A monostable acoustic metamaterial for broadband low frequency soundabsorption ∗Xianhui LI(1) , Tuo XING, Junjuan ZHAO, Xiaoling GAI
Beijing Municipal Institute of Labor Protection, Taoranting Road 55, Beijing 100054, PR China

Abstract
This paper presents a monostable acoustic metamaterial for achieving a broadband sound absorption in low
frequencies. The proposed system is realized by placing a flexible panel with a magnetic proof mass in asymmetric magnetic field. An equivalent circuit model of such a system is presented. It is shown that the soundabsorption peak significantly shifts downwards with the increasing magnetic field. In the nonlinear regime,
the jump phenomenon is observed in the frequency sweeping test. Numerical simulation demonstrates that theproposed design can realize a broadband low frequency sound absorption in the deep subwavelength scale.
Keywords: Sound absorption, acoustic metamaterial, monostable

1INTRODUCTION

Sound absorption is an important topic in noise control engineering. Traditional sound absorption materials include porous materials and fiber materials. They are mostly effective in the middle and high frequency ranges.
While in the low frequency range, resonant structures are often used to gain better sound absorption performances. However, their applications in real practice are quite limited due to the large back cavities involved invery low frequencies. Some researchers proposed to resolve the problem by place a flexible back wall or adding
flexible structures in the absorbers (1, 2). Others designed compound absorbers using multiple layers of microperforated panels or membranes to broaden the absorption bandwidth in the low frequencies (3-5). However,these complex structures have only limited sound absorption capability in the deep subwavelength scale.
In recent years, acoustic metamaterials (AMMs) have drawn a lot of attention in the noise control community
due to their unusual properties. Among various implementations of AMMs, membrane- and plate-type AMMs
are particularly important for noise control (6, 7). They are relatively simple and light-weighted, having theability to actively tune the absorption characteristics. When properly designed, this type of AMMs can achievea high sound absorption in the deep subwavelength scale. Unfortunately, it is narrow-banded in nature due tothe causality constraint on the minimal structural thickness (8).
In this paper, a monostable acoustic metamaterial is proposed and its equivalent circuit model is presented. It
is shown that the sound absorption peak significantly shifts downwards with the increasing magnetic field. In
the nonlinear regime, a jump phenomenon is observed in the frequency sweeping test. Numerical simulation
demonstrates that the proposed design can realize a broadband low frequency sound absorption in the deepsubwavelength scale.

2

MODELING

The proposed monostable absorber, illustrated in Figure 1, consists of a plate-type AMM with a centered magnetic proof mass. Two external magnets are symmetrically positioned in coaxial with the proof mass, resulting
an attractive force on each side of it. For such a configuration, the proof mass and one of the two external
magnets would snap together if their gap is small, which changes the overall boundary conditions of the systemand disables the system. Care should be taken to adjust the external magnets away from the proof mass so that∗(1) Email: lixianh@vip.sina.com

4491

the elastic force of the plate is always larger than the magnetic attraction. Meanwhile, the gap should be smallenough to ensure the strong coupling between the plate and the external magnets for better performances.

Figure 1. Diagram of the monostable absorber
The monostable absorber can be modeled by an equivalent circuit, as shown in Figure 2.

Figure 2. Equivalent circuit of the monostable absorberSuppose a plane wave with an amplitude of pin incident on the absorber from above, it drives the AMM witha blocked pressure 2pin , which vibrates with the air loadings from both sides, together with the loading due tothe magnetic interaction between the proof mass and the external magnets. The impedance of the air above isρc and that of the air in the cavity is Zcav = − jρcctg(ωD/c), where ρ is the air density, c the air sound speed,ω the radian frequency, and D the depth of the cavity.
The impedance of the AMM isjω( φn (x) dS)2ZAMM = S ∑22n Mn (ωn − ω + 2 jζn ωn ω)
R

−1,(1)

where S is the surface area of the plate, Mn , ωn , and ζn are the modal mass, natural frequency, and modal
damping ratio of the nth mode of the AMM, respectively. Assuming that the dynamics of the AMM is dominated by a single mode φi , Equation 1 reduces toZAMM ≈ S

4492

Mi (ωi2 − ω 2 + 2 jζi ωi ω)R.
jω( φi (x) dS)2(2)The magnetic force between the proof mass and the external magnets is approximated byfNL ≈ −k1 x − k3 x3 ,(3)where x is the displacement of the proof mass, k1 and k3 are the linear and nonlinear stiffness coefficients
derived from the magnetic force. The corresponding impedance is
k3
k1−ZNL =jωS jωSv̄
ωβ2,(4)

where v̄ its surface averaged velocity. The magnetic loading is connected to the circuit through a transformation
element with the transformation ratio
β = hφi i /φi,0 ,(5)where φi,0 and hφi i are the center and the surface averaged values of the ith modal shape, respectively.
When pin is small enough, the monostable absorber can be considered as a linear resonator. The magneticinteraction works as a negative stiffness as in Reference (9). The effective impedance from the air cavity andthe magnetic interaction isk1
ρc2 1+ 2 2,(6)Ze f f = Zcav + k1 /( jωSβ 2 ) ≈jω D ρc β S
where the low frequency approximation ωD/c  1 is made. Note that k1 < 0 due to the symmetrical setupof the external magnets, its effect on the monostable absorber is equivalent to a cavity with a larger depth−1De f f = 1/D + k1 /ρc2 β 2 S . Hence the absorption peaks shift downwards to the lower frequencies.
To simulate the dynamics of the monostable absorber in the time domain, the above equations in the frequency
domain are rewritten as the following equationM ẍ(t) + (2ζi ωi M + ρcβ 2 S)ẋ(t) + K + ρc2 β 2 S/D + k1 x(t) + k3 x3 (t) = 2β Spin (t),(7)2 , and K = Mω 2 .
where the dot denotes the time derivative, M = Mi /φi,0i
For the case of moderate acoustic and magnetic fields applying on the absorber, the system is weakly nonlinear
and the dissipated acoustic power is approximated by

Pdiss = 1/2(2ζi ωi M) |ẋ|2 ,(8)where |ẋ| denotes the velocity amplitude of the proof mass.
The acoustic power incident on the absorber is
Pin = Sp2in /2ρc.

(9)

The sound absorption coefficient of the absorber is defined as
α = Pdiss /Pin = 2ρcζi ωi M |ẋ/pin |2 /S.

3(10)VALIDATION

To validate the above model, an absorber is made up of a aluminum disk of radius 34 mm and thickness 0.28 mm,backed by a rigid cylindrical cavity of depth 49 mm. The material has a density of 2700 kg/m3 , Young’s modulus 69 GPa, Poisson’s ratio 0.33, and modal damping ratio 0.08. The density of the air is 1.21 kg/m3 and its
sound speed 340 m/s. On each side of the disk, a proof mass of weight 0.183 g is placed at the center. Two4493

Table 1. Absorption peaks of the monostable absorber
Gap [mm]
k1 [N/m]Predicted [Hz]Measured [Hz]3.2-1540274
276

3.7-913297
296

6.0-170
320
312

∞0327
328

external magnets are symmetrically positioned in coaxial with the proof mass. The attractive force on each side
is measured with a dynamometer and the effective stiffnesses are obtained by curve fitting.
Firstly, the absorber is installed in a impedance tube and tested according to ISO 10534-2 (10). In the linear
regime, the interaction between the proof mass and the external magnets leads to a negative stiffness effect. The
predicted peak frequencies are in good agreement with the measured data, which is shown in Table 1.
To investigate the nonlinear dynamics of the monostable absorber, a frequency sweeping test is conducted witha sweep rate of 18 Hz/s. The driving sound pressure is monitored with a microphone in front of the absorber,
which is about 90 dB in the test. The back cavity is removed so that the velocity of the absorber can bemeasured by a laser vibrometer. The results from the upward and downward sweep are shown in Figure 3,
which clearly shows a jump phenomenon due to a softening nonlinearity. In the numerical simulation, thefollowing parameters k1 = −0.13 N/mm, k3 = −290 kN/mm3 , ζi = 0.01, and pin = 0.048 Pa are used. Thesimulation results are shown in Figure 4. It should be noted that the above parameters are not obtained fromthe experimental setup and used for illustration only. Further work is needed for the purpose of validation.

Figure 3. Frequency response of the velocity on a point near the proof mass: upward sweep, black line;
downward sweep, red line
The last example illustrates a broadband low frequency absorber in a deep subwavelength scale. The structure
parameters remains the same except that the back cavity depth is reduced to 5 mm. The gap between theexternal magnets and the proof mass is chosen to be 1.4 mm, which has the effective stiffnesses k1 = −23 N/mm
and k3 = −45.7 N/mm3 . The nonlinearity is negligible for the current configuration. As shown in Figure 5, thepeak absorption frequency drastically shifts downwards from 723 Hz to 161 Hz. The relative bandwidth is alsobroadened due to the symmetrically positioned magnets.

4494

Figure 4. Frequency response of the velocity of the proof mass: (a) upward sweep, (b) downward sweep

Figure 5. Absorption coefficients of the absorbers with (red line) and without (blue line) the external magnets

4495

4CONCLUSIONSThis paper proposed a monostable acoustic metamaterial which places a plate-type AMM with a magnetic
proof mass in a symmetric magnetic field. Under the assumption that the absorber is dominated by a single
mode, the equivalent circuit model is obtained by connecting the nonlinear impedance to that of the AMM
through a transformation element. It is shown that the sound absorption peak significantly shifts downwards
with the increasing magnetic field. In the nonlinear regime, a jump phenomenon due to a softening nonlinearity
is observed in the frequency sweeping test. Numerical simulation demonstrates that the proposed design canrealize a broadband low frequency sound absorption in a deep subwavelength scale.

ACKNOWLEDGEMENTS
This work is supported by Beijing Natural Science Foundation (1172007, 1182011), Natural Science Foundationof China (11704035, 11604015, 11174041), Xicheng District Excellent Talent Project (20180029), Beike Scholar
Project (BS201901) and Innovation Project (PXM2019-178304-000003) of BJAST.

REFERENCES[1] Lee, Y. Y.; Lee, E. W. M. Widening the sound absorption bandwidths of flexible micro-perforated curved
absorbers using structural and acoustic resonances, International Journal of Acoustics and Vibration, International Journal of Mechanical Sciences, Vol 49 (8), 2007, pp 925-934.
[2] Bravo, T.; Maury, C.; Pinhède, C. Sound absorption and transmission through flexible micro-perforated
panels backed by an air layer and a thin plate, The Journal of the Acoustical Society of America, Vol 131(5), 2012, pp 3853-3863.
[3] Zhang, Z. M.; Gu, X.T. The theoretical and application study on a double layer microperforated soundabsorption structure, Journal of Sound and Vibration, Vol 215 (3), pp 399-405.
[4] Sakagami, K.; Fukutani, Y.; Yairi, M.; Morimoto, M. A theoretical study on the effect of a permeable
membrane in the air cavity of a double-leaf microperforated panel space sound absorber, Applied Acoustics,
Vol 79, 2014, pp 104-109.
[5] Sakagami, K.; Fukutani, Y.; Yairi, M.; Morimoto, M. Sound absorption characteristics of a double-leaf
structure with an MPP and a permeable membrane, Applied Acoustics, Vol 76, 2014, pp 28-24.
[6] Yang, Z.; Mei, J.; Yang, M.; Chan, N.; Sheng, P. Membrane-type acoustic metamaterial with negative
dynamic mass, Physical Review Letters, 2008, 101:204301-4.
[7] Huang, T. Y.; Shen, C.; Jing Y. Membrane- and plate-type acoustic metamaterials, The Journal of the
Acoustical Society of America, Vol 139 (6), 2016, pp 3240–3250.
[8] Yang, M.; Sheng, P. Sound Absorption Structures: From Porous Media to Acoustic Metamaterials, Annual
Review of Materials Research, Vol 47, 2017, pp 83-114.
[9] Zhao, J.; Li, X.; Wang, Y.; Wang W.; Zhang, B.; Gai, X. Membrane acoustic metamaterial absorbers withmagnetic negative stiffness, The Journal of the Acoustical Society of America, Vol 141 (2), 2017, pp 840846.
[10] International Organization for Standardization, ISO-10534-2: Determination of Sound Absorption Coefficient and Impedance in Impedance Tubes, Geneva, 1998.

4496