The impact of damping on the sound transmission loss of locally resonant metamaterial plates

Journal of Sound and Vibration 461 (2019) 114909

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The impact of damping on the sound transmission loss of locally resonant metamaterial plates Lucas Van Belle a,b, ∗ , Claus Claeys a,b , Elke Deckers a,b , Wim Desmet a,b a b

KU Leuven, Department of Mechanical Engineering, Division PMA, Celestijnenlaan 300 - Box 2420, 3001, Heverlee, Belgium DMMS Lab, Flanders Make, Belgium

article info

abstract

Article history: Received 11 October 2018 Revised 4 August 2019 Accepted 17 August 2019 Available online 23 August 2019 Handling Editor: I. Lopez Arteaga

Vibro-acoustic locally resonant metamaterials with structural stop band behaviour can lead to a strongly increased sound transmission loss in a targeted frequency range. This work assesses the impact of damping in the constituents of metamaterial plates on their acoustic insulation performance by means of infinite periodic and finite structure modelling. Besides applying the hybrid Wave Based - Finite Element unit cell method for infinite plates and the Finite Element Method for finite plates, qualitative dispersion curve based predictions are extended to quantitative sound transmission loss approximations by introducing a dispersion curve based equivalent plate method. Both an idealised and a realisable locally resonant metamaterial plate are analysed. Damping in the resonators in particular is found to have an important impact in and around the stop band, reducing the sound transmission loss peak, but improving the subsequent dip and reducing resonant transmission in a broadening frequency range around the stop band. The damping influenced sound transmission loss predictions for the realisable locally resonant metamaterial plate are experimentally validated by means of insertion loss measurements. It is shown that, by including damping in the infinite periodic structure modelling, acoustic insulation performance predictions with improved accuracy are obtained. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Locally resonant metamaterial Damping Stop band Sound transmission loss Unit cell modelling Insertion loss

1. Introduction In the search for novel lightweight and compact solutions for noise and vibration reduction, vibro-acoustic locally resonant metamaterials (LRMs) have recently emerged [1]. Regular lightweight materials suffer from an impaired noise and vibration attenuation performance due to their typically increased stiffness-to-mass ratio. Vibro-acoustic LRMs exhibit stop band behaviour for elastic wave propagation, which are frequency ranges in which no free wave propagation is allowed in the structure [2]. This enables achieving a strongly improved noise and vibration attenuation in a targeted frequency range. The resonance-based stop bands in LRMs can be obtained by adding or embedding resonant structures to or in a flexible host structure on a sub-wavelength scale [3–5]. A resonance-based stop band for the acoustically relevant out-of-plane bending waves can generate a frequency zone of increased sound transmission loss (STL), strongly outperforming the acoustic mass-law. This has been investigated for among others LRM plates [6–13], double-walls [14–16], sandwich panels [15,17], solids [1,18] and enclosures [19]. In this

∗ Corresponding author. KU Leuven, Department of Mechanical Engineering, Division PMA, Celestijnenlaan 300 - box 2420, 3001, Heverlee, Belgium. E-mail address: [email protected] (L. Van Belle). URL: http://www.mech.kuleuven.be/ (L. Van Belle). https://doi.org/10.1016/j.jsv.2019.114909 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

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work, LRM plates composed of a flat plate host structure with sub-wavelength, periodically attached resonators are considered. The resonance-based nature of LRMs makes them especially suitable for vibro-acoustic problems in distinct frequency ranges, in particular in the hard-to-address low-frequency region, while their sub-wavelength nature allows preserving a compact design. However, their typical narrowband performance currently hinders the applicability of LRMs for broadband vibroacoustic problems [20]. Different strategies have been proposed to broaden the frequency range of improved STL, for instance combining resonators tuned to different frequencies [6,12,13,15], using parallel arranged or stacked LRM partitions [7,12,13,21] or transitioning to active LRMs using e.g. tunable piezoelectric resonators [22,23]. Also damping has been found to potentially broaden the frequency range of vibro-acoustic attenuation. Damping is present in all structures, in particular in plastic or rubberlike materials often used in LRM realisations. The effect of damping on the vibration attenuation performance of LRMs has been widely investigated [4,24–30]. Especially damping in the resonators has been found to broaden the frequency range of vibration attenuation at the expense of a reduced peak attenuation performance inside the stop band. The effect of damping on the STL of vibro-acoustic LRMs has been less extensively investigated. Wester et al. [18] analysed the influence of damping in the resonator on the normal incidence STL of an LRM slab composed of rubber coated steel inclusions in an epoxy hosting matrix, by means of a simplified 2 degree-of-freedom mass-spring-based model. Resonator damping was found to decrease the tuned STL peak, but to improve the subsequent STL dip. Good agreement was obtained between the predicted STL and measured normal incidence STL in an impedance tube. Hall et al. [12] performed a similar analysis of an LRM partition using the same model and also reported good agreement between predicted and measured diffuse field STL for a larger partition. The simplified model, however, discards the dynamics of the host structure and the interactions between the resonators, and does not account for damping in the host structure. In Refs. [6,15], damping was included in the STL analysis of infinite LRM plates with periodically attached mass-spring resonators, showing a rounding of the STL peak and dip, but its effect was not explicitly investigated. Recently, Wang et al. [8] analysed the effect of damping in the resonator and host structure on the STL of an infinite LRM plate with periodically attached mass-spring resonators. The STL peak and dip around the stop band were found to be smoothed by the resonator damping, while damping in the host structure was found to only affect the coincidence region away from the stop band. The modelling of LRMs is generally based on infinite periodic structure representations, often relying on unit cell (UC) modelling, to predict the structural wave propagation and vibration attenuation by means of dispersion curves, or to predict the STL of LRM partitions. When considering practically realisable LRMs, however, finite dimensions also have to be accounted for. In case of sound transmission, this leads to resonant transmission due to structural modes as well as to diffraction effects due to the aperture containing the finite plate [31,32]. It is important to assess the influence of these finite structure effects and to verify the representativeness of the damping influenced STL predictions resulting from infinite periodic structure modelling. In current literature, mainly simplified models or infinite and idealised LRM partitions have been considered to analyse the effect of damping on the STL, while the comparison and validation of the damping influenced STL predictions with the acoustic insulation performance of LRM partitions of representative dimensions is limited, but necessary [33]. In view of practically realisable LRM partitions, this work investigates the impact of damping on the STL of a realisable LRM plate and compares it with an idealised design in order to gain insight in the effects of damping, to obtain more accurate vibro-acoustic performance predictions and to also assess the broadening of the acoustic insulation performance. The impact of damping in the LRM constituents on the STL of infinite periodic and finite LRM plates is compared, damping influenced dispersion curves for bending wave propagation are related to qualitative and quantitative acoustic insulation performance predictions and the damping influenced STL predictions for the realisable LRM plate are experimentally validated by means of insertion loss (IL) measurements on a manufactured sample of representative dimensions. This paper is organised as follows. Section 2 gives a brief overview of the analysis methods used in this work to predict the acoustic insulation performance of infinite and finite LRM plates. In section 3, the impact of damping on the STL of an idealised and a realisable LRM plate is analysed. In section 4, the damping influenced acoustic insulation performance of the realisable LRM plate is experimentally validated. Section 5 summarises the main conclusions. 2. Methodology After introducing the problem setting, this section briefly explains the applied analysis methods to predict the acoustic insulation performance of LRM plates. 2.1. Problem description In this work, LRM plates with 2D periodicity are considered (Fig. 1a). Resonators are added to a flat plate host structure on a sub-wavelength scale, with spatial period Lx and Ly in the xy-plane, and can be arbitrarily complex. For the sound transmission problem, the LRM plate is coupled to semi-infinite acoustic domains above and below the structure. The LRM plate is excited from one side by an acoustic plane wave with amplitude A and wave vector ka = (kax , kay , kaz ) = − ka (sin𝜃 cos𝜓, sin𝜃 sin𝜓, cos𝜃 ), with elevation angle 𝜃 and azimuth angle 𝜓 . The acoustic wavenumber in the surrounding acoustic medium with soundspeed ca and density 𝜌a is given by ka = 𝜔∕ca , while time harmonic behaviour with ej𝜔t dependence is considered, with j the imaginary unit (j2 = − 1), angular frequency 𝜔 and time t. The considered acoustic medium is air with ca = 340 m∕s and 𝜌a = 1.225 kg∕m3 . Although periodicity is not strictly required for resonance-based

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Fig. 1. The sound transmission problem for a 2D periodic LRM plate, consisting of a plate with sub-wavelength, periodically added resonators, represented schematically by mass-spring resonators (a), and corresponding vibro-acoustic UC with boundaries of the semi-unbounded acoustic domains at infinity indicated by Γa∞ (b).

stop band behaviour [1,19,34], it allows for infinite periodic structure modelling by considering a vibro-acoustic UC model (Fig. 1b) and application of the Bloch-Floquet theorem [2,35]. 2.2. Dispersion curve analysis The structural wave propagation in the infinite periodic LRM plate is analysed by means of dispersion curves, calculated using a structural LRM UC. The UC is modelled using the Finite Element Method (FEM), leading to following time harmonic equations of motion: D(𝜔)q = f,

(1)

with generalised displacements and forces q and f and the frequency dependent dynamic stiffness matrix D(𝜔), which is in general complex in the presence of damping. Bloch-Floquet periodicity boundary conditions are applied, which relate q and f on either side of the UC by means of the complex propagation constants 𝜇x = kx Lx and 𝜇 y = ky Ly , with structural wavenumbers kx and ky , leading to a dispersion eigenvalue problem in 𝜔 and (𝜇x , 𝜇 y ) [36]: A(𝜔, 𝜇x , 𝜇y )q(red) = 0,

(2)

with q(red) a reduced set of generalised displacements and A a general matrix depending on 𝜔 and (𝜇x , 𝜇 y ). To determine stop band behaviour, dispersion curves for free wave propagation are calculated for the undamped structure along the irreducible Brillouin contour (IBC) using the 𝜔(𝜇) solution approach to the dispersion eigenvalue problem. To account for damping, the 𝜇(𝜔) solution approach is also applied in this work [5,26,36]. √ By comparing the trace propagation constant 𝜇a =

(kax ∕Lx )2 + (kay ∕Ly )2 of an incident acoustic plane wave with the in-

vacuo bending wave dispersion curves, the uncoupled vibro-acoustic performance can be qualitatively assessed. Coincidence frequencies are found as intersections between 𝜇 a and the bending wave dispersion curves, while bending wave solutions falling within the radiation cone, bounded by 𝜇 a for grazing incidence (𝜃 = 90◦ ), lead to efficient acoustic radiation [31,37,38]. For a homogeneous, isotropic plate, the lowest coincidence frequency occurs for grazing incidence and is denoted the critical coincidence frequency fc . In Refs. [6,9,17,39], this qualitative approach was applied to dispersion curves for undamped LRM plates obtained with the 𝜔(𝜇) approach. In this work, the 𝜇(𝜔) dispersion curves for damped LRM plates will also be used to analyse the influence of damping in and around the stop band. This qualitative approach does not predict actual STL levels and does not account for vibro-acoustic interaction. 2.3. Infinite plate STL 2.3.1. Hybrid Wave Based - Finite Element UC method In current literature, the STL of infinite periodic LRM partitions is most often calculated using analytical approaches, considering idealised LRM plates with periodic mass-spring additions [6,7,9,15]. To calculate the STL of more general infinite periodic LRM partitions with arbitrarily complex resonators, FE-based UC modelling has been used, mostly applying perfectly matched layers (PMLs) on the acoustic boundaries of vibro-acoustic FE UC models [10,22]. In this work, the hybrid Wave Based - Finite Element (WB-FE) UC method recently introduced by Deckers et al. [11] is adopted. The hybrid WB-FE UC method represents the infinite periodic vibro-acoustic system by a single, vibro-acoustic UC model (Fig. 1b). The arbitrarily complex LRM UC is discretised using FEM and is coupled to two semi-unbounded periodic acoustic WB domains, above and below the structure, along a flat interface in the xy-plane. The pressure field in the acoustic WB domains is described using an expansion of wave functions that inherently fulfil the governing Helmholtz equation, the Sommerfeld

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radiation condition and the Bloch-Floquet periodicity boundary conditions. By including an acoustic FE subdomain in the FE UC domain, the direct vibro-acoustic interaction between resonator and surrounding acoustic medium can be accounted for. Using a direct hybrid WB-FE coupling strategy along the flat hybrid interface, a system of equations is obtained, which is solved for a given incident plane wave to the nodal degrees of freedom in the FE domain and wave function contributions in the WB domain. Using the latter, the sound power transmission coefficient 𝜏 𝜃 ,𝜓 is obtained, from which the STL𝜃 ,𝜓 = − 10log10 (𝜏 𝜃 ,𝜓 ) is calculated. In a diffuse field, plane waves are considered to be incident from all directions with equal probability and random phase [31]. The diffuse field sound power transmission coefficient 𝜏 d is calculated by averaging 𝜏 𝜃 ,𝜓 over 𝜃 and 𝜓 [31]: 𝜃

2𝜋

𝜏d =

∫0

∫0 l 𝜏𝜃,𝜓 sin 𝜃 cos 𝜃 d𝜃 d𝜓

2𝜋

∫0

𝜃

∫0 l sin 𝜃 cos 𝜃 d𝜃 d𝜓

,

(3)

with 𝜃 l the limiting elevation incidence angle. The 𝜃 l is often restricted to 78◦ , referred to as field incidence, to obtain better agreement with STL measurements in reverberation chambers [31,32,40]. 2.3.2. Equivalent plate method The hybrid WB-FE UC method can be computationally expensive for large UC models. Therefore, an equivalent infinite plate method is proposed to approximate the STL of infinite LRM plates. For resonators added on a sub-wavelength scale and with dimensions much smaller than the acoustic wavelength in the frequency range of interest, the infinite periodic LRM plate can be represented by an infinite plate with an equivalent dynamic mass density 𝜌eq for which the STL can be analytically calculated, corresponding to the effective medium methods in Refs. [6,7,15]. Contrary to an often used analytical calculation of 𝜌eq for idealised LRM plates with periodic mass-spring resonators, in this work, 𝜌eq is obtained for LRM plates with periodic, arbitrarily complex resonant additions by post-processing the in-vacuo 𝜇(𝜔) dispersion curves of the bending waves inside the first Brillouin zone [2]. Besides offering a fast STL approximation, this also allows relating the 𝜇(𝜔) dispersion curves for (damped) LRM plates to quantitative STL predictions and enables the impact of damping on the STL to be interpreted by means of 𝜌eq . Assume an LRM plate of thickness h, with sub-wavelength periodically added resonators. Consider, for generality, the thick plate bending wave dispersion relation kC (𝜔) of an isotropic, homogeneous plate of thickness h [37]:

𝜌 𝜌 ] 12𝜔2 𝜌(1 − 𝜈 2 ) 𝜔4 𝜌2 + − + =0 (4) 𝜅G E Eh2 𝜅 GE with the bending wavenumber kC , density 𝜌, complex Young’s modulus E = E0 (1 + j𝜂 ) with Young’s modulus E0 and structural damping loss factor 𝜂 , complex shear modulus G = E∕(2(1 + 𝜈 )), Poisson’s coefficient 𝜈 , and 𝜅 = 5∕(6 − 𝜈 ) the shear correction coefficient [41]. Rewriting Eq. (4) in terms of 𝜌 gives: k4C − 𝜔2 k2C

𝜌=

[

𝜔2 k2C (E

+ 𝜅 G) +

12𝜔2 𝜅 G(1−𝜈 2 ) h2

√[

−

−𝜔2 k2C (E + 𝜅 G) −

12𝜔2 𝜅 G(1−𝜈 2 ) h2

]2

2𝜔4

− 4𝜔4 k4C 𝜅 GE

.

(5)

For a damped, isotropic and homogeneous plate with loss factor 𝜂 , Eq. (5) leads to the real-valued 𝜌 of the plate. Considering the in-plane propagation direction 𝜓 , the kC (𝜔) in Eq. (5) is now replaced by the propagative (or spatially decaying) bending wave dispersion solution kb (𝜔, 𝜓 ) obtained with the 𝜇 (𝜔) approach in the first Brillouin zone, with Re(kb ) ≥ 0 and Im(kb ) ≤ 0 considering the wave propagation convention ejk·r ej𝜔t with wave vector k and position vector r. This results in the equivalent dynamic mass density 𝜌eq :

𝜌eq = 𝜌 ∣kC =kb (𝜔,𝜓 ) .

(6)

For LRM plates, this 𝜌eq can become complex-valued, providing insight in the behaviour of the equivalent dynamic mass in and around the stop band and the effect of damping thereon. The acoustic surface impedance of the equivalent infinite plate subjected to an acoustic plane wave incident at angles (𝜃 ,𝜓 ) is written in terms of this 𝜌eq [31,37]:

⎛ ⎜1 − Zeq,𝜃,𝜓 = j𝜔𝜌eq h ⎜ ⎜ ⎜ ⎝

[ ] ( ) 𝜔2 𝜌eq ⎞ h2 2 sin2 𝜃 1 + E + k − ⎟ a 12𝜌eq 𝜔2 (1−𝜈 2 ) 12 𝜅G 𝜅G ⎟. [ ( ) ] 𝜔2 𝜌eq h2 E ⎟ 1+ k2a sin2 𝜃 − ⎟ 12 𝜅G 𝜅G ⎠ k2a sin2 𝜃 Eh2

(7)

Using Zeq,𝜃 ,𝜓 , the sound power transmission coefficient and STLeq,𝜃 ,𝜓 of the equivalent plate are obtained [6,37]:

𝜏eq,𝜃,𝜓 =

1

| |1 + | |

2 Zeq,𝜃,𝜓 cos 𝜃 | 2𝜌a ca

| | |

, STLeq,𝜃,𝜓 = −10log10 (𝜏eq,𝜃,𝜓 ).

(8)

Since this equivalent plate method uses the in-vacuo dispersion curves, it does not account for fluid loading effects or the direct vibro-acoustic interaction between the infinite periodic structure and surrounding acoustic medium. Moreover, effective medium methods are only applicable in the long wavelength limit, while the combination of the isotropic plate assumption with the solutions only inside the first Brillouin zone also discards higher order Bloch waves [6].

L. Van Belle et al. / Journal of Sound and Vibration 461 (2019) 114909

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Fig. 2. Finite plate FEM-AML model with upper (red mesh) and lower (green mesh) acoustic halfspace separated by a symmetry plane which represents an infinite, rigid baffle (light green rectangle) (a) and excited by a set of distributed plane waves (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

2.4. Finite plate STL Finite dimensions have to be considered for practically realisable LRM partitions. In this work, the STL of an n × m UCs sized finite LRM plate with simply supported boundary conditions is calculated by means of a strongly coupled vibro-acoustic FE model using the software LMS VIRTUAL .LAB. The structural FE model of the finite LRM plate is strongly coupled to two semiunbounded acoustic domains on either side, separated by a symmetry plane representing an infinite, rigid baffle (Fig. 2a). The acoustic halfspaces are modelled using acoustic FE domains with Automatically Matched Layer (AML) non-reflecting boundary conditions [42]. The plate is excited on the incident side by a set of uncorrelated, distributed plane waves which represent an acoustic diffuse field excitation (Fig. 2b). A guideline is to consider at least 12 plane waves to obtain a good diffuse field representation and to limit the computational cost [42]. The coupled vibro-acoustic system is solved using the S YSNOISE direct vibro-acoustic solver and the STL is obtained from the sound power incident on and transmitted through the LRM plate. 3. Numerical analysis In this section, the idealised and realisable LRM plate designs are introduced and the acoustic insulation performance is first qualitatively assessed by means of dispersion curves. This is followed by the analysis of the impact of damping on the STL of the infinite periodic LRM plates. Next, the finite plate STL is investigated and compared with the infinite periodic structure STL. Eventually, the STL predictions for experimental validation are summarised. 3.1. LRM design An idealised and realisable LRM plate are considered. Both LRM plates consist of a 4 mm thick PMMA (Table 1) plate host structure with periodically added resonators (Fig. 3). For the realisable LRM plate, a beam-shaped PMMA resonator is considered [26], targeting the bending waves around 550 Hz by means of its first out-of-plane bending mode (Fig. 4b). For the relative mass addition of the resonators to the host structure, ratios between e.g. 0.2 and 19 have been considered in Ref. [6], between 0.2 and 2 in Ref. [18] and between 5.6 and 89.3 and higher in Ref. [23], showing that an increase of the mass ratio increases and broadens the STL improvement [6,18,23]. In this work, an added mass ratio below 1 is considered appropriate, as the added mass does not exceed the host structure mass. A mass ratio of 0.5 is aimed for, which also allows for the comparison with a 6 mm thick PMMA plate in section 4. To satisfy the sub-wavelength requirement for resonance-based stop band behaviour, a 0.03 × 0.03 m UC size is chosen. This leads to a half-wavelength frequency f𝜆/2 = 4282 Hz for the bending waves in the host structure, below which resonance-based bending wave stop band behaviour is ensured [4]. To obtain the targeted mass addition with reasonably sized resonators, two resonators are attached per UC using a Loctite® 406™ adhesive, with a 180◦ rotation symmetry and equidistant spacing in x- and y- direction (Fig. 3a). Based on a first numerical design, resonators were manufactured for the experimental validation of section 4. An average resonance frequency fres = 563 Hz was measured for the first out-of-plane bending mode for 10 laser cut resonators, glued with their base to a shaker. The measured average resonator dimensions are shown in Fig. 4a and a resulting actual mass ratio of 0.47 is obtained. The updated resonator FE model, consisting of 48 quadratic solid elements, predicts a resonance frequency for the first, out-of-plane bending mode at 562 Hz (Fig. 4b) and for the second, in-plane bending Table 1 Material parameters for PMMA.

Young’s modulus E0 [GPa]

Mass density 𝜌 [kg∕m3 ]

Poisson’s ratio 𝜈 [−]

Damping loss factor 𝜂 [−]

4.85

1188.38

0.31

0.05

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Fig. 3. UCs and corresponding IBCs of the realisable (a) and idealised (b) LRM plate.

mode at 1074 Hz (Fig. 4c). Application of the half power bandwidth method to the measured resonator responses leads to a damping loss factor 𝜂 = 0.05 [43]. The same 𝜂 is considered for both host structure and resonator, by means of a complex Young’s modulus E0 (1 + j𝜂 ). For the idealised LRM plate, mass-spring resonators are added to the host structure with 0.03 × 0.03 m periodicity (Fig. 3b), tuned to fres = 563 Hz, which satisfies the sub-wavelength requirement, and add 50% of mass. Corresponding to the realisable LRM design, structural damping with a loss factor up to 0.05 will be considered in both the host structure and resonators, by means of a complex Young’s modulus E0 (1 + j𝜂 h ) and spring stiffness k(1 + j𝜂 r ), respectively. In what follows, the influence of damping will be primarily analysed by means of the idealised LRM design, while differences arising from the real, non-zero resonator geometry in the realisable LRM design will be discussed. For both designs, the interest lies in the frequency range in and around the stop band, considering frequencies up to 1500 Hz. Since fc = 7499 Hz for the 4 mm PMMA host structure, regular coincidence effects are not considered. 3.2. Dispersion curve analysis The stop band behaviour of the undamped LRM plates is assessed by applying the 𝜔(𝜇) approach along their IBC (Fig. 5). The FE UC of the idealised LRM plate consists of 10 × 10 quadratic quadrilateral shell elements for the host structure, while the analytical equations of motion of the resonator are added to the FE dynamic stiffness matrix. The FE UC model of the realisable LRM plate consists of 96 quadratic hexahedral solid elements for the two resonators combined and 120 quadratic quadrilateral shell elements for the plate host structure, including a 2 mm neutral plane offset and connected with a nodal shell-solid coupling. The acoustic trace propagation constant is added for different 𝜃 for qualitative comparison, as well as the bending wave dispersion curves of the undamped, bare host structure. For the idealised LRM plate, a bending wave stop band is obtained around fres , between 560–688 Hz. The band gap width varies slightly with 𝜓 , from 557–688 Hz along OA (𝜓 = 0◦ ) to 560–688 Hz along OB (𝜓 = 45◦ ), due to the difference in interaction distance between adjacent resonators and periodicity of Re(𝜇). For the realisable LRM plate, a bending wave stop band is obtained between 557–615 Hz (Fig. 5b), around the tuned first resonator mode. The stop band is narrower as compared to the idealised resonator, due to the partly added non-resonant mass (e.g. the foot and part of the beam) and the lower effective resonator mass, with a calculated out-of-plane modal mass efficiency of 57% for its first mode. The band gap width varies more outspokenly with 𝜓 , from 546–615 Hz along OA (𝜓 = 0◦ ) to 557–615 Hz along OC (𝜓 = 90◦ ), also due to the directionally dependent interaction between the realisable resonator mode and the bending waves [26]. For both LRM plates, the repeated bending wave after the stop band causes an additional low-frequency coincidence zone, due to the stop band tuning below fc [39]. Contrary to the incidence angle dependency of the oblique coincidence frequency for a homogeneous isotropic plate [31], the additional low-frequency coincidence intersections occur in a narrow frequency region with a limited incidence angle dependency for the LRM plates, as well as for normal incidence. At higher frequencies after the stop band, the bending wave

Fig. 4. Realisable resonator dimensions in mm (a) and numerically predicted first, out-of-plane bending mode at 562 Hz (b) and second, in-plane bending mode at 1074 Hz (c), for a fixed resonator base.

L. Van Belle et al. / Journal of Sound and Vibration 461 (2019) 114909

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Fig. 5. Dispersion curves calculated using the 𝜔(𝜇 ) approach along the IBC for the undamped idealised (a) and realisable (b) LRM plates (solid lines), with acoustic trace propagation constant for incidence angles 𝜃 (dashed lines), obtained bending wave stop band (shaded grey) and bending wave dispersion curves of the undamped bare host structure (dash-dotted lines).

evolves to the bending wave of the bare host structure, due to the anti-phase resonator motion after resonance. For the realisable LRM plate, the second resonator mode also leads to a directional bending wave band gap, with absence of band gap behaviour for 𝜓 = 0◦ [26], after which an in-plane directionally dependent new zone of efficient radiation results. To gain insight in the impact of damping in and around the stop band, the idealised LRM plate is first considered. The 𝜇(𝜔) approach is applied for frequencies f = 1 − 1 − 1500 Hz (Fig. 6). The propagative or decaying bending wave dispersion curves are analysed along a single propagation direction 𝜓 = 0◦ , corresponding to OA along the IBC. The effect of damping in the host structure and resonator is assessed by varying 𝜂 h and 𝜂 r between 0 and 0.05. As discussed in Ref. [26], the 𝜇 (𝜔) approach finds the same freely propagating wave solutions for the undamped UC as the 𝜔(𝜇) approach (Fig. 6a). Inside the stop band, strongly spatially decaying bending waves occur, which also intersect with the acoustic trace propagation constants. Damping in the host structure mainly affects the solutions outside the stop band, increasing the bending wave attenuation (Fig. 6b). Only a limited onset of the closing of the stop band occurs near the stop band edges. For damping in the resonator, a strong closing of the dispersion curves by means of a complex 𝜇 loop occurs [26–28], reducing the peak attenuation inside the stop band and causing the bending waves to become attenuated in a broadening zone around the stop band (Fig. 6c). The intersections with the acoustic trace propagation constant right after the stop band also disappear. The attenuation away from the stop band is more limited as compared to host structure damping. Damping in both constituents combines these effects (Fig. 6d). The influence on the acoustic radiation is analysed further by means of the difference between the propagative part Re(𝜇) of the bending wave dispersion curves and the acoustic trace propagation constant 𝜇 a for grazing incidence (𝜃 = 90◦ ) (Fig. 7) [39]. For the propagative solutions inside the shaded region, corresponding to the radiation cone, Re(𝜇) − 𝜇 a < 0, which leads to efficient acoustic radiation [38]. For the LRM plate without damping, the cut-on freely propagating bending wave right after

Fig. 6. Bending wave dispersion curves of the idealised LRM plate calculated using the 𝜇(𝜔) approach along 𝜓 = 0◦ for 𝜂 h = 0, 𝜂 r = 0 (a), 𝜂 h = 0.05, 𝜂 r = 0 (b), 𝜂 h = 0, 𝜂 r = 0.05 (c) and 𝜂 h = 0.05, 𝜂r = 0.05 (d), coloured according to log10 (|Im(𝜇 )|), and with acoustic trace propagation constant for incidence angles 𝜃 (dashed lines).

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Fig. 7. Difference between Re(𝜇 ) of the bending wave solutions of the idealised LRM plate (Fig. 6) and 𝜇a for 𝜃 = 90◦ , for 𝜂 h = 0, 𝜂 r = 0 (a), 𝜂 h = 0.05, 𝜂 r = 0 (b), 𝜂 h = 0, 𝜂 r = 0.05 (c) and 𝜂 h = 0.05, 𝜂 r = 0.05 (d), coloured according to log10 (|Im(𝜇 )|). The zone of efficient acoustic radiation is shaded in grey.

the stop band indicates a zone of efficient radiation. Near the start of the stop band, the positive difference Re(𝜇) − 𝜇 a becomes large, strongly reducing the radiation efficiency [39]. For the LRM plate with damping in the host structure only, the stop band is limitedly affected and the bending wave right after the stop band still indicates a zone of efficient radiation (Fig. 7b). While the bending wave attenuation has increased right before the stop band, an outspoken positive peak difference Re(𝜇) − 𝜇 a still exist near the start of the stop band. Both for the undamped LRM plate and for the LRM plate with damping in the host structure only, part of the solutions inside the stop band are also located in the zone of efficient acoustic radiation (Fig. 7a and b). Given their strong spatial attenuation, however, coincidence is not expected to emerge given its resonant nature. Damping in the resonator shows potential to alleviate the zone of efficient acoustic radiation right after the stop band due to the strong closing of the bending wave stop band and the outspoken broadening of the zone of attenuation (Fig. 7c and d). However, the positive peak difference Re(𝜇) − 𝜇 a near the start of the stop band is also reduced, increasing the radiation efficiency compared to the undamped LRM plate and the LRM plate with damping in the host structure only. For the realisable LRM plate, with 𝜂 = 0.05 for both the resonator and host structure, damped dispersion curves are calculated for frequencies f = 1 − 1 − 1500 Hz along 𝜓 = 0◦ , 45◦ and 90◦ to assess the more outspoken directional dependency as well as the directional band gap (Fig. 8). The closing of the bending wave dispersion curves around the tuned stop band, mainly attributed to the damping in the resonator, increases the bending wave attenuation before and after the stop band and also causes the coincidence intersections right after the stop band to disappear. The host structure damping mainly contributes to the increased wave attenuation away from the stop band. The realisable LRM design leads to an outspoken directionally dependent closing of the dispersion curves, which is strongest along 𝜓 = 90◦ . This causes the peak distance Re(𝜇 ) − 𝜇a near

Fig. 8. Dispersion curves of the realisable LRM plate with damping calculated using the 𝜇(𝜔) approach along 𝜓 = 0◦ (a), 45◦ (b) and 90◦ (c), coloured according to log10 (|Im(𝜇 )|), with acoustic trace propagation constant for incidence angles 𝜃 (dashed lines).

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Fig. 9. STL of the infinite idealised LRM plate for 𝜂 h = 0, 𝜂 r = 0 (a), 𝜂 h = 0.05, 𝜂 r = 0 (b), 𝜂 h = 0, 𝜂 r = 0.05 (c) and 𝜂 h = 0.05, 𝜂 r = 0.05 (d), for incidence angles 𝜃 , along 𝜓 = 0◦ , calculated using the hybrid WB-FE UC method.

the start of the tuned stop band of the bending wave dispersion curves to be the largest along 𝜓 = 0◦ . However, near the end of the tuned stop band, Re(𝜇) − 𝜇 a is also the smallest along this direction. Around the second resonator mode, the resonator damping causes nearly closed band gaps, with the strongest remaining Re(𝜇)-variation and expected STL influence along 𝜓 = 90◦ . 3.3. Infinite plate STL 3.3.1. Hybrid Wave Based - Finite Element UC analysis The impact of damping on the STL of the infinite periodic, idealised LRM plate is calculated using the hybrid WB-FE UC method for frequencies f = 1 − 1 − 5000 Hz. The same 𝜓 = 0◦ as for the 𝜇 (𝜔) dispersion curves is analysed and different 𝜃 are considered. The structural FE UC model of previous section is used, while for the WB domains a truncation factor Np = 2 is applied [11]. As before, the 𝜂 h and 𝜂 r are varied between 0 and 0.05 (Fig. 9). For the undamped UC, a strong STL peak occurs inside the stop band at 561 Hz, near fres (Fig. 9a). The peak is followed by a strong STL dip, related to the introduced coincidence intersections for the different 𝜃 right after the stop band around 688 Hz. The STL peak occurs where the distance between Re(𝜇) and the acoustic trace propagation constants becomes large (Fig. 7a). The intersections with the strongly spatially decaying bending wave solutions inside the stop band do not lead to additional coincidence STL dips, since the high wave attenuation suppresses any resonant transmission effects, corresponding to the absence of STL dips inside the stop band as observed in e.g. Refs. [6,8,9,15]. Damping in the host structure leads to a limited reduction of the STL peak (Fig. 9b), corresponding to Ref. [8]. The coincidence dips after the stop band show a minor STL increase compared to the undamped LRM plate. Outside the stop band, the increased bending wave attenuation does not affect the non-resonant sound transmission [31,37]. For damping in the resonator, the STL in and around the stop band is strongly influenced (Fig. 9c), as found in Refs. [8,12,18]. The STL in the coincidence zone right after the stop band is improved, corresponding to the increased bending wave attenuation and improved Re(𝜇 ) − 𝜇 a , and the avoided coincidence intersections. However, also the STL peak is reduced, corresponding to the reduced Re(𝜇) − 𝜇 a peak (Fig. 7c). The STL peak frequency shifted slightly down to 560 Hz, while the STL dip frequencies shifted up, e.g. to 691 Hz for 𝜃 = 0◦ and to 696 Hz for 𝜃 = 80◦ . For damping in both resonator and host structure, mainly damping in the resonator governs the STL peak and dip around the tuned stop band (Fig. 9d). For the realisable LRM plate, the resonators are considerably smaller than the governing acoustic wavelength in the frequency range of interest. To calculate the damping influenced STL of the infinite periodic LRM plate with 𝜂 = 0.05, the coupling between the LRM plate and surrounding acoustic domains is hence simplified and a pure structural FE UC model is first considered (Fig. 10a and b). The WB domains are coupled to the FE shell faces along the entire UC plate surface on either side, discarding direct vibro-acoustic interaction with the resonators. The STL is calculated for frequencies f = 1 − 1 − 1500 Hz and varying incidence angles 𝜃 . To assess the influence of the outspoken directional bending wave attenuation, 𝜓 is also varied between 0◦ , 45◦ and 90◦ (Fig. 11). An STL peak at 554 Hz is obtained for all 𝜓 , near fres = 562 Hz, followed by a damped STL dip right after the stop band. As observed in the preceding analysis, the STL dip and peak are mainly governed by the resonator damping. Contrary to the clear in-plane directional dependency of the width and closing of the stop band (Fig. 8), the STL peak and dip show only little variation with 𝜓 . Between 1000–1100 Hz, the STL is only slightly influenced by the directional bending wave band gap and shows an inplane dependent STL peak-dip. Whereas the tuned omnidirectional stop band leads to an STL peak-dip for all 𝜃 , the STL peak-dip related to the directional band gap only emerges for oblique incidence since the underlying resonator mode is not excited for a normally incident acoustic plane wave. In order to obtain an STL increase, an omnidirectional resonance-based bending wave

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Fig. 10. Structural FE UC domain with highlighted (blue) host structure (a) and resonators (b) and vibro-acoustic FE UC domain with highlighted host structure (c) and resonators (d) and acoustic FE subdomain shaded in grey. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 11. STL of the infinite realisable LRM plate for 𝜓 = 0◦ (a), 45◦ (b) and 90◦ (c) for incidence angles 𝜃 , calculated using the hybrid WB-FE UC method with the structural FE UC model.

stop band is thus desired, generated by sub-wavelength arranged resonators that generate a net non-zero out-of-plane force at resonance. To assess the influence of the vibro-acoustic coupling between the realisable resonators and surrounding acoustic medium, a coupled vibro-acoustic FE UC model is now considered, consisting of the same structural FE domain, strongly coupled to an acoustic FE domain (Fig. 10c and d). The acoustic FE domain has a height of 18.74 mm and is modelled with 984 quadratic hexahedral acoustic elements. The WB domain on the incidence side is coupled to the flat acoustic boundary, while the WB domain on the transmission side is coupled to the flat shell boundary. The STL is compared for normal and oblique (𝜃 = 70◦ , 𝜓 = 90◦ ) incidence (Fig. 12). Around the tuned stop band, the STL peak-dip is nearly unaffected. At higher frequencies, the directional band gap leads to a slightly more outspoken STL peak-dip behaviour for oblique incidence, since the second resonator mode is better excited by the acoustic waves when the vibro-acoustic interaction is included. The good agreement justifies omitting the coupling between the sub-wavelength resonators and the surrounding acoustic medium for the considered LRM design in the frequency range of interest. 3.3.2. Equivalent plate analysis The equivalent plate method is applied to the idealised LRM plate, using the 𝜇(𝜔) bending wave solutions of Fig. 6 for different 𝜃 (Fig. 13). Good agreement is obtained with the results from the hybrid WB-FE UC method (Fig. 9), especially for damping in the resonator. The omitted fluid loading effects are of limited influence. For the LRM plate without damping (Fig. 13a) and with damping in the host structure only (Fig. 13b), the STL peak is underestimated. This is caused by the periodicity in Re(𝜇) of the wave solutions, which results in a bifurcation of the 𝜇 (𝜔) dispersion curves inside the stop band at the edge of the first Brillouin zone (Fig. 6a), limiting the maximum value of Re(𝜇) [44] and thus of 𝜌eq and the STL. For host structure damping, the solutions still attain this boundary (Fig. 6b). For sufficient resonator damping, the strong closing of the dispersion curves causes the bending wave solutions inside the stop band to no longer attain this boundary (Fig. 6c and d), reducing this discrepancy. The impact of damping in the host structure and resonator on the STL of the infinite idealised LRM plate is further investigated by analysing 𝜌eq . To reduce the influence of the complex bifurcations at the edges of the Brillouin zone, 𝜌eq is calculated for 𝜓 = 45◦ . The real and imaginary parts of 𝜌eq , related to the inertia and attenuation of the structure respectively [45], are

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Fig. 12. STL of the infinite realisable LRM plate for normal (a) and oblique (𝜃 = 70◦ , 𝜓 = 90◦ ) (b) incidence, obtained using the hybrid WB-FE UC method with the structural (solid line) and vibro-acoustic (dashed line) FE UC model.

Fig. 13. STL of the infinite idealised LRM plate for 𝜂 h = 0, 𝜂 r = 0 (a), 𝜂 h = 0.05, 𝜂 r = 0 (b), 𝜂 h = 0, 𝜂 r = 0.05 (c) and 𝜂 h = 0.05, 𝜂 r = 0.05 (d), for incidence angles 𝜃 , along 𝜓 = 0◦ , calculated using the equivalent plate method.

compared, as well as the magnitude |𝜌eq |. Additionally, 𝜌eq is compared with the host structure density 𝜌h and with the static mass density 1.5𝜌h of the LRM plate (Fig. 14). To assist the following discussion, the STL is calculated using the equivalent plate method for 𝜓 = 45◦ and 𝜃 = 40◦ (Fig. 14d). For the undamped LRM plate, 𝜌eq is real outside the stop band. Inside the stop band, 𝜌eq is complex since Im(𝜌eq ) is nonzero, indicating the attenuation due to the Fano-type interference [44,46], with Im(𝜌eq ) reaching a peak magnitude around the start of the stop band, near fres . At low frequencies, below the stop band, Re(𝜌eq ) and thus |𝜌eq | equals 1.5𝜌h , while after the stop band, Re(𝜌eq ) and thus |𝜌eq | evolves to 𝜌h . This corresponds to the in-phase and anti-phase motion between host structure and resonators before and after the stop band [39], causing the STL to correspond to the STL of an equivalent plate with static mass density 1.5𝜌h and 𝜌h , respectively. Near the start of the stop band, Re(𝜌eq ) strongly increases and reaches a peak, followed by a rapid transition around fres to strongly negative values, characteristic of the local resonance mechanism, and is negative until the end of the stop band [6,7,47]. Considering the total dynamic mass density |𝜌eq |, a peak results near fres , strongly exceeding 1.5𝜌h and leading to the mass law outperforming, strong STL peak. Near the end of the stop band, |𝜌eq | is very small, enabling the low-frequency coincidence and the STL dip after the stop band. These results agree with the observations in Refs. [6,7]. Damping in the host structure leaves 𝜌eq largely unaffected. Since the 𝜂 h is included in E in Eq. (5), it does not contribute to a complex 𝜌eq away from the stop band. The frequency range of non-zero Im(𝜌eq ) is limitedly broadened around the stop band, while the dip magnitude slightly decreases. Damping makes the distinction between attenuation due to dissipation and Fano-type interference vague. Around the stop band, the Re(𝜌eq ) peak and dip magnitude slightly reduce. The resulting |𝜌eq | peak is slightly reduced and |𝜌eq | dip is slightly improved. This leads to a slightly lower STL peak and improved STL dip around the stop band. However, the |𝜌eq | peak still strongly outperforms the mass law, while the dip still enables coincidence right after the stop band.

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Fig. 14. Re(𝜌eq ) (a), Im(𝜌eq ) (b) and |𝜌eq | (c) and corresponding STL for 𝜃 = 40◦ (d) for the infinite idealised LRM plate for different 𝜂 h and 𝜂 r , calculated using the equivalent plate method along 𝜓 = 45◦ .

Damping in the resonator strongly impacts 𝜌eq , mainly in and around the stop band. The frequency range of non-zero Im(𝜌eq ) broadens outspokenly, while the Im(𝜌eq ) dip magnitude greatly decreases. The sharp Re(𝜌eq ) peak-dip transition is strongly smoothed. The resulting strong |𝜌eq | peak reduction significantly reduces the mass law outperforming STL peak. The |𝜌eq | dip, on the other hand, is strongly improved, improving the STL dip right after the stop band. It also causes the intersections between the bending wave solution and trace propagation constants to disappear, adding to the improvement of the STL dip after the stop band. Corresponding to the STL peak and dip frequencies, the |𝜌eq | peak and dip frequencies have slightly decreased and increased, respectively. Damping in both constituents combines the above effects, while mainly resonator damping influences the stop band region. The applicability of the equivalent plate method to the realisable LRM plate is investigated. While good agreement with the hybrid WB-FE UC method was obtained for the idealised LRM plate, the realisable LRM plate hosts resonators with a non-zero geometry and leads to an outspoken directional bending wave attenuation performance. Using the bending wave solutions obtained with the 𝜇(𝜔) approach (Fig. 8), the STL is calculated along propagation directions 𝜓 = 0◦ , 45◦ and 90◦ and compared with the hybrid WB-FE UC method (Fig. 15). Overall, the STL trend is similar to the results obtained with the hybrid WB-FE UC method, using the same FE UC model. However, some outspoken differences are observed. Around the tuned stop band, the in-plane directional dependency of the STL peak and dip obtained with the equivalent plate method is considerably stronger than predicted by the hybrid WBFE UC method. This indicates that the STL is determined by the forced response of the infinite periodic structure due to the distributed acoustic excitation, rather than by a single, possibly spatially decaying, bending wave propagating along 𝜓 . The best agreement is found along 𝜓 = 90◦ , for which the narrowest band gap and lowest bending wave attenuation are

Fig. 15. Comparison between normal and oblique (𝜃 = 60◦ ) incidence STL calculated for the infinite realisable LRM plate using the hybrid WB-FE UC method (solid lines) and equivalent plate method (dashed lines) for 𝜓 = 0◦ (a), 45◦ (b) and 90◦ (c).

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predicted. Along this direction, the damping influenced STL peak and dip around the targeted stop band are well approximated. The equivalent plate method also predicts an in-plane directional dependency for 𝜃 = 0◦ . However, as in this case (kx , ky ) = (0, 0) is imposed, no 𝜓 -dependency is expected, as found with the hybrid WB-FE UC method. This can be caused by the combination of an isotropic plate equation with the anisotropic LRM plate bending wave dispersion curves along the imposed 𝜓 , instead of imposing a distributed excitation. Around 1000–1100 Hz, the directional small STL peak and dip variation related to the directional band gap is retrieved, but the equivalent plate method again overestimates the magnitude and incorrectly predicts a directional dependency for 𝜃 = 0◦ . The equivalent plate approach allows for fast STL approximations and assessment of the influence of damping, while it also provides physical insight by means of 𝜌eq . However, discrepancies emerge in particular for outspoken directionally dependent bending wave propagation due to the underlying simplified and uncoupled representation.

3.4. Finite plate STL In this section, the STL of a finite idealised LRM plate counterpart is analysed. A simply supported 0.6 × 0.42 m LRM plate is considered, consisting of 20 × 14 UCs. The LRM plate is modelled with 4480 linear quadrilateral shell elements, with 20 × 14 attached mass-spring resonators, modelled using CBUSH and CONM2 elements. The acoustic domains are discretised with 4480 linear hexahedral acoustic elements each. To obtain a more accurate diffuse field representation, a set of 67 uncorrelated distributed acoustic plane waves is used as opposed to the guideline of 12 plane waves. The diffuse field STL is calculated for the undamped LRM plate and for damped LRM plates, by varying 𝜂 h and 𝜂 r between 0, 0.005 and 0.05 in the frequency range f = 1 − 1 − 1500 Hz (Fig. 16). The undamped LRM plate shows the STL peak and dip related to the bending wave stop band (Fig. 16a). Inside the stop band, no structural modes occur and the transmission is non-resonant, while the structural modes outside the stop band cause resonant STL dips. The high modal density right before and after the stop band is related to the highly dispersive bending wave mode before and after the stop band [39]. The impact of damping in the constituents corresponds well to the observations for the infinite LRM plate. In addition, the resonant transmission due to structural modes is strongly affected. For damping in the host structure, the increased wave attenuation outside the stop band strongly attenuates the modal resonant STL dips, with an increasing improvement with frequency (Fig. 16b). Damping in the resonator causes freely propagating modes before and after the stop band to disappear in a broadening frequency range around the stop band. Together with the broadening zone of attenuation around the stop band, the modal resonant STL dips before and after the stop band are strongly improved in a broadening frequency range (Fig. 16c). Further away from the stop band, the improvement is lower as compared to damping in the host structure, due to the higher wave attenuation away from the stop band for the latter (Fig. 6). The diffuse field STL predictions for the infinite and finite LRM plates without damping and with 𝜂 h = 0.05, 𝜂 r = 0.05 and 𝜂 h , 𝜂 r = 0.05 are compared (Fig. 17). For the predictions with the hybrid WB-FE UC method, Eq. (3) is used with 𝜃 l = 78◦ and d𝜃 = 1◦ . Given the diffuse field representation by a set of incidence angles up to a given 𝜃 l , the STL is also calculated with 𝜃 l = 72◦ and 84◦ to provide an interval of diffuse field STL predictions. For different 𝜃 l , the trend of the STL and the STL peak and dip frequencies is retained, but the STL decreases with increasing 𝜃 l .

Fig. 16. Diffuse field STL calculated using FEM-AML for the undamped finite LRM plate with stop band limits (black lines) (a) and for the finite LRM plate with damping in host structure (b), resonator (c) and both host structure and resonator (d).

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Fig. 17. Comparison of the diffuse field STL using the hybrid WB-FE UC method for the infinite ( ) and the FEM-AML method for the finite ( ) LRM plate for 𝜂 h = 0, 𝜂 r = 0 (a), 𝜂h = 0.05, 𝜂r = 0 (b), 𝜂 h = 0, 𝜂 r = 0.05 (c) and 𝜂h = 0.05, 𝜂r = 0.05 (d). For the infinite LRM plate, additional STL bounds for 𝜃 l = 72◦ , 84◦ are ). added (

Fig. 18. Comparison of the ΔSTL between the diffuse field STL of the LRM and bare host structure with 𝜂 h = 0.05, using the hybrid WB-FE UC method ( ) and the ) for 𝜂 h = 0, 𝜂 r = 0 (a), 𝜂 h = 0.05, 𝜂 r = 0 (b), 𝜂 h = 0, 𝜂 r = 0.05 (c) and 𝜂 h = 0.05, 𝜂 r = 0.05 (d). For the infinite LRM plate, additional FEM-AML method ( ΔSTL bounds for 𝜃 l = 72◦ , 84◦ are added ( ).

Very good agreement between the infinite and finite LRM plate STL predictions is obtained in and around, as well as outside the stop band. Inside the stop band, no structural modes are present and the STL of the finite plate corresponds well to the non-resonant STL predicted for the infinite LRM plate. The structural modes outside the stop band cause fluctuations in the finite plate STL around the predicted STL for the infinite LRM plate. Some discrepancies are observed, mainly in the lower frequency range. On the one hand, the STL of the finite plate shows a stiffness region, followed by some outspoken structural modal behaviour. On the other hand, diffraction effects due to the finite aperture also cause the STL predictions for the infinite plate to underestimate the finite plate performance [32,40]. To reduce discrepancies arising from finite aperture diffraction effects, a ΔSTL is also compared. In view of the experimental validation in the upcoming section, the ΔSTL = STLLRM − STLbare is calculated between the STL of LRM plate and the STL of the 4 mm bare host structure with 𝜂 h = 0.05 (Fig. 18). Compared to the STL, the variation of the ΔSTL with 𝜃 l for the infinite LRM plate is reduced and an even better agreement between the predictions for the finite and infinite LRM plates is obtained. The predicted impact of damping on the STL of the infinite periodic LRM plate corresponds well to its finite counterpart. 3.5. STL predictions for experimental validation For the experimental validation in the upcoming section, the damping influenced diffuse field STL of the infinite periodic realisable LRM plate with 𝜂 = 0.05 is calculated with the hybrid WB-FE UC method, using the structural FE UC model. Additionally, the STL is calculated for an infinite 4 mm and 6 mm bare PMMA plate with 𝜂 = 0.05. The 4 mm bare plate corresponds

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Fig. 19. Diffuse field STL predictions for the realisable LRM plate and the 4 mm and 6 mm bare plates for 𝜃 l = 78◦ , with STL bounds for 𝜃 l = 72◦ , 84◦ , calculated using the hybrid WB-FE UC method (a) and ΔSTL between the LRM and 4 mm bare plate (b).

to the LRM plate host structure and the static mass of the 6 mm bare and the LRM plate is approximately equal. The STL is calculated for frequencies f = 50 − 5 − 1500 Hz and Eq. (3) is applied with d𝜃 = 3◦ , d𝜓 = 3◦ and 𝜃 l = 78◦ . Additional STL bounds for 𝜃 l = 72◦ and 84◦ are also calculated (Fig. 19a). Around the tuned stop band, an STL peak of 36.1 dB is obtained at 555 Hz, outperforming the 4 mm and 6 mm bare plates, followed by an STL dip of 13.9 dB at 620 Hz. Below the stop band, towards low frequencies, the STL corresponds to the STL of the 6 mm bare plate since both panels have a comparable static mass. After the stop band, at higher frequencies, the STL approaches the STL of the 4 mm bare plate due to the out-of-phase motion between host structure and resonators. Since the realisable resonators also add some non-resonant mass, the STL still exceeds the STL of the 4 mm bare plate. Between 1000–1100 Hz, the directional band gap causes a small peak and dip in the diffuse field STL, due to the contribution of oblique incident waves. Between 𝜃 l = 72◦ and 84◦ , the targeted STL peak varies from 37.1 dB to 34.8 dB, while the STL dip varies from 14.5 dB to 13.3 dB. In the considered frequency range, the variation with respect to the diffuse field STL calculated for 𝜃 l = 78◦ is approximately ±1.5 dB. To obtain a more robust comparison of the acoustic insulation performance, the ΔSTL = STLLRM − STLbare is calculated between the diffuse field STL predictions for the LRM and the 4 mm bare plate (Fig. 19b). Compared to the STL, the ΔSTL dip shows a limited variation between −7.2 dB and −8.3 dB, with a value of −7.9 dB for 𝜃 l = 78◦ , while the variation on the ΔSTL peak of 15.3 dB for 𝜃 l = 78◦ is negligible. Besides being more robust against the choice for 𝜃 l , the ΔSTL can also be compared with IL measurement results. 4. Experimental validation Since a realisable LRM design is considered, the damping influenced STL predictions for the infinite realisable LRM plate are verified with the measured acoustic insulation performance of a manufactured LRM plate. In what follows, the experimental setup and the manufactured plate are first described, after which the IL measurements are presented and compared with the STL predictions. 4.1. Measurement setup and manufactured sample The acoustic insulation performance of a manufactured LRM plate is characterised by means of IL measurements on the KU Leuven Soundbox (Fig. 20a) [48]. This setup consists of a small cabin, which is composed of five reinforced concrete walls, sealed off by a 35 mm thick aluminium front wall. The impervious walls are skewed in order to obtain a uniform distribution of the natural frequencies of the acoustic modes in the interior 0.83 m3 cavity. Diffuse field conditions are met from 1600 Hz onwards. The front wall has an A2-sized (420 × 594 mm) transmission opening onto which a test sample can be clamped with a 10 mm thick aluminium clamping frame using 52 bolts. The Soundbox does not lend itself to STL measurements. Instead, the IL of the sample is obtained by measuring the radiated acoustic power Πopen through the open transmission opening and the radiated acoustic power Πclosed through the transmission opening closed with the mounted sample, for a speaker excitation in the corner of the cabin: IL = 10log10

Πopen . Πclosed

(9)

The sound power is measured using a B&K SOUND INTENSITY PP PROBE TYPE 2681 with a 12 mm spacer, applying a roving probe measurement procedure. The valid measurement range is 125–6300 Hz, with a frequency resolution of 1 Hz. An LMS SCADAS MOBILE is used in combination with the LMS T EST .L AB 16 software for data acquisition. Three different test samples are considered: the LRM plate, a 4 mm bare PMMA plate and a 6 mm bare PMMA plate. The plates have a total size of 640 × 860 mm, with an A2-sized wetted surface and an outer border containing a two-row bolt hole

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Fig. 20. The KU Leuven Soundbox [48] (a) with the LRM plate (b) clamped onto the front wall transmission opening (c).

pattern for the clamping frame. For the LRM plate, the wetted surface hosts a total of 532 resonators in 14 × 19 UCs centred on the plate, with the resonator beams oriented along the long side of the panel (Fig. 20b). The samples are clamped to the front wall with a fixed bolt torque of 30 Nm to allow for repeatable measurements (Fig. 20c). The IL describes the sound power reduction that is achieved with the test sample for a given source. However, the IL carries information of the test setup and the impedance of the test sample can influence the acoustic field. Moreover, the diffusivity limit is above the frequency range of the STL predictions, which causes the acoustic field in the Soundbox to be influenced by acoustic modes in the frequency range of interest. To reduce the influence of the acoustic cabin modes and to compare the IL measurements to the STL predictions, a difference ΔIL = ILLRM − ILbare is calculated between the measured IL of the LRM and 4 mm bare plate. 4.2. Measurement results and comparison An IL peak is obtained for the LRM plate near 534 Hz, around the start of the tuned stop band, exceeding the IL of the 4 mm and 6 mm bare plate (Fig. 21a). This is followed by an IL dip around 639 Hz, around the end of the stop band, which lies below the IL of both bare plates. In the considered frequency range, the narrowband spectra show many smaller peaks and dips, due to the modally dominated low-frequency range of the setup [48]: structural modes of the plates lead to resonant IL dips, while the acoustic modal behaviour of the small cabin is non-negligible. Acoustic cavity modes can in the closed configuration also lead to off-resonance forced structural modes or can coincide with structural modes. The ΔIL reduces the influence of the acoustic modes on the evaluation of the acoustic insulation performance (Fig. 21b). The IL peak-dip emerges as a well-defined ΔIL peakdip between 543 Hz and 603 Hz, slightly below the predicted STL peak and dip frequencies. In the stop band, the presence of modal ΔIL peaks and dips is less outspoken, since no structural modes are allowed in the LRM plate. The remaining peaks and dips in the stop band relate to the structural modes of the 4 mm bare plate, which are rather damped, and differences in the forced responses by the acoustic cavity modes. Around 1000–1100 Hz, a smaller peak-dip originates from the directional band gap.

Fig. 21. Measured IL of the LRM, 4 mm bare and 6 mm bare plate (a) and ΔIL between the LRM and 4 mm bare plate (b), with targeted stop band (vertical lines).

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Fig. 22. Measured IL of the LRM, 4 mm bare and 6 mm bare plate, averaged in 1∕24 octave bands (a), and ΔIL between the LRM and 4 mm bare plate (b), with targeted stop band (vertical lines).

Fig. 23. Comparison of the calculated diffuse field ΔSTL with the narrowband (a) and 1∕24 octave band (b) ΔIL between the LRM and the 4 mm bare plate.

To facilitate the interpretation, the IL is averaged in 1∕24 octave bands and a corresponding ΔIL is calculated (Fig. 22). Around the tuned stop band, this results in an IL peak of 35.8 dB at 538.6 Hz and an IL dip of 13.7 dB at 640.1 Hz. For the ΔIL, a peak and dip of +13.6 dB and −7.6 dB occur at 538.6 Hz and 604.3 Hz, respectively. Before the stop band, the positive ΔIL indicates the higher static and dynamic mass of the LRM plate as compared to the bare 4 mm plate host structure. After the stop band, the small but positive ΔIL highlights the contribution of the non-resonant added mass in the LRM plate. The narrowband and 1∕24 octave band ΔIL are compared to the predicted, damping influenced diffuse field ΔSTL for the infinite LRM plate, showing very good agreement (Fig. 23). Both the damping influenced ΔIL peak and dip frequency and amplitudes are well predicted as well as the slope inside the targeted stop band, despite the complex vibro-acoustic measurement environment and the non-diffuse field in the Soundbox. The ΔIL in 1∕24 octave bands allows for a more clear comparison, without strongly affecting the peak and dip amplitude around the stop band. Around 1000–1100 Hz, the ΔIL shows a more outspoken peak-dip behaviour related to the directional band gap as compared to the predicted ΔSTL. This can be related to the slight skewness of the laser cut resonators, which may lead to a better coupling of the in-plane bending resonator mode with the bending waves in the host structure. The slightly lower ΔIL peak and dip amplitudes can be caused by deviations in periodicity and scatter on the resonator dimensions and attachment. Additional differences between predicted ΔSTL and measured ΔIL can arise from the difference between infinite periodic structure assumptions and the finiteness of the Soundbox, including the finite aperture and structural and acoustic modes. Especially at low frequencies, the outspoken structural and acoustic modal behaviour as well as the valid measurement range of the intensity probe lead to stronger variations of the ΔIL. 5. Conclusion In this work, the influence of damping on the acoustic insulation performance of an idealised and realisable LRM plate with resonance-based bending wave stop band behaviour was investigated. The acoustic insulation performance was qualitatively assessed using dispersion curves, the STL was predicted using the hybrid WB-FE UC method for infinite and using FEM-AML for finite LRM plates and compared, and a dispersion curve based equivalent plate method was introduced. Mainly damping in the resonator was found to influence the STL of the infinite periodic structure in and around the stop band. Resonator damping reduces the peak STL performance and improves the additional coincidence STL dip after the stop band, while slightly shifting the STL peak and dip frequencies respectively down and up. Damping in the host structure leaves the

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stop band frequency range largely unaffected. The STL peak-dip for the realisable LRM plate was found to have a less outspoken directional dependency than found for its structural wave attenuation. Using the introduced equivalent plate method, good agreement was obtained with the STL predictions for the infinite periodic LRM plate using the hybrid WB-FE UC method, especially in presence of sufficient resonator damping. Discrepancies are attributed to the isotropic plate assumption and the periodicity of the dispersion curves, while an overestimation of the directional STL performance can occur due to the uncoupled vibro-acoustic nature of the method, discarding the distributed acoustic excitation. The equivalent plate method also provided insight in the effect of damping on the STL by means of 𝜌eq . The reduced STL peak around the start of the stop band and the improved STL dip after the stop band, mainly outspoken for resonator damping, correspond respectively to a reduced and increased dynamic mass compared with the undamped LRM plate. The impact of damping on the STL of a finite, idealised LRM plate was investigated and compared with the predictions for the infinite LRM plate. Good agreement was obtained, especially in and around the targeted stop band. Damping in the host structure mainly reduces resonant sound transmission outside the stop band. Damping in the resonator does not broaden the STL peak, but leads to reduced resonant transmission STL dips caused by structural modes in a broadening frequency range around the stop band along with an improved STL of the coincidence dip right after the stop band, at the cost of a reduced peak STL. The damping influenced diffuse field STL predictions obtained with the hybrid WB-FE UC method for the infinite realisable LRM plate were experimentally validated by means of IL measurements on an LRM plate of representative dimensions. The good agreement between predictions and measurements underlines the influence of damping on the acoustic insulation performance and demonstrates that, by including damping in infinite periodic structure LRM modelling, in particular resonator damping, acoustic insulation performance predictions with improved accuracy can be obtained. Acknowledgements The Research Foundation - Flanders (F.W.O.) is gratefully acknowledged for its support. The research of L. Van Belle is funded by a doctoral grant from the Research Foundation - Flanders (11ZH817N). E. Deckers is a postdoctoral fellow of the Research Foundation - Flanders (12D2618N). The authors gratefully acknowledge the KU Leuven Research Fund for their support through an IOF-Leverage project. 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