A lightweight low-frequency sound insulation membrane-type acoustic metamaterial

A novel membrane-type acoustic metamaterial with a high sound transmission loss (STL) at low frequencies (⩽500Hz) was designed and the mechanisms were investigated by using negative mass density theory. This metamaterial’s structure is like a sandwich with a thin (thickness=0.25mm) lightweight flexible rubber material within two layers of honeycomb cell plates. Negative mass density was demonstrated at frequencies below the first natural frequency, which results in the excellent low-frequency sound insulation. The effects of different structural parameters of the membrane on the sound-proofed performance at low frequencies were investigated by using finite element method (FEM). The numerical results show that, the STL can be modulated to higher value by changing the structural parameters, such as the membrane surface density, the unite cell film shape, and the membrane tension. The acoustic metamaterial proposed in this study could provide a potential application in the low-frequency noise insulation.


I. INTRODUCTION
Lightweight materials are often used in aerospace and automotive industry, 1 however, limited by the mass law, these materials have a fatal disadvantage.The low frequency sound insulation performance is very poor.In order to reduce the low frequency noise, a relatively thicker material is needed for the traditional sound insulation material.The optimal design of lightweight and high STL of the material would therefore usually require design trade-offs.
The concept of band-gap materials has been examined to deal with the difficult problems of noise insulation, especially the locally resonant crystal proposed by Liu et al. 2 Those materials showed a high STL at low frequencies, which is attributed to the negative effective elastic constants during the vibration.Subsequently, a great many locally resonant sonic materials were investigated by many researchers, [3][4][5][6][7][8][9][10][11] and some achievements have been obtained, but seldom have these materials been used in engineering applications due to the heavy weight.Recently, the membrane-type acoustic metamaterials were proposed, [12][13][14][15][16][17][18] which had broadened the realm of elastic wave characteristics achievable by phononic crystals.By attaching an adjustable small mass block onto the membrane with clamped boundaries, a negative mass density can be obtained at a specific frequency band, thus realizing the total reflection of low-frequency sound and breaking the mass law. 12The effects of the membrane and mass properties on the transmission loss have been investigated, [13][14][15][16][17][18] and the sound insulation at low frequencies can be tuned by varying the acoustic metamaterial structure.Layered on no-mass-attached membrane structures with negative mass density below a cut-off frequency have been demonstrated by Lee et al. 19 Yao et al. 20 pointed out that a rectangular solid waveguide with clamped boundary condition could get negative dynamic mass below the first natural frequency of flexural wave, and then elastic metamaterials with broadband negative-mass density were designed.Varanasi et al. 21roposed a panel consisting of an array of cellular unit structures without undue mass penalty which possess a high STL at low frequencies.Sui et al. 22 studied a lightweight yet sound-proof honeycomb acoustic metamaterial which has a good performance for the low frequencies sound proof and his works enlighten the research of this paper.
During the present work, a novel membrane-type acoustic metamaterial was studied.Compared to Sui's works, minor changes have been made in the structure.In order to fix the membrane better, the latex rubber is clamped between two layers of honeycomb plates.This structure can ensure that the membrane has the enough vibration space and it is convenient for users to install.Subsequently, the mechanisms of low-frequency sound insulation of this acoustic metamaterial were successfully investigated by using negative mass density theory, and the low-frequency sound insulation performance of this acoustic metamaterial can be improved by enlarging the negative-mass band.At last, the effects of different structural parameters of the membrane on the sound-proofed performance at low frequencies were discussed by using finite element method and the results were discussed in detail.The results could provide a possible research method for this type acoustic metamaterial.

A. Sample construction
Figure 1 shows the novel acoustic metamaterial proposed in this paper.The latex rubber is clamped between two layers of honeycomb plate.The honeycomb core, as the role of the rigid frame, is made from acrylonitrile-butadiene-styrene copolymer and the static density is ρ h =1040kg/m 3 .The dimensions are h z =10mm, l=3.65mm, t=0.07mm.This structure of honeycomb core makes its out-of-plane effective Young's modulus is considerably greater than the isotropic in-plane moduli. 23The membrane material, as the roles of mass and stiffness, is latex rubber with thickness h m =0.25mm and the mass density is ρ m =1000kg/m 3 .The Young's modulus and Poisson's ratio for the latex rubber are E m = 7MPa, υ m =0.49, respectively.The mass per unit area of the metamaterial composite is only 1.1611kg/m 2 .
Figure 1 shows that the structure is similar to the structure of Sui's 22 and the only difference between two structures is the position of latex rubber.Unlike the Sui's structure, the new structure could ensure the membrane has the enough vibration space and installation on both sides.

B. Finite element analysis
The sound transmission measurements of the transmission loss for the structure were conducted by using FEM (COMSOL Multiphysics).In this study, the acoustic-structural coupled module was used to analyze the STL behavior of the membrane-type acoustic metamaterials.In this paper, the actual size of the computational model was established according to the real structure of the designed cells and even the thicknesses of the honeycomb were considered in the simulation.The computational model has two calculation domains, the air domain and the solid domain.The frequency-domain sound Helmholtz equation 9 for the air domain is where p is the sound pressure, ω is the angular frequency, c is the sound velocity in air.
For the solid domain, thin membrane can be abstracted into thin plate.And it is useful to consider the bending wave of a thin solid elastic membrane satisfying the biharmonic equation, 24 where z is the normal displacement of the membrane, f = p i + p r − p t is force for the surface of a membrane, with p i , p r , p t representing incident sound pressure, reflected sound pressure and transmission sound pressure respectively, D = Eh 3 /12 1 − υ 2 is the flexural rigidity and h is the thickness of the membrane, with ρ, E, and ν being the mass density, Young's modulus, and Poisson's ratio, respectively.
The STL is defined as where w in and w tr denote the incoming power and the transmission power of the sound, respectively.By varying the excitation frequencies of the incident waves, the STL can be obtained.

III. RESULTS
The acoustic metamaterial studied in this paper, the honeycomb plates are just the supporting role.According to the characteristics of the simulation and the experiment, the plane wave is perpendicular to hit on the acoustic metamaterial, the vibration of the thin film is mainly in the vertical direction, and the out-of-plane effective Young's modulus of the honeycomb cell structure is greater than the isotropic in-plane moduli.
1][22] Figure 2 shows the calculation STL profile (black dotted line) for the membrane-type proposed in this paper.The calculation FIG. 2. The calculated STL (black dotted line) compared with experimental (red dotted line), simulation results (pink dotted line) and the mass law (blue dotted line).
AIP Advances 6, 025116 (2016) results were compared with the experiment results (red dotted line) and simulation results (pink dotted line) of reference 22 for the same structural parameters and boundary conditions.It can be seen from the characteristic curve that there is a STL dip frequency occurring at 1162Hz, before which the STL decreased with the increase of frequency, and then rose up afterwards.The STL of the mass law prediction for a thin uniform limp panel was calculated by using equation (4), 13 and the calculation result was also shown in the figure (blue dotted line).
where ω = 2π f ,ρ s the surface density of the acoustic metamaterial, and ρ 0 and c are the density and speed of sound in the surrounding fluid, respectively.The STL was very high ( 20dB) at low frequency, especially at ultra-low frequency ( 100Hz), when compared with the mass law prediction.
The STL characteristics of the numerical results share the common features with those of experiment and simulation results of reference 22.The STL dip frequencies of the experiment and the simulation results of the reference 22 are 1100Hz and 1160Hz, respectively, and they are both very closed to the result calculated by using FEM of this paper.It can been seen that, in spite of these assumptions the numerical predictions agree very well with the experiments, and the simulation results can capture major trends of the STL behavior.

A. Mechanisms of the low-frequency sound insulation
According to the results of the experiment and simulation, the overall sound insulation property of this acoustic metamaterial is determined by the sound insulation property of the unit cell, which is similar to the locally resonant phononic crystal.So, parallel with the research on the locally resonant phononic crystal, only a cell unit (a single layer membrane) of the acoustic metamaterial was considered in this investigation.The experiment and simulation results showed that, the low frequency sound insulation interval mainly concentrated under the first natural frequency of a unit single cell membrane, so the film can be simplified to a single-degree-of-freedom mass-spring vibration system.According to the related literature, [19][20][21][22] the vibration characteristics of the single-degree-of-freedom mass-spring vibration system can be described by using the mass-spring lattice system with infinite one-dimensional periodic cycles.
Now consider a single membrane model and a one-dimensional periodic infinite film system.Based on these assumptions, they can be simplified as the mass-spring structure showed in Fig. 3(a) and 3(b) respectively.
For the infinite membrane system, as shown in figure 3(b), the differential equation of motion for the nth mass, whose displacement is denoted by u n , is 20 The steady state solution for equation (5), under the excitation of the angular frequency of ω and using Bloch's condition, can be written as where q is Bloch wave vector.The dispersion relation of the system which is derived from equations ( 5) and ( 6) can be expressed as The effective mass of a single layer membrane can be defined as It can be seen from equation ( 8) that the effective mass of the proposed structure is negative below the frequency ω 0 .The experiment and theory 20 had proved that, the effective mass for a single unit cell membrane model, as shown in figure 3(a), can be written as where ω c is the first natural frequency of membrane bending vibration.
In order to verify the effective mass defined by equation ( 9) for the membrane with clamped boundary conditions, the effective mass density of the film with fixed boundaries was also calculated by using FEM (COMSOL Multiphysics).The effective dynamic mass of the system can be defined as 12 where ⟨ p⟩ is the resultant force of the membrane, ⟨ āz ⟩ denoting the volume integration of accelerations.
Figure 4(a) shows the calculation results of the effective dynamic mass density (black dotted line) and the average out-of-plane displacement (blue dotted line) of the membrane system.It can be seen from the curve that effective dynamic mass density increased from an extremely low value to the static  mass density of the metamaterial as frequency increased and a wide band gap of the negative effective mass was opened below 1162Hz.According to equation ( 5), the cut-off frequency ω c that separates the positive and the negative effective mass is the first natural frequency of membrane bending vibration.Across the resonant frequency, the vibration direction of the membrane is reversed, resulting in a phase change in the normal displacement (blue dotted line) of the film, as shown in Fig. 4.
In the negative-mass band region, the propagation constant will be purely imaginary, resulting in the evanescent wave mode in the transmission direction.The magnitude of the negative effective mass increases as the frequency decreases and the decay length of the waves in transmission direction will be greatly shortened, and giving rise to the high STL at low frequencies.
The average normal velocity (red dotted line) was also calculated as shown in figure 4(b).There was a peak of the average normal velocity at frequency 1162Hz.Before the peak frequency, the average normal velocity increased with the increasing of frequency as a function of exponential form, and then decreased after the peak.At low frequencies ( 500Hz), the average normal velocity approached zero, which meant that the membrane was "quasi stationary" in the normal direction, and the sound wave was hard to pass through.At the peak frequency, the effective mass density was zero, which meant "nothing" in the transmission direction of the acoustic wave, and so, the sound was easy to pass through, thus would generate a STL dip.The frequencies of the average normal velocity peak and the zero effective mass density were coincidence.

B. Analysis of structural parameters on the low-frequency sound insulation
It can be found that the cut-off frequency, the first natural frequency of the membrane influences the negative-mass band directly, as shown in Fig. 4. In order to study the influence of the thickness, geometry shape, lattice constant and membrane tension on the STL, several groups of computational models of the acoustic metamaterials, with different film surface density, membrane shapes and membrane tensions were calculated.
Figure 5 exhibits the calculation results varying the frequencies with different surface density of membrane applied to the acoustic metamaterials.It can be easily seen that the STL characteristics for all samples shared the common features and the predicted STL started from a high value at 0Hz, decreased as frequencies increased before the STL dip frequency of each case.The relationship between the film surface density and the negative-mass band width, the STL at 500Hz were shown in Fig. 5(b).For the ρ s =1.01111kg/m 2 film, the negative mass density band was only 535Hz; when the surface density reached 1.31111kg/m 2 , the band width rose to 1795Hz and the STL of each at 500Hz were 1.28dB and 11.68dB respectively.
It is useful to consider the bending vibration of a thin solid elastic membrane satisfying the bending vibration equation of a thin plate.This hexagonal "thin plate" can be simplified to a thin circular plate with fixed boundary conditions whose resonance frequency is proportionate to the surface density of the plate.So, the frequency of STL dip increases with the increase of the surface density and the negative-mass band is expanded, which has been confirmed by the  For simplicity while without losing generality, the influence of different boundary forms of the unit cell on the STL was only studied.So, the several common shapes were considered in this paper without considering their global structures.
Figure 6 demonstrates the STL profile varying with the geometrical shapes.For convenience, all the shapes used in this study were normal polygon without considering their global structures.In this study, the shapes of equilateral triangle, square, pentagonal, hexagon and circle of the membrane were considered under the condition of equal area.It can be easily seen that the STL profile for the equal-area membrane structure exhibited only a single STL dip, and the STL of the triangle film at low frequencies was higher than that of the circular one.The negative-mass band and the STL at 500Hz both decreased as the number of the edges increased.
The frequency range of low-frequency STL studied in this paper is mainly under first natural frequency, and the membrane system can be simplified to a mass-spring vibration system.The first natural frequency of the film is where K e and M e are equivalent stiffness and mass respectively.For fixed area of a model, the change of model geometry shape caused the change in the perimeter of the membrane.When the membrane vibrates, the elastic potential energy density is mainly concentrated in the perimeter region, 24 and the longer the perimeter is, the larger the energy density is.The increase of the elastic potential energy density means the higher of the K e , and according to equation (11), the first natural frequency and the negative-mass band of the film both increase to a higher value, thus a good STL at low frequencies of the membrane can be obtained.In order to obtain a higher STL at low frequencies, a relative longer boundary can be adopted to obtain a lager elastic potential energy density in the perimeter region.
The effects of membrane tension on STL frequency response are shown in Fig. 7.It can be seen that, the first natural frequency of a unit cell membrane increased obviously as the membrane tension increased, thus greatly expanding the negative-mass band.With the membrane tension rising form 0 MPa to 2 MPa the negative mass band width increased about five times, and the STL at 500Hz increased 2.4 times.More importantly, the increase in membrane tension did not increase the film surface density.
The changes in the membrane tension shift the equivalent stiffness K e .According to the equation ( 11), the larger K e is, the larger the first natural frequency will become.Therefore, in order to improve the low frequency STL of thin film materials, the membrane tension can be appropriately increased.

V. CONCLUSIONS
A novel membrane-type acoustic metamaterial with a lightweight rubber material and stiffness and two layers of honeycomb cell structure were designed.It is suggested that the membrane showed a good performance for sound insulation at low frequencies ( 500Hz), and the mechanisms of sound insulation of the acoustic metamaterial in the low frequency region were successfully explained by the negative mass density.The negative mass density of the membrane was demonstrated at frequencies below the cut-off frequency, and enlarging the negative-mass bandwidth can raise the STL at low frequencies.The cut-off frequency was related to the first natural frequency of a unit cell membrane which can be tuned to specific values by varying the membrane properties.The effects of different membrane structural parameters on the sound-proofed performance at low frequencies were investigated by suing FEM.The numerical results showed that, the location and the magnitude of the STL dip and the negative-mass band width can be tuned by varying the membrane properties, including but not limited to film surface density, membrane geometry shape and membrane tension.
The structures proposed in this study have potential applications in low-frequency noise reduction.Moreover, when a large size of the metamaterial is applied, the whole stiffness of the skeleton structure must be considered, and the low-frequency sound insulation performance will be declined.So, in order to obtain a high STL at low frequencies, the metamaterial could be fabricated with some small parts with size, such as 100 × 100mm 2 , and these small parts could be located in the regions where the low-frequency noise is most intense.This method is similar to the traditional sound insulation method, which can achieve the low-frequency noise reduction by passing the mass and damping blocks on the intense noise regions.

FIG. 3 . 5 Lu
FIG. 3. (a) The equivalent simplified mass-spring model of a single membrane; (b) the equivalent simplified mass-spring lattice system of membrane systems with infinite one-dimensional periodic cycles.m, G and K are the equivalent mass and stiffness of membrane and the equivalent stiffness of the air between two membranes, respectively;.

FIG. 4 .
FIG. 4. (a) The calculated effective dynamic mass of the membrane (black dotted line) as defined in the paper, together with the averaged normal displacement (blue dotted line); (b) The average normal velocity of the membrane.

FIG. 5 .
FIG. 5. (a) Calculation results of STL varying with the frequencies with different surface density of the membrane applied to the structure, (b) the negative-mass band width (black dotted line) and the STL (blue dotted line) at 500Hz of each case varying with the surface density, respectively.

FIG. 6 .
FIG. 6.The results of STL of different geometry shape varying with the frequencies under the condition of equal area, (b) the first natural frequencies (black dotted line) and the STL (blue dotted line) at 500Hz of different models.

FIG. 7 .
FIG. 7. (a) The results of STL varying with the frequencies under the condition of different membrane tension, (b) the first natural frequencies (black dotted line) and the STL (blue dotted line) at 500Hz of different models varying with the membrane tension.