Broadband low-frequency sound absorption by periodic metamaterial resonators embedded in a porous layer

Journal of Sound and Vibration 461 (2019) 114922

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Broadband low-frequency sound absorption by periodic metamaterial resonators embedded in a porous layer Xing-Feng Zhu a, b, Siu-Kit Lau a, *, Zhenbo Lu c, Wonju Jeon d a

Department of Architecture, School of Design and Environment, National University of Singapore, 4 Architecture Drive, 117566, Singapore b Jiangsu Key Laboratory on Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing, 210023, China c Temasek Laboratories, National University of Singapore, 117411, Singapore d Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 34141, South Korea

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 March 2019 Received in revised form 21 August 2019 Accepted 21 August 2019 Available online 22 August 2019 Handling Editor: L. Huang

Broadband absorption of the audible sound wave at low frequency has been achieved by using periodic acoustic metamaterial resonators (AMRs) embedded inside a porous layer. A single AMR embedded in a porous layer could reach perfect absorption (PA) at the resonance frequency, and it can be easily tuned by adjusting the inner radius of AMR. With four AMRs in the porous layer, a high absorption (>80%) is obtained in the frequency range from 180 Hz to 550 Hz, while the thickness of the porous layer is only 1 =10 of the relevant wavelength at 300 Hz. The broadband and high absorption performances are due to the interferences of the low-frequency resonances of the AMRs and the energy trapping between the AMRs and the rigid backings. The finite element simulations are experimentally validated. Moreover, the broadband low-frequency absorption is robust under various oblique incidence even at large incident angles. The effects of the acoustic parameters of the porous layer on the absorption properties are also discussed. The absorbers should have high potential for the practical applications in buildings, aircrafts and automobiles due to their ease of fabrication, ultra-thin, and robust high-efficiency. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Acoustic metamaterial Low-frequency sound absorption Broadband Porous material

1. Introduction The absorption of low-frequency sound has been a challenging task because of the inherently weak dissipation for classic sound absorbing materials [1]. Porous materials are widely used as sound-absorbing materials in noise and vibration control. When porous materials are exposed to the incident sound waves, the acoustic energy of the air-particle oscillations is partly converted into heat due to thermal and viscous losses in the pores within the materials and walls of the interior pores. They are different from other homogenous materials in which there are two media for the sound propagation, the solid material itself and the air in the pores of the material. The bulk acoustical properties of porous materials are conventionally specified by a set of physical parameters, such as the flow resistivity, porosity, tortuosity, and viscous and thermal characteristic lengths. These physical parameters which provide the link between the acoustical and material properties can be measured directly in the experiments [2].

* Corresponding author. Department of Architecture, School of Design and Environment, National University of Singapore, Singapore. E-mail address: [email protected] (S.-K. Lau). https://doi.org/10.1016/j.jsv.2019.114922 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

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To interpret the acoustic behavior of porous absorbing materials, various theoretical models, grouped into empirical and phenomenological models, have been proposed. In general, these models aim to derive the characteristic acoustic impedance and the propagation constant of a wave as functions of macroscopic parameters like flow resistivity, porosity, tortuosity, etc. Only air flow resistivity is required in empirical models while additional parameters are required in phenomenological models. These are simple models for fast approximation using power-law relations through the best fitting of a large amount of experimental data for various porous absorbers [3e5]. Although the empirical models are highly advantageous as they only need a single parameter, they are only suitable for one type of material and the certain frequency ranges. The phenomenological models have been applied to porous materials for the relation between porous absorbing material with sound. These models have high accuracy but need more physical parameters of the material to be measured beforehand. The greater the number of physical parameters required, the higher the accuracy of the model is for the determination of the acoustical characteristics. Johnson et al. [6] used the dynamic tortuosity in their model based on the limit behavior at zero and infinite frequencies to express the dynamic viscous diffusion. Champoux and Allard [7] extended Johnson et al.'s work and showed that an additional thermal characteristic length was required to express the dynamic bulk modulus of the fluid in the pores. Johnson-Champoux-Allard (JCA) [8] model correlated the effective density and the effective bulk modulus of a porous material to all the five macroscopic parameters by assuming the porous material as acoustically rigid (motionless) over a wide range of frequencies. Recently, the efforts have led to many advances in the use of natural fibers to address the noise control with sustainable solutions [9]. Investigation of the relationship between the acoustic properties, the integral transport properties and the microstructure of the porous media is under development [10]. Conventional methods for absorbing acoustic waves at low frequencies have to rely on the use of some bulky and heavy weight materials such as porous and fibrous materials [11e13] or perforated structures with a back cavity [14,15], which are less practical for controlling the low-frequency noise. The emergence of acoustic metamaterials has provided a new way to design the novel materials or structures for sound absorption because they can support deep-resonance modes and offer a large density of states at low frequencies. Many high-efficiency acoustic metamaterial absorbers have been proposed and tested, such as Helmholtz resonator (HR) [16e19], acoustic membrane [20e25], acoustic metasurface [26e29], split-tuberesonator [30,31], lossy resonant plate [32], labyrinthine acoustic metamaterial [33,34], and optimal sound absorber [35]. However, most metamaterial absorbers are in an inherently narrow band in nature since high absorption would necessarily occur only in the vicinity of their resonant frequencies. In recent years, acoustic absorbers based on periodic inclusions embedded in porous materials have been proved capable of widening the absorption band. The effect on the absorption properties of non-resonant or resonant inclusions in a porous layer has been studied in two or three dimensions [36e40]. However, their high-absorption bandwidth is also in the middleor high-frequency ranges (>1000 Hz) and very limited. Therefore, it is desirable to develop an ultra-thin absorber with the broadband performance of low-frequency absorption (<500 Hz). In this paper, we present the numerical and experimental investigations on the broadband low-frequency acoustic absorption based on a porous layer with embedded periodic acoustic metamaterial resonators (AMRs). This broadband high-efficiency absorption originates from the coupling of the lowfrequency resonance modes and the trapped mode. The low-frequency resonance frequency can be easily tuned by adjusting the inner radius of AMR. The influence of the angle of the incident plane wave and the acoustic parameters of the porous layer are is further examined by using finite element method (FEM). Finally, we verify the design by experiment.

2. Finite element method The finite element method (FEM) is a numerical method that serves as a powerful instrument to predict the acoustic behavior of layers or multilayers of porous materials. FEM is widely used for complex geometry and multi-physics coupled problems. In present study, numerical simulation is performed by using Thermoviscous Acoustics, Frequency Domain Interface of the FEM commercial software COMSOL Multiphysics 5.4. The linearized Navier-Stokes equation, the continuity equation, and the energy equation are solved in the thermoacoustic domain, and the Helmholz equation is solved in the acoustic domain. The time-dependent term is eiut where u is the angular frequency. In thermoacoustic domain, the motion of a viscous compressible Newtonian fluid, including the energy equation, is governed by the set of equations as follows [41,42].

dr þ rðV,uÞ ¼ 0; dt

r

du ¼ rðV,sÞ þ F; dt

rCp

dT dp  ap T ¼ V,q þ 4 þ Q ; dt dt

(1)

(2)

(3)

velocity u, temperature T, and density r are the dependent variables,
 s ¼ pI þ mðVu þ ðVuÞ Þ  23 m  mB ðV,uÞI with m and mB are Dynamic viscosity and Bulk viscosity, Cp is the heat capacity at where

pressure

p,

T

X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

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constant pressure, ap is the coefficient of thermal expansion, 4 is the viscous dissipation function, F is a volume force and Q is a heat source. The three equations are the continuity equation, the momentum equation (the Navier-Stokes equation), and the energy equation, respectively. In the acoustic domain, the Helmholtz equation governing the sound pressure distribution can be written as:


 1 u2 p V,  Vp  2 ¼ 0; r rc

(4)

where r is the density, c is the speed of sound, p is the acoustic pressure, qd is the dipole domain source and Qm is monopole domain source. Floquet periodic boundary conditions, sound hard boundary, and perfectly matched layer (PML) are applied in the acoustic domains in the present study. For Floquet periodicity, also known as Bloch periodicity, is used to model infinite periodic structures with non-normal incident pressure fields. Each field (F) satisfies the Floquet-Bloch relation: Fðx þ LÞ ¼ FðxÞeik,L , where k is the incident wave vector and L is the spatial periodicity. For example, an oblique incident wave with wave vector k illuminates a subunits array and only one subunit which is periodic need analyze when Floquet periodic boundary condition is used in the model. The sound hard boundary (Wall) is a boundary at which the normal component of the acceleration (and vp thus the velocity) is zero: vn ¼ 0. The PML provided absorption of propagating waves without introducing reflections from the interface between the PML and the physical domain. The porous material can be described as an effective fluid, which is modeled based on the JCA model [8]. The effective density re and modulus ke of the porous material can be obtained as

re ¼

r0 a∞  4

1þi

 un FðuÞ ; u

(5)

gP0 k1   0.  1 ; e ¼  4 g  ðg  1Þ 1 þ i uc ðPr uÞ GðPr uÞ where FðuÞ ¼

(6)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2ffi a∞ 1  ihr0 u s24L and GðPr uÞ ¼ 1  ihr0 Pr u 20 a∞ 0 are the correction functions, r0 is the density of air, s 4L

Pr is the Prandtl number, P0 is the atmospheric pressure, g is the specific heat ratio, h is the dynamic viscosity of the fluid, un ¼ s4 r0 a∞ is the angular Biot frequency, and

0

uc ¼ rs0 a4∞ is the adiabatic cross-over angular frequency. The other parameters of the porous material are the tortuosity a∞, the porosity 4, the flow resistivity s, the viscous characteristic length L, the thermal 0 0 0 characteristic length L , and the thermal resistivity s ¼ 8a∞ h=4L . Here, the acoustical parameters of the air-saturated 0 4 porous material used are a∞ ¼ 1:07, 4 ¼ 0:96, s ¼ 2843 Nsm , L ¼ 273 mm, and L ¼ 672 mm [38,43]. 0

Fig. 1. (a) Cross-sectional configuration of the acoustic metamaterial resonator (AMR). (b) The schematic of absorption unit (AU) backed with a rigid wall.

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3. Absorption unit description Fig. 1(a) illustrates the cross section of an acoustic metamaterial resonator (AMR), which consists of two 180 -twisted split rings [32,44,45], denoted by SR1 and SR2. The thickness of rings and the width of splits are t ¼ 1 mm and w ¼ 1 mm. The inner radii of the SR1 and SR2 are R1 ¼ 39 mm and R2 ¼ 37 mm, respectively. The absorption unit (AU) is constructed by embedding the AMR into the air-saturated porous layer, as shown in Fig. 1(b). The slip orientation of the AMR is along the y direction, and the inner volume and the splits of the AMR are filled with air. The thickness and length of AU are D ¼ 112 mm and L ¼ 94 mm, respectively. The AUs are arranged periodically along the x direction and backed by a rigid wall. We assume that all walls are hard boundaries due to the large impedance mismatch between air and solid materials. The mass density and sound speed of the air are r0 ¼ 1:21kg/m3 and c0 ¼ 343 m/s, respectively. Floquet periodic boundary conditions are imposed in the x direction and PML is applied in the acoustic domain to model the open boundary condition. The incident acoustic plane wave is modeled as background pressure field and its amplitude is fixed at 1.0 Pa, corresponding to sound pressure level 94 dB (pref ¼ re 20 mPa). The complex transmission coefficient t and reflection coefficient r are achieved from simulation results, and the absorption coefficient a is then calculated as a ¼ 1  T  R, in which T ¼ jtj2 and R ¼ jrj2 . Fig. 2(a) shows the a, R, and T of the absorption system for normal incidence wave. T is zero because of the rigid backing in the simulation. R has a sharp resonant dip (almost zero) at 102 Hz, and thus a is almost unity at this frequency, meaning that perfect absorption (PA) is achieved. The PA frequency is consistent with previous report [32], which verifies the correction of our simulation. It is noted that the acoustic wavelength l at 102 Hz is 30 times larger than the thickness of the AU. The distribution of the acoustic pressure field jP=P0 j is plotted at the resonance frequency (102 Hz) in Fig. 2(b). P is the scattered acoustic pressure and P0 is the pressure of the incident wave. The AMR resonance corresponds to a maximum value of the pressure field located inside the AMR. We also plot the pressure field at the frequency beyond the resonance for comparison, such as 70 Hz, in Fig. 2(c). The difference of acoustic pressure between the two split rings (SRs) at 102 Hz is much larger than that at 70 Hz, indicated in Fig. 2(b) and (c). The absorption of a D-thick porous layer (green dash-dot-dot line) is also plotted in the inset. a is lower than 2% at 102 Hz, indicating that the absorption of the porous layer is very weak. The AMR plays an essential role in achieving the PA. The origin of the absorption can be essentially explained by critically coupling the inherent dissipation factor to the inherent leakage factor in the AU. In such cases, one can consider the resonator as an open system with quasi-modes characterized by fuzzy energy levels of a finite width [46]. The time dependence of the field j is not purely harmonic anymore and can be described as an oscillator with damping,

d2 j dt

2

þ ures Q 1

dj þ u2res j ¼ 0; dt

(7)

where ures is the resonant angular frequency and the dimensionless Q characterizes the total losses in the resonator. Here,

Fig. 2. (a) The absorption (A), reflection (R), and transmission (T) of the AU. The green dash-dot-dot line in the inset shows the absorption of the single porous layer with a thickness of D. Acoustic pressure field jP=P 0 j at (b) 102 Hz, (c) 70 Hz, and (d) 478 Hz P is the scattered acoustic pressure and P 0 is the pressure of the incident wave. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

1 Q 1 ¼ Q 1 diss þ Q leak ≪1;

5

(8)

where Qdiss and Qleak are the dissipation factor and leakage factor, respectively. The dissipation factor and leakage factor can be turned by changing the length L and the thickness D of AU, respectively. In our AU model, the total transmittance T ¼ 0 and the resonant reflection coefficient is then given by

 Rres ¼ 

1 Q 1 diss  Q leak 1 Q 1 diss þ Q leak

2 2 :

(9)

We can make the leakage factor equal to the dissipation factor (Qdiss ¼ Qleak ) by carefully decorating L and D to achieve Rres ¼ 0. Therefore, the incident wave can be closed to totally absorbed by the proposed AU and the PA is achieved. Although we employed a low resistivity foam in the present study, the nearly PA at low frequency can be realized in general porous materials including high resistivity foams. We have tested several common porous materials with different flow resistance s (not show here) and found that the scheme proposed here could show good generality in various porous materials by decorating the geometric parameters of the system. At higher frequency, another high absorption at about 475 Hz in Fig. 2(a) is due to the trapped mode, which is excited by the periodicity when the structure is placed against a rigid wall [36e38]. Fig. 2(d) shows the acoustic pressure distribution at trapped mode frequency, where the rigid wall is placed at the top of the figure. This clearly exhibits a maximum on the side of the rigid wall and a minimum on the side of the air, which is typical of a trapped mode. The influence of the oblique incident angle q is investigated for the AU in Fig. 3. The incident angle q is defined as the intersection angle between the oblique incident waves and the normal direction of the boundary of the porous layer, as shown in Fig. 1(b). It can be seen that the absorption peak associated with the AMR resonance is almost not affected by the variation of q because the resonance frequency is independent on the excited way of the resonator. The maximum absorption at the resonance frequency is slightly deteriorated at large angles, which can also be seen from the distribution of the pressure field for q ¼ 40 in the inset. The difference of acoustic pressure amplitude between the two SRs is a little smaller for q ¼ 40 than that for q ¼ 0 in Fig. 2(b). On the other hand, the absorption peak associated with the trapped mode increases with q. That is because the trapped mode frequency increases as the projection of the distance between the inclusion and the rigid backing decreases on the wave vector direction for oblique incidence with an angle q [36,37]. 4. Design and broadband absorption Although the PA and trapped peaks at low-frequencies (<500 Hz) can be obtained by our AU, the two absorption peaks are separated, and the bandwidth is generally narrow. To further extend the relative absorption bandwidth, a basic strategy is to use more units in parallel, which can increase the density of resonant and trapped modes. These units should have high absorption at tunable resonance frequency over a wider frequency range. Therefore, we first adjust the geometric parameter of the AMR to turn the absorption peak in the AU. Fig. 4(a) shows the absorption tuned by adjusting the inner radius R2 of SR2. The PA frequencies appear at 124, 182, and 256 Hz for R2 ¼36.5, 34.5 and 30.5 mm, respectively. The trapped mode frequency is about 475 Hz and nearly not affected by the variation of R2 . To understand the absorption detailed above, the normalized

Fig. 3. Absorption coefficient of the AU at different incident angle q. The inset shows the acoustic pressure field jP=P 0 j at 102 Hz for q ¼ 40.

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X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

Fig. 4. (a) Absorption coefficient of the AU while varying R2 . (b) Real parts and (c) imaginary parts of relative impedance. The arrow indicates the position of the perfect absorption peak frequency.

impedance Zs =Z0 are compared for different R2 . The Zs =Z0 is derived as

Zs Z0

¼ r 1c0 0

ps us ,

where ps and us are the average sound

pressure and velocity on the boundary between the AU and air. To achieve PA, the absorber impedance should match well to that of air, which means that the imaginary part of the normalized impedance goes to zero and the real part goes to unity simultaneously. Fig. 4(b) and (c) show the real and imaginary parts of the normalized impedance, respectively. The real and imaginary parts of the normalized impedance cross 1 and 0 at 124, 182, and 256 Hz for R2 ¼ R2 ¼36.5, 34.5 and 30.5 mm respectively, corresponding to the frequencies of the resonance absorption peaks. Thus, the impedance of the system is perfectly matched to that of air at the resonance frequency. For the trapped mode absorption peak, the real and imaginary parts of impedance is slightly mismatching to the impedance of air and this mismatching induces the decreased absorbance as observed. The sound absorption spectrum as a function of R2 is shown in Fig. 5. The resonance absorption peak shifts to higher frequency with decreasing R2 and PA can be obtained at the resonance frequency over a wide range of R2 . The bandwidth of the resonance peak increases with decreasing R2 , and the resonance mode and trapped mode are coupled well

Fig. 5. Absorption spectrum of the AU as a function of R2 .

X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

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when R2 reaches 25 mm. Therefore, decreasing R2 can increase the resonance frequency and enable it to reach the frequency of the trapped mode. It has been shown above that adjusting R2 can offer a tunability of the resonance absorption but keep the trapped mode absorption unchanged. This leads to the design of a“supercell” composed of several AUs with different R2 . Different configurations are possible, but here we discuss the supercell composed of four AUs. The thickness of the supercell is kept equal to D ¼ 112 mm, but the periodicity of the supercell is now L ¼ 376 mm. The R2 of these AMRs are 34.5, 33, 30.5 and 25 mm, corresponding to the resonance frequencies 182, 214, 256 and 326 Hz, respectively. Fig. 6(a) shows the absorption of the supercell. A large absorption coefficient can be maintained in a wide frequency band because the resonance modes and the trapped mode are close enough to excite some coupled modes in the supercell. In this case, five absorptive peaks [denoted by red arrows in Fig. 6(a)] are found, which correspond to the four resonance absorption peaks and the trapped mode. The absorption coefficient value reaches about 0.8 in a frequency range between 180 Hz and 550 Hz, which is relatively wide bandwidth in low-frequency. We further calculate the absorption coefficients of the four AUs supercell for the width of splits w ¼ 2 mm as depicted by the green dotted line in Fig. 6(a). Compared with the case of w ¼ 1 mm, the frequency of the resonance absorption peaks become higher while that of trapped absorption peak keeps equivalent. The broadband performance of the supercell is robust when the width of splits w changes. The distribution of the acoustic pressure field jP=P0 j is plotted at the second resonance frequency of 210 Hz and the trapped frequency of 465 Hz in Fig. 6(b) and (c), respectively. It is obvious that a maximum of pressure field is located inside the AMR with R2 ¼ 33 mm at a frequency of 210 Hz corresponding to the second absorption peak, while at the trapped mode frequency the pressure maximum is mostly located between the AMRs and the rigid wall at the top of Fig. 6(c). As the spatial dimension of our system is very small, the absorption broadband performance is expected to be robust under oblique incidence. Fig. 7 shows the absorption coefficient for varying the incident angle q. It could be seen that the frequencies of the four absorption peaks are nearly unchanged and the trapped mode frequency increases with q. Increasing incident angle q increases the amplitudes of both resonance and trapped absorption peaks, which is due to the larger initial absorption of the porous layer. Thus, the broadband absorption is fulfilled in a wide range of incident angle. In reality, a typical sound field does not consist of a single plane wave but a large number of waves with various amplitudes, phases and directions. To find the sound absorption effect of such complicated sound fields, we consider the random incident absorption coefficient under a uniformly distributed sound incidence. The theoretical random incidence absorption coefficient on an infinitely large surface can be calculated as follows [47]:

Fig. 6. Absorption coefficient of the supercell composed of four AUs. The green dotted line denotes the absorption coefficient for the supercell while varying w ¼ 2 mm. Acoustic pressure field jP=P 0 j for the supercell at (b) 210 Hz and (c) 465 Hz. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

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X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

Fig. 7. Absorption coefficient of the supercell at different incident angle q.

arand ¼

Z 0

p 2

aðqÞsinð2qÞdq;

(10)

where aðqÞ is the oblique incidence absorption coefficient at an incidence angle q, that can be achieved from the numerical results. Fig. 8 shows the calculated random incidence absorption coefficient of the supercell absorber. We extend the frequency range to 80-10 k Hz to show the influence of embedded metamaterials on the absorption performance not only in low frequency range but also in medium-to-high frequency range. The random incidence absorption (red dash line) of a porous material layer alone with the same thickness is also plotted for comparison. The proposed absorber layer exhibits enhanced sound absorption performance in the low-frequency range compared to the porous layer with the same total thickness without any inclusions. In the medium-to-high frequency range, the random incidence absorption coefficients have high value except a dip in the frequency range of 1200 Hze1700 Hz as shown in Fig. 8, but the minimum coefficient value in this range still can reach 0.65. Compared to the absorption of porous material, our absorber can enhance the absorption at low frequency part and almost maintain the performance at medium-to-high frequency part despite the sacrifice of sound absorption in a limited frequency range. We further investigate the influence of the parameters of porous material because the parameters of foam are often difficult to predict before its polymerization. Sensitivity analysis of the absorption band is essential with regards to the acoustic parameters of the porous layer. Each parameter of the porous foam is varied individually while keeping the other 0 constant. As the absorption peaks are weakly dependent on a variation of 4 (¼[0.9, 1.0]), L (¼[250 mm, 400 mm]), and L (¼[450 mm, 800 mm)) (results are not shown), only the influence of a∞ and s are plotted in Fig. 9(a) and (b), respectively. When tortuosity a∞ increases from 0.99 to 1.15, the resonance absorption frequencies are not affected, while the trapped frequency decreases since the sound speed in the material decreases. The amplitudes of both resonances and trapped peaks can keep

Fig. 8. Random incidence absorption coefficient of the supercell. The red dashed line presents the random incidence absorption of a porous material layer with the same thickness. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

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Fig. 9. Absorption coefficient of the supercell while varying porous material parameters of (a) tortuosity a∞ and (b) flow resistivity s.

unchanged by the variation of a∞ . The flow resistivity s influences both the resonance and trapped peaks. When s increases, the amplitudes slightly decrease, while only the trapped frequency increases as shown in Fig. 9(b). These results indicate that the broadband and high absorption performances are stable in a range of porous material parameters. 5. Experiment validation Fig. 10(a) shows the acoustic measurement system, where a loudspeaker is installed at one end of the square tube (100 mm  100 mm) and the sample is placed at the other end. All the acquisition and control signals with a sampling rate of 20 kHz are operated by the NI PCI-6251. Two microphones of model 130E20 referred as ‘Mic1’ and ‘Mic2’ are used for measuring the sound pressure inside the tube. The sound pressures measured by the microphones are then processed with a calculation program based on two-microphone method [48] which calculates the absorption coefficient. The AMRs can be obtained by the three-dimensional (3D) printing prototype, as shown in Fig. 10(b). The four AMRs are labeled as I, II, III, and IV with R2 ¼ 34.5, 33, 30.5 and 25 mm, which are the same as those in the supercell. By embedding the AMR into the polyurethane foam, the experimental sample is fabricated as shown in Fig. 10(c). The top and side covers of the sample can prevent sample deformation and reduce energy leakage. The measured absorption coefficients of the four samples are plotted as solid lines in Fig. 11. The black, red, blue, and green lines represent the absorption coefficients of the samples with AMR I, II, III, and

Fig. 10. (a) The schematic of the acoustic measurement system. (b) Photograph of four AMRs labeled as I, II, III, and IV with R2 ¼ 34.5, 33, 30.5 and 25 mm, respectively. (c) Photograph of the experimental sample (the left is the sample without foam).

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X.-F. Zhu et al. / Journal of Sound and Vibration 461 (2019) 114922

Fig. 11. Measured absorption coefficients of the four samples are plotted with the solid lines in the same colors of AMR labels in Fig. 9(b). The absorption peaks are marked by numbers. The simulated absorption coefficients (dashed-circle lines) are also shown for comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

IV respectively. It's obvious that the samples can achieve high absorption peaks at the resonance modes and trapped mode. The measured resonance peak frequency is increased by decreasing R2 while the trapped mode frequency is unchanged, which confirms the previous prediction. The simulated absorption coefficients (dashed-circle lines) of the four samples are also shown for comparison. The resonance absorption peaks for the four samples are marked by number 1, 2, 3, and 4, which match well with the simulated results except that the measured absorption peak is slightly lower than the simulated one. The trapped absorption peaks marked by number 5 are in the vicinity of 400 Hz, which are also a little lower than the simulated results. These discrepancies could be attributed to the imperfections in the sample manufacturing and acoustic parameters of porous material. Therefore, the absorber introduced in this work can be considered as validated. 6. Conclusion We have designed and fabricated an ultra-thin absorber composed of a porous layer with embedded multiple AMRs and demonstrated the broadband low-frequency absorption of the absorber. High absorptions can be achieved at resonance mode and trapped mode in simulations and experiments. The resonances of each AMRs and the trapped mode are close enough to excite coupled modes and to keep a significant absorption in a wide frequency band. It is found that a high absorption (>80%) is obtained in the frequency range from 180 Hz to 550 Hz, while the thickness of the layer is only 1 =10 of the relevant wavelength at 300 Hz. The high broadband absorption can be observed regardless the incident angle and porous material parameters. Our work opens the possibilities of designing compact acoustic absorbers for low-frequency range, which has great potential applications in buildings, aircrafts and automobiles. Acknowledgment The authors wish to acknowledge the funding support by the Singapore's Ministry of Education Academic Research Fund Tier 1 (WBS R-295-000-157-114) and the National Natural Science Foundation of China (11704193). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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