Control and broadening of multiple noise frequencies using an assembly of sub-metamaterials connected by membranes for aircraft noise mitigation Tenon Charly Kone 1 National Research Council Canada, Flight Research Laboratory 1200 Montreal Road, Ottawa, ON, K1A 0R6, Canada. Sebastian Ghinet 2 National Research Council Canada, Flight Research Laboratory 1200 Montreal Road, Ottawa, ON, K1A 0R6, Canada. Raymond Panneton 3 CRASH, Centre de Recherche Acoustique-Signal-Humain, Université de Sherbrooke, 2500 Boul. de l’université, J1K 2R1, Sherbrooke, Québec, Canada. Zacharie Laly 4 CRASH, Centre de Recherche Acoustique-Signal-Humain, Université de Sherbrooke, 2500 Boul. de l’université, J1K 2R1, Sherbrooke, Québec, Canada. Christopher Mechefske 5 Department of Mechanical and Materials Engineering, Queen's University, Kingston, ON, K7L 3N6, Canada. Anant Grewal 6 National Research Council Canada, Flight Research Laboratory 1200 Montreal Road, Ottawa, ON, K1A 0R6, Canada.

ABSTRACT The simultaneous control of multiple tonal frequencies and broadband noise, at low frequencies is a challenge for the aerospace, ground transportation and building industries. The technologies proposed in the literature, using layered porous materials with embedded Helmholtz resonators (HR) with a structured neck, exhibited considerable potential when tuned at tonal, multi-tonal, or 1 TenonCharly.Kone@nrc-cnrc.gc.ca 2 Sebastian.Ghinet@nrc-cnrc.gc.ca 3 Raymond.Panneton@USherbrooke.ca 4 Zacharie.Laly@USherbrooke.ca 5 Chris.Mechefske@queensu.ca 6 Anant.Grewal@nrc-cnrc.gc.ca

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narrow frequency bands. However, the resonance frequencies due to the structured necks of the metamaterial are narrow. Our recent investigations have shown that parallel arrangements of several structured metamaterials separated by membranes can broaden the resonance frequencies and increase the number of resonance frequencies to be controlled. This paper presents a parallel assembly of four sub-metamaterials structured and separated by membranes. Each of these sub-metamaterials is also a serial assembly of a periodic unit cell with a half-neck + cavity + half-neck configuration. The metamaterial is embedded in a layer of fiberglass. Comsol's coupled sound pressure and shell approaches in the frequency domain were used to predict the sound absorption coefficient of the metamaterial. The results obtained show a widening of the absorption frequency peaks and the appearance of additional frequencies due to the membranes.

1. INTRODUCTION

Aircraft cabin comfort is an important design aspect for any aircraft manufacturer. The entire travel experience for passengers and aircrew on-board an aircraft is highly determined by the comfort, efficiency and level or diversity of services available in the cabin. The overall aim of any innovative approach adopted by aircraft manufacturers is to offer travelers the quietest, most comfortable and attractive cabin environment possible. The long acoustic wavelengths associated with the low frequencies of interest determine the main design constraints for the problem under study. In the low frequency range, the available volume to integrate the noise insulation is a serious limitation for conventional materials which require large volume to improve the sound absorption. Therefore, innovative thin and light-weight acoustic material designs, efficient at low frequencies need to be developed. One solution to this problem is the use of thin acoustic metamaterial [1-3] based on a structural material composed of a periodic unit cell (PUC) assembly. These new concepts of thin geometry were first proposed by Leclaire et al. [4], and improved by Dupont et al. [5]. The proposed design comprised a perforated material for which the main perforations were connected to an array of periodically spaced thin annular dead-end pores with respect to the lateral size. This solution consisted of connected dead-end pores, i.e., thin cavity resonators, on a main tubular pore to create dead-end porosity materials. One of the advantages of this metamaterial design was to shift the resonance frequencies of the metamaterial (absorption peaks) towards the low-frequencies. This can be explained by an increase in the effective compressibility of the material. In continuation of the work of Dupont et al., Kone et al. [6,7] have proposed two types of metamaterials composed of a superposition of cylindrical PUCs. The PUC was a composition of a half-neck + a cavity + a second half-neck. The cavity and the half-necks were all cylindrical. The first metamaterials studied had a PUC composed of a co-axial cylinder (centered neck metamaterial). The second metamaterial with off-centered necks had the two half-necks of each PUC diametrically opposed. The number of the resonance frequencies of these metamaterials was directly dependent on the number of PUCs assembled in series. Moreover, like the metamaterial presented by Dupont et al. [5] the first resonance frequency was shifted towards the low frequencies as compared to the resonance frequency of a conventional resonator. The offset position of the half-necks of the off- centered neck metamaterial made it possible to increase the visco-thermal losses and the propagation path of the wave (increased tortuosity). Thus, a shift of the resonant frequencies of the metamaterial from the decentered neck towards the low frequencies compared to the centered neck metamaterial was obtained. Despite the good low-frequency acoustic performance of metamaterials proposed by Dupont et al. [5], and later by Kone et al.[7] the resonance frequencies are still relatively narrow. For extended frequency range of resonances at low frequency, Kone et al. [8] proposed another thin acoustic metamaterial based on a parallel assembly [8,9] of four sub-metamaterials of off-centered metamaterial type. Each of the four sub-metamaterials has been designed such that their lowest resonant frequencies are close. Thus their assembly made it possible to widen the first resonant frequency while attenuating up to 70% of the noise. Although the other resonant frequencies were also widened, the metamaterial was able to attenuate these frequencies by 40%.

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During our recent investigations [10,11], we recognized that it is possible to widen the absorption ranges while maintaining good acoustic attenuation by embedding the metamaterials in a layer of porous materials such as fiberglass. In addition, the preliminary work of Zacharie et al. [12] showed that the integration of a membrane in the metamaterials introduced other resonant frequencies. This article proposes a new thin acoustic metamaterial resulting from a parallel assembly of 4 sub- metamaterials separated by membranes, all embedded in a layer of fiberglass. Each sub-metamaterial is a serial assembly of periodic unit cells. Comsol's coupled sound pressure and shell approaches in the frequency domain were used to predict the sound absorption coefficient of the metamaterial. 2. MATERIALS The objective of this paper is to propose thin acoustic metamaterials permitting simultaneous control at multi-tonal low frequency and broadband noise for aeronautical applications. The studied metamaterials have cylindrical shapes (Figure 1). They are produced from an assembly of four sub- metamaterials, optionally separated by a membrane and all embedded in a layer of fiberglass of thickness 𝑡 𝑝 and length 𝐿 𝑝 . Each sub-metamaterial is a serial assembly of a finite number of PUCs. A PUC is a superposition of a half-neck + a cavity and a half neck. The two half-necks were cylindrical with diameter 𝑑 and thickness ℓ . The cavity of a PUC corresponded to a quarter of a cylinder of diameter 𝐷 𝐻𝑅 and thickness ℎ . Two metamaterials, one with centered necks of the sub- metamaterial and one with off-centered neck of sub-metamaterial are proposed. The two metamaterials differ according to the position of the half-necks on the cavities. In a section of the metamaterial (plane perpendicular to the axis of rotation) the neck of the sub-metamaterial 𝑖 was identified by its polar position (𝑟 𝑖 , 𝜃 𝑖 ) .

Figure 1: Assembled metamaterials section. 2.1. Centered neck sub-metamaterial The first metamaterial studied was the centered neck sub-metamaterials (Figure 2c). Each sub- metamaterial had - cylindrical co-axial necks. The half-necks constituting each PUC of the sub- metamaterial (Figure 2a) are positioned at the center of the cavity faces perpendicular to the axis of the metamaterial according to the polar coordinates given in Table 1. The thickness ℎ of the cavity of each PUC was identical. Similarly the thickness ℓ/2 of the half neck of each PUC was identical. Each sub-metamaterial was composed of N PUCs assembled in series (Figure 2b). Half-necks have been added to the ends of each sub-metamaterial. Finally, the 4 sub-metamaterials were assembled in parallel and embedded in a fiberglass layer. Thus the total thickness of the metamaterial was 𝐿 𝑝 = 𝑁(ℓ+ ℎ) + ℎ+ 𝑒 with 𝑒 being the thickness of the rigid walls (Figure 2c).

Table 1 : Neck polar coordinates of 4 sub-metamaterial numbered from 1 to 4

𝒊 𝟏 𝟐 𝟑 𝟒 𝒓 𝒊 𝐷 𝐻𝑅 /5

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𝜽 𝒊 𝜋4 ⁄ 3𝜋4 ⁄ 5𝜋4 ⁄ 7𝜋4 ⁄

(a) (b) (c) Figure 2: Centered neck metamaterial. (a) Four PUCs of sub-metamaterial assembly, (b) Four sub- metamaterial assembly without fiberglass and (c) Four sub-metamaterials assembly with fiberglass. 2.2. Off-Centered neck sub-metamaterial The second studied metamaterial had an identical shape as the centered neck metamaterial but the necks were not co-axial (Figure 3a). The position in polar coordinate (𝑟 𝑖 , 𝜃 𝑖 ) of the half-necks before and after the cavity of the PUC of each sub-metamaterial (Figure 3b) is shown in the Table 2. Table 2: Polar coordinate of two half-Necks of the PUC according of sub-metamaterial number 𝑖 with 𝐷′ 𝐻𝑅 = (𝐷−𝑑 𝑖 −𝑒)/2 .

Half neck before PUC cavity Half neck after PUC cavity 𝒊 𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒 𝒓 𝒊 𝐷′ 𝐻𝑅 /2 𝜽 𝒊 11𝜋12 ⁄ 3𝜋4 ⁄ 5𝜋4 ⁄ 7𝜋4 ⁄ 𝜋12 ⁄ 3𝜋12 ⁄ 5𝜋12 ⁄ 7𝜋12 ⁄

(a) (b) (c) Figure 3: Off-centered neck metamaterial. (a) Four PUCs of sub-metamaterial assembly, (b) Four sub-metamaterial assembly without fiberglass and (c) Four sub-metamaterials assembly with fiberglass. 2.1. Centered neck sub-metamaterial with side wall membranes The objective of this configuration was to study the effect of a membrane on the design of the two metamaterials described in the previous sections. For this, membranes of the same thickness 𝑒𝑚 were used to separate the four cavities resulting from the parallel assembly of the four sub-metamaterial PUCs (Figure 4). Thus these membranes replaced the rigid walls of the previous configurations.

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Titanium beta-21S with a Young's modulus 𝐸= 105 × 10 9 𝑃𝑎 and Poisson's ratio 𝜈= 0.33 was used as the material of the membrane.

Figure 4: Center neck metamaterial incorporating cavity separation membranes and no fiberglass. 3. MODELING OF THE METAMATERIAL Comsol Multiphysics was used for the prediction of the absorption coefficient of the metamaterial backed by a rigid wall at normal incidence. The frequency domain sound pressure modulus was used. The walls of each PUC modelled as rigid. The Johnson Champoux-Allard (JCA) [13] model was used to represent the acoustics in the necks, the cavities (slit) and the fiberglass. The parameters required for Comsol's JCA model for a circular neck, cavity and fiberglass are given in Table 3. When the membranes were included in the metamaterials, they were modeled using the Multiphysics shell module from Comsol. Fixed constraints were imposed on the center line of the metamaterial, whereas on the other extremities of the membranes, a free boundary condition was imposed. Table 3: Johnson-Champoux-Allard (JCA) [13] parameters of the cavity and neck and fiberglass, where 𝜂 is the dynamic viscosity of air.

Materials Viscous and thermal characteristic lengths 𝚲

Viscous and thermal characteristic lengths Λ’

Tortuosity (   )

Static airflow resistivity 𝝈(𝑷 𝒂 . 𝒔𝒎 𝟐 ) ⁄

Open porosity 𝚽 (%)

Cavity h (𝑚𝑚) h (𝑚𝑚) 1 12𝜂ℎ 2 Φ ⁄ 100 Neck 𝑑 𝑖 /2(𝑚𝑚) 𝑑 𝑖 /2 (𝑚𝑚) 1 32𝜂𝑑 2 Φ ⁄ 100 Fiberglass 85 (𝜇𝑚) 170 (𝜇𝑚) 1 20 709 85 4. RESULTS The Comsol FEM approach was used to predict the normal incidence sound absorption coefficients of the metamaterials described in the previous section. The geometric parameters used for the design of each sub-metamaterial are given in - Table 4. The design of each metamaterial consisted of 8 PUCs. Table 4: Values of geometric parameters o f metamaterials for numerical calculation.

Neck Cavity Fiberglass Rigid wall 𝒊 𝟏 𝟐 𝟑 𝟒 𝒉 𝑫 𝑯𝑹 𝒕 𝒑 𝑫 𝒑 𝒆 𝒅 𝒊 2.75 2.075 2.4 1.73 2.59 40.0 9.0 60.0 1 𝓵 2.59 4.1. Centered neck sub-metamaterial The sound absorption coefficients at normal incidence of the centered neck metamaterial with or without a layer of fiberglass are shown in Figure 5. As previously demonstrated in ref. [11] the band of first resonant frequency of absorption at low frequencies has been widened. The band central frequency was 𝑓 0 = 340 Hz. The width of the resonant frequency band was ∆𝑓= 180 Hz. The sound

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absorption coefficient within the resonance frequency band was of at least 50% without the fiberglass layer and at least 35% with the fiberglass layer. For frequencies above 430 Hz the average noise absorption was 30% for the metamaterial without the fiberglass layer. Embedding the metamaterial within a layer of fiberglass increases the acoustic efficiency of the high-frequency metamaterial. Thus, for frequencies between 430 Hz and 850 Hz the noise absorption coefficient with the fiberglass layer was between 35% and 65% . Moreover, above 850 Hz the sound absorption coefficient of the metamaterials embedded in a layer of fiberglass was greater than 65% .

Figure 5: Centered neck metamaterial sound absorption coefficient predictions using the FEM calculations with and without fiberglass. Black curve corresponds to the metamaterial without fiberglass and red curve corresponds to the metamaterial with fiberglass. 4.2. Off centered neck sub-metamaterial The sound absorption coefficients (SAC) at normal incidence of the off centered neck metamaterial with or without the layer of fiberglass are shown in - Figure 6. As for the centered neck metamaterial, a broadening of the low frequency absorption frequency band was observed with the off centered neck metamaterial. The central resonant frequency was 𝑓 0 = 278 Hz. The width of the resonant frequency band was ∆𝑓= 125 Hz. The SAC within the resonance frequency band was at least 45% without the fiberglass layer and at least 35% with the fiberglass layer. This broadening of the range was due to the parallel assembly of the four sub-metamaterials. The four sub-metamaterials have been designed so that their first resonance frequencies are very close to one another [11]. It can be observed that embedding the metamaterial in a layer of fiberglass resulted in an increase in the absorption efficiency of metamaterial up to 65% for the frequencies above 850 Hz.

33333 33 35 U9[94yJ009 oAsNooy punog 2500 +500 "000 0 eer a

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‘500 +000 1500 2000 2500 ny

Figure 6: Off centered neck metamaterial sound absorption coefficient predictions using the FEM calculations with and without fiberglass. Black curve corresponds to the metamaterial without fiberglass and red curve corresponds to the metamaterial embedded in fiberglass. 4.3. Comparison between centered neck and off centered neck metamaterials Sound absorption coefficients at normal incidence of centered neck and off centered neck metamaterial configurations embedded in a layer of fiberglass are shown in- Figure 7. It can be observed that the central resonant frequency 𝑓 0 of the off centered neck metamaterial was shifted by approximately 55 Hz towards the low frequencies when compared to the centered neck metamaterial. This shift is due to the increase in the acoustic wave propagation path within the metamaterials [11]. Moreover, the resonance frequency band ∆𝑓= 125 Hz of the off centered metamaterial was narrowed and less sound absorption was observed in comparison to the centered metamaterial ( ∆𝑓= 180 Hz). However, both metamaterials had a similar sound absorption performance at high frequencies.

Figure 7: Sound absorption coefficient predictions using the FEM calculations Black curve corresponds of centered neck metamaterial and red curve corresponds to off center neck metamaterial. 4.4. Centered neck sub-metamaterial with membranes The objective of this study was to investigate the effect of a membrane integration as side walls in the two designs of the metamaterials presented in the previous sections. Only the metamaterial with a centered neck was used for this study. Figure 8 shows the sound absorption coefficient at normal incidence of the centered neck metamaterial with and without the membrane of thicknesses 𝑒𝑚= 0.5 mm, 1 mm and 1.5 mm mm (Figure 8). The predicted sound absorption coefficients with or without the membranes were the same regardless of the membrane thickness 𝑒𝑚= 1 mm and 1.5 mm at low frequency. Little change was observed at high frequency. For the metamaterial with membrane thickness less than 1 mm, the resonance frequency band was narrowed and its attenuation performance improved. Its upper limit was shifted towards the low frequency by about 30 Hz. At high frequencies a resonant frequency was obtained due to the vibration of the thin membrane. So from 𝑒𝑚= 1 mm- the membrane was assimilated into a rigid wall at low frequency because of the small cavity thickness ( ℎ= 3.59 mm). Only the thin membrane had an influence on the acoustic behavior of the proposed metamaterials. These preliminary results on the investigation of membranes influence in these types of metamaterials demonstrated that further -in-depth studies will be necessary.

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+500 2000 2500 Frequency (Hz) +1000 500

Figure 8: Centered neck metamaterial sound absorption coefficient predictions using the FEM calculations with and without membrane. Black curve corresponds to the metamaterial without membrane, red curve: membrane of 𝑒𝑚 = 1 mm; curve blue: membrane 𝑒𝑚= 0.5 mm and green curve: membrane 𝑒𝑚= 1.5 mm. 5. CONCLUSIONS This study presented two configuration of thin acoustic metamaterials resulting from an assembly of four sub-metamaterials embedded in a layer of fiberglass. Each sub-metamaterial was also a series assembly of periodic unit cells composed of a half-neck + cavity + half-neck. These two metamaterials differed by the positioning with respect to the cavity surface of the two half-necks. For each of the configurations, the sound absorption coefficient at normal incidence was simulated with Comsol Multiphysics. The results showed that these metamaterials could attenuate broadband noise by more than 40% above 230 Hz. Moreover, around of the first resonance frequency, the attenuation could reach 50%. At high frequencies the attenuation could reach more than 65%. Moreover, this study has shown the sound absorption performance improvement potential of the integration of a membrane in the design of these types of metamaterials 6. REFERENCES 1. Auregan Y. & Leroux M. Experimental evidence of an instability over an impedance wall in a

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