Broadband low-frequency membrane-type acoustic metamaterials with multi-state anti-resonances

Applied Acoustics 159 (2020) 107078

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Broadband low-frequency membrane-type acoustic metamaterials with multi-state anti-resonances Guojian Zhou a,b, Jiu Hui Wu a,⇑, Kuan Lu c, Xiujie Tian b, Wei Huang b, Keda Zhu b a

School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structure, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China Gissing Tech. Co., Ltd., Wuxi 214191, People’s Republic of China c School of Mechanical and Electrical Engineering, North University of China, Taiyuan 030051, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 10 May 2019 Received in revised form 21 August 2019 Accepted 29 September 2019

Keywords: Membrane-type acoustic metamaterials Multi-state anti-resonance Broadband Low-frequency Large-scale

a b s t r a c t In this paper, a well-designed low-frequency membrane-type acoustic metamaterial (MAM) with continuous multi-state anti-resonance modes is proposed, which may effectively broaden a sound attenuation zone in low-frequency regime and produce single-negative effective parameter characteristic. Firstly, four lightweight MAM samples with distinct resonator distributions are purposefully designed, based on the low-order vibration characteristics of membrane and the design concept of dynamic balance. Then, the realization principle of the multi-state anti-resonance modes and the regulation mechanism of the sound transmission loss (STL) are progressively compared and investigated. Among them, a cross-like resonator easy to achieve dynamic balance is capable of offsetting and broadening the STL bandwidth, and eliminates node-circle-type oscillation mode. Additionally, a MAM sample whose subresonators are improved in distributions is proposed, wherein the structure and material parameters of the sub-resonators are symmetrical and interlaced. This sample shows 15 hybrid low-order antiresonance modes within the STL bandwidth, and achieves an excellent ability to insulate the broadband noise between 48 Hz and 736 Hz. Furthermore, its STL performance could be proved, between 72 Hz and 560 Hz, by a homogeneous ethylene vinyl acetate copolymer plate. Secondly, the coupling characteristics between this sample and the sound field are revealed in detail, through the analyses of effective parameters, surface-averaged normal displacement, coupled kinetic energy and the coupled vibration behaviors. Finally, the low-frequency STL performance of a large-scale MAM plate sample with one supercell as a periodic unit is verified by simulation and experiment. This paper proposes a lightweight, flexible and ultrathin distributed MAM structure that can produce continuous multi-state anti-resonance modes, which may promote the engineering application investigations of the broadband low-frequency MAMs to some extent. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction It is well known that traditional acoustic materials, such as rubber, felt, foam, etc. [1,2], have been powerless to control low-frequency noise in the lightweight applications [3–7]. Over decades, studies have shown that MAMs with negative effective parameter can manipulate and control sound waves in a way that is not possible for ordinary acoustic materials [8–13], which provide a turning point to improve the low-frequency noise in the field of passive noise control. Consistent with the vibration characteristics prevalent in physical structures, the vibration properties of MAMs are determined by their structural and material ⇑ Corresponding author. E-mail address: [email protected] (J.H. Wu). https://doi.org/10.1016/j.apacoust.2019.107078 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.

parameters [4,7,10,13–18]. Therefore, multi-resonators are introduced into a vibration system in a reasonable way, which may greatly enrich the hybrid vibration modes of the MAM units [12,14–19]. Literatures have shown that MAMs with multiresonator arrays can create a multi-peak STL profile compared to a single-resonator structure [3,4,6,12,13,15,17,18,20–23], especially at the anti-resonance frequencies. That is, an alternative of introducing multi-STL peaks or broadening the STL bandwidth can be achieved by the reasonable design of the distributed resonators attached to the membrane structure. However, the antiresonance modes of the traditional MAMs is intermittently created within a wide-frequency range. These discontinuous antiresonance modes are the main reason why the STL bandwidth is often narrower in the low-frequency regime [14,17,21]. In addition, the existing acoustic metamaterials have more or less

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weaknesses that restrict their applied researches, such as the large and heavy structures, rigid frameworks, fast-aging membrane materials (rubber), unstable membrane tension, additional heavy mass resonators, as well as the complex structures [3–5,7,10,12,1 4,17,18,24–37]. Therefore, designing an ultrathin and flexible MAM unit with the multi-parameter resonators that generates the continuous multi-state anti-resonance modes, could be considered as an effective way to improve the STL bandwidth and amplitude, as well as to meet the practical application needs. The resonators in the unit when excited by a certain frequency may show the opposite vibration behavior with each other. The entire unit can be harmonized into a dynamic balance state, and its average displacement will also appear as a minimum (even zero) [20]. Accordingly, this dynamic balance mode will make the propagated sound power to be greatly localized into the resonators to implement energy transfer [5,8–10,18,28]. Among which, the excitation frequency and vibration mode are called antiresonance frequency and anti-resonance mode, respectively [9,10,12,17,18,38,39]. In addition, we know that the low-order anti-resonance modes are first presented as the dipole and quadrupole patterns. If a resonator structure that may achieve both the modes is designed, the low-frequency STL bandwidth can be broadened to some extent. Furthermore, if additional hybrid resonators are introduced in the MAM unit, the two modes may be derived into more anti-resonance modes. Therefore, it would be possible to improve the STL bandwidth. These ideas are the outlines of this paper. In what follows, taking the low-order vibration characteristics of membrane as the cut-in point of this study, and combining with the design concept of dynamic balance, at first, we purposefully propose four lightweight and flexible MAM samples with distinct resonator distributions. And then the STL performances of these samples are numerically simulated and experimentally verified, respectively. Secondly, implementation principles of the multistate anti-resonance modes and the regulation mechanisms of the STL bandwidth are gradually investigated. Specifically, the relationships between the vibration characteristics and the STL bandwidths, from the sample I with a homogeneous membrane and the sample II with a lumped resonator, are first analyzed in Section 3.1. In Section 3.2, the sample III with a cross-shaped resonator structure is designed, which shows both the dipole and quadrupole modes, and achieves the offset and broadening of the STL bandwidth into low-frequency band. Furthermore, a welldesigned sample IV with distributed hybrid resonators is proposed. It shows that 15 hybrid anti-resonance modes are within the studied frequency regime, which makes the whole unit to be continuously in the dynamic balance states, and improving the STL bandwidth effectively. Additionally, in Section 3.3, the coupling characteristics between the sample IV and the sound field are analyzed in detail through the surface-averaged normal displacement (d_z), coupled kinetic energy (Wk), and the coupled displacement contours. In addition, the effective parameters of the sample IV, dynamic effective mass (Meff) and dynamic effective stiffness (Keff), also implement the negative parameter characteristics of the metamaterials. Finally, in Section 4, the low-frequency STL performance of a large-scale MAM plate sample with the supercell IV as a periodic unit is verified by simulation and experiment. The supercell structures proposed in this paper may enrich the studies of lightweight MAMs with multi-state anti-resonances to some extent.

2. Samples and methods We know that the structural morphology of acoustic metamaterials directly determines their coupled vibration modes, which in

turn remarkably interfere with the transmission characteristics of the sound waves [14]. Hence we design four distinct ultra-thin MAM prototypes that will be used to progressively reveal the regulation mechanisms of the multi-state anti-resonance modes on the low-frequency sound waves. Sample I, as shown in Fig. 1a, is a pure polyimide (PI) membrane with a thickness of h = 0.2 mm. The sample II shown in Fig. 1b is a common MAM sample with a lumped resonator, in which a circular metal mass platelet is distributed on the center of the PI membrane. Fig. 1c shows a MAM sample III in which a flexible cross-like ethylene vinyl acetate copolymer (EVA) grid is fixed on the center of the PI membrane. Based on the sample III, a novel sample IV with distributed resonators is designed in Fig. 1d, and with a surface density of 1.93 kg/m2. Specifically, the four subcell metal platelets are symmetrically attached to the membrane areas between each pair of cross-like swing arms, which divide the supercell into 12 symmetrical subcells with different parameters, including 4 swing-arm areas, 4 platelet areas, and 4 membrane areas at the end of the swing arms. This design ensures that the structure and material parameters of the sub-resonators are symmetrical and interlaced. The EVA framework, metal platelets and cross-like EVA grid in these samples are all glued onto a surface of the PI membrane by an adhesive, wherein a flexible annular EVA framework with a thickness of 2 mm served as a support. Moreover, a homogeneous EVA plate with a thickness of 0.94 mm is fabricated as a comparison sample, and it has the same surface density as the sample IV, as seen in Fig. 1e. Furthermore, the structural and material parameters of the above samples are shown in Table 1. The STL (Exp_STL), the transmission (Exp_T) and reflection (Exp_R) coefficients are measured by the Brüel & Kjær 4206-T acoustic impedance tube system (ASTM E2611-09) as shown in Fig. 1f [40], in which the sound source is the plane waves, and the experimental principle is the transfer matrix method. Moreover, Meff (Exp_Meff) and Keff (Exp_Keff) are reversed calculation based on the measured Exp_T and Exp_R [39]. In addition, the eigenmodes, STL (CAE_STL), Meff (CAE_Meff), Keff (CAE_Keff), d_z, Wk and coupled vibration behaviors of the metamaterial units can be investigated in this work, by using the acoustic-structure interaction module of the finite element (FE) software COMSOL Multiphysics. COMSOL analysis software is a commercial finite element software for multiphysics simulation analysis, wherein the acoustic-structure interaction module not only can macroscopically analyze the characteristics such as the structural vibrations, sound wave propagations and coupling behaviors, but also comprehensively calculate the mechanics, acoustic and energyrelated parameters (such as coupled kinetic energy, strain energy, etc.) in a microscopic manner. In addition, many literatures indicate that it has been widely adopted to characterize acoustic metamaterials [3,5,41]. First, an acoustic-structure interaction model of a single supercell containing the air and solid domains is built in COMSOL. Second, the boundary of the air domain is set as a sound-hard boundary in the pressure acoustics interface. In addition, an incident pressure field (IPF) with an amplitude of 1 Pa is set in the plane wave radiation node, which is applied to the incident surface (IS) of the IPF as an excitation condition. Simultaneously, the transmitted pressure field (TPF) is terminated by a perfectly matched layer (PML) absorbing boundary condition to avoid the multi-reflections of sound waves. Then, the boundary of the MAM structure is set as the fixed constraint in the solid mechanics interface. Finally, the MAMs are meshed by using a quadrilateral mesh with Lagrange quadratic elements with a maximum mesh size of 10 mm [42]. Therefore, the FE models in the paper have enough degrees of freedom to describe the lowfrequency modal information. The Fig. 1g shows the acousticstructure interaction FE model of the sample IV. Moreover, the boundary conditions and meshing methods of the samples I, II

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Fig. 1. (a–d) The measurement samples with different resonator structures; (e) the homogeneous EVA plate; (f) the Brüel & Kjær 4206-T impedance tube test system with the inner diameter of 100 mm; (g) the acoustic-structure interaction FE model of sample IV with sound-hard boundary conditions.

Table 1 The structural and material parameters of the MAM samples. Parameters PI membrane EVA grid EVA frame Metal platelet

Thickness (mm) 0.2 2 2 1.8

Diameter (mm) 100 / 100 11.2

Width (mm) / 4 5 /

Length (mm) / 40 / /

and III are the same as those of the sample IV. The Young’s modulus, mass density, Poisson’s ratio and damping loss factor of the PI membrane, metal platelet and EVA material within the MAM samples are (1.42 GPa, 1100 kg/m3, 0.36, and 0.04), (200 GPa, 7800 kg/m3, 0.33, and zero) and (170 MPa, 2050 kg/m3, 0.45, and 0.09), respectively. In addition, in the computer aided engineering (CAE) model, the air density is 1.29 kg/m3, and the sound speed in air is 340 m/s. The calculated STL is described as


2
 1 jhPin ij ¼ 10lg STL ¼ 10lg ; s jhP out ij

ð1Þ

wherein s is the transmission coefficient of the sound energy, Pin and Pout are the incident and transmitted sound pressure respectively, the amplitude of Pin is set as 1 Pa, h i and | | denote the average and the moduli of the parameter respectively. 3. Result analysis and experimental verification 3.1. Modal characteristics and STL bandwidth 3.1.1. Sample I As evident in Fig. 2a, the STL data show that the simulated STL bandwidth is between 120 Hz (A1) and 496 Hz (A3), with an STL

Modulus (Pa) 1.42 * 10 1.7 * 108 1.7 * 108 2 * 1011

9

Density (kg/m3)

Poisson’s ratio

Loss factor

1100 2050 2050 7800

0.36 0.45 0.45 0.33

0.04 0.09 0.09 /

peak of 29.6 dB at 424 Hz (A2), and the experimented STL bandwidth is between 136 Hz and 448 Hz, with an STL peak of 21.9 dB at 488 Hz. The simulated and experimented STL results show that there are certain deviations in both STL amplitudes and frequencies. First, these deviations can be understood because the uncertainties that are induced during the processing and experimenting of the MAM samples cannot be totally covered in the theoretical models, and the damping characteristics generated during these processes are unavoidable and difficult to be quantified. Therefore, the experimented STL amplitudes will have the slight attenuation. Alternatively, the mass and stiffness of the adhesive layer between the PI membrane and the sub-resonators (cross-like EVA grid and metal platelets) within the experimental samples always affect Meff and Keff in the overlapping areas, and they are difficult to be solved quantitatively. Besides, the difference in the constraint strength of the experimental and simulation units can also cause some offset in Keff. Therefore, there will be a small amount of drift in the STL peak and valley frequencies. Despite there are some inevitable factors during the processing and experimenting of the MAM samples, which make the simulation conditions difficult to strictly correspond to the experimental conditions. However, the simulated concur with the measured results, including the trend, peak and trough frequencies of the STL curves, which prove the effectiveness and correctness of the CAE model and numerical method. Therefore, the CAE results could be used to

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Fig. 2. (a) The measured (dashed curve) and simulated (solid curve) STLs (black curves) and d_z (cyan curve) of the sample I; (b) the simulated eigenmode contours within the STL bandwidth, wherein the cyan arrows denote their vibration directions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

qualitatively characterize the physical properties and the coupling characteristics of the MAMs. It is to be noted that the frequency band between the first and second STL troughs herein is designated as the STL bandwidth, the corresponding first and second STL trough frequencies are defined as the lower and upper limits of the STL bandwidth, respectively. The modal shapes and natural frequencies within the STL bandwidth are shown in Fig. 2b. Among them, the first-order lumped resonance mode that cannot be eliminated causes the nearcomplete sound energy be transmitted at point A1 [18,20,43,44]. The lower limit (f1) of the STL bandwidth is defined by the firstorder effective mass (M1) and effective stiffness (K1) in the whole pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vibration region (f 1 ¼ K 1 =M 1 =2p, f1 is the first-order natural frequency). Since no additional mass is on the membrane surface of the sample I, a small M1 makes f1 not low enough (only 120 Hz). At point A2, the out-of-phase hybridization of two nearby eigenmodes makes the resonance mode of the membrane to be decoupled [5,18]. It can be seen that the sample I exhibits two loworder anti-resonance modes in Fig. 2b in the STL peak band, namely dipole and quadrupole vibrations. Owing to the law of action and reaction, the vibrational amplitudes of the mutually antiresonant subcells in the supercell are equal in magnitude but of opposite sign, thus achieving the force balance [1]. The d_z curve in Fig. 2a indicates that the deformation components of the positive vibrations are counteracted by the negative vibrations, such that d_z  0. Therefore, these interactive synergistic behaviors between subcells achieve the dynamic balance of the structural vibration. Meanwhile, the subcells vibrating in the negative direction can radiate the reflected sound waves that are opposite to each other with the incident sound waves, which causes the incident pressure sound fields are counteracted by the reflected pressure sound fields of the same amplitude and opposite phase, so that the forward propagating sound energy is greatly attenuated. Therefore, the coupling cancellation behaviors of such echoes allow the sound field on both sides of the supercell to achieve the dynamic balance of the sound field propagation. Hence, the attenuation of sound energy is perfectly realized by the synergistic coupling behaviors between the sound waves and the antiresonant resonators, which is the mechanism by which the transmitted sound energy is largely attenuated at the anti-resonance frequency [6,8–10,34]. However, it is worth noting that the upper limit of the STL bandwidth is terminated by the first node-circletype oscillation mode at point A3. The node-circle-type modal shape is a special vibration mode of a homogeneous circular plate

structure, and the node-circle frequency (x) can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 x ¼ Ra2 12ðEh =qh. It can be seen that x is related to the radius 1t2 Þ (R), thickness (h), density (q), Young’s modulus (E), Poisson’s ratio (ʋ) and mode shape coefficient (a) of the circular plate. Therefore, x can be regulated by these structural and material parameters. Additionally, the reason for the low STL amplitude is attributed to a small energy-storage capacity that caused by a membrane material with low potential energy, resulting in incomplete exchange of the sound energy.

3.1.2. Sample II Firstly, although the first-order lumped resonance mode cannot be eliminated, its frequency can be transferred. As indicated in Fig. 3a, the STL bandwidth position of the sample II is shifted into a lower band between 40 Hz (B1) and 296 Hz (B3). The mass platelet attached to the surface of the membrane effectively decreases the eigenfrequency of the resonator, which makes the lower limit of the STL bandwidth to be transferred to the ultra-lowfrequency [45]. Secondly, the anti-resonance modes of the sample II are blocked by the node-circle-type oscillation mode that appears in advance, which shows that there are only two dipole modes and no quadrupole mode in the STL bandwidth, so that its STL bandwidth is terminated at B3 quickly. Therefore, the energystorage bandwidth of the sample II is narrower (only 256 Hz) compared with that of the sample I. In addition, the increased elastic potential energy improves the exchanging efficiency of sound energy, so that the STL peak to be boosted to 38.6 dB at 128 Hz (B2), which is 9 dB higher than that of the sample I. Sample II also verified that, the MAMs with a lumped mass have excellent lowfrequency sound insulation performance, but the STL bandwidth is always narrow in the low-frequency regime [10,20]. From the analyses above, a structure that can achieve both the low-order dipole and quadrupole modes could broaden the lowfrequency STL bandwidth. Increasing the mass resonators can transfer the STL bandwidth to a lower frequency band, and could increase the STL peak, but the STL bandwidth is narrow. In addition, it is feasible that the low-frequency STL bandwidth is further broadened if the node-circle-type oscillation mode is suppressed. Accordingly, it can be considered as an effectual way to improve the STL bandwidth and amplitude by designing a MAM structure that can produce continuous and multi-state anti-resonance modes.

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Fig. 3. (a) The measured and simulated STL curves of the sample II; (b) the eigenmode contours within the STL bandwidth of the sample II.

3.2. Implementations of continuous multi-state anti-resonance modes 3.2.1. Sample III According to the low-order anti-resonance characteristics of the samples I and II, a symmetrical cross-shaped structure is skillfully employed to design the sample III. As seen in Fig. 4a, the STL bandwidth of sample III is significantly broadened to 400 Hz (between 64 Hz and 464 Hz), with an STL peak of 38.5 dB at 256 Hz (C2). First, as with the sample II, the added center mass makes the STL bandwidth of the sample III shifted to the lower band and the STL amplitude is substantially increased compared to the sample I. Second, the sample III shows four consecutive anti-resonance modes in Fig. 4b, which illustrates that the low-order opposite vibration behaviors that help to regulate the vibration into dynamic balance mode can be easily achieved by a symmetrical cross-swing-arm structure. Therefore, the STL bandwidth of sample III is significantly broadened compared to the sample II. Moreover, the STL peak (38.5 dB) of sample III is close to that of sample II, and owing to their approximate effective potential energy (the mass of crosslike EVA grid is equal to that of the metal platelet). Finally, Fig. 4b shows that the node-circle-type oscillation mode can be perfectly suppressed by the swing arms of the cross-like resonator. Nevertheless, co-resonance modes in the membrane areas between each pair of swing arms break the continuous anti-resonance modes of the sample III, which cut off the STL bandwidth at 464 Hz (C3). The above studies show that the symmetrical cross-like swingarm structure can effortlessly create the low-order anti-resonance

modes of both the dipole and quadrupole, which is an ingenious idea to broaden the low-frequency STL bandwidth. Furthermore, the cross-distribution feature of the cross-like resonator can effectively eliminate the node-circle-type oscillation mode. Therefore, the lightweight sample III shows a good low-frequency sound insulation characteristic to a certain extent. However, whether the coresonance modes in the membrane areas that limit the upper limit of the STL bandwidth can be further hybridized into the multi-state anti-resonance modes is a key issue that determines whether the STL bandwidth can be further broadened. 3.2.2. Sample IV At present, many scholars regard widening the low-frequency STL bandwidth as a major goal for one of their scientific research breakthroughs. Without exception, further broadening the STL bandwidth of sample III is also the focus of this paper. The above analyses show that the STL bandwidth and peak are related to the continuity of the anti-resonant modal and the effective potential energy of the metamaterial unit, respectively. Therefore, based on the design ideas of samples II and III, the sample IV with hybrid and distributed resonators is proposed. Specifically, the four subcell metal platelets are symmetrically attached to the membrane areas between each pair of cross-like swing arms, which divide the supercell into 12 symmetrical subcells with different parameters, including 4 swing-arm areas, 4 platelet areas, and 4 membrane areas at the end of the swing arms, wherein the structure and material parameters of the sub-resonators are symmetrical

Fig. 4. (a) The measured and simulated STL results of the sample III; (b) the eigenmode contours within the STL bandwidth of the sample III.

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and interlaced. As shown in Fig. 5a, the STL bandwidth and peak of the sample IV are greatly improved within the low-frequency regime, compared to the samples I, II and III. Specifically, the STL peak is increased to 48.1 dB at 408 Hz (D2), and the STL bandwidth is markedly broadened to 688 Hz (between 48 Hz and 736 Hz), which are significantly better than that of the sample III. In addition, the experimental verification indicates that the STL of sample IV, between 72 Hz and 560 Hz, is much higher than that of the homogenous EVA plate with the same surface density, at a maximum of 24 dB and an average of 8.2 dB. The multiple symmetrically interlaced sub-resonators make the sample IV easy to create multi-state anti-resonance modes and achieve the broadband dynamic balances in the low-frequency range. As illustrated in Fig. 5b, as many as 15 anti-resonance modes are continuously generated without any one of the node-circletype oscillation mode being within the STL bandwidth. These continuous anti-resonance modes are occurred among the cross-like EVA grid and the metal platelets, such as the opposite vibrations from two metal platelets or four (at 98 Hz, 102 Hz, 152 Hz, 408 Hz, 433 Hz, 443 Hz, 515 Hz), from the ends of the EVA grid (at 298 Hz, 302 Hz, 355 Hz, 588 Hz), and from between the EVA grid and the metal platelets (at 224 Hz, 542 Hz, 560 Hz, 571 Hz). It can be clearly seen from these anti-resonance modes that the vibrational amplitudes of the mutually anti-resonant subcells in the supercell are equal in magnitude but of opposite sign, thus achieving the force balance [1]. The opposite synergy behaviors of the cross-like resonator and metal platelets can significantly enrich the decoupling modes between the supercell and the incident sound waves, and can realize multi-set of dynamic balances.

So that the STL bandwidth can be greatly broadened. Among them, at 408 Hz (D2), the four metal platelets simultaneously exhibit the strongest anti-resonance modes in the lateral direction, which provides an extremely favorable exchange path for the transfer of the sound energy to the localized kinetic energy, so the STL reaches to a peak. As analyzed above, the four metal platelets introduced can synergistically transform the co-resonance modes in the membrane areas of the sample III into a plurality of localized anti-resonance modes, thereby extending the upper limit of the STL bandwidth. The design of the sample IV effectively validates the widening effect of the continuous multi-state anti-resonance modes on the STL bandwidth. It also shows that the dipole and quadrupoletype vibrations can be further hybridized into the multi-state anti-resonance modes. Through the analyses of the above four samples, we should subjectively know what the vibration characteristics corresponding to the structural forms of such MAM units are, and what the sound insulation performances corresponding to the vibration modes are. These may directionally guide us to design some multi-state anti-resonant structures. 3.3. Coupling characteristics of sample IV The structural forms and material distributions of the resonators will directly determine their eigenmodes of the MAMs, which will inevitably affect the anti-resonant coupling characteristics between the units and the sound fields. Further, the multistate anti-resonance modes will result in multiple anti-resonant coupling behaviors. Accordingly, only the well-designed

Fig. 5. (a) The measured and simulated STL results of the sample IV, wherein the purple STL curve is the experimental data from a homogeneous EVA plate with thickness of 0.94 mm; (b) the eigenmode contours within the STL bandwidth of the sample IV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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multi-state anti-resonance modes can have the beneficial tuning effects on the incident sound fields. These coupling characteristics and tuning mechanisms can be revealed by the factors such as d_z, Meff, Keff, and Wk respectively, which are not independent of each other. Therefore, the coupling characteristics and the generation mechanisms of the broadband, low frequency and high STL amplitude of the sample IV could be revealed in detail based on these factors. In addition, the extremely consistent experimented and simulated STL curves shown in Fig. 5a indicate that the simulation model of the sample IV could be used to reveal its physical properties. As evident in Fig. 6a, Meff and Keff curves show that the sample IV has the single negative effective parameter characteristic that is common to the conventional metamaterials. The simulated results show that there are two negative Meff intervals, namely 48 Hz and 416 Hz–744 Hz, which are in good agreement with the retrieved results (48 Hz and 400 Hz–736 Hz) [23]. Moreover, there are two negative Keff intervals, namely 56 Hz–408 Hz and 744 Hz, respectively, which are consistent with the trend and direction of the retrieved results (72 Hz–408 Hz and 752 Hz). Furthermore, the curves of Meff and Keff generate the same ‘‘zero value conversions” at points D1 and D3, and an ‘‘extreme value conversion” at point D2, respectively. Moreover, the directions of Meff and Keff are completely correlated to the direction of d_z shown in Fig. 6b. When the direction of d_z is changed, Meff and Keff will also, accordingly, alternate between positive and negative, and vice versa. This is because Meff / ha_zi1  (x2d_z)1 and Keff / hd_zi1. Meff and Keff are bound to undergo a dramatic extreme reversion between a positive and a negative while d_z passes through 0. d_z  0, Meff and Keff are at extremums, and it is considerably difficult that the MAM unit be shaken by an excitation, and it also proves that the structure in dynamic balance hardly radiates sound energy. At point D1 in Fig. 6b, the maximum Wk means that the vibration of the cross-like resonator creates a strongly lumped coupling resonance behavior with the shear waves, and the incident sound energy has hardly any reverse dissipation mode. As shown in the coupling vibration contour at point D1 in Fig. 6c, wherein the red

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arrows and the streamlines exhibit a large amount of sound energy travels upward through the center area. The coupling mode at point D1 indicates that the lower limit of STL bandwidth is determined by the effective parameter characteristics of the central region. If the bending stiffness in this area is decreased, both the negative Keff and d_z bands will be moved to the left on the frequency axis, and the reverse coupling vibration behaviors will also be generated earlier. Therefore, the lower limit of the STL bandwidth could be moved to a lower frequency [46]. Near point D2, the coupling characteristics of the unit are characterized by the anti-vibration in its central EVA grid area and the co-vibrations in its surrounding membrane regions. Numerous deformation components cause the positive and the negative displacements of the subcells to be counteracted each other, which is expressed as d_z  0. Therefore, these interactive synergistic behaviors between subcells achieve the dynamic balance of the structural vibration. Meanwhile, the subcells vibrating in the negative direction can radiate the reflected sound waves that are opposite to each other with the incident sound waves. Then, the incident pressure sound fields are counteracted by the reflected pressure sound fields of the same amplitude and opposite phase, so that the forward propagating sound energy is greatly attenuated. Therefore, the coupling cancellation behaviors of such echoes allow the sound field on both sides of the supercell to achieve the dynamic balance of the sound field propagation. The synergistic operations of the two dynamic balances make the sound energy to be dramatically attenuated and unable to propagate forward. Therefore, the concept connotations of dynamic balance described in this manuscript include both the dynamic balances of the structure and the sound field. When the MAM unit and the sound field are coupled into the dynamic balance state, the incident sound energy is attenuated in two ways. One is that a portion of the sound energy is transferred into the elastic potential energy of the anti-resonant resonators, which is then locally dissipated and can be clarified by Wk  0. Wk  0 means that a large amount of the kinetic energy is almost completely confined inside the unit by the anti-resonant resonators and transformed into the elastic potential energy. The other is that

Fig. 6. (a) The retrieved (dashed curves) and simulated (solid curves) curves of Meff (red) and Keff (blue); (b) the relationships between the Wk (purple dashed curve), d_z (cyan solid curve), and STL (black solid curve); (c) the coupling vibration behaviors between the sound field and the unit at points D1, D2, and D3, including the distribution and direction of the sound intensity (the red arrows) and the coupling displacement (the cyan arrows), and the flow forms of the sound energy (the streamlines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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the incident sound waves are counteracted by the reflected sound waves, which can be illustrated by a vortex phenomenon of the sound energy at point D2 in Fig. 6c. Meanwhile, these two attenuation modes of sound energy are the typical local energydissipation mechanisms of the anti-resonance-type acoustic metamaterials, which ensure the maximum STL amplitude. After point D2, STL shows a downward trend until the second STL trough (at point D3). The dynamic balance state of the unit is gradually broken and translated into the co-coupled vibration behavior (d_z is positive). In addition, as shown in Fig. 6a, Meff tends to be 0 from the minimum, and Keff to 0 from the maximum, meaning that the elastic strain energy is gradually transferred to kinetic energy, which is verified by the slowly increasing Wk curve in Fig. 6b. Therefore, the ability of the MAM unit to control sound waves is gradually disintegrated. At point D3, the anti-resonance intensity in the center region is weaker than the resonance in the membrane region between every two mass platelets, and the red arrows in Fig. 6c indicate that a large amount of the sound energy is gathered and transmitted in the membrane regions. This coupling mode indicates that the upper limit of STL bandwidth is determined by the effective parameters in the membrane regions without additional mass. This resonance mode will be delayed if the constraint stiffness in these regions are increased, so the upper limit of STL bandwidth will be further shifted to the high frequencies. From the above analyses, the decoupling behaviors between the multi-state anti-resonance modes and the sound waves make the sample IV in the dynamic balance states within a broadband. In addition, the lower and upper limit of the STL bandwidth depend on the coupled vibration behaviors within the particular areas. The distributions of the effective parameters directly affect the vibration modes of the unit. Therefore, it is possible to regulate the STL bandwidth and peak into a desired frequency band as long as the structural and material parameters within the corresponding area are reasonably designed. For the sample IV, the STL amplitude and bandwidth would be further tailored by optimizing the thickness and the Young’s modulus of the membrane, the length and width of the cross-like resonator, or the distributions and the material properties of the mass platelet resonators.

are parametrically scanned, and the transmission coefficients (s) in different incident angles are further calculated. Then, an averaged STL of LMAM is obtained by using the statistical average method. Specifically, the incident angles between 0° and 90° are scanned at 2° interval, and s corresponding to these incident angles are calculated, respectively. The s of every incident angle (n) can be expressed as

s1 ¼

jhpout 1 ij2 2

jhpin 1 ij

;

s2 ¼

jhpout 2 ij2 2

jhpin 2 ij

;   ;

sn ¼

jhpout n ij2 jhpin n ij2

:

ð2Þ

Then, the average s (hsi) is given by

hsi ¼

s1 þ s2 þ    þ sn n

:

ð3Þ

Therefore, the averaged STL (hSTLi) can be calculated as


 1 : hSTLi ¼ 10lg hsi

ð4Þ

4.1.2. Experimental samples and methods According to the supercell periodic model, a LMAM plate sample consisting of a periodic array of square unit cells is manufactured, and with an area of 0.6  0.6 m2 and a surface density of 2.3 kg/m2, as shown in Fig. 7c. In addition, to illustrate the lowfrequency STL performance of LMAM, a large-scale homogeneous EVA (LEVA) plate sample with a surface density of 2.4 kg/m2 as shown in Fig. 7d is selected as a reference. Then, a large-scale plate sample is placed in the test window between the reverberation chamber and the anechoic chamber. Then, based on the ISO 15186-1 standard [48], the STL can be measured using the sound insulation experiment system shown in Fig. 7e. The measured STL can be obtained by

  Wi STL ¼ 10lg W ¼ 10lgW i  10lgðSt It Þ t p2 S  ¼ 10lg 4iqci  10lgðIt Þ  10lgðSt Þ

2      p p2 ¼ 10lg p2i þ 10lg SSti þ 10lg I ref4qc  10lg I It ref

ref

ð5Þ

ref

¼ Lp  Li  6:08 4. STL performance of large-scale MAM (LMAM) panel 4.1. Models and methods 4.1.1. Simulation model and methods Based on the structure of MAM-IV, a square supercell unit with a lattice constant of L (L = 90 mm) is built in COMSOL as shown in Fig. 7a. Second, the supercell acoustic-structure interaction FE model as shown in Fig. 7b is established, including IPF, LMAM, and TPF, and then the background pressure field (BPF) and PML are built on the outside of IPF and TPF [41], respectively. Then, Floquet periodic boundary conditions based on the Bloch’s theorem are applied to relate the two opposite boundaries of both the air and structure domains [42,47]. Finally, a BPF with an amplitude of 1 Pa is established on the outside of IPF as the excitation condition [41], which ensures that the sound pressure level within IPF is equal everywhere and avoids the multi-reflections of sound waves. The simulation model of LMAM is established in the diffuse sound field condition. First, the characteristics of the diffused sound field are that the sound energy density is the same at all positions in the sound field, the directions are approximately arbitrary, and the energies are not attenuated, which can be considered as a statistically averaged uniform sound field. Additionally, it is well known that the BPFs with different incident directions can produce different reflections and transmissions. Therefore, based on the energy superposition principle, the incident angles of BPF

wherein, Wi and Wt are the sound powers incident on the test sample and transmitted through the sample, respectively. Si is the total area of the measurement surface, St is the area of the test sample under test, and Si = St = 0.36 m2. Pi, Pref, It and Iref are the incident sound pressure, the reference sound pressure (Pref = 2e5 Pa), the transmitted sound intensity and the reference sound intensity (Iref = 1e12 W/m2), respectively. The ambient temperature is 22 °C, the sound speed in air is c = 343.4 m/s, the air density is q = 1.18 kg/m3, so the characteristic specific acoustic impedance qc = 405.2 Ns/m3. Lp is average sound pressure level in a source room. Li is the average sound intensity level over the measurement surface in the receiving room. 4.2. Simulation and experiment results As shown in Fig. 8a, it can be found that there is an obvious STL peak band within the low- frequency regime below 500 Hz, and the experimental results are in good agreement with the CAE results within the studied frequency regime. In addition, the experimental verification indicates that the STL of LMAM sample, between 80 Hz and 350 Hz, is much higher than that of LEVA plate, at a maximum of 17 dB and an average of 5.2 dB. Thus, the good low-frequency sound insulation performance of this MAM structure is further verified. However, it needs to be clarified that the STL peak band of the LMAM sample shown in Fig. 8a is shifted to a lower frequency

G. Zhou et al. / Applied Acoustics 159 (2020) 107078

9

L

Fig. 7. (a) A square supercell unit with a lattice constant of L; (b) an acoustic-structure interaction FE model of LMAM with Floquet periodic boundary conditions; (c) a LMAM plate sample and (d) a LEVA plate sample fixed in the sound insulation window; (e) a standard large-scale sound insulation measurement system.

regime than that of the small sample IV shown in Fig. 5a. The first, the frequency band in which the STL peak is located has a certain relationship with the magnitude of L. Since the increase in L will cause the effective stiffness in the vibration region to be decreased, and conversely the effective stiffness will be increased. Therefore, the magnitude of L will cause the offset of the anti-resonance frequency, so that the STL peak band will also shift. Then, as long as L is reasonably designed, it is possible that the STL bandwidth is regulated to a desired frequency band. The blue curve in Fig. 8b shows that the STL peak and bandwidth of the LMAM with the lattice constant decreased to L = 80 mm are shifted to the right, and the other parameters are kept constant. The second, the analyses of the coupling characteristics in Section 3.3 show that the lower limit of STL bandwidth is determined by the effective parameters in the central region of the unit. The boundary conditions of each unit in LMAM are periodic, and the constraint strength of every unit is weaker than that of the small- scale sample IV. Moreover, the frameworks around the resonators are the flexible EVA material. Therefore, the effective stiffness around every resonator is also decreased, so the lower limit of the STL bandwidth of the LMAM is moved to the left.

Additionally, the upper limit of the STL bandwidth is determined by the effective parameters in the membrane regions without additional mass. The areas of the membrane without additional mass in every periodic unit are larger than that in the sample IV, so the effective constraint stiffness of these membrane areas is weakened. Therefore, the upper limit of the STL bandwidth of the LMAM is also shifted to the left, and the STL peak frequency is also offset to the left to some extent. Conversely, when the Young’s modulus of the membrane is increased to E_M = 4e9 Pa, the pink STL curve shown in Fig. 8b indicates that both the STL peak and the upper limit are shifted to the right, and the STL bandwidth is also greatly broadened. It also shows that the Young’s modulus of the membrane has a high sensitivity to the STL bandwidth. Alternatively, when the thicknesses of the metal platelets are decreased to h_Fe = 1 mm, it is equivalent to the effective mass of the circular resonators being decreased. It creates the STL peak frequency to move to the right, as shown by the green curve in Fig. 8b. In addition, the amplitude of the STL peak of the LMAM is lower than that of the sample IV. This is because the anti-resonant coupling strength between the resonators and membrane is weakened

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G. Zhou et al. / Applied Acoustics 159 (2020) 107078

Fig. 8. (a) The simulated and measured STL data from LMAM (red curves) and LEVA (black curves) plate samples; (b) the STL data of the three LMAMs with different parameters, the metal platelets with the thickness of h_Fe = 1 mm, the membrane with the Young’s modulus of E_M = 4e9 Pa, and the periodic unit with the lattice constant of L = 80 mm. Among them, the solid curves are the CAE results, and the dashed curves are the experimental results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

by the periodic boundary conditions, which hinders the exchange efficiency of the sound energy. On the other hand, the sound waves in the reverberation field are obliquely incident on the surface of the sample, rather than normal. Thereby, the refractive properties of the oblique incident increase the transmission coefficient of the large-scale MAM sample. The above analyses show that the investigations of the vibration and coupling characteristics of the small sample MAMs can effectively guide the design of the LMAMs. Next, the quantitative design relationships between the parameters of the large-scale periodic unit and the small-scale single unit, the correlations between the STL characteristics, and the coupling characteristics between the framework and the membrane need to be further studied. 5. Conclusions The low-order vibration characteristics of the membrane are taken as the starting point of this paper. Combined with the design concept of dynamic balance, we progressively investigate the realization principle of the multi-state anti-resonance modes and its regulation mechanism on the STL bandwidth. Firstly, the correlations between modal characteristics and STL bandwidth are analyzed by using two common metamaterial samples, I and II. Secondly, according to these two low-order anti-resonance characteristics, a symmetrical cross-shaped structure that is easy to achieve dynamic balances is skillfully employed to design the sample III that could achieve both low-order dipole and quadrupole vibration modes. It constrains the node-circle-type oscillation mode, and creates the offset and widening of the STL bandwidth. Further, on the basis of the sample III, a well-designed sample IV shows 15 continuous anti-resonance modes, and achieves the dynamic balances in the broadband between 72 Hz and 560 Hz. Among them, the four metal platelets advantageously transform the co-resonance modes of the membrane region into the antiresonance modes. This design manner effectively verifies the widening effect of the multi-state anti-resonance modes on the STL bandwidth. It is also demonstrated that the dipole and quadrupole-type vibration modes can be further hybridized into more anti-resonance modes. Then, within the STL peak bandwidth, the coupling characteristics between the sample IV and the sound field are comprehensively analyzed by the effective parameters, average normal displacement, and coupled kinetic energy. Finally, the low-frequency STL performance of the LMAM plate sample

with the sample IV as the periodic unit is verified by simulation and experiment. This paper will be able to, to a certain extent, enrich the multiple decoupling designs between the mass platelets and the membrane, and may promote the investigations of the lightweight MAMs with multi-state anti-resonances. Moreover, the designs and verifications of the LMAM samples may have some guiding significance for engineering application investigations. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51705395, China Postdoctoral Science Foundation under Grant No. 2018T111042, and the Fundamental Research Funds for the Central Universities. References [1] Bolton JS, Shiau N-M, Kang YJ. Sound transmission through multi-panel structures lined with elastic porous materials. J Sound Vib 1996;191:317–47. [2] Chen J-S, Chen Y-B, Chen H-W, Yeh Y-C. Bandwidth broadening for transmission loss of acoustic waves using coupled membrane-ring structure. Mater Res Express 2016;3:105801. [3] Zhang Y, Wen J, Zhao H, Yu D, Cai L, Wen X. Sound insulation property of membrane-type acoustic metamaterials carrying different masses at adjacent cells. J Appl Phys 2013;114:063515. [4] Naify CJ, Chang C-M, McKnight G, Scheulen F, Nutt S. Membrane-type metamaterials: transmission loss of multi-celled arrays. J Appl Phys 2011;109:104902. [5] Mei J, Ma G, Yang M, Yang Z, Wen W, Sheng P. Dark acoustic metamaterials as super absorbers for low-frequency sound. Nat Commun 2012;3:756. [6] Yang Z, Dai HM, Chan NH, Ma GC, Sheng P. Acoustic metamaterial panels for sound attenuation in the 50–1000 Hz regime. Appl Phys Lett 2010;96:041906. [7] Ma F, Wu JH, Huang M, Zhang W, Zhang S. A purely flexible lightweight membrane-type acoustic metamaterial. J Phys D Appl Phys 2015;48:175105. [8] Cummer SA, Christensen J, Alù A. Controlling sound with acoustic metamaterials. Nat Rev Mater 2016;1:16001. [9] Yang M, Ma G, Yang Z, Sheng P. Coupled membranes with doubly negative mass density and bulk modulus. Phys Rev Lett 2013;110:134301. [10] Yang Z, Mei J, Yang M, Chan NH, Sheng P. Membrane-type acoustic metamaterial with negative dynamic mass. Phys Rev Lett 2008;101:204301. [11] Ma G, Sheng P. Acoustic metamaterials: from local resonances to broad horizons. Sci Adv 2016;2:e1501595.

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