© The Author(s), Published by EDP Sciences, 2024

1 Introduction

Thin film acoustic metamaterial is a kind of acoustic control material with special structure and performance, which has the ability to absorb acoustic waves efficiently in low frequency range [1]. In many fields, such as architecture, automobiles, aerospace and so on, the control and absorption of low-frequency sound waves has become an important issue. In the low frequency range [2–4], the frequency of sound wave is low and the wavelength is long, so the corresponding material structure is needed to adapt to this wavelength. Generally speaking, for acoustic wave absorbing materials [5], when the absorption frequency is low, the traditional thickness and porosity affected by noise [6] often cannot achieve effective absorption effect. Thin film acoustic metamaterial [7, 8] is a new solution, which can effectively solve the issue of absorbing low-frequency sound waves. This material is usually composed of multilayer films, each of which has specific acoustic characteristics [9–11]. By adjusting the thickness, porosity, contact area and other parameters of the film, the low-frequency attenuation signal can be absorbed efficiently. In order to achieve efficient absorption of low-frequency sound waves, researchers began to pay attention to thin-film acoustic metamaterials. These materials have special structures and complex acoustic characteristics, and the absorption of low-frequency attenuation signals can be realized by adjusting the parameters such as the thickness, porosity and contact area of the materials.

In the aspect of low-frequency attenuation signal absorption, many scholars have made research. Goldsberry et al. proposed the use of equal set analysis to simulate the propagation law of sound waves in acoustic metamaterials. By studying the influence of acoustic metamaterials’ parameters on sound absorption performance, it was found that the sound absorption coefficient of this acoustic metamaterial was insensitive to the changes of material thickness and aperture under the action of airflow [12]. Malléjac et al. designed an acoustic metamaterial with a density close to zero, and analyzed the hidden space in the material through experiments. After adjusting the density, the maximum absorption bandwidth of 0.8 can be widened to 188% [13]. Ramírez-Solana et al. changed the stiffness of Helmholtz resonator (HR) and designed elastic HR tubes arranged periodically. Their research found that the resonant frequency of elastic HR was much lower than that of rigid HR for a single Helmholtz cavity unit with the same size [14]. Raval et al. proved the anti-parallel diode function of one-dimensional nonreciprocal acoustic metamaterials, and verified the key function of the diode through experiments [15]. In addition, they also show that the increase of the ratio of depth to area and cross section of the resonant cavity leads to the decrease of the resonant frequency of the system.

In the performance analysis of thin film acoustic metamaterials, this paper adopts a finite element numerical simulation method based on implicit dynamic analysis. This method utilizes ANSYS software to discretize the sound wave field and stress field, accurately capturing the absorption process of low-frequency attenuation signals, while considering the interaction between the film and air and nonlinear vibration behavior. By adjusting parameters such as film thickness, density, and tension, the absorption effect of thin film acoustic metamaterials on low-frequency attenuation signals is calculated, and the absorption changes of low-frequency attenuation signals under various influencing factors are verified.

2 Materials and methods

2.1 Preparation of thin film acoustic metamaterials

2.1.1 Raw materials and equipment

Silica gel: Shenzhen Silica Gel Co., Ltd.; Curing agent: Sichuan Fangchen Chemical Co., Ltd.; Quality block: Tianjin Xina Intelligent Technology Co., Ltd.; Frame: Cangzhou Zhongjian Precision Instrument Co., Ltd. Spin coater: Beijing Tongde Venture Technology Co., Ltd.

2.1.2 Preparation process

In order to study the properties of thin-film acoustic metamaterials, it is necessary to ensure the integrity of the thin film, and there can be no large number of bubbles or uneven thickness, so it is necessary to strictly control the film processing links. The following is the film processing process:

1. Weighing silica gel and curing agent according to the weight ratio of silica gel to curing agent = 100:3;

2. Choose a regular and smooth flat-bottomed container as the device for stirring silica gel and curing agent, choose a 100 mL measuring cup as the container, and use flat and wrinkle-free sulfuric acid paper as the substrate. Weighing a certain weight of silica gel, adding curing agent according to the weight ratio, fully stirring and standing for 5 min, and then pouring the glue solution onto the sulfuric acid paper substrate;

3. Turn on the power supply, set the low gear to 500 r/min for 60 s and the high gear to 600 r/min for 30 s, and start the spin coater for spin coating.

4. Take out the spin-coated film and let it stand for a period of time. After the surface is dried, peel off the film from the sulfuric acid paper substrate to obtain the film needed for the test.

5. A weight block is positioned at the center of both the front and back sides of the processed flexible membrane, which constitutes the membrane acoustic metamaterial to be studied in this paper.

6. Installing a metal frame on the outer side of the film to obtain the film-type acoustic metamaterial.

2.2 Test methods

2.2.1 Test parameter setting

(1) Basic law of low frequency attenuated signal absorption

The specific steps for absorbing low-frequency attenuated signals are as follows:

1. Definition of low-frequency attenuation signal absorption: The low-frequency attenuation signal absorption (SS) is calculated using its definition formula, which represents the portion of sound waves absorbed after passing through the material.

2. Identify key parameters: Determine the key parameters that affect the absorption of low-frequency attenuated signals, including the mass of the unit area absorption structure, air characteristic impedance, structural characteristic impedance, etc. And obtain the specific values of these parameters according to the formula.

3. Calculation result analysis: Based on the law of mass action and the definition of low-frequency attenuation signal absorption, calculate the low-frequency attenuation signal absorption.

Before using finite element model to analyze the performance of thin-film acoustic metamaterials, the basic law of low-frequency attenuation signal absorption is clarified. Generally, the sound intensity transmission coefficient is not used when dealing with the problem of low-frequency attenuation signal absorption t₁ [16], but with its reciprocal

$$L=10\lg \frac{{\eta }_1}{t_1}.$$(1)

In the process of high-frequency oscillation [17], the middle layer of air is generally solid material, so their characteristic impedance is much larger than that of air, so it exists L ≤ 1. Considering that the thickness of thin-film acoustic metamaterial and the wavelength of sound waves, the law of mass action is defined by combining the absorption of low-frequency attenuation signal as shown in Formula (2):

$$T=\frac{\lg {\left(\omega M/2\mathrm{R}\right)}^2}{L}.$$(2)

In the formula (2), M is absorb the mass of the structure for the unit area, R is that characteristic impedance of air, ω is the structural characteristic impedance.

(2) Calculation of acoustic parameters

The specific steps for obtaining acoustic parameters are as follows:

1. Clarify acoustic parameters: Determine the acoustic parameters that need to be calculated, including reflection coefficient, absorption coefficient, and acoustic resistivity.

2. Clarify the relationship between various acoustic parameters: the absorption coefficient is defined as the absorbed energy divided by the incident sound energy, and the reflection coefficient is defined as the amplitude of the reflected sound wave divided by the amplitude of the incident sound wave, etc.

3. Result analysis: Based on the definitions and relationships of various acoustic parameters, establish corresponding calculation formulas to calculate acoustic parameters such as reflection coefficient, absorption coefficient, and acoustic resistivity.

Four parameters have an impact on the performance of thin-film acoustic metamaterials, as prescribed by the law of mass action: the absorption of low-frequency attenuation signals, the mass of absorbing structure per unit area, the air characteristic impedance and the structural characteristic impedance. Among them, the proper design and processing of low-frequency attenuation signal absorption can increase the absorption ability of acoustic metamaterials to low-frequency signals. The quality of the unit area absorption structure directly influences the sound absorption effectiveness of the overall material. The characteristic impedance of air can adjust the characteristic impedance between material and air, thus affecting the reflection and absorption of sound. Structural characteristic impedance can adjust the structural characteristic impedance of materials and provide effective absorption and attenuation of low-frequency signals. Therefore, when exploring the performance of thin-film acoustic metamaterials [18], this paper chooses reflection coefficient, sound absorption coefficient and acoustic resistivity to describe its sound absorption effect.

It is considered that when the sound waves in the air are incident on the surface, part of the sound energy is reflected by the surface of the material or structure, and the remaining part propagates inside the material or structure, and then is absorbed. Therefore, this paper defines the sound absorption coefficient as the ratio of absorbed energy to incident sound energy, denoted by α, and is calculated by Formula (3):

$$\alpha =\left(E_i-E_r\right)E_i.$$(3)

Among them, E_i and E_r respectively expressed as incident acoustic energy and reflected acoustic energy.

In the process of sound absorption of materials [19], the reflection coefficient is defined as the ratio of the amplitude of reflected sound waves to the amplitude of incident sound waves, which is expressed by Formula (4):

$$r=p_r/p_i.$$(4)

In the formula (4), p_i and p_r respectively representing the amplitudes of the reflected sound wave. The relationship between sound absorption coefficient and reflection coefficient is shown in Formula (5):

$$\alpha =1-r.$$(5)

Formula (6) is acoustic impedance and the definition of Z:

$$Z=p\left(x\right)/u\left(x\right).$$(6)

In the formula (6), p(x) and u(x) respectively indicating that in the medium sound pressure and particle velocity at x. The relationship between acoustic impedance ratio Z and reflection coefficient r is Formula (7):

$$Z=(1+r)/(1-r)\mathrm{*T}$$(7)

It is not simply necessary to conduct numerical research by changing one characteristic parameter at a time, but to conduct comprehensive parameter research on the model, because the low-frequency attenuation signal absorption performance of acoustic metamaterials is affected by multiple parameters such as porosity, thickness, density, size, tension, frame material, and contact area between the mass block and the film, and there may be interactions between these parameters. Conducting comprehensive parameter research can systematically evaluate the impact and interaction of various parameters on performance, ensuring the comprehensiveness and reliability of the results. In addition, this multi parameter research method provides rich data support for subsequent numerical optimization. By adjusting the values of each parameter, better performance combinations can be explored to improve the low-frequency attenuation signal absorption performance of the material.

2.2.2 Test preparation

Using the reflection coefficient, absorption coefficient, and acoustic resistivity obtained above as inputs, the ANSYS finite element analysis software is used to simulate the finite element model and complete the experimental preparation. The side length of the thin-film acoustic metamaterial prepared in this paper has been fixed, but when the model is used for analysis, it should be adjusted in the finite element software according to the requirements of experimental simulation. The thickness of the thin-film acoustic metamaterial prepared in this paper is also adjusted and set according to the requirements of experiments. Figure 1 illustrates the structure of the thin-film acoustic metamaterial, where a frame is securely attached to the outer surface of the thin film, with the mass block positioned at the center.

Figure 1 Structure of thin film acoustic metamaterials.

Membrane acoustic metamaterial is a composite structure composed of supporting frame, membrane and additional mass. In practical application, supporting conditions and configuration methods have an important influence on its acoustic characteristics. When considering supporting strips and adopting complex configuration, analytical method is difficult to deal with effectively, and other calculation methods need to be developed. The finite element method and a large number of finite element analysis software provide convenience for the finite element analysis of. In order to accurately simulate the vibration and absorption characteristics of thin film acoustic metamaterials during the construction of finite element models, the following boundary conditions were set:

1. Fixed constraints: Set fixed constraints around the film and aluminum frame to ensure the stability of the structure during simulation and avoid unnecessary vibration interference.

2. Radiation boundary conditions: During simulation, radiation boundary conditions are set for the incident and exit ends, which simulate the propagation characteristics of sound waves in air and ensure the accuracy of the calculation of the incident and reflected energy of sound waves.

The finite element model mainly includes four types of elements: (1) Fluid elements composed of air layers; (2) A solid unit consisting of a mass block and a supporting structure; (3) Nonlinear element composed of thin film structure; (4) Fluid-solid coupling unit [20] at the interface between fluid and structure (including film). When constructing a model of thin film acoustic metamaterials in ANSYS, this paper utilizes interface elements and contact pairs to achieve coupling connections between different types of components, especially nonlinear components. The interface unit is defined at the material interface to handle the acoustic interaction between fluid and solid, while the contact pair simulates the mechanical behavior between surfaces that may separate but come into contact. For the connection between nonlinear thin films and surrounding solid structures, if deformation remains continuous, displacement and stress continuity can be achieved through shared nodes. If there is a possibility of separation, the contact pair algorithm is used to calculate and apply the contact force as the boundary condition to ensure that the model accurately reflects the actual physical interaction. The specific finite element model is shown in Figure 2.

Figure 2 Finite element model.

The model mesh is divided into a free mesh tetrahedral mesh. Due to the presence of quality blocks, the grid in the central area is relatively dense to ensure the accuracy of calculations. The density of the specific mesh is automatically adjusted according to the finite element software, but it is ensured that there is sufficient mesh density in critical areas such as mass blocks. For thin film acoustic metamaterials, the model includes complex structures such as fluid elements, solid elements, nonlinear elements, and fluid solid coupling elements, and the model has been finely meshed, with a relatively high number of degrees of freedom solved at 100,000 degrees of freedom.

In order to analyze the vibration form of the thin film metamaterial, the fixed constraints are set around the thin film and aluminum frame, and the radiation boundary conditions at the entrance and exit ends are set during the simulation. The grid is divided into free tetrahedral grids, and the results of model grid division are shown in Figure 3.

Figure 3 Thin-film acoustic metamaterial meshing.

Detailed calculation is carried out by using the finite element model of thin film acoustic metamaterial. In Figure 3, the grid in the central area is dense, mainly because the area is a mass block, so the grid is dense.

2.2.3 Test process

The finite element model is employed to analyze the following test procedure:

(1) Investigation of the absorption effect of thin film acoustic metamaterials with varying porosity.

The porosity of the thin-film acoustic metamaterial is generally in the range of 70%–95%. In the finite element model, the porosity of the thin-film acoustic metamaterial is calculated as the ratio of the total volume of internal pores to the overall volume of the entire structure, and is calculated by Formula (8):

$$K=V_a/V_m.$$(8)

In the formula (8), K represents porosity, V_a is the total pore volume of the thin-film acoustic metamaterial, V_m represents the total volume of thin-film acoustic metamaterials. In the finite element software, the changes of absorption coefficient of thin-film acoustic metamaterials prepared in this paper are simulated at 75%, 85% and 95% porosity to absorb low-frequency attenuation signals. In the experiment, the porosity of thin film acoustic metamaterials is precisely controlled. By monitoring and calculating the ratio of the total pore volume to the total volume of the material structure, the preparation process parameters such as the ratio of silica gel to curing agent, spin coating speed, and time are adjusted to ensure that the pore distribution is uniform and meets the preset values, thereby achieving precise control of porosity.

(2) Absorption effect with different thickness

The thickness will also affect the effect of absorbing low-frequency attenuation signals. The finite element software is used to simulate the absorption coefficient of absorbing low-frequency attenuation signals when the thickness of thin-film acoustic metamaterials prepared in this paper is 5 mm, 8 mm and 11 mm respectively, and the simulation results are analyzed in depth.

(3) Absorption effect with different densities

The density of thin-film acoustic metamaterials directly impacts their absorption effectiveness. The absorption curves of low-frequency attenuation signals prepared in this paper are simulated with finite element software when the film surface densities are 0.15 kg/m³, 0.25 kg/m³ and 0.35 kg/m³, respectively.

(4) The absorption effect of thin film acoustic metamaterials with different size.

Under the premise of not changing other parameters, the finite element software is employed to simulate the change of absorption of low-frequency attenuation signals when the side lengths of the thin-film acoustic metamaterials prepared in this paper are 16 mm, 21 mm and 26 mm respectively, and the simulation test results are calculated by computer.

(5) Absorption effect of thin film acoustic metamaterials with different tension.

Without changing other parameters in the simulation test of thin-film acoustic metamaterials, the finite element software is employed to simulate the effect of the thin-film acoustic metamaterials prepared in this paper when the tension is 110 N/m, 130 N/m and 160 N/m respectively, and the change curve is drawn by computer statistical test results.

(6) The influence of parameter errors on the absorption effect of thin-film acoustic metamaterial.

The membrane in the membrane acoustic metamaterial is nonmetallic, and its Young’s modulus and Poisson’s ratio are usually obtained by bending method, indentation method and expansion method, and the measurement accuracy of different test methods is different. Because the material parameters of the film need to be input in the calculation of the absorption of low-frequency attenuation signals, it is necessary to explore the influence of different values of the film materials on the calculation results of the absorption of low-frequency attenuation signals.

Young’s modulus values of thin-film acoustic metamaterials are set to 3.5e9 GPa, 2.1e9 GPa and 1.1e9 GPa, respectively, and the changes of low-frequency attenuation signal absorption of thin-film acoustic metamaterials under different Young’s modulus are simulated by finite element software.

Set the Poisson’s ratio of thin-film acoustic metamaterials to 0.25, 0.35 and 0.45 respectively, and use finite element software to simulate the change of low-frequency attenuation signal absorption of thin-film acoustic metamaterials under different Poisson’s ratios.

(7) The influence of frame structure on the absorption effect.

Frame structure is a key component of thin-film acoustic metamaterials, and photosensitive resin, aluminum, steel, etc. are generally selected. These materials belong to hard materials, among which photosensitive resin is a lightweight material with low density and is suitable for automobiles. The basic parameters of frame materials are shown in Table 1.

The data in Table 1 are input into the finite element software to simulate the absorption of three different frame materials for the low-frequency attenuation signals of the thin-film acoustic metamaterials prepared in this paper.

(8) The influence of the contact area between the mass and the membrane on the absorption effect of thin-film acoustic metamaterials.

According to the analysis of previous modes, in the first mode, the mass drives the membrane to vibrate up and down, so the contact area between the mass and the membrane will also affect the sound insulation of the unit. Here, three kinds of mass blocks are designed in this paper (to make the volume of each mass block equal), and the contact area with the film is increasing on the premise of ensuring the equal mass (the contact areas are 3.5 mm², 4.5 mm² and 5.5 mm² respectively). Among them, the mass of A2 mass block is slightly smaller than that of the other two mass blocks, but the extremely small 0.02 mm³ can be ignored. The finite element software is used to calculate the absorption of low-frequency attenuation signals by thin-film acoustic metamaterials under three different contact areas.

3 Result analysis

3.1 Low-frequency attenuation signal absorption results of thin-film acoustic metamaterials with different porosity

By independently changing specific parameters such as porosity, thickness, density, size, or tension of thin film acoustic metamaterials while keeping other parameters constant, it is possible to precisely adjust the relevant process conditions of this parameter during the preparation process, while fixing all other parameters constant. Then, a finite element model can be used for simulation analysis to separately evaluate the specific impact of this parameter change on the low-frequency attenuation signal absorption performance. The finite element simulation results of absorption coefficient of thin-film acoustic metamaterials absorbing low-frequency attenuation signals under three different porosities are shown in Figure 4.

Figure 4 Simulation results of absorption coefficient under different porosity.

As can be seen from the curve changes in Figure 4, with the increase of the porosity of the thin-film acoustic metamaterial prepared in this paper, the absorption coefficient of the same structure gradually increases, and the absorption coefficient of the thin-film acoustic metamaterial shows a rapid rising trend in the low frequency band (30–110 KHz). After the frequency increases, the absorption coefficient of the low-frequency attenuation signal tends to be stable. For the same structure, the increase of the porosity of the thin-film acoustic metamaterial will lead to a slight improvement in the low-frequency sound absorption performance of the material, which is greatly improved in the slightly higher frequency range. When the porosity is 95%, the effect of absorbing the low-frequency attenuation signal is the best. For the same homogeneous thin-film acoustic metamaterial, when other parameters are unchanged, the increase of porosity adaptation will lead to the improvement of the overall low-frequency attenuation signal absorption performance of the material, because the increase of porosity makes the internal structure of porous sound-absorbing materials more complicated, and the number of collisions between low-frequency attenuation signals and the surface of the holes increases during the process of penetrating into the material along these pores, which leads to more friction between the low-frequency attenuation signals and the inside of the structure and converts acoustic energy into heat energy; At the same time, when the low-frequency attenuation signal propagates in the material structure, it will reflect and refract many times, which will strengthen the vibration of the hole wall and increase the interference noise elimination. It is worth noting that the porosity should be controlled within a certain range, too large or too small will affect its low-frequency attenuation signal absorption performance. Some studies show that when the porosity exceeds a certain limit, the average absorption coefficient of thin-film acoustic metamaterials will decrease, because when the porosity is too large, the interior of the material structure is too sparse, and the low-frequency attenuation signal is not easy to reflect many times when it propagates inside, and more acoustic energy will directly pass through the material; When the porosity of the material is too small, the material will become dense if the flow resistance is too large. Generally speaking, the permeability of the material will decrease, and its low-frequency attenuation signal absorption performance will also decrease. Therefore, this paper should reasonably design the material structure and choose a better porosity according to the specific frequency spectrum requirements, so as to make its low-frequency attenuation signal absorption performance optimal.

3.2 Low-frequency attenuation signal absorption results of thin-film acoustic metamaterials with different thicknesses

In the finite element software, the changes of absorption coefficient of 5 mm, 8 mm and 11 mm thin film acoustic metamaterials with low frequency attenuation are simulated, and the test results are shown in Figure 5.

Figure 5 Effect of thin film acoustic metamaterials of different thicknesses on absorption coefficients of low frequency attenuated signals.

With the increase of the thickness of the thin-film acoustic metamaterial prepared in this paper, the sound absorption coefficient of the same sound absorption structure in low frequency band (30 KHz–110 KHz) increases gradually. In addition, looking at each structure separately, the absorption coefficient of the highest low-frequency attenuation signal gradually increases with the increase of the thickness and tends to shift to the low frequency direction. When the frequency changes slightly, the absorption performance of the low-frequency attenuation signal is greatly improved, and there are many peaks and valleys in the absorption coefficient change curve. This is because with the increase of the thickness of the material, the propagation distance of the incident sound wave in the material increases. Increasing the thickness of the thin-film acoustic metamaterial prepared in this paper can improve the absorption effect incident sound waves and ensure the absorption performance of high-frequency sound waves. It is easy to draw the conclusion from Figure 5 that the influence of material thickness on the absorption effect is greater than that of porosity, because the increase of material thickness is equivalent to the extension of the length of pore channels, and the folding times of low-frequency attenuation signals penetrating into the material along these pores are greatly increased, resulting in greater acoustic energy loss, so the increase of thickness is obviously beneficial to improving the absorption performance of low-frequency attenuation signals of thin-film acoustic metamaterials. Of course, this does not mean that the thicker the material is, the better the sound absorption effect will be. When the thickness of the thin-film acoustic metamaterial increases to a certain length, the low-frequency attenuation signal absorption performance of the material is stable, that is, the average low-frequency attenuation signal absorption coefficient increases gradually, reaching the limit low-frequency attenuation signal absorption coefficient. Continuing the thickness of the thin-film acoustic metamaterial, the improvement of the low-frequency attenuation signal absorption performance is no longer obvious. In addition, the increase of the thickness increases the total volume of the structure, which has limitations in some practical applications.

3.3 Low-frequency attenuation signal absorption results of thin-film acoustic metamaterials under density difference

The effect curve of absorbing low-frequency attenuation signals under different densities of thin-film acoustic metamaterials is shown in Figure 6.

Figure 6 Illustrates the impact of various densities of thin-film acoustic metamaterials on the absorption of low-frequency attenuated signals.

Figure 6 illustrates the sound insulation curves of the thin-film acoustic metamaterials prepared in this study when other parameters remain constant, with film areal densities of 0.15 kg/m³, 0.25 kg/m³ and 0.35 kg/m³ respectively. From the Figure 6, it can be observed that as the density of thin-film acoustic metamaterials increases, the first absorption valley frequency (f1) and absorption peak frequency (f2) of the entire material change minimally, while the second absorption valley frequency (f3) shifts towards lower frequencies. Based on the previous analysis, it is evident that the absorption valley at f3 is attributed to the negligible vibration amplitude of the thin-film acoustic metamaterial at the location of additional mass, whereas the vibrating system comprising the remaining part of the thin-film acoustic metamaterial (with its own mass and tension force) resonates under the excitation of incident sound waves. This resonance phenomenon can be approximated as an elastic structure resonance problem involving a region (at the location of additional mass) and a boundary with fixed constraints.

3.4 Low-frequency attenuation signal absorption results of thin-film acoustic metamaterials under scale differences

The absorption of low-frequency attenuation signals by thin-film acoustic metamaterials with different sizes is shown in Figure 7.

Figure 7 Absorption results of low frequency attenuation signal of thin film acoustic metamaterials of different sizes.

From Figure 7, it can be observed that with an increase in the dimensions of the thin-film acoustic metamaterials prepared in this study, the overall sound insulation performance of the acoustic metamaterial rapidly decreases. This is attributed to the fact that the low-frequency sound insulation of the thin-film acoustic metamaterial arises from the local resonance caused by the “spring-mass” system composed of the additional mass and the film. When the area of the thin-film acoustic metamaterial increases, the proportion of the localized resonance region decreases, leading to a decline in the absorption performance of low-frequency attenuated signals. It can be imagined that when the thin-film acoustic metamaterial becomes infinitely large, the impact of the localized resonance caused by the additional mass on the overall sound insulation of the film can be negligible.

3.5 Low-frequency attenuation signal absorption results of thin-film acoustic metamaterials under different tension

In the finite element software, the effects of absorbing low-frequency attenuation signals by thin-film acoustic metamaterials with tensile forces of 110 N/m, 130 N/m and 150 N/m are simulated, and the test results are shown in Figure 8.

Figure 8 Absorption results of low frequency attenuation signal of thin film acoustic metamaterials under tension difference.

From Figure 8, it can be observed that, with other parameters remaining constant, the sound insulation curves of the thin-film acoustic metamaterials are plotted for different tension forces applied to the film, namely 110 N/m, 130 N/m, and 160 N/m. From the graph, it is evident that as the tension force applied to the thin-film acoustic metamaterial increases, the characteristic frequencies of the entire material shift towards higher frequencies.

3.6 Influence of parameter error on absorption effect of thin-film acoustic metamaterials

Young’s modulus is set to 3.5e9 GPa, 2.1e9 GPa and 1.1e9 GPa respectively, and the absorption of low-frequency attenuation signals of thin-film acoustic metamaterials under these three Young’s modulus is simulated by finite element software. The test results are shown in Figure 9.

Figure 9 Absorption of low-frequency attenuated signal under different Young’s elastic modulus.

As can be seen from Figure 9, the reduction of Young’s modulus of the thin film material prepared in this paper will lead to the sound insulation curve of the thin film acoustic metamaterial moving to the low frequency, and the bandwidth at the absorption peak of the low frequency attenuation signal will be narrowed. Young’s modulus is reduced from 3.5e9 GPa to 1.1e9 GPa, the frequency of sound insulation peak is moved from 30 Hz to 110 Hz, and the bandwidth with sound insulation volume greater than 45 dB is also greatly reduced.

Set Poisson’s ratios to 0.25, 0.35 and 0.45 respectively, and use finite element software to simulate the absorption of low-frequency attenuation signals of thin-film acoustic metamaterials under these three Poisson’s ratios.

The influence of different Poisson’s ratios on the sound insulation capacity is shown in Figure 10. As can be seen from Figure 10, with the increase of Poisson’s ratio from 0.25 to 0.35, the absorption curves of low-frequency attenuation signals of thin-film acoustic metamaterials almost coincide, with a peak value of 65 dB, and the whole curve slightly shifts to high frequency.

Figure 10 Absorption of low-frequency attenuated signals at different Poisson ratios.

To sum up, both Young’s modulus and Poisson’s ratio of thin-film acoustic metamaterials have great influence on the low-frequency attenuation signal absorption of thin-film acoustic metamaterials. Therefore, it is very important to obtain accurate material Young’s modulus and Poisson’s ratio for calculating the absorption of low-frequency attenuation signals.

3.7 Influence of frame material on absorption effect

Considering the material of the frame, when steel, aluminum and photosensitive resin are selected as the frame respectively, the absorption capacity of thin-film acoustic metamaterials to absorb low-frequency attenuation signals is calculated, and the absorption capacity curve is shown in Figure 11.

Figure 11 Absorption of low frequency attenuation signal by frame material selection.

As can be seen from Figure 11, under the premise of ensuring the rigidity of the frame and providing sufficient rigidity, the difference of frame materials has little influence on the absorption of low-frequency attenuation signals of the unit. The sound insulation curves of the three frame materials are almost identical in the required frequency band, but slightly different at the sound insulation peak. At 30 Hz, the sound insulation capacity of photosensitive resin is about 30 dB, that of aluminum is 29 dB, and that of steel is 31 dB. However, the sound insulation in other frequency bands of the absorption curve of low-frequency attenuation signal is that steel is slightly larger than aluminum, and aluminum is slightly larger than photosensitive resin, and the difference is within 1 dB. Therefore, under the premise of meeting the stiffness, the frame can be made of lighter materials.

3.8 Analysis of the influence of the contact area between the mass and the membrane on the absorption effect

The effect of thin-film acoustic metamaterials on absorbing low-frequency attenuation signals under three contact areas of 3.5 mm², 4.5 mm² and 5.5 mm² is studied, and the absorption results are simulated by finite element software. The absorption curve is shown in Figure 12.

Figure 12 Absorption effect of mass block contact area on low frequency attenuated signal.

From Figure 12, it can be seen that as the contact area between the mass block and the film increases, the first sound insulation valley of the film type acoustic metamaterial absorption curve hardly moves, while the sound insulation in the other frequency bands moves towards high frequencies, and the sound insulation peak and absorption bandwidth greater than 20 dB slightly increase. From this, it can be seen that increasing the contact area between the mass block and the film is beneficial for low-frequency attenuation signal absorption, provided that the mass of the mass block remains unchanged.

The specific results obtained from the experiment are shown in Table 2.

4 Conclusion

The porosity, thickness, density, size and tension of thin-film acoustic metamaterials will affect the absorption effect of thin-film acoustic metamaterials to absorb low-frequency attenuation signals. Through calculation and experimental analysis, the most practical porosity, thickness, density, size and tension that are most suitable for thin-film acoustic metamaterials to absorb low-frequency attenuation signals are determined. The influence factors of absorption capacity of thin-film acoustic metamaterials are analyzed. The analysis shows that the frame mainly plays a supporting role, but has little influence on sound insulation. The thin film is the main influence factor of sound insulation of metamaterials, and the contact area between mass blocks and the film and the number of mass blocks will also affect the ability of metamaterials to absorb low-frequency attenuation signals. The incidence angle of sound waves can improve the sound insulation effect of metamaterials as a whole. The conditions for achieving the optimal low-frequency attenuation signal absorption performance of thin film acoustic metamaterials include a porosity of 95%, a thickness of 11, a variable length of 16 mm, a tension of 160 N/m, and a contact area between the mass block and the film of 5.5 mm². These conditions not only optimize the material’s absorption effect on low-frequency sound waves, but also demonstrate the possibility of achieving efficient sound wave control by adjusting the material’s microstructure and parameters. These research results are crucial for promoting the development of acoustic metamaterials, providing scientific basis and practical guidance for designing more efficient sound absorbing materials, and further expanding the application prospects of acoustic metamaterials in fields such as architecture, automotive, aerospace, etc.

In the future, work will be carried out in the following research directions:

1. Dynamic characteristics and stability analysis: In the future, we will delve into the performance stability of thin film acoustic metamaterials under long-term use or dynamic loads, including fatigue life, durability testing, etc., to evaluate their long-term reliability in practical engineering applications.

2. Optimization design and exploration of new materials: Based on current research results, further optimization of parameters such as pore structure, material composition, and thickness of thin films will be carried out in the future to improve the absorption efficiency and bandwidth of low-frequency attenuation signals. Meanwhile, exploring the potential application of new high-performance materials in thin film acoustic metamaterials.

3. Multi band and wideband absorption research: In the future, the working frequency range of thin film acoustic metamaterials will be expanded to achieve sound wave absorption from low frequency to high frequency and even ultra wideband, in order to meet the needs of different application scenarios.

Acknowledgments

National Natural Science Foundation of China (51675350); Scientific Research Project of the Education Department of Liaoning Province (202007141); Project supported by the Natural Science Foundation of Liaoning Province (2019-MS-176).

Conflicts of interest

There are no conflicts to declare.

Data availability statement

All data that support the findings of this study are included within the article.

Ethical compliance

There are no researches conducted on animals or humans.

References

All Tables

All Figures

	Figure 1 Structure of thin film acoustic metamaterials.
In the text

	Figure 2 Finite element model.
In the text

	Figure 3 Thin-film acoustic metamaterial meshing.
In the text

	Figure 4 Simulation results of absorption coefficient under different porosity.
In the text

	Figure 5 Effect of thin film acoustic metamaterials of different thicknesses on absorption coefficients of low frequency attenuated signals.
In the text

	Figure 6 Illustrates the impact of various densities of thin-film acoustic metamaterials on the absorption of low-frequency attenuated signals.
In the text

	Figure 7 Absorption results of low frequency attenuation signal of thin film acoustic metamaterials of different sizes.
In the text

	Figure 8 Absorption results of low frequency attenuation signal of thin film acoustic metamaterials under tension difference.
In the text

	Figure 9 Absorption of low-frequency attenuated signal under different Young’s elastic modulus.
In the text

	Figure 10 Absorption of low-frequency attenuated signals at different Poisson ratios.
In the text

	Figure 11 Absorption of low frequency attenuation signal by frame material selection.
In the text

	Figure 12 Absorption effect of mass block contact area on low frequency attenuated signal.
In the text