. 4. The resonance peak P3 at the cutoff frequency of the bandgap and other resonance peaks within other passband ranges are determined by the resonance modes of the plate. Around the resonance peaks P1, P2, and P3, there are some minimal points in the frequency response function, such as T1, T2, and T3 in the figure, corresponding to the anti-resonance points under different modes. At the anti-resonance point, the reaction force generated by the resonance unit is exactly opposite to the vibration of the plate, and the plate vibration is suppressed to the maximum, resulting in a minimum of vibration. It can also be observed from the figure that there is no vibrational attenuation in the in-plane bandgap range, which is due to the fact that the point excitation in the z-direction mainly excites the in-plate bending wave. Figure 10(b) shows the frequency response function of finite PMMs at Δp = 50 kPa. A comparison of the results at the two gauge pressures shows that the PMMs possess the largest bending wave bandgap without inflation, and the bandgap shifts to lower frequencies with inflation, the bending wave bandgap range decreases, but the complete bandgap range increases.

FIG. 10.

FRF in the finite-sized PMM structures consisting of 10 × 10 unit cells. (a) Δp = 0 kPa. (b) Δp = 50 kPa. The color blocks represent the extent of the bandgap, with gray representing the complete bandgap interval, gray plus yellow representing the in-plane bandgap, and gray plus purple representing the out-of-plane bandgap.

FIG. 10.

FRF in the finite-sized PMM structures consisting of 10 × 10 unit cells. (a) Δp = 0 kPa. (b) Δp = 50 kPa. The color blocks represent the extent of the bandgap, with gray representing the complete bandgap interval, gray plus yellow representing the in-plane bandgap, and gray plus purple representing the out-of-plane bandgap.

Close modal

In conclusion, the low-frequency bandgap characteristics of the pneumatic metamaterial plate are verified by finite element simulation calculations. Theoretically, as long as the low-frequency characteristic line spectrum of the equipment falls within the range of the complete bandgap, and actively changes the range of the bandgap according to the change of operating conditions, it is possible to achieve low-frequency broadband vibration suppression for various wave propagations simultaneously.

A pneumatically driven metamaterial plate composed of airbag local resonators was designed in this study, enabling the shifting of the bandgap to lower frequencies through air pressure adjustment. The band structure of PMMs under different gauge pressures and geometric parameters was computed using the finite element method, analyzing the formation and regulation mechanisms of the bandgap. The conclusions are as follows:

  • The bandgap characteristics of pneumatic metamaterials can be tuned by air pressure regulation. The mechanism underlining bandgap control is the alteration of the natural frequency of the airbag local resonator by adjusting the air pressure, thereby realizing active bandgap control.

  • With the increase in gauge pressure, the bandgap of bending waves moves toward lower frequencies, but accompanied by a decrease in bandwidth. Bending waves spanning from 22 to 121 Hz can be attenuated effectively over the air pressure adjustment range (0–50 kPa).

  • Increasing the air pressure not only decreases the opening frequency of the bandgap but also significantly broadens the complete bandgap. Specifically, an increment in gauge pressure from 0 to 50 kPa leads to a drop in the opening frequency of the complete bandgap from 36 to 22 Hz, while the relative bandwidth increases from 0.52 to 0.85.

The resonant components composed of airbags and weight can be further integrated with other devices, thus designing multi-mechanism local resonators with abundant control methods. For example, piezoelectric materials can be introduced into the internal space of the airbags, thus constructing pneumatically and piezoelectrically driven micro-electromechanical local resonators. Moreover, resonant components can be embedded into honeycomb structures, thereby developing structures with integrated lightweight, load-bearing, and functional features. It is worth mentioning that pneumatic metamaterials have limited response speeds to external stimuli. As pneumatic tuning requires the flow of gas within air chambers, the response speed of gas flow is generally slower compared to electromagnetic stimuli, such as electric or magnetic fields. This may limit the practical effectiveness of pneumatic metamaterials in applications that require rapid responses.

The authors acknowledge the financial support provided by National Major Special Basic Research (Grant No. J2019-II-0013-0033) and Basic Strengthening Plan Key Basic Research (Grant No. 2020-JCJQ-ZD-204).

The authors have no conflicts to disclose.

Yingjie Zhang: Conceptualization (equal); Data curation (equal); Investigation (equal); Software (equal); Writing – original draft (equal). Wei Xu: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Zhimin Chen: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Junqiang Fu: Visualization (equal); Writing – review & editing (equal). Lihang Yin: Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
Y.
Huang
,
J.
Li
,
W.
Chen
, and
R.
Bao
, “
Tunable bandgaps in soft phononic plates with spring-mass-like resonators
,”
Int. J. Mech. Sci.
151
,
300
313
(
2019
).
2.
P.
Jiao
,
J.
Mueller
,
J. R.
Raney
,
X. Y.
Zheng
, and
A. H.
Alavi
, “
Mechanical metamaterials and beyond
,”
Nat. Commun.
14
,
6004
(
2023
).
3.
J.
Qi
,
Z.
Chen
,
P.
Jiang
,
W.
Hu
,
Y.
Wang
,
Z.
Zhao
et al, “
Recent progress in active mechanical metamaterials and construction principles
,”
Adv. Sci.
9
,
2102662
(
2022
).
4.
K.
Wang
,
J.
Zhou
,
C.
Cai
,
D .
Xu
, and
G.
Wen
, “
Review of low-frequency elastic wave metamaterials
,”
Chin. J. Theor. Appl. Mech.
54
,
2678
2694
(
2022
).
5.
H.
Peng
,
P.
Frank Pai
, and
H.
Deng
, “
Acoustic multi-stopband metamaterial plates design for broadband elastic wave absorption and vibration suppression
,”
Int. J. Mech. Sci.
103
,
104
114
(
2015
).
6.
H.
Peng
and
P.
Frank Pai
, “
Acoustic metamaterial plates for elastic wave absorption and structural vibration suppression
,”
Int. J. Mech. Sci.
89
,
350
361
(
2014
).
7.
Z. H.
He
,
Y. Z.
Wang
, and
Y. S.
Wang
, “
Active feedback control of effective mass density and sound transmission on elastic wave metamaterials
,”
Int. J. Mech. Sci.
195
,
106221
(
2021
).
8.
W.
Jiang
,
M.
Yin
,
Q.
Liao
,
L.
Xie
, and
G.
Yin
, “
Three-dimensional single-phase elastic metamaterial for low-frequency and broadband vibration mitigation
,”
Int. J. Mech. Sci.
190
,
106023
(
2021
).
9.
X.
Xu
,
M. V.
Barnhart
,
X.
Fang
,
J.
Wen
,
Y.
Chen
, and
G.
Huang
, “
A nonlinear dissipative elastic metamaterial for broadband wave mitigation
,”
Int. J. Mech. Sci.
164
,
105159
(
2019
).
10.
Z.
Li
,
H.
Hu
, and
X.
Wang
, “
A new two-dimensional elastic metamaterial system with multiple local resonances
,”
Int. J. Mech. Sci.
149
,
273
284
(
2018
).
11.
A.
Nateghi
,
L.
Van Belle
,
C.
Claeys
,
E.
Deckers
,
B.
Pluymers
, and
W.
Desmet
, “
Wave propagation in locally resonant cylindrically curved metamaterial panels
,”
Int. J. Mech. Sci.
127
,
73
90
(
2017
).
12.
J.
Zhou
,
L.
Dou
,
K.
Wang
,
D.
Xu
, and
H.
Ouyang
, “
A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams
,”
Nonlinear Dyn.
96
,
647
665
(
2019
).
13.
Q.
Lin
,
J.
Zhou
,
H.
Pan
,
D.
Xu
, and
G.
Wen
, “
Numerical and experimental investigations on tunable low-frequency locally resonant metamaterials
,”
Acta Mech. Solida Sin.
34
,
612
623
(
2021
).
14.
S.
Taniker
and
C.
Yilmaz
, “
Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges
,”
Int. J. Solids Struct.
106–107
,
129
138
(
2017
).
15.
N. M. M.
Frandsen
,
O. R.
Bilal
,
J. S.
Jensen
, and
M. I.
Hussein
, “
Inertial amplification of continuous structures: Large band gaps from small masses
,”
J. Appl. Phys.
119
,
124902
(
2016
).
16.
X. L.
Xie
,
G. M.
Dong
,
F.
Lin
,
Z. Y.
Zhang
, and
X. Y.
Wen
, “
Development of vibration control technologies for marine power and gearing systems
,”
Chin. J. Eng. Sci.
24
,
193
202
(
2022
).
17.
K. J.
Yi
,
Y. Y.
Chen
,
R.
Zhu
, and
G. L.
Huang
, “
Electromechanical active metamaterials and their applications in controlling elastic wave propagation
,”
Chin. Sci. Bull.
67
,
1290
1304
(
2022
).
18.
C.
El Helou
,
P. R.
Buskohl
,
C. E.
Tabor
, and
R. L.
Harne
, “
Digital logic gates in soft, conductive mechanical metamaterials
,”
Nat. Commun.
12
,
1633
(
2021
).
19.
J. A.
Jackson
,
M. C.
Messner
,
N. A.
Dudukovic
,
W. L.
Smith
,
L.
Bekker
et al, “
Field responsive mechanical metamaterials
,”
Sci. Adv.
4
,
eaau6419
(
2018
).
20.
H.
Gu
,
Q.
Boehler
,
H.
Cui
,
E.
Secchi
,
G.
Savorana
et al, “
Magnetic cilia carpets with programmable metachronal waves
,”
Nat. Commun.
11
,
2637
(
2020
).
21.
A.
Grant
and
C.
O’Dwyer
, “
Thickness-based dispersion in opal photonic crystals
,”
ECS Meeting Abstr.
MA2022-02
,
1304
(
2022
).
22.
C.
Yuan
,
X.
Mu
,
C. K.
Dunn
,
J.
Haidar
,
T.
Wang
, and
H.
Jerry Qi
, “
Thermomechanically triggered two-stage pattern switching of 2D lattices for adaptive structures
,”
Adv. Funct. Mater.
28
,
1705727
(
2018
).
23.
C.
Yang
,
M.
Boorugu
,
A.
Dopp
,
J.
Ren
,
R.
Martin
et al, “
4D printing reconfigurable, deployable and mechanically tunable metamaterials
,”
Mater. Horiz.
6
,
1244
1250
(
2019
).
24.
W.
Matysiak
,
T.
Tański
, and
W. M.
Smok
, “
Morphology and structure characterization of crystalline SnO2 1D nanostructures
,”
Photonics Lett. Poland
12
,
70
(
2020
).
25.
D.
Yu
,
G.
Hu
,
W.
Ding
,
Y.
Yang
, and
J.
Hong
, “
Zero-thermal-expansion metamaterial with broadband vibration suppression
,”
Int. J. Mech. Sci.
258
,
108590
(
2023
).
26.
Y.
Chen
and
L.
Jin
, “
Geometric role in designing pneumatically actuated pattern-transforming metamaterials
,”
Extreme Mech. Lett.
23
,
55
66
(
2018
).
27.
A.
Rafsanjani
,
L.
Jin
,
B.
Deng
, and
K.
Bertoldi
, “
Propagation of pop ups in kirigami shells
,”
Proc. Natl. Acad. Sci. U. S. A.
116
,
8200
8205
(
2019
).
28.
L. F.
Lin
,
Z. Q.
Lu
,
L.
Zhao
,
Y. S.
Zheng
,
H.
Ding
, and
L. Q.
Chen
, “
Vibration isolation of mechatronic metamaterial beam with resonant piezoelectric shunting
,”
Int. J. Mech. Sci.
254
,
108448
(
2023
).
29.
G.
Wang
,
J.
Cheng
,
J.
Chen
, and
Y.
He
, “
Multi-resonant piezoelectric shunting induced by digital controllers for subwavelength elastic wave attenuation in smart metamaterial
,”
Smart Mater. Struct.
26
,
025031
(
2017
).
30.
K.
Yi
,
G.
Matten
,
M.
Ouisse
,
E.
Sadoulet-Reboul
,
M.
Collet
, and
G.
Chevallier
, “
Programmable metamaterials with digital synthetic impedance circuits for vibration control
,”
Smart Mater. Struct.
29
,
035005
(
2020
).
31.
X.
Li
,
Y.
Chen
,
G.
Hu
, and
G.
Huang
, “
A self-adaptive metamaterial beam with digitally controlled resonators for subwavelength broadband flexural wave attenuation
,”
Smart Mater. Struct.
27
,
045015
(
2018
).
32.
M. I. N.
Rosa
and
M.
Ruzzene
, “
Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions
,”
New J. Phys.
22
,
053004
(
2020
).
33.
A.
Sasmal
,
N.
Geib
,
B. I.
Popa
, and
K.
Grosh
, “
Broadband nonreciprocal linear acoustics through a non-local active metamaterial
,”
New J. Phys.
22
,
063010
(
2020
).
34.
J.
Ma
,
B.
Franco
,
G.
Tapia
,
K.
Karayagiz
,
L.
Johnson
et al, “
Spatial control of functional response in 4D-printed active metallic structures
,”
Sci. Rep.
7
,
46707
(
2017
).
35.
S.
Akbari
,
A. H.
Sakhaei
,
S.
Panjwani
,
K.
Kowsari
,
A.
Serjouei
, and
Q.
Ge
, “
Multimaterial 3D printed soft actuators powered by shape memory alloy wires
,”
Sens. Actuators, A
290
,
177
189
(
2019
).
36.
X.
Huang
,
K.
Kumar
,
M. K.
Jawed
,
A.
Mohammadi Nasab
,
Z.
Ye
et al, “
Highly dynamic shape memory alloy actuator for fast moving soft robots
,”
Adv. Mater. Technol.
4
,
1800540
(
2019
).
37.
J. H.
Lee
,
Y. S.
Chung
, and
H.
Rodrigue
, “
Long shape memory alloy tendon-based soft robotic actuators and implementation as a soft gripper
,”
Sci. Rep.
9
,
11251
(
2019
).
38.
Z.
Zhao
,
K.
Wang
,
L.
Zhang
,
L. C.
Wang
,
W. L.
Song
, and
D.
Fang
, “
Stiff reconfigurable polygons for smart connecters and deployable structures
,”
Int. J. Mech. Sci.
161–162
,
105052
(
2019
).
39.
N.
Yang
,
M.
Zhang
, and
R.
Zhu
, “
3D kirigami metamaterials with coded thermal expansion properties
,”
Extreme Mech. Lett.
40
,
100912
(
2020
).
40.
P.
Polygerinos
,
S.
Lyne
,
W.
Zheng
,
L. F.
Nicolini
,
B.
Mosadegh
et al, “
Towards a soft pneumatic glove for hand rehabilitation
,” in
2013 IEEE/RSJ International Conference on Intelligent Robots and System
(
IEEE
,
2013
), pp.
1512
1517
.
41.
C.
Laschi
,
M.
Cianchetti
,
B.
Mazzolai
,
L.
Margheri
,
M.
Follador
, and
P.
Dario
, “
Soft robot arm inspired by the octopus
,”
Adv. Rob.
26
,
709
727
(
2012
).
42.
H. T.
Lin
,
G. G.
Leisk
, and
B.
Trimmer
, “
GoQBot: A caterpillar-inspired soft-bodied rolling robot
,”
Bioinspiration Biomimetics
6
,
026007
(
2011
).
43.
S.
Seok
,
C. D.
Onal
,
K. J.
Cho
,
R. J.
Wood
,
D.
Rus
, and
S.
Kim
, “
Meshworm: A peristaltic soft robot with antagonistic nickel titanium coil actuators
,”
IEEE/ASME Trans. Mechatron.
18
,
1485
1497
(
2013
).
44.
N. G.
Cheng
,
M. B.
Lobovsky
,
S. J.
Keating
,
A. M.
Setapen
,
K. I.
Gero
et al, “
Design and analysis of a robust, low-cost, highly articulated manipulator enabled by jamming of granular media
,” in
2012 IEEE International Conference on Robotics and Automation
(
IEEE
,
2012
), pp.
4328
4333
.
45.
T.
Wang
,
L.
Ge
, and
G.
Gu
, “
Programmable design of soft pneu-net actuators with oblique chambers can generate coupled bending and twisting motions
,”
Sens. Actuators, A
271
,
131
138
(
2018
).
46.
Z.
Yuan
and
J.
Ju
, “
Tunable triangular cellular structures by pneumatic control of dual channel actuators
,” in
Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition
,
2017
.
47.
E. G.
Barnwell
,
W. J.
Parnell
, and
I.
David Abrahams
, “
Antiplane elastic wave propagation in pre-stressed periodic structures; tuning, band gap switching and invariance
,”
Wave Motion
63
,
98
110
(
2016
).
48.
A.
Rafsanjani
,
Y.
Zhang
,
B.
Liu
,
S. M.
Rubinstein
, and
K.
Bertoldi
, “
Kirigami skins make a simple soft actuator crawl
,”
Sci. Rob.
3
,
eaar7555
(
2018
).
49.
Y. S.
Narang
,
J. J.
Vlassak
, and
R. D.
Howe
, “
Mechanically versatile soft machines through laminar jamming
,”
Adv. Funct. Mater.
28
,
1707136
(
2018
).
50.
Q.
Pan
,
S.
Chen
,
F.
Chen
, and
X.
Zhu
, “
Programmable soft bending actuators with auxetic metamaterials
,”
Sci. China Technol. Sci.
63
,
2518
2526
(
2020
).
51.
X.
Tan
,
B.
Wang
,
S.
Zhu
,
S.
Chen
,
K.
Yao
et al, “
Novel multidirectional negative stiffness mechanical metamaterials
,”
Smart Mater. Struct.
29
,
015037
(
2020
).
52.
S.
Li
,
J. J.
Stampfli
,
H. J.
Xu
,
E.
Malkin
,
E. V.
Diaz
et al, “
A vacuum-driven origami ‘magic-ball’ soft gripper
,” in
2019 International Conference on Robotics and Automation (ICRA)
(
IEEE
,
2019
), pp.
7401
7408
.
53.
W.
Kim
,
J.
Byun
,
J. K.
Kim
,
W. Y.
Choi
,
K.
Jakobsen
et al, “
Bioinspired dual-morphing stretchable origami
,”
Sci. Rob.
4
,
eaay3493
(
2019
).
54.
S.
Ning
,
F.
Yang
,
C.
Luo
,
Z.
Liu
, and
Z.
Zhuang
, “
Low-frequency tunable locally resonant band gaps in acoustic metamaterials through large deformation
,”
Extreme Mech. Lett.
35
,
100623
(
2020
).
55.
F. M.
Foong
,
C. K.
Thein
,
B. L.
Ooi
, and
D.
Yurchenko
, “
Increased power output of an electromagnetic vibration energy harvester through anti-phase resonance
,”
Mech. Syst. Signal Process.
116
,
129
145
(
2019
).
56.
Y. Y.
Zhang
,
N. S.
Gao
, and
J. H.
Wu
, “
New mechanism of tunable broadband in local resonance structures
,”
Appl. Acoust.
169
,
107482
(
2020
).
57.
L.
Fan
,
Y.
He
,
X.
Chen
, and
X.
Zhao
, “
Elastic metamaterial shaft with a stack-like resonator for low-frequency vibration isolation
,”
J. Phys. D: Appl. Phys.
53
,
105101
(
2020
).
58.
K.
Lu
,
G.
Zhou
,
N.
Gao
,
L.
Li
,
H.
Lei
, and
M.
Yu
, “
Flexural vibration bandgaps of the multiple local resonance elastic metamaterial plates with irregular resonators
,”
Appl. Acoust.
159
,
107115
(
2020
).
59.
S.
Ning
,
Z.
Yan
,
D.
Chu
,
H.
Jiang
,
Z.
Liu
, and
Z.
Zhuang
, “
Ultralow-frequency tunable acoustic metamaterials through tuning gauge pressure and gas temperature
,”
Extreme Mech. Lett.
44
,
101218
(
2021
).
60.
X. H.
Liu
,
N.
Chen
,
J. R.
Jiao
, and
J.
Liu
, “
Pneumatic soft phononic crystals with tunable band gap
,”
Int. J. Mech. Sci.
240
,
107906
(
2023
).
61.
X.
Tan
,
S.
Chen
,
B.
Wang
,
J.
Tang
,
L.
Wang
et al, “
Real-time tunable negative stiffness mechanical metamaterial
,”
Extreme Mech. Lett.
41
,
100990
(
2020
).
62.
L.
Yin
,
W.
Xu
,
Z.
Hu
,
M.
He
, and
C.
Li
, “
Research on damping and vibration absorption performance of air spring dynamic vibration absorber
,”
J. Vib. Eng. Technol.
12
,
3771
(
2023
).
63.
H. J.
Zhao
,
H. W.
Guo
,
M. X.
Gao
,
R. Q.
Liu
, and
Z. Q.
Deng
, “
Vibration band gaps in double-vibrator pillared phononic crystal plate
,”
J. Appl. Phys.
119
,
014903
(
2016
).
64.
Y.
Tian
,
J. H.
Wu
,
H.
Li
,
C.
Gu
,
Z.
Yang
et al, “
Elastic wave propagation in the elastic metamaterials containing parallel multi-resonators
,”
J. Phys. D: Appl. Phys.
52
,
395301
(
2019
).
65.
K.
Lu
,
J.
Wu
,
L.
Jing
,
N.
Gao
, and
D.
Guan
, “
The two-degree-of-freedom local resonance elastic metamaterial plate with broadband low-frequency bandgaps
,”
Appl. Phys.
50
,
095104
(
2017
).
66.
F.
Walter
, Computing and Controlling the Volume of a Cavity, COMSOL (
2014
), https://www.comsol.com/blogs/computing-controlling-volume-cavity?setlang=1; accessed 3 February 2014.
67.
H.
Zhang
,
D.
Shi
, and
Q.
Wang
, “
An improved Fourier series solution for free vibration analysis of the moderately thick laminated composite rectangular plate with non-uniform boundary conditions
,”
Int. J. Mech. Sci.
121
,
1
20
(
2017
).
68.
L.
He
and
Y. L.
Zhao
, “
Theory and design of high-pressure and heavy-duty air spring for naval vessels
,”
J. Vib. Eng.
26
,
886
(
2013
).
69.
J. H.
Wu
,
S. W.
Zhang
, and
L.
Shen
, “
Low-frequency vibration characteristics of periodic spiral resonators in phononic crystal plates
,”
J. Mech. Eng.
49
,
62
69
(
2013
).
70.
J. C.
Hsu
, “
Local resonances-induced low-frequency band gaps in two-dimensional phononic crystal slabs with periodic stepped resonators
,”
J. Phys. D: Appl. Phys.
44
,
055401
(
2011
).
71.
J. C.
Hsu
and
T. T.
Wu
, “
Lamb waves in binary locally resonant phononic plates with two-dimensional lattices
,”
Appl. Phys. Lett.
90
,
201904
(
2007
).
72.
W.
Xiao
,
G. W.
Zeng
, and
Y. S.
Cheng
, “
Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs
,”
Appl. Acoust.
69
,
255
261
(
2008
).