# Research on Low-Frequency Noise Control Based on Fractal Coiled Acoustic Metamaterials

Authors: Hongyu Cui; Chengtao Liu; Haoming Hu
Journal: Shock and Vibration
 (2022)

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## Abstract

In this paper, a self-similar fractal coiled acoustic metamaterial is designed for the low-frequency noise control problem by combining a perforated plate and a coiled back cavity structure. Based on the multiphysics field coupling method and thermoviscous acoustic theory, the effect of structural parameter changes on the sound absorption performance of the metamaterial structure is investigated using finite element analysis. The sound energy dissipation mechanism of the metamaterial structure is studied. Finally, 3D printing technology is used to prepare the metamaterial model structure, and the low-frequency sound absorption performance of the metamaterial structure is tested through structural sound absorption performance experiments, which verify that the metamaterial has subwavelength sound absorption performance.


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## Body

## 1. Introduction

Noise pollution has become an important source of environmental pollution, and low-frequency noise has become a major problem for noise control because of its strong penetration ability and slow attenuation [1]. Traditional acoustic materials such as composite panels and rock wool are limited by the law of mass density and can only be used to achieve effective control of low-frequency noise by increasing the volume of the material; thus, for structures with limited space, traditional acoustic materials have difficulty meeting the actual engineering needs. The acoustic metamaterials proposed in recent years can achieve subwavelength sound absorption and insulation, providing new solutions for low-frequency noise control problems [2–4].Some scholars have defined materials with acoustic manipulation functions that common materials in nature do not possess as acoustic metamaterials. In 2000, Liu et al. [3] proposed the concept of acoustic metamaterials, designed acoustic metamaterials with negative energy density using soft-coated materials and high-density solid cores, and achieved subwavelength sound insulation [5]. Subsequently, some researchers proposed acoustic metamaterials such as thin-film-type acoustic metamaterials, coiled acoustic metamaterials, and Helmholtz cavity acoustic metamaterials [6–9]. Based on the resonant scattering phenomenon of electromagnetic Mie scattering theory, some researchers proposed Mie resonant metamaterial structures, in which hard boundary materials are used to design acoustic propagation channels, reduce the effective phase velocity, and increase the refractive index of acoustic waves propagating in the channels [10, 11]. In 2012, Liang et al. constructed two-dimensional coiled-space acoustic metamaterials by changing the “z” channel to change the effective phase velocity in the material to achieve low-frequency sound insulation [12]. In 2013, Frenzel et al. proposed three-dimensional coiled-space acoustic metamaterials and prepared a metamaterial model for experiments to verify that the designed metamaterials had subwavelength sound insulation [13]. In 2015, Cheng et al. constructed a coiled structure with a circular fan shape to achieve a strong reflection of low-frequency acoustic waves [14]. In 2016, Zhang et al. achieved perfect absorption in a wide frequency band by designing three-dimensional single-port labyrinth acoustic metamaterials [15]. In 2016, Miniaci et al. discovered a spider web geometry for local resonant acoustic metamaterials with the advantage of low-frequency elastic wave manipulation [16]. In 2017, Krushynska et al. of the University of Turin designed novel labyrinth acoustic metamaterials with hybrid dispersion properties based on the spider web structure to achieve low-frequency noise control [17]. In 2017, Liu et al. tuned the bandwidth by varying the cross-sectional area of coiled metamaterial channels to achieve perfect absorption of acoustic waves near 400 Hz [18].Coiled acoustic metamaterials can achieve effective control of low- and medium-frequency noise, but the sound absorption in the low-frequency range is still unsatisfactory; thus, many scholars have launched studies combining microperforated plates with coiled channels. In 2016, Yong and Assouar [19] combined small holes with a coiled back cavity to achieve low-frequency sound absorption performance at 125 Hz. In 2018, Huang et al. [20] combined an inner Jack with a coiled back cavity to achieve perfect sound absorption at 141 Hz. In 2018, Wang et al. [21] combined a perforated plate with a coiled back cavity to achieve low-frequency sound absorption at 258 Hz by adjusting the cavity spacer. The metamaterial structure of a microperforated plate combined with a coiled back channel was observed to have low-frequency sound absorption performance.In this paper, self-similar fractal theory [22] is applied to the design of coiled channels, and new coiled acoustic metamaterials are designed based on a microperforated plate with a coiled structure. The acoustic waves have a lower equivalent sound velocity in the designed metamaterials than in air and a higher refractive index, which can generate multiorder resonant mode phenomena. Numerical analysis of the sound absorption performance of the metamaterial structures with different fractal dimensions is carried out using the multiphysics field coupling method, and the effect of structural parameter changes on the low-frequency sound absorption effect of the metamaterials is investigated. Finally, a model structure of the metamaterials is prepared by 3D technology, and the sound absorption performance of the designed metamaterial structure is experimentally studied.


## 2. Self-Similar Fractal Coiled Acoustic Metamaterials

A geometric shape that can be divided into several parts such that each part has the same statistical characteristics as the whole is called a fractal, and the principle of self-similarity is an important principle of fractal geometry. Self-similarity means that an object is similar to a part of itself. Applying self-similarity fractal geometry to the design of coiled channels can significantly extend the acoustic wave propagation path and improve the refractive index of metamaterials.


### 2.1. Coiled Acoustic Metamaterial Structure Design

In this paper, a new acoustic metamaterial structure is designed by combining a perforated plate with a self-similar fractal coiled back-cavity structure, and the designed coiled acoustic metamaterial structure is shown in Figure1. The structure of the coiled acoustic metamaterials is shown in Figure 1(a), which consists of three perforated layers, three coiled channel layers, and one closed layer, totalling seven layers. The specific structure form and dimensions of each layer are shown in Figure 1(b). The overall thickness of the designed metamaterials is 50 mm, of which the thickness of each perforated plate is b = 5 mm, the thickness of each coiled channel layer is c = 10 mm, and the thickness of the closed plate is 5 mm. Figure 2 shows the structural form of coiled acoustic metamaterials based on the Hilbert fractal for each order, in which Figures 2(a)–2(i) show the structural form of the 1st- to 3rd-order Hilbert fractals. When the three coiled channel layers are shown in Figure 1, the 1st-, 2nd-, or 3rd-order structural forms are adopted as shown in Figures 2(a)–2(c), and they constitute the corresponding 1st-, 2nd-, or 3rd-order coiled acoustic metamaterial. When an acoustic wave is transmitted from the first perforated plate into the interior of the structure, it propagates along the coiled channels of the self-similar fractal coiled back cavity structure, which leads to a decrease in the effective phase velocity of the propagating acoustic wave and an increase in the refractive index.Figure 1
Coiled acoustic metamaterial structure diagram. (a) Coiled acoustic metamaterial structure. (b) Internal structure of metamaterials.
(a)(b)Figure 2
Structural form of coiled acoustic metamaterials of each order based on the Hilbert fractal. (a) First-order Hilbert fractal. (b) Second-order Hilbert fractal. (c) Third-order Hilbert fractal. (d) First-order central perforated plate. (e) Second-order central perforated plate. (f) Third-order central perforated plate. (g) First-order perforated plate. (h) Second-order perforated plate. (i) Third-order perforated plate.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 2.2. Coiled Acoustic Metamaterial Absorption Coefficient Analysis

As sound waves propagate in coiled metamaterials with small thickness and width of the coiled channel, both thermal and viscous losses cause attenuation of the sound waves in the channel. Thus, to obtain accurate results matching the actual results, the effect of thermoviscous loss on the sound wave propagation needs to be considered when performing simulation calculations.Assuming that the medium in the coiled channel is the ideal case of a uniform and continuous medium without loss, the dependent variables are shown as follows:(1)p=p0+Δpeiωt,v=v0+Δveiωt,T=T0+ΔTeiωt,where p is the pressure, v is the velocity field, T is the temperature, and the term with variable Δ in the equation indicates the acoustic variable.By inserting (1) into the transient control equation with the higher-order terms omitted, the sound field is assumed harmonic in time, and the control equation for acoustic wave propagation in the frequency domain with the addition of thermoviscous loss is finally obtained [23, 24].The continuity equation for the thermoviscous acoustic model in the steady-state background is given as follows:(2)iωρ+ρ0∇v=0,where ρ0 is the background density, ω is the angular frequency, v is the velocity, and ∇ is the nabla operator.The momentum equation is formulated as follows:(3)iωρ0v=∇−pI+μ∇v+∇vT+μB−23μ∇vI,where μ is the dynamic viscosity, μB is the bulk viscosity, and I is the 3 × 1 identity matrix.The equation for the conservation of energy is given as follows:(4)iωρ0CpT−T0α0p=−∇−k∇T+Q,where Cp is the constant pressure heat capacity, k is the thermal conductivity, α0 is the coefficient of thermal expansion (isobaric), and Q is the possible heat source.The relationship between the medium density and medium pressure and temperature is shown as follows:(5)ρ=ρ0βTp−α0T,where βT is the compression rate (isothermal).The joint equations (2)–(5) correspond to the solution in the frequency domain of the thermoviscous acoustics, which can be solved to obtain the sound pressure p after considering the thermoviscous loss.The acoustic impedance is usually used to express the sound absorption coefficientα. The ratio of the sound pressure when sound waves pass through a plane and the sound flux through the interface is the acoustic impedance of the material. The expression is given as follows:(6)Z=pu=R+jZ=R+jωm+ωCa,where Z is the acoustic impedance, p is the sound pressure through the plane, u is the particle velocity, R is the acoustic resistance, m is the acoustic mass, and Ca is the acoustic compliance.When an acoustic wave is vertically incident on a metamaterial surface, the reflection coefficientr can be expressed in terms of acoustic impedance Z as follows:(7)r=Z−ρ0c0Z+ρ0c0,where ρ0c0 is the air characteristic impedance.Combining (6) and (7), the absorption coefficient in terms of the acoustic impedance can be obtained as follows:(8)α=1−r2=1−Z−ρ0c0Z+ρ0c02.


## 2.1. Coiled Acoustic Metamaterial Structure Design

In this paper, a new acoustic metamaterial structure is designed by combining a perforated plate with a self-similar fractal coiled back-cavity structure, and the designed coiled acoustic metamaterial structure is shown in Figure1. The structure of the coiled acoustic metamaterials is shown in Figure 1(a), which consists of three perforated layers, three coiled channel layers, and one closed layer, totalling seven layers. The specific structure form and dimensions of each layer are shown in Figure 1(b). The overall thickness of the designed metamaterials is 50 mm, of which the thickness of each perforated plate is b = 5 mm, the thickness of each coiled channel layer is c = 10 mm, and the thickness of the closed plate is 5 mm. Figure 2 shows the structural form of coiled acoustic metamaterials based on the Hilbert fractal for each order, in which Figures 2(a)–2(i) show the structural form of the 1st- to 3rd-order Hilbert fractals. When the three coiled channel layers are shown in Figure 1, the 1st-, 2nd-, or 3rd-order structural forms are adopted as shown in Figures 2(a)–2(c), and they constitute the corresponding 1st-, 2nd-, or 3rd-order coiled acoustic metamaterial. When an acoustic wave is transmitted from the first perforated plate into the interior of the structure, it propagates along the coiled channels of the self-similar fractal coiled back cavity structure, which leads to a decrease in the effective phase velocity of the propagating acoustic wave and an increase in the refractive index.Figure 1
Coiled acoustic metamaterial structure diagram. (a) Coiled acoustic metamaterial structure. (b) Internal structure of metamaterials.
(a)(b)Figure 2
Structural form of coiled acoustic metamaterials of each order based on the Hilbert fractal. (a) First-order Hilbert fractal. (b) Second-order Hilbert fractal. (c) Third-order Hilbert fractal. (d) First-order central perforated plate. (e) Second-order central perforated plate. (f) Third-order central perforated plate. (g) First-order perforated plate. (h) Second-order perforated plate. (i) Third-order perforated plate.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 2.2. Coiled Acoustic Metamaterial Absorption Coefficient Analysis

As sound waves propagate in coiled metamaterials with small thickness and width of the coiled channel, both thermal and viscous losses cause attenuation of the sound waves in the channel. Thus, to obtain accurate results matching the actual results, the effect of thermoviscous loss on the sound wave propagation needs to be considered when performing simulation calculations.Assuming that the medium in the coiled channel is the ideal case of a uniform and continuous medium without loss, the dependent variables are shown as follows:(1)p=p0+Δpeiωt,v=v0+Δveiωt,T=T0+ΔTeiωt,where p is the pressure, v is the velocity field, T is the temperature, and the term with variable Δ in the equation indicates the acoustic variable.By inserting (1) into the transient control equation with the higher-order terms omitted, the sound field is assumed harmonic in time, and the control equation for acoustic wave propagation in the frequency domain with the addition of thermoviscous loss is finally obtained [23, 24].The continuity equation for the thermoviscous acoustic model in the steady-state background is given as follows:(2)iωρ+ρ0∇v=0,where ρ0 is the background density, ω is the angular frequency, v is the velocity, and ∇ is the nabla operator.The momentum equation is formulated as follows:(3)iωρ0v=∇−pI+μ∇v+∇vT+μB−23μ∇vI,where μ is the dynamic viscosity, μB is the bulk viscosity, and I is the 3 × 1 identity matrix.The equation for the conservation of energy is given as follows:(4)iωρ0CpT−T0α0p=−∇−k∇T+Q,where Cp is the constant pressure heat capacity, k is the thermal conductivity, α0 is the coefficient of thermal expansion (isobaric), and Q is the possible heat source.The relationship between the medium density and medium pressure and temperature is shown as follows:(5)ρ=ρ0βTp−α0T,where βT is the compression rate (isothermal).The joint equations (2)–(5) correspond to the solution in the frequency domain of the thermoviscous acoustics, which can be solved to obtain the sound pressure p after considering the thermoviscous loss.The acoustic impedance is usually used to express the sound absorption coefficientα. The ratio of the sound pressure when sound waves pass through a plane and the sound flux through the interface is the acoustic impedance of the material. The expression is given as follows:(6)Z=pu=R+jZ=R+jωm+ωCa,where Z is the acoustic impedance, p is the sound pressure through the plane, u is the particle velocity, R is the acoustic resistance, m is the acoustic mass, and Ca is the acoustic compliance.When an acoustic wave is vertically incident on a metamaterial surface, the reflection coefficientr can be expressed in terms of acoustic impedance Z as follows:(7)r=Z−ρ0c0Z+ρ0c0,where ρ0c0 is the air characteristic impedance.Combining (6) and (7), the absorption coefficient in terms of the acoustic impedance can be obtained as follows:(8)α=1−r2=1−Z−ρ0c0Z+ρ0c02.


## 3. Performance Analysis of Sound Absorption by Self-Similar Fractal Coiled Acoustic Metamaterials

### 3.1. Fractal Coiled Metamaterial Physical Field Model

This paper analyses the sound absorption performance of fractal coiled acoustic metamaterials based on the multiphysics field coupling software COMSOL Multiphysics. To fully consider the thermoviscous loss of sound propagation in small channels, the internal coiled channel regions of the structure are selected as thermoviscous acoustic regions, the regions on both sides of the metamaterial model are taken as pressure acoustic regions, the metamaterial model regions are set as solid mechanics regions, and the multiphysics field coupling module is prepared. The established multiphysics field coupling model is shown in Figure3. In the calculation, the incident sound wave perpendicular to the structure surface is a plane wave, and the sound pressure is 1 Pa. The structure of the metamaterials is American red oak with a density of 630 kg/cm3. The value of Young’s modulus is 12.4 GPa, and that of Poisson’s ratio is 0.3. First-order, second-order, and third-order coiled acoustic metamaterial finite element models are established, and the model structure is set to periodic boundary conditions. The sound absorption performance of the metamaterial model structure in the frequency range of 30–1600 Hz is analysed.Figure 3
Multiphysics field coupling model.In this paper, the perfectly matched layer (PML) is divided by the sweeping method, the number of layers is set to 8, and the remainder is described by free tetrahedron elements. In order to ensure the accuracy of the calculation results, the size of the largest cell in all cell grids is not greater than 1/6 of the minimum wavelength, and the size of the smallest cell is not greater than the minimum boundary size of the model.


### 3.2. Analysis of the Sound Absorption Performance of Coiled Acoustic Metamaterials

#### 3.2.1. Effect of Fractal Order on the Sound Absorption Performance of Metamaterials

For the self-similar fractal coiled acoustic metamaterial structure designed in this paper, based on the overall metamaterial model shown in Figure1, the coiled channel layers adopt the 1st-, 2nd-, or 3rd-order coiled channels as shown in Figure 2, and the corresponding 1st-, 2nd-, and 3rd-order self-similar fractal coiled acoustic metamaterial finite element models are established to calculate the sound absorption characteristics of the different order metamaterial structures in the range of 30–1600 Hz. The sound absorption coefficient curves of each order metamaterial are shown in Figure 4.Figure 4
Curve of the sound absorption coefficients of metamaterials of each order.As seen from Figure4, in the 30–1600 Hz band, the first-order self-similar fractal coiled acoustic metamaterial has three absorption peaks with centre frequencies at 214 Hz, 725 Hz, and 1013 Hz; the second-order self-similar fractal coiled acoustic metamaterial has seven absorption peaks with centre frequencies at 119 Hz, 359 Hz, 591 Hz, 833 Hz, 1077 Hz, 1291 Hz, and 1562 Hz; and the third-order self-similar fractal coiled acoustic metamaterial has 12 absorption peaks with centre frequencies at 69 Hz, 212 Hz, 352 Hz, 485 Hz, 627 Hz, 760 Hz, 905 Hz, 1046 Hz, 1175 Hz, 1319 Hz, 1460 Hz, and 1576 Hz. As the self-similar fractal order increases, the path length of the coiled channel increases, the refractive index of the metamaterial increases, multiorder resonance modes can be excited, the number of absorption peaks for the self-similar fractal coiled acoustic metamaterial increases, the wavelength of the acoustic wave that can be manipulated by the metamaterial increases, and the centre frequency of the first absorption peak moves to a lower frequency.


#### 3.2.2. Coiled Acoustic Metamaterial Sound Dissipation Mechanism

Coiled acoustic metamaterials use spatial folding to make the metamaterials have a higher refractive index than the background medium, which excites multiorder resonance modes, causing rapid particle motion inside the metamaterials and generating large sound energy loss.To visualize the sound absorption mechanism of the designed self-similar fractal coiled acoustic metamaterials, the sound pressure distribution at absorption peak frequencies of the self-similar fractal coiled acoustic metamaterials of orders 1–3 is selected for analysis. Figures5(a)–5(c) show the typical sound pressure distributions of the first-order metamaterial at 214 Hz, 725 Hz, and 1013 Hz, Figures 5(d)–5(f) show the typical sound pressure distributions of the second-order metamaterial at 119 Hz, 833 Hz, and 1569 Hz, and Figures 5(g)–5(i) show the typical sound pressure distributions of the third-order metamaterial at 69 Hz, 905 Hz, and 1576 Hz.Figure 5
Sound pressure distribution for each order metamaterial structure. (a) 214 Hz overall sound pressure distribution. (b) 725 Hz single-layer sound pressure distribution. (c) 1013 Hz single-layer sound pressure distribution. (d) 119 Hz overall sound pressure distribution. (e) 833 Hz single-layer sound pressure distribution. (f) 1569 Hz single-layer sound pressure distribution. (g) 69 Hz overall sound pressure distribution. (h) 905 Hz single-layer sound pressure distribution. (i) 1576 Hz single-layer sound pressure distribution.
(a)(b)(c)(d)(e)(f)(g)(h)(i)According to the sound pressure distribution diagrams of each order metamaterial in Figure5, the complex channels inside the metamaterials cause the acoustic waves to have a slower sound velocity and a higher refractive index compared with the background medium. The acoustic waves have opposite phases in the channels and produce a multiorder resonance mode, such as the overall monopole resonance mode at low frequencies for each order and multipole resonance modes at high frequencies for each layer. Additionally, the channel paths become longer, and the channel refractive index increases due to their higher order, making the manipulated acoustic wave longer in wavelength and lower in resonance frequency.Based on the analysis of the sound pressure distributions of the designed metamaterials, the particle velocity distributions at an absorption peak frequency and a nonabsorption peak frequency are selected for analysis. Figures6(a) and 6(b) show the centre plane particle velocity distributions at 214 Hz and 400 Hz for the 1st-order metamaterial. Figures 6(c) and 6(d) show the centre plane particle velocity distributions at 119 Hz and 200 Hz for the 2nd-order metamaterial. Figures 6(e) and 6(f) show the centre plane particle velocity distributions at 69 Hz and 150 Hz for the 3rd-order metamaterial.Figure 6
Particle velocity distributions of the metamaterial structures of each order. (a) 214 Hz centre plane particle velocity distribution. (b) 400 Hz centre plane particle velocity distribution. (c) 119 Hz centre plane particle velocity distribution. (d) 200 Hz centre plane particle velocity distribution. (e) 69 Hz centre plane particle velocity distribution. (f) 150 Hz centre plane particle velocity distribution.
(a)(b)(c)(d)(e)(f)According to the particle velocity distributions in Figure6, due to the generation of the multiorder resonance mode phenomenon, the antiphase vibration of the acoustic wave in the channel makes the particles in the channel move rapidly, and the velocity is larger at smaller acoustic pressures, which generates a larger sound loss in the smaller channel.Based on the analysis of the designed metamaterial particle velocity distribution maps, the thermoviscous power density distribution maps at the same frequencies are selected for analysis. Figures7(a)–7(f) show the thermoviscous power density distribution maps of the 1st- to 3rd-order metamaterials. Figures 7(a) and 7(b) show the thermoviscous power density distribution maps at 214 Hz and 400 Hz for the 1st-order metamaterial. Figures 7(c) and 7(d) show the thermoviscous power density distribution maps at 119 Hz and 200 Hz for the 2nd-order metamaterial. Figures 7(e) and 7(f) show the thermoviscous power density distribution maps at 69 Hz and 150 Hz for the 3rd-order metamaterial.Figure 7
Overall coiled channel thermoviscous power consumption density distributions of each order metamaterial structure. (a) 214 Hz overall thermoviscous power consumption density distribution. (b) 400 Hz overall thermoviscous power consumption density distribution. (c) 119 Hz overall thermoviscous power consumption density distribution. (d) 200 Hz overall thermoviscous power consumption density distribution. (e) 69 Hz overall thermoviscous power consumption density distribution. (f) 150 Hz overall thermoviscous power consumption density distribution.
(a)(b)(c)(d)(e)(f)The thermoviscous power density distributions of the metamaterials in Figure7 show that the maximum thermoviscous power densities at the first absorption peak frequency for orders 1–3 are 0.72 W/m3, 3.07 W/m3, and 4.69 W/m3, and the maximum thermoviscous power densities at the nonabsorption peak frequency are 9.06 × 10–4 W/m3, 8.06 × 10–3 W/m3, and 0.04 W/m3. The thermoviscous power density at the absorption peak frequency is much larger than that at the nonabsorption peak frequency. Thus, the designed metamaterials clearly have a higher refractive index than the background medium, which can generate multiorder resonance modes such that the metamaterials have a higher velocity of particle motion inside the metamaterials and thus generate a larger sound loss.


### 3.3. Effect of Channel Layer Width and Thickness on the Sound Absorption Performance

Changes in the structural parameters of the coiled channel layers of the metamaterials alter the volume of the cavity inside the coiled channels, which affects the refractive index of the metamaterials and the thermoviscosity of the channels. Therefore, a change in the structural parameters of the coiled channel layers will affect the sound absorption performance of the metamaterials. For the structure of the metamaterials in Figure1, the second-order self-similar coiled channel layer is selected to construct the acoustic metamaterials, and the effects of channel width a and channel layer thicknesses b and c on the acoustic absorption performance of the metamaterials are investigated.


#### 3.3.1. Effect of Channel Width on the Sound Absorption Performance

As the channel width decreases, the air volume inside the structure decreases, and the thermoviscous loss of the structure increases. To make the metamaterials effectively control noise at lower frequencies, the relationship between the channel width and the sound absorption performance of the metamaterials is analysed. The channel widths shown in Figure2(b) are set to 3 mm, 4 mm, and 5 mm for numerical simulations, and the influence of the change in the channel width on the sound absorption performance of the metamaterials is analysed, as shown in Figure 8.Figure 8
Sound absorption characteristics curves of metamaterials at different channel widths.According to Figure8, as the channel width decreases, the refractive index of the metamaterial increases, the wavelength of the acoustic wave that the metamaterial can modulate increases, the resonant frequencies move to lower frequencies, and the absorption peaks decrease. Over 1300–1600 Hz, a new sound absorption peak appears at 1412 Hz when the channel width is 3 mm.To investigate the cause of the absorption peak at 1412 Hz, modal analysis and sound insulation characteristics analysis are conducted for the 3 mm channel width metamaterial. Figure9(a) shows the structural vibration mode of the metamaterial structure at 1412 Hz, and Figure 9(b) shows the sound insulation characteristics curve of the metamaterial structure in the frequency range of 1200–1600 Hz. According to Figure 9(a), an inherent vibration mode of the metamaterial structure appears at 1412 Hz, and according to Figure 9(b), a new absorption peak is formed at 1412 Hz in the sound insulation characteristics curve due to the sound transmission caused by resonance.Figure 9
Sound insulation characteristics curve and vibration mode of metamaterials. (a) Sound insulation characteristics curve. (b) Vibration mode at 1412 Hz.
(a)(b)To investigate why the sound absorption peaks of the metamaterial decrease with decreasing channel width, the centre plane particle velocities of the metamaterials with 4 mm and 3 mm channel widths are analysed, and Figure10 shows the centre plane particle velocity distributions of the 4 mm and 3 mm metamaterials at the peak sound absorption frequency.Figure 10
Centre plane particle velocity distributions of 4 mm and 3 mm metamaterials. (a) 4 mm particle velocity distribution. (b) 3 mm particle velocity distribution.
(a)(b)According to Figure10, a reduction in the channel width leads to a reduction in the acoustic energy entering the interior of the metamaterial, a weakening of the resonance intensity, a decrease in the particle velocity, and a decrease in the structural absorption coefficient.


#### 3.3.2. Effect of Channel Layer Thickness on the Sound Absorption Performance

A change in the channel layer thickness affects the air volume inside the structure and has an impact on the sound absorption performance of the metamaterial. To enable the metamaterials to effectively control noise at lower frequencies, the relationship between the channel thickness and the sound absorption performance of the metamaterials is analysed, and the influence of the channel layer thickness on the sound absorption performance of the metamaterials is investigated by setting different channel layer thicknessesb and c for the model structure in Figure 1 while keeping the overall thickness constant at 50 mm. The designed channel layer thicknesses of the metamaterial structure are shown in Table 1.Table 1
Combinations of channel layer thicknesses.
Channel layer thicknessb (mm)Channel layer thicknessc (mm)Combination 1510Combination 2411Combination 3312The sound absorption coefficient analysis is performed for several combinations as shown in Table1, and the results shown in Figure 11 are obtained.Figure 11
Curves of sound absorption characteristics of acoustic metamaterials with different channel thicknesses.According to the calculation results in Figure11, with an increase in the channel thickness, the wavelength of the acoustic wave that can be modulated decreases, and the resonant frequencies move towards higher frequencies. Thus, an increase in the channel layer thickness causes the centre frequency of the absorption peaks to move to a higher frequency.


### 3.4. Influence of the Number of Channel Layers on the Sound Absorption Performance

An increase in the number of coiled channel layers can extend the sound propagation distance, increase the refractive index of the metamaterial, and affect the multiorder resonance mode phenomenon. To enable the metamaterials to effectively control noise at lower frequencies, the relationship between the number of metamaterial channel layers and the sound absorption performance is therefore analysed. Based on the structure of the three-layer second-order coiled channel shown in Figure1, the numbers of coiled channel layers and perforated plate layers are increased to form four-layer and five-layer self-similar fractal coiled acoustic metamaterials with thicknesses of 65 mm and 80 mm, and the sound absorption performance at 30–400 Hz in terms of the sound absorption coefficient curves is shown in Figure 12.Figure 12
Sound absorption coefficient curves of metamaterials with different numbers of layers.According to Figure12, after adding more coiled channel layers, the refractive index of the metamaterial increases, the wavelength of the acoustic wave that can be manipulated by the metamaterial increases, the resonant frequencies move to lower frequencies, the centre frequency of the lowest absorption peak moves from 119 Hz to 89 Hz and 71 Hz, and the centre frequencies of the other absorption peaks all substantially move to lower frequencies.


## 3.1. Fractal Coiled Metamaterial Physical Field Model

This paper analyses the sound absorption performance of fractal coiled acoustic metamaterials based on the multiphysics field coupling software COMSOL Multiphysics. To fully consider the thermoviscous loss of sound propagation in small channels, the internal coiled channel regions of the structure are selected as thermoviscous acoustic regions, the regions on both sides of the metamaterial model are taken as pressure acoustic regions, the metamaterial model regions are set as solid mechanics regions, and the multiphysics field coupling module is prepared. The established multiphysics field coupling model is shown in Figure3. In the calculation, the incident sound wave perpendicular to the structure surface is a plane wave, and the sound pressure is 1 Pa. The structure of the metamaterials is American red oak with a density of 630 kg/cm3. The value of Young’s modulus is 12.4 GPa, and that of Poisson’s ratio is 0.3. First-order, second-order, and third-order coiled acoustic metamaterial finite element models are established, and the model structure is set to periodic boundary conditions. The sound absorption performance of the metamaterial model structure in the frequency range of 30–1600 Hz is analysed.Figure 3
Multiphysics field coupling model.In this paper, the perfectly matched layer (PML) is divided by the sweeping method, the number of layers is set to 8, and the remainder is described by free tetrahedron elements. In order to ensure the accuracy of the calculation results, the size of the largest cell in all cell grids is not greater than 1/6 of the minimum wavelength, and the size of the smallest cell is not greater than the minimum boundary size of the model.


## 3.2. Analysis of the Sound Absorption Performance of Coiled Acoustic Metamaterials

### 3.2.1. Effect of Fractal Order on the Sound Absorption Performance of Metamaterials

For the self-similar fractal coiled acoustic metamaterial structure designed in this paper, based on the overall metamaterial model shown in Figure1, the coiled channel layers adopt the 1st-, 2nd-, or 3rd-order coiled channels as shown in Figure 2, and the corresponding 1st-, 2nd-, and 3rd-order self-similar fractal coiled acoustic metamaterial finite element models are established to calculate the sound absorption characteristics of the different order metamaterial structures in the range of 30–1600 Hz. The sound absorption coefficient curves of each order metamaterial are shown in Figure 4.Figure 4
Curve of the sound absorption coefficients of metamaterials of each order.As seen from Figure4, in the 30–1600 Hz band, the first-order self-similar fractal coiled acoustic metamaterial has three absorption peaks with centre frequencies at 214 Hz, 725 Hz, and 1013 Hz; the second-order self-similar fractal coiled acoustic metamaterial has seven absorption peaks with centre frequencies at 119 Hz, 359 Hz, 591 Hz, 833 Hz, 1077 Hz, 1291 Hz, and 1562 Hz; and the third-order self-similar fractal coiled acoustic metamaterial has 12 absorption peaks with centre frequencies at 69 Hz, 212 Hz, 352 Hz, 485 Hz, 627 Hz, 760 Hz, 905 Hz, 1046 Hz, 1175 Hz, 1319 Hz, 1460 Hz, and 1576 Hz. As the self-similar fractal order increases, the path length of the coiled channel increases, the refractive index of the metamaterial increases, multiorder resonance modes can be excited, the number of absorption peaks for the self-similar fractal coiled acoustic metamaterial increases, the wavelength of the acoustic wave that can be manipulated by the metamaterial increases, and the centre frequency of the first absorption peak moves to a lower frequency.


### 3.2.2. Coiled Acoustic Metamaterial Sound Dissipation Mechanism

Coiled acoustic metamaterials use spatial folding to make the metamaterials have a higher refractive index than the background medium, which excites multiorder resonance modes, causing rapid particle motion inside the metamaterials and generating large sound energy loss.To visualize the sound absorption mechanism of the designed self-similar fractal coiled acoustic metamaterials, the sound pressure distribution at absorption peak frequencies of the self-similar fractal coiled acoustic metamaterials of orders 1–3 is selected for analysis. Figures5(a)–5(c) show the typical sound pressure distributions of the first-order metamaterial at 214 Hz, 725 Hz, and 1013 Hz, Figures 5(d)–5(f) show the typical sound pressure distributions of the second-order metamaterial at 119 Hz, 833 Hz, and 1569 Hz, and Figures 5(g)–5(i) show the typical sound pressure distributions of the third-order metamaterial at 69 Hz, 905 Hz, and 1576 Hz.Figure 5
Sound pressure distribution for each order metamaterial structure. (a) 214 Hz overall sound pressure distribution. (b) 725 Hz single-layer sound pressure distribution. (c) 1013 Hz single-layer sound pressure distribution. (d) 119 Hz overall sound pressure distribution. (e) 833 Hz single-layer sound pressure distribution. (f) 1569 Hz single-layer sound pressure distribution. (g) 69 Hz overall sound pressure distribution. (h) 905 Hz single-layer sound pressure distribution. (i) 1576 Hz single-layer sound pressure distribution.
(a)(b)(c)(d)(e)(f)(g)(h)(i)According to the sound pressure distribution diagrams of each order metamaterial in Figure5, the complex channels inside the metamaterials cause the acoustic waves to have a slower sound velocity and a higher refractive index compared with the background medium. The acoustic waves have opposite phases in the channels and produce a multiorder resonance mode, such as the overall monopole resonance mode at low frequencies for each order and multipole resonance modes at high frequencies for each layer. Additionally, the channel paths become longer, and the channel refractive index increases due to their higher order, making the manipulated acoustic wave longer in wavelength and lower in resonance frequency.Based on the analysis of the sound pressure distributions of the designed metamaterials, the particle velocity distributions at an absorption peak frequency and a nonabsorption peak frequency are selected for analysis. Figures6(a) and 6(b) show the centre plane particle velocity distributions at 214 Hz and 400 Hz for the 1st-order metamaterial. Figures 6(c) and 6(d) show the centre plane particle velocity distributions at 119 Hz and 200 Hz for the 2nd-order metamaterial. Figures 6(e) and 6(f) show the centre plane particle velocity distributions at 69 Hz and 150 Hz for the 3rd-order metamaterial.Figure 6
Particle velocity distributions of the metamaterial structures of each order. (a) 214 Hz centre plane particle velocity distribution. (b) 400 Hz centre plane particle velocity distribution. (c) 119 Hz centre plane particle velocity distribution. (d) 200 Hz centre plane particle velocity distribution. (e) 69 Hz centre plane particle velocity distribution. (f) 150 Hz centre plane particle velocity distribution.
(a)(b)(c)(d)(e)(f)According to the particle velocity distributions in Figure6, due to the generation of the multiorder resonance mode phenomenon, the antiphase vibration of the acoustic wave in the channel makes the particles in the channel move rapidly, and the velocity is larger at smaller acoustic pressures, which generates a larger sound loss in the smaller channel.Based on the analysis of the designed metamaterial particle velocity distribution maps, the thermoviscous power density distribution maps at the same frequencies are selected for analysis. Figures7(a)–7(f) show the thermoviscous power density distribution maps of the 1st- to 3rd-order metamaterials. Figures 7(a) and 7(b) show the thermoviscous power density distribution maps at 214 Hz and 400 Hz for the 1st-order metamaterial. Figures 7(c) and 7(d) show the thermoviscous power density distribution maps at 119 Hz and 200 Hz for the 2nd-order metamaterial. Figures 7(e) and 7(f) show the thermoviscous power density distribution maps at 69 Hz and 150 Hz for the 3rd-order metamaterial.Figure 7
Overall coiled channel thermoviscous power consumption density distributions of each order metamaterial structure. (a) 214 Hz overall thermoviscous power consumption density distribution. (b) 400 Hz overall thermoviscous power consumption density distribution. (c) 119 Hz overall thermoviscous power consumption density distribution. (d) 200 Hz overall thermoviscous power consumption density distribution. (e) 69 Hz overall thermoviscous power consumption density distribution. (f) 150 Hz overall thermoviscous power consumption density distribution.
(a)(b)(c)(d)(e)(f)The thermoviscous power density distributions of the metamaterials in Figure7 show that the maximum thermoviscous power densities at the first absorption peak frequency for orders 1–3 are 0.72 W/m3, 3.07 W/m3, and 4.69 W/m3, and the maximum thermoviscous power densities at the nonabsorption peak frequency are 9.06 × 10–4 W/m3, 8.06 × 10–3 W/m3, and 0.04 W/m3. The thermoviscous power density at the absorption peak frequency is much larger than that at the nonabsorption peak frequency. Thus, the designed metamaterials clearly have a higher refractive index than the background medium, which can generate multiorder resonance modes such that the metamaterials have a higher velocity of particle motion inside the metamaterials and thus generate a larger sound loss.


## 3.2.1. Effect of Fractal Order on the Sound Absorption Performance of Metamaterials

For the self-similar fractal coiled acoustic metamaterial structure designed in this paper, based on the overall metamaterial model shown in Figure1, the coiled channel layers adopt the 1st-, 2nd-, or 3rd-order coiled channels as shown in Figure 2, and the corresponding 1st-, 2nd-, and 3rd-order self-similar fractal coiled acoustic metamaterial finite element models are established to calculate the sound absorption characteristics of the different order metamaterial structures in the range of 30–1600 Hz. The sound absorption coefficient curves of each order metamaterial are shown in Figure 4.Figure 4
Curve of the sound absorption coefficients of metamaterials of each order.As seen from Figure4, in the 30–1600 Hz band, the first-order self-similar fractal coiled acoustic metamaterial has three absorption peaks with centre frequencies at 214 Hz, 725 Hz, and 1013 Hz; the second-order self-similar fractal coiled acoustic metamaterial has seven absorption peaks with centre frequencies at 119 Hz, 359 Hz, 591 Hz, 833 Hz, 1077 Hz, 1291 Hz, and 1562 Hz; and the third-order self-similar fractal coiled acoustic metamaterial has 12 absorption peaks with centre frequencies at 69 Hz, 212 Hz, 352 Hz, 485 Hz, 627 Hz, 760 Hz, 905 Hz, 1046 Hz, 1175 Hz, 1319 Hz, 1460 Hz, and 1576 Hz. As the self-similar fractal order increases, the path length of the coiled channel increases, the refractive index of the metamaterial increases, multiorder resonance modes can be excited, the number of absorption peaks for the self-similar fractal coiled acoustic metamaterial increases, the wavelength of the acoustic wave that can be manipulated by the metamaterial increases, and the centre frequency of the first absorption peak moves to a lower frequency.


## 3.2.2. Coiled Acoustic Metamaterial Sound Dissipation Mechanism

Coiled acoustic metamaterials use spatial folding to make the metamaterials have a higher refractive index than the background medium, which excites multiorder resonance modes, causing rapid particle motion inside the metamaterials and generating large sound energy loss.To visualize the sound absorption mechanism of the designed self-similar fractal coiled acoustic metamaterials, the sound pressure distribution at absorption peak frequencies of the self-similar fractal coiled acoustic metamaterials of orders 1–3 is selected for analysis. Figures5(a)–5(c) show the typical sound pressure distributions of the first-order metamaterial at 214 Hz, 725 Hz, and 1013 Hz, Figures 5(d)–5(f) show the typical sound pressure distributions of the second-order metamaterial at 119 Hz, 833 Hz, and 1569 Hz, and Figures 5(g)–5(i) show the typical sound pressure distributions of the third-order metamaterial at 69 Hz, 905 Hz, and 1576 Hz.Figure 5
Sound pressure distribution for each order metamaterial structure. (a) 214 Hz overall sound pressure distribution. (b) 725 Hz single-layer sound pressure distribution. (c) 1013 Hz single-layer sound pressure distribution. (d) 119 Hz overall sound pressure distribution. (e) 833 Hz single-layer sound pressure distribution. (f) 1569 Hz single-layer sound pressure distribution. (g) 69 Hz overall sound pressure distribution. (h) 905 Hz single-layer sound pressure distribution. (i) 1576 Hz single-layer sound pressure distribution.
(a)(b)(c)(d)(e)(f)(g)(h)(i)According to the sound pressure distribution diagrams of each order metamaterial in Figure5, the complex channels inside the metamaterials cause the acoustic waves to have a slower sound velocity and a higher refractive index compared with the background medium. The acoustic waves have opposite phases in the channels and produce a multiorder resonance mode, such as the overall monopole resonance mode at low frequencies for each order and multipole resonance modes at high frequencies for each layer. Additionally, the channel paths become longer, and the channel refractive index increases due to their higher order, making the manipulated acoustic wave longer in wavelength and lower in resonance frequency.Based on the analysis of the sound pressure distributions of the designed metamaterials, the particle velocity distributions at an absorption peak frequency and a nonabsorption peak frequency are selected for analysis. Figures6(a) and 6(b) show the centre plane particle velocity distributions at 214 Hz and 400 Hz for the 1st-order metamaterial. Figures 6(c) and 6(d) show the centre plane particle velocity distributions at 119 Hz and 200 Hz for the 2nd-order metamaterial. Figures 6(e) and 6(f) show the centre plane particle velocity distributions at 69 Hz and 150 Hz for the 3rd-order metamaterial.Figure 6
Particle velocity distributions of the metamaterial structures of each order. (a) 214 Hz centre plane particle velocity distribution. (b) 400 Hz centre plane particle velocity distribution. (c) 119 Hz centre plane particle velocity distribution. (d) 200 Hz centre plane particle velocity distribution. (e) 69 Hz centre plane particle velocity distribution. (f) 150 Hz centre plane particle velocity distribution.
(a)(b)(c)(d)(e)(f)According to the particle velocity distributions in Figure6, due to the generation of the multiorder resonance mode phenomenon, the antiphase vibration of the acoustic wave in the channel makes the particles in the channel move rapidly, and the velocity is larger at smaller acoustic pressures, which generates a larger sound loss in the smaller channel.Based on the analysis of the designed metamaterial particle velocity distribution maps, the thermoviscous power density distribution maps at the same frequencies are selected for analysis. Figures7(a)–7(f) show the thermoviscous power density distribution maps of the 1st- to 3rd-order metamaterials. Figures 7(a) and 7(b) show the thermoviscous power density distribution maps at 214 Hz and 400 Hz for the 1st-order metamaterial. Figures 7(c) and 7(d) show the thermoviscous power density distribution maps at 119 Hz and 200 Hz for the 2nd-order metamaterial. Figures 7(e) and 7(f) show the thermoviscous power density distribution maps at 69 Hz and 150 Hz for the 3rd-order metamaterial.Figure 7
Overall coiled channel thermoviscous power consumption density distributions of each order metamaterial structure. (a) 214 Hz overall thermoviscous power consumption density distribution. (b) 400 Hz overall thermoviscous power consumption density distribution. (c) 119 Hz overall thermoviscous power consumption density distribution. (d) 200 Hz overall thermoviscous power consumption density distribution. (e) 69 Hz overall thermoviscous power consumption density distribution. (f) 150 Hz overall thermoviscous power consumption density distribution.
(a)(b)(c)(d)(e)(f)The thermoviscous power density distributions of the metamaterials in Figure7 show that the maximum thermoviscous power densities at the first absorption peak frequency for orders 1–3 are 0.72 W/m3, 3.07 W/m3, and 4.69 W/m3, and the maximum thermoviscous power densities at the nonabsorption peak frequency are 9.06 × 10–4 W/m3, 8.06 × 10–3 W/m3, and 0.04 W/m3. The thermoviscous power density at the absorption peak frequency is much larger than that at the nonabsorption peak frequency. Thus, the designed metamaterials clearly have a higher refractive index than the background medium, which can generate multiorder resonance modes such that the metamaterials have a higher velocity of particle motion inside the metamaterials and thus generate a larger sound loss.


## 3.3. Effect of Channel Layer Width and Thickness on the Sound Absorption Performance

Changes in the structural parameters of the coiled channel layers of the metamaterials alter the volume of the cavity inside the coiled channels, which affects the refractive index of the metamaterials and the thermoviscosity of the channels. Therefore, a change in the structural parameters of the coiled channel layers will affect the sound absorption performance of the metamaterials. For the structure of the metamaterials in Figure1, the second-order self-similar coiled channel layer is selected to construct the acoustic metamaterials, and the effects of channel width a and channel layer thicknesses b and c on the acoustic absorption performance of the metamaterials are investigated.


### 3.3.1. Effect of Channel Width on the Sound Absorption Performance

As the channel width decreases, the air volume inside the structure decreases, and the thermoviscous loss of the structure increases. To make the metamaterials effectively control noise at lower frequencies, the relationship between the channel width and the sound absorption performance of the metamaterials is analysed. The channel widths shown in Figure2(b) are set to 3 mm, 4 mm, and 5 mm for numerical simulations, and the influence of the change in the channel width on the sound absorption performance of the metamaterials is analysed, as shown in Figure 8.Figure 8
Sound absorption characteristics curves of metamaterials at different channel widths.According to Figure8, as the channel width decreases, the refractive index of the metamaterial increases, the wavelength of the acoustic wave that the metamaterial can modulate increases, the resonant frequencies move to lower frequencies, and the absorption peaks decrease. Over 1300–1600 Hz, a new sound absorption peak appears at 1412 Hz when the channel width is 3 mm.To investigate the cause of the absorption peak at 1412 Hz, modal analysis and sound insulation characteristics analysis are conducted for the 3 mm channel width metamaterial. Figure9(a) shows the structural vibration mode of the metamaterial structure at 1412 Hz, and Figure 9(b) shows the sound insulation characteristics curve of the metamaterial structure in the frequency range of 1200–1600 Hz. According to Figure 9(a), an inherent vibration mode of the metamaterial structure appears at 1412 Hz, and according to Figure 9(b), a new absorption peak is formed at 1412 Hz in the sound insulation characteristics curve due to the sound transmission caused by resonance.Figure 9
Sound insulation characteristics curve and vibration mode of metamaterials. (a) Sound insulation characteristics curve. (b) Vibration mode at 1412 Hz.
(a)(b)To investigate why the sound absorption peaks of the metamaterial decrease with decreasing channel width, the centre plane particle velocities of the metamaterials with 4 mm and 3 mm channel widths are analysed, and Figure10 shows the centre plane particle velocity distributions of the 4 mm and 3 mm metamaterials at the peak sound absorption frequency.Figure 10
Centre plane particle velocity distributions of 4 mm and 3 mm metamaterials. (a) 4 mm particle velocity distribution. (b) 3 mm particle velocity distribution.
(a)(b)According to Figure10, a reduction in the channel width leads to a reduction in the acoustic energy entering the interior of the metamaterial, a weakening of the resonance intensity, a decrease in the particle velocity, and a decrease in the structural absorption coefficient.


### 3.3.2. Effect of Channel Layer Thickness on the Sound Absorption Performance

A change in the channel layer thickness affects the air volume inside the structure and has an impact on the sound absorption performance of the metamaterial. To enable the metamaterials to effectively control noise at lower frequencies, the relationship between the channel thickness and the sound absorption performance of the metamaterials is analysed, and the influence of the channel layer thickness on the sound absorption performance of the metamaterials is investigated by setting different channel layer thicknessesb and c for the model structure in Figure 1 while keeping the overall thickness constant at 50 mm. The designed channel layer thicknesses of the metamaterial structure are shown in Table 1.Table 1
Combinations of channel layer thicknesses.
Channel layer thicknessb (mm)Channel layer thicknessc (mm)Combination 1510Combination 2411Combination 3312The sound absorption coefficient analysis is performed for several combinations as shown in Table1, and the results shown in Figure 11 are obtained.Figure 11
Curves of sound absorption characteristics of acoustic metamaterials with different channel thicknesses.According to the calculation results in Figure11, with an increase in the channel thickness, the wavelength of the acoustic wave that can be modulated decreases, and the resonant frequencies move towards higher frequencies. Thus, an increase in the channel layer thickness causes the centre frequency of the absorption peaks to move to a higher frequency.


## 3.3.1. Effect of Channel Width on the Sound Absorption Performance

As the channel width decreases, the air volume inside the structure decreases, and the thermoviscous loss of the structure increases. To make the metamaterials effectively control noise at lower frequencies, the relationship between the channel width and the sound absorption performance of the metamaterials is analysed. The channel widths shown in Figure2(b) are set to 3 mm, 4 mm, and 5 mm for numerical simulations, and the influence of the change in the channel width on the sound absorption performance of the metamaterials is analysed, as shown in Figure 8.Figure 8
Sound absorption characteristics curves of metamaterials at different channel widths.According to Figure8, as the channel width decreases, the refractive index of the metamaterial increases, the wavelength of the acoustic wave that the metamaterial can modulate increases, the resonant frequencies move to lower frequencies, and the absorption peaks decrease. Over 1300–1600 Hz, a new sound absorption peak appears at 1412 Hz when the channel width is 3 mm.To investigate the cause of the absorption peak at 1412 Hz, modal analysis and sound insulation characteristics analysis are conducted for the 3 mm channel width metamaterial. Figure9(a) shows the structural vibration mode of the metamaterial structure at 1412 Hz, and Figure 9(b) shows the sound insulation characteristics curve of the metamaterial structure in the frequency range of 1200–1600 Hz. According to Figure 9(a), an inherent vibration mode of the metamaterial structure appears at 1412 Hz, and according to Figure 9(b), a new absorption peak is formed at 1412 Hz in the sound insulation characteristics curve due to the sound transmission caused by resonance.Figure 9
Sound insulation characteristics curve and vibration mode of metamaterials. (a) Sound insulation characteristics curve. (b) Vibration mode at 1412 Hz.
(a)(b)To investigate why the sound absorption peaks of the metamaterial decrease with decreasing channel width, the centre plane particle velocities of the metamaterials with 4 mm and 3 mm channel widths are analysed, and Figure10 shows the centre plane particle velocity distributions of the 4 mm and 3 mm metamaterials at the peak sound absorption frequency.Figure 10
Centre plane particle velocity distributions of 4 mm and 3 mm metamaterials. (a) 4 mm particle velocity distribution. (b) 3 mm particle velocity distribution.
(a)(b)According to Figure10, a reduction in the channel width leads to a reduction in the acoustic energy entering the interior of the metamaterial, a weakening of the resonance intensity, a decrease in the particle velocity, and a decrease in the structural absorption coefficient.


## 3.3.2. Effect of Channel Layer Thickness on the Sound Absorption Performance

A change in the channel layer thickness affects the air volume inside the structure and has an impact on the sound absorption performance of the metamaterial. To enable the metamaterials to effectively control noise at lower frequencies, the relationship between the channel thickness and the sound absorption performance of the metamaterials is analysed, and the influence of the channel layer thickness on the sound absorption performance of the metamaterials is investigated by setting different channel layer thicknessesb and c for the model structure in Figure 1 while keeping the overall thickness constant at 50 mm. The designed channel layer thicknesses of the metamaterial structure are shown in Table 1.Table 1
Combinations of channel layer thicknesses.
Channel layer thicknessb (mm)Channel layer thicknessc (mm)Combination 1510Combination 2411Combination 3312The sound absorption coefficient analysis is performed for several combinations as shown in Table1, and the results shown in Figure 11 are obtained.Figure 11
Curves of sound absorption characteristics of acoustic metamaterials with different channel thicknesses.According to the calculation results in Figure11, with an increase in the channel thickness, the wavelength of the acoustic wave that can be modulated decreases, and the resonant frequencies move towards higher frequencies. Thus, an increase in the channel layer thickness causes the centre frequency of the absorption peaks to move to a higher frequency.


## 3.4. Influence of the Number of Channel Layers on the Sound Absorption Performance

An increase in the number of coiled channel layers can extend the sound propagation distance, increase the refractive index of the metamaterial, and affect the multiorder resonance mode phenomenon. To enable the metamaterials to effectively control noise at lower frequencies, the relationship between the number of metamaterial channel layers and the sound absorption performance is therefore analysed. Based on the structure of the three-layer second-order coiled channel shown in Figure1, the numbers of coiled channel layers and perforated plate layers are increased to form four-layer and five-layer self-similar fractal coiled acoustic metamaterials with thicknesses of 65 mm and 80 mm, and the sound absorption performance at 30–400 Hz in terms of the sound absorption coefficient curves is shown in Figure 12.Figure 12
Sound absorption coefficient curves of metamaterials with different numbers of layers.According to Figure12, after adding more coiled channel layers, the refractive index of the metamaterial increases, the wavelength of the acoustic wave that can be manipulated by the metamaterial increases, the resonant frequencies move to lower frequencies, the centre frequency of the lowest absorption peak moves from 119 Hz to 89 Hz and 71 Hz, and the centre frequencies of the other absorption peaks all substantially move to lower frequencies.


## 4. Model Experiments

For the form of the metamaterial structure designed in this paper, 3D printing technology is used to prepare the model structure, and a standing wave tube test experiment is used to test the sound absorption performance of the metamaterial structure. The experimental test equipment consists of the SW422 impedance tube test system shown in Figure13. The diameter of the impedance tube is 100 mm, and the frequency range of this test is 80–400 Hz. The test system includes a circular impedance tube, a sound pressure calibrator, a power amplifier, a VA-Lab test system, and a microphone. The test results are measured by the two-microphone transfer function method, and the sound absorption performance curve of the test structure is obtained and analysed.Figure 13
Test system.Two methods are used for 3D printing technology. Stereolithography (SLA) technology and photosensitive resin material are selected for layered printing, as shown in Figures14(a) and 14(b), and fused deposition modelling (FDM) technology and polylactic acid (PLA) material are selected for integrated printing, as shown in Figure 14(c). This technology can ensure a preparation error of 0.1 mm.Figure 14
Overall model and local structure of metamaterials. (a) Silicone rubber glue model. (b) Cyanoacrylate glue model. (c) Overall model.
(a)(b)(c)Since the experimental test equipment used in this paper is a circular impedance tube with a diameter of 100 mm, the length and width of the metamaterial structure are reduced in equal proportion and prepared in a circular structure by 3D printing technology. Figures15(a)–15(d) show the structure of each layer of the metamaterial model. Due to the limitations of the preparation process, two materials and three preparation methods are selected. Since the resin material cannot be printed in an integrated manner, both the first and second models are printed in layers, and the structure of each layer of the metamaterial model shown in Figures 15(a)–15(d) is printed. The first model is connected with soft silicone rubber, the second model is connected with hard cyanoacrylate glue, and the third model is prepared by integrated printing of PLA material. The overall thickness of the metamaterial model structure is 50 mm, the channel width is 3 mm, the thickness of each coiled channel layer is 10 mm, and the thickness of both perforated and closed plates is 5 mm. Figures 14(a)–14(c) show the overall structure and local structure of the third-order metamaterial models. Figure 14(a) shows the overall structure and local structure of the silicone rubber adhesive model; glue overflows when silicone rubber is used as an adhesive. Figure 14(b) shows the overall structure and local structure of the cyanoacrylate glue adhesive model; there are interlayer gaps and model preparation errors when cyanoacrylate glue is used as an adhesive. Figure 14(c) shows the overall structure and local structure of the integrated model; the surface of the PLA integrated model is rougher than those of the other models.Figure 15
Structure of each layer of the model. (a) Centre perforated plate. (b) Coiled channel plate. (c) Perimeter perforated plate. (d) Closure plate.
(a)(b)(c)(d)A multiphysics field coupled model is used to analyse the sound absorption performance of the structure for the reduced size model structures. The sound absorption performance curves from the experimental tests and calculated by the finite element method are shown in Figure16. According to the sound absorption performance curve of the metamaterials in Figure 16, the designed metamaterial structures have low-frequency sound absorption performance, and the experimental and numerical simulation results have the same trend, but the absorption coefficient has some deviation. When the model structure shown in Figure 14(a) is used for the experiments, due to overflowing glue and failure to satisfy rigid boundary conditions, the absorption coefficient in the low-frequency band decreases and is less than that of the cyanoacrylate glue preparation and the integrated preparation, but the effect in the high-frequency band is better than that of the cyanoacrylate glue preparation and the integrated preparation. When the model structure shown in Figure 14(b) is used for the experiments, due to the model preparation errors and the existence of interlayer gaps, the absorption coefficient in the low-frequency band decreases, the absorption peak frequencies shift to higher frequencies compared with the simulation results and other model structures, and the absorption performance in the high-frequency band deteriorates. When the model structure shown in Figure 14(c) is used for the experiments, in the low-frequency band, since the surface roughness is much smaller than the acoustic wavelength, it has little effect on the low-frequency sound absorption, and the experimental effect is the closest to the simulation results. This structure is better than the silicone rubber preparation and cyanoacrylate glue preparation. In the high-frequency band, the sound absorption performance is weaker than that of the silicone rubber preparation and cyanoacrylate glue preparation.Figure 16
Comparison of experimental results and simulation results.


## 5. Summary

To achieve effective control of low-frequency noise, this paper combines microperforated plates, Hilbert fractals, and coiled acoustic metamaterials to design a metamaterial structural form suitable for low-frequency noise control. The designed acoustic metamaterial model is analysed for its sound absorption performance. According to the analysis results, the sound absorption performance of the metamaterial structure is influenced by the self-similar fractal order, channel width, channel thickness, and number of coiled channel layers. The sound absorption performance of the experimental models was tested using the standing wave tube test method. According to the results, the designed metamaterial structure has a lower sound velocity and a higher refractive index than the background medium, which can produce a multiorder resonance mode effect, and has good low-frequency sound absorption performance. The metamaterials still possess a good sound absorption effect at low frequencies, which provides a means for ultralow-frequency noise control and a new direction for subsequent low-frequency noise control research.



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*Source: 2083563-2022-03-26.xml*