Fluid-structural coupling in metamaterial plates for vibration and noise mitigation in acoustic cavities

International Journal of Mechanical Sciences 152 (2019) 151–166

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Fluid-structural coupling in metamaterial plates for vibration and noise mitigation in acoustic cavities A. Aladwani a, A. Almandeel a, M. Nouh b,∗ a b

Manufacturing Engineering Technology Department, College of Technological Studies, Shuwaikh 70654, Kuwait Mechanical and Aerospace Engineering Department, University at Buffalo (SUNY), Buffalo, New York 14260, USA

a r t i c l e

i n f o

Keywords: Metamaterial Bandgaps Fluid-structure interaction

a b s t r a c t Locally resonant metamaterials exhibit sub-wavelength tunable bandgaps that can be exploited for vibroacoustic mitigation. The present work investigates the use of metamaterial plates for the simultaneous control of structural vibration and acoustic sound radiation in an adjacent acoustic cavity. We adopt a coupled fluid-structure finite element model based on a variational mathematical framework in terms of structural displacement and fluid pressure to capture the vibroacoustic characteristics of the coupled system and shed light onto the spatial average pressure levels inside the cavity. The model is used to predict and distinguish between structural and fluid modes within frequency ranges of interest. Furthermore, differences between the metamaterial’s structural response in the presence and lack of fluid coupling are explained. The pressure changes inside the cavity are discussed in relation to the frequency bandgap range predicted theoretically via a dispersion analysis for a couple of different metamaterial designs. Results obtained from the numerical analysis can be used to set design guidelines to optimally tune locally resonant metamaterials to achieve prescribed acoustic properties in the fluid component of such coupled systems.

1. Introduction Elastic metamaterials are artificially engineered structures which provide unique vibroacoustic characteristics owing to their hallmark attributes in wave attenuation, filtering, and guidance [1,2]. Such metamaterials have been shown to provide unique and robust solutions to challenges in acoustic sensing [3], directional noise cancellation [4], topological sound insulators [5,6] as well as sound deceleration [7]. The growing interest in metamaterials stems from their ability to exhibit tunable bandgaps, or frequency ranges within which excitations are not able to penetrate the structural medium of the metamaterial [8,9]. Bandgaps in elastic metamaterials take place by virtue of Bragg scattering effects in phononic crystals [10–12] or by utilizing locally resonant systems and exploiting the negative effective mass effect [13–16]. Phononic crystals comprise a periodic arrangement of materials or geometry which create a set of impedance mismatches intended to disrupt incident wave propagations via destructive interferences. The mechanisms of bandgap formation in phononic crystals and locally resonant metamaterials are radically different and have been investigated separately in recent literature [17,18]. Bandgaps emerging in phononic crystals take place at a wavelength scale and, consequently, require impractically large structures to tune such bandgaps to lower frequencies. On the other hand, bandgaps in locally resonant (LR) metamaterials are size-independent ∗

and mechanically tunable and have thus been extensively used recently to induce bandgaps which can be tuned to as low as < 1 kHz. Furthermore, the integration of such LR metamaterials with piezoelectric materials and negative capacitance shunting have been shown to successfully control the width, location, and directivity of such bandgaps within the frequency spectrum [19,20]. The use of Metamaterial Plates (MMPs) to achieve extraordinary acoustic transmission loss has recently received attention [21,22]. Notable recent efforts have included metamaterials with mass arrays placed in acoustic chambers [23], LR metamaterials to optimize sound absorption characteristics [24], as well as topographic metasurfaces for underwater applications [25]. On the opposite hand, metamaterials have been also used to enhance sound emission in acoustic cavities [26,27]. However, the influence of fluid coupling on the structural response of locally resonant metamaterials as well as its effect on the spatial pressure field of the fluid itself is yet to be investigated. The objective of this work is to extend the design of MMPs for the purpose of simultaneously controlling the sound radiation inside acoustic cavities as well as the attenuation of structural wave propagation. To do so, we study a finite 2-D locally resonant metamaterial plate in contact with an adjacent 3D fluid cavity of a known size and introduce a fluid-structural coupling framework to depict changes in the cavity’s average spatial pressure levels in the context of emerging bandgaps. This problem resembles, to a great extent, many of the existing practical vibroacoustic applications,

Corresponding author. E-mail address: mnouh@buffalo.edu (M. Nouh).

https://doi.org/10.1016/j.ijmecsci.2018.12.048 Received 11 September 2018; Received in revised form 16 December 2018; Accepted 28 December 2018 Available online 29 December 2018 0020-7403/© 2018 Elsevier Ltd. All rights reserved.

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 1. (a) A locally resonant metamaterial plate (LR MMP) and (b) the corresponding unit cell.

including automobile acoustic cabin/flexible body panels and helicopter acoustic cabin/flexible fuselage systems. A generic variational mathematical formulation is adopted in terms of structural displacement and fluid pressure and the finite element method is used to model the coupled MMP-cavity system. Following this, a numerical analysis is carried out to validate the dispersion predictions and develop design guidelines. The paper is organized into six sections. Following the introduction, the MMP design as well as the different geometric parameters of the coupled system are outlined. Next, the finite element formulation of the fluid-structure coupled problem is detailed, followed by the wave propagation (dispersion) analysis and band structure calculations. A set of numerical results is then presented with special attention given to the cavity’s pressure distribution inside, outside, and in the proximity of the bandgap tuning frequencies. Finally, the conclusions of the current work are summarized. 2. A metamaterial plate coupled with acoustic cavity The study of one and two dimensional locally resonant metamaterials which exhibit mechanically tunable bandgaps have received considerable attention lately [28–31]. The emphasis here is placed on extending the modeling framework for such metamaterials to accommodate fluid-structure interactions. The latter will enable us to utilize such metamaterial structures for the attenuation of sound radiation inside acoustic cavities. To start, we first show a flexural plate with an attached array of local mass-spring resonators as a typical configuration of a 2-D metamaterial (Fig. 1a). The resonators act as distributed vibration absorbers and their integration with the host plate (also referred to as the plate substructure) constitutes an ideal periodic structure with finite number of periodic unit cells (Fig. 1b). The lateral dimensions along the x and y coordinates for the plate substructure and a single unit cell are denoted as (Lx , Ly ) and (𝐿𝑐𝑥 , 𝐿𝑐𝑦 ), respectively, with a thickness h. On the other hand, we consider a 3-D acoustic cavity which is completely filled with an acoustic fluid. The cavity is assumed to be completely rigid except from one side which is formed by a flexible conventional (plain) plate as shown in Fig. 2a. The dimensions of the cavity along the x, y and z coordinates are denoted as Lx , Ly , and Lz , respectively. The modeling of this type of coupled systems is well-documented in the literature [32–35]. Next, the LR metamaterial plate of Fig. 1a is used to replace the plain plate that is coupled with the acoustic cavity of Fig. 2a (see Fig. 2b). The aim here is to study the effect of each plate (plain vs. metamaterial) on the vibroacoustic characteristics of the fluid-structure coupled system.

Fig. 2. The fluid-structure coupled system with: (a) conventional plate and (b) metamaterial plate.

pressible and barotropic) which is contained in a cavity that is bounded by a flexible structure. The domains occupying the structure and the interior fluid are denoted as ΩS and ΩF , respectively, and the interface between them by Σ. Fig. 3 shows a typical coupled system of general geometry.

3. Finite element formulation of the coupled fluid-structure problem

3.1. Variational formulation

In this section, a generic variational formulation of the coupled fluidstructure interaction problem is reviewed [36,37]. Here, we are interested in the linear oscillations of an acoustic fluid (i.e. inviscid, com-

The mechanical boundary conditions constitute of the part Γu of the structure exterior boundary that is subjected to a prescribed mechanical displacement 𝑢𝑑𝑖 whereas the remaining part Γ𝜎 is that which corre152

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

displacement ui and the fluid pressure p. Multiplying Eq. (1) by 𝛿ui and Eq. (2) by 𝛿p, then applying Green’s formula, and utilizing the boundary conditions (Eqs. (3)–(7)) and the constitutive and kinematic Eqs. (8) and (9) leads to the following generic representation of the variational equations 𝜕 2 𝑢𝑖 𝑐𝑖𝑗𝑘𝑙 𝜖𝑘𝑙 𝑑𝑣 + 𝜌𝑆 𝛿𝑢 𝑑𝑣 − 𝑝𝑛 𝛿𝑢 𝑑𝑠 = 𝐹 𝑑 𝛿𝑢 𝑑𝑠, (10) ∫Ω𝑆 ∫Ω𝑆 𝜕𝑡2 𝑖 ∫Σ 𝑖 𝑖 ∫Γ𝜎 𝑖 𝑖 𝜕 2 𝑢𝑖 𝜕2 𝑝 1 1 𝑝,𝑖 𝛿𝑝,𝑖 𝑑𝑣 + 𝛿𝑝 𝑑𝑣 + 𝑛 𝛿𝑝 𝑑𝑠 = 0 2 2 2 𝑖 ∫ ∫ ∫ 𝜌𝐹 Ω𝐹 𝜌𝐹 𝑐𝐹 Ω𝐹 𝜕 𝑡 Σ 𝜕𝑡

(11)

3.2. Finite element discretization The system shown in Fig. 2b consists of a metamaterial plate that is coupled with acoustic cavity. Since the variational Eqs. (10) and (11) were derived for systems of general geometry, they can be used here. For simplicity, the plate substructure is assumed to be in a state of plane-stress and is modeled using the Kirchhoff-Love plate assumptions. The fluid pressure inside the cavity is assumed to be varying spatially along the x, y and z coordinates. The structural domain is discretized using two-dimensional quadrilateral four-node finite elements. Each node has three degrees of freedom which are the transverse mechanical displacement and its derivatives (rotations). On the other hand, the fluid domain is discretized using three dimensional hexahedric eight-node finite elements where the nodal pressure is the only degree of freedom. Introducing the global vectors U and P of mechanical and fluid degrees of freedom, respectively, the following discretized equations of the coupled problem can be written in matrix form [36,37] [ ]{ } [ ]{ } { } 𝐌𝑢𝑢 𝟎 𝐔̈ 𝐊𝑢𝑢 −𝐂𝑢𝑝 𝐔 𝐅 + = (12) 𝐂𝑇𝑢𝑝 𝐌𝑝𝑝 𝟎 𝐊𝑝𝑝 𝐏 𝟎 𝐏̈

Fig. 3. A general fluid-structure coupled problem.

sponds to a prescribed force density 𝐹𝑖𝑑 . The fluid boundary conditions constitute of the part Γp of the fluid that is subjected to a prescribed fluid pressure pd whereas the remaining part Σ is that which defines the fluid-structure interface. The local equations of the coupled problem are given by Refs. [36,37] 𝜎𝑖𝑗,𝑗 = 𝜌𝑆

𝑝,𝑖𝑖 =

𝜕 2 𝑢𝑖 𝜕𝑡2

1 𝜕2 𝑝 𝑐𝐹2 𝜕𝑡2

in Ω𝑆 ,

(1)

in Ω𝐹

(2)

where Muu and Kuu are the mechanical mass and stiffness matrices, Mpp and Kpp are the fluid mass and stiffness matrices, Cup is the fluidstructure coupling matrix and F is the applied mechanical load vector.

Eqs. (1) and (2) are the well-known elastodynamic and Helmholtz equations, respectively (subscript “i” can take x, y or z directions whereas “,i” denotes a partial differentiation). These equations are supplemented by the following mechanical and fluid boundary conditions 𝜎𝑖𝑗 𝑛𝑆𝑗 = 𝐹𝑖𝑑

on Γ𝜎 ,

(3)

𝑢𝑖 = 𝑢𝑑𝑖

on Γ𝑢 ,

(4)

𝜎𝑖𝑗 𝑛𝑆𝑗 = 𝑝𝑛𝑖

on Σ,

(5)

𝑝 = 𝑝𝑑

on Γ𝑝 ,

(6)

on Σ

(7)

3.3. Reduced order model Introducing the transformations 𝐔 = 𝚽𝑢 𝐡𝑢 and 𝐏 = 𝚽𝑝 𝐡𝑝 into Eq. (12) and premultiplying the first and second rows by 𝚽u and 𝚽p , respectively yields [̄ ]{ } [ ]{ } { } ̈𝐡𝑢 ̄ 𝑢𝑢 ̄ 𝑢𝑝 𝐡𝑢 𝐌𝑢𝑢 𝟎 𝐊 𝐅̄ −𝐂 + = (13) ̄𝑇 ̄ 𝑝𝑝 ̈𝐡 ̄ 𝑝𝑝 𝐂 𝐌 𝐊 𝟎 𝐡𝑝 𝟎 𝑝 𝑢𝑝

𝑝,𝑖 𝑛𝑖 = −𝜌𝐹

𝜕 2 𝑢𝑖 𝜕𝑡2

𝑛𝑖

where 𝚽u and 𝚽p are the mass normalized modal matrices associated with the structure and fluid, respectively, hu and hp are their corresponding vectors of modal coordinates. The mass normalized vectors and matrices given in Eq. (13) can be expressed in terms of those in Eq. (12) as ̄ uu = 𝚽𝑇 𝐌uu 𝚽𝑢 , 𝐌 𝑢

where 𝜎 ij denote the stress tensor components, p is the interior fluid pressure, 𝜌S is the structure mass density, 𝜌F is the fluid mass density, and cF is the speed of sound in the fluid. In addition, 𝑛𝑆𝑗 is the unit normal external to ΩS whereas ni is the unit normal external to ΩF . Moreover, in order to set a well-posed coupled problem, Eqs. (1)–(7) must be supplemented by appropriate initial conditions. The stress-strain and strain-displacement relations for a linear elastic structure can be expressed by 𝜎𝑖𝑗 = 𝑐𝑖𝑗𝑘𝑙 𝜖𝑘𝑙 ,

𝜖𝑘𝑙 =

) 1( 𝑢 + 𝑢𝑙,𝑘 2 𝑘,𝑙

̄ pp = 𝚽𝑇 𝐌pp 𝚽𝑝 𝐌 𝑝

(14)

̄ uu = 𝚽𝑇 𝐊uu 𝚽𝑢 , 𝐊 𝑢

̄ pp = 𝚽𝑇 𝐊pp 𝚽𝑝 𝐊 𝑝

(15)

̄ up = 𝚽𝑇 𝐂up 𝚽𝑝 , 𝐂 𝑢

𝐅̄ = 𝚽𝑇𝑢 𝐅

(16)

For a harmonic input excitation at a frequency 𝜔 such that 𝐅̄ = 𝐅̄ 0 e𝑗𝜔𝑡 , the output solution is assumed also in harmonic form as 𝐡𝑢 = 𝐇0𝑢 e𝑗𝜔𝑡 ,

𝐡𝑝 = 𝐇0𝑝 e𝑗𝜔𝑡

(17)

By defining the structure and fluid impedance matrices, respectively, ̄𝑢 = 𝐊 ̄ 𝑢𝑢 − 𝜔2 𝐌 ̄ 𝑢𝑢 and 𝐙 ̄𝑝 = 𝐊 ̄ 𝑝𝑝 − 𝜔2 𝐌 ̄ 𝑝𝑝 , substituting Eq. (17) into as 𝐙 Eq. (13), and carrying out some manipulations, we arrive at the following relations for the output modal amplitudes ( )−1 ̄ 𝑢𝑝 𝐙 ̄ −1 𝐂 ̄𝑇 ̄ 𝑢 − 𝜔2 𝐂 𝐅̄ 0 𝐇0𝑢 = 𝐙 (18) 𝑢 𝑢𝑝

(8)

(9)

where 𝜖 kl denote the strain tensor components and cijkl denote the elastic material constants. It can be seen from Eqs. (1) through (7) that the unknown variables of the coupled problem are chosen as the mechanical

( )−1 ̄ −1 𝐂 ̄𝑇 𝐙 ̄ 𝑢𝑝 𝐙 ̄ −1 𝐂 ̄𝑇 ̄ 𝑢 − 𝜔2 𝐂 𝐅̄ 0 𝐇0𝑝 = 𝜔2 𝐙 𝑝 𝑢𝑝 𝑢 𝑢𝑝 153

(19)

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

4. Dispersion analysis and band structure computation

4.3. 𝜔(k) Solution

By considering the metamaterial plate of Fig. 1a, it can be seen that it consist of a finite number of identical unit cells. Each unit cell is in the form of a SDOF vibration absorber that is attached to a rectangular plate-like structure as shown in Fig. 1b. This unit cell configuration is helpful in studying the wave propagation characteristics of the periodic plate. According to Bloch’s theorem, waves in crystals can be decomposed into a plane wave component and a periodic amplitude. This decomposition allows the analysis of an infinitely repeating array of unit cells by considering just a single unit cell. By modeling a single unit cell, band structures can be obtained. These diagrams provide important information about how a single frequency wave propagates along a certain direction in an infinite plate.

To compute the frequency band structure, the 𝜔(k) formulation is used here. This solution is straightforward as it involves stepping through the reciprocal lattice space (Brillouin zone) and solving for the frequencies of wave propagation at each wave vector of interest [38]. Substituting Eq. (23) into Eq. (21) and premultiplying by P† , the Bloch equations of motion are obtained as ( ) 𝐏† (𝐤) 𝐊𝑐𝑢𝑢 − 𝜔2 𝐌𝑐𝑢𝑢 𝐏(𝐤)𝐔̂ 𝑐 = 𝟎, (25) where P† denotes the Hermitian transpose of the Bloch periodicity matrix. 5. Numerical validations This section presents some numerical examples for the analysis of fluid-structure coupled systems. Firstly, the free vibration problem of a conventional plate that is coupled with an acoustic cavity is analyzed. The conventional plate-cavity system will act as a datum for our comparison purposes later on. Then, a similar analysis is conducted for the MMP-cavity system. Following this, we study the vibroacoustic characteristics of both systems described by the displacement and pressure frequency response functions (FRFs) for two different metamaterial designs in order to asses the influence of using MMPs on the performance of such coupled systems. Finally, we shed light onto the cavity’s spatial pressure distribution inside and outside the bounds of the LR frequency bandgap.

4.1. Unit cell model The dispersion analysis begin by writing the finite element equations of motion of a free unit cell as 𝐌𝑐𝑢𝑢 𝐔̈ 𝑐 + 𝐊𝑐𝑢𝑢 𝐔𝑐 = 𝟎,

(20)

where 𝐌𝑐𝑢𝑢 and 𝐊𝑐𝑢𝑢 are the mechanical mass and stiffness matrices of the unit cell and Uc is the free DOF vector of the unit cell. Assuming plane time harmonic solution, the above equation can be rewritten as ( 𝑐 ) (21) 𝐊𝑢𝑢 − 𝜔2 𝐌𝑐𝑢𝑢 𝐔𝑐 = 𝟎, where 𝜔 is the frequency of oscillation. By applying Bloch boundary conditions, the eigenvalue problem of Eq. (21) can be solved for the natural frequencies of the free structure.

5.1. Free vibration analysis and band structures We consider the free vibration problem of a conventional plate that is coupled with a 3-D acoustic cavity as shown in Fig. 2a. The cavity is completely filled with air and its walls are rigid except from the top side, which is made of a flexible Plain Aluminum Plate (PAP) that is clamped from its all edges. The geometric, material and fluid parameters of the coupled system are given in Table 1. Eigenfrequencies for the (i) 3-D rigid acoustic cavity alone, (ii) Clamped plate alone, and (iii) Clamped plate coupled with acoustic cavity are listed in Table 2. A selective number of mode shapes of the coupled system are shown in Fig. 4a. The finite element mesh used to obtain the eigenfrequencies considers 20 × 16 × 16 finite elements. The unit cell lateral dimensions which correspond to the considered mesh are such that 𝐿𝑐𝑥 = 30mm and 𝐿𝑐𝑦 = 25 mm. It is worthwhile to mention that the last column of Table 2 states the type of coupled mode where “S” denotes a structural predominant mode whereas “F” denotes a fluid predominant mode. Next, we repeat a similar process for the second case, namely, the proposed MMP coupled with the 3-D acoustic cavity (Fig. 2b). The geometric, material and fluid parameters are the same as those given in Table 1. However, since mass-spring vibration absorbers are attached to the PAP, we need values for the mass m and stiffness k of the

4.2. Bloch boundary conditions According to Bloch’s theorem, elastic waves in a periodic medium can be expressed as 𝑇

𝐮(𝐱, 𝐤) = 𝐮̄ (𝐱, 𝐤)e𝑗𝐤 𝐱 ,

(22)

where x and k denote the position and wave vectors, respectively [38]. The first term on the right hand side of Eq. (22) denotes the periodic amplitude whereas the second term represents the plane wave. According to Ref. [39], it can be shown that Bloch’s theorem can be expressed as a relationship between the boundaries of the unit cell. For a 2-D periodic structure, this can be achieved by dividing the mechanical degrees of freedom given by the free DOF vector Uc into nine groups such that { }𝑇 𝐔𝑐 = 𝐔𝐼 𝐔𝐵 𝐔𝑇 𝐔𝐿 𝐔𝑅 𝐔𝐿𝐵 𝐔𝑅𝐵 𝐔𝐿𝑇 𝐔𝑅𝑇 where the subscripts I, B, T, L and R denote internal, bottom, top, left and right nodes, respectively. These nine groups can be mapped using a linear transformation, denoted by P which is parametrized by the wave numbers 𝜇x and 𝜇 y such that

Table 1 Geometric, material and fluid parameters of the coupled system.

𝐔𝑐 = 𝐏(𝐤)𝐔̂ 𝑐 , (23) { } 𝑇 where 𝐔̂ 𝑐 = 𝐔𝐼 𝐔𝐵 𝐔𝐿 𝐔𝐿𝐵 is the periodic DOF vector of the unit cell and P is called the Bloch periodicity matrix. It can be expressed as [39] ⎡𝐈 ⎢𝟎 ⎢ ⎢𝟎 ⎢𝟎 ⎢ 𝐏(𝐤) = ⎢𝟎 ⎢𝟎 ⎢ ⎢𝟎 ⎢𝟎 ⎢ ⎣𝟎

𝟎 𝐈 𝐈 e 𝜇𝑦 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝐈 𝐈 e 𝜇𝑥 𝟎 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝟎 𝟎 𝐈 𝐈 e 𝜇𝑥 𝐈 e 𝜇𝑦 (

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ) ⎥ 𝐈 e 𝜇𝑥 + 𝜇𝑦 ⎦

Geometric parameters

Value

Dimensions: Lx × Ly × Lz (mm3 ) Thickness: h (mm)

600 × 400 × 480 3.0

Material parameters (Aluminum)

Value

Mass density: 𝜌S (kg/m ) Young’s modulus: YS (GPa) Poisson’s ratio: 𝜈

2700 70 0.3

Fluid parameters (Air)

Value

3

(24)

Mass density: 𝜌F (kg/m ) Speed of sound: cF (m/s) 3

154

1.2 340

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 4. Mode shapes of the fluid-structure coupled system with: (a) PAP, (b) MMP (Design #1), and (c) MMP (Design #2). Mode shape number, frequency, and type are marked below each mode.

SDOF vibration absorbers. It has been shown that it is possible to design such absorbers for the purpose of creating band structures [28– by tuning the natural frequency 30,40,41]. In fact, this can be achieved √ of the vibration absorbers (𝜔𝑛 = 𝑘∕𝑚) to match the MMP bandgap frequency of main interest. Two designs are considered in our analysis: (i) 𝜔𝑛 = 450 Hz and (ii) 𝜔𝑛 = 550 Hz. In both designs, the lumped mass is chosen such that 𝑚 = 0.18𝑀𝑐 where Mc is the mass of the plate substructure for a single unit cell. The metamaterial plate is modeled using 20 × 16 finite elements such that the vibration absorbers are attached to the plate elements nodes. Tables 3 and 4 list the eigenfrequencies of the coupled system for the first and second designs, respectively. A selective number of mode shapes using both designs are shown in Fig. 4b and c. Finally, by solv-

ing Eq. (25) for each metamaterial design, two band structures can be created as shown in Fig. 5a and b, with a close-up of the 0 − 1 kHz frequency range shown in Fig. 5c and d, respectively. From Fig. 5c, it is found that a bandgap is created for the first design which spans the region 450 < 𝜔 < 488.8 Hz whereas the second design (Fig. 5d) produces a bandgap in the region 550 < 𝜔 < 597.4 Hz. These bandgaps are denoted by the shaded regions in Fig. 5. 5.2. Frequency response analysis The aim of this section is to estimate the influence of using MMPs on the vibration characteristics of the structure alone as well as the sound radiation characteristics of the fluid-structure coupled system. Firstly, 155

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 5. Band structures for the MMPs: (a) Design #1 (𝜔𝑛 = 450 Hz) and (b) Design #2 (𝜔𝑛 = 550 Hz). (c) and (d) represent closeups of (a) and (b), respectively, revealing the LR bandgap range.

Table 4 Eigenfrequencies (Hz) of the coupled system (with MMP Design #2).

Table 2 Eigenfrequencies (Hz) of the coupled system (with PAP). Mode

1

Cavity alone

Plate alone

Mode

Coupled system

Type

1 2

123.82 190.68

3 4

303.17 303.97

5

363.68

6 7

460.75 467.58

8 9

574.59 615.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

124.36 189.03 284.42 301.24 302.30 356.38 361.90 426.08 450.36 463.99 466.06 511.82 555.02 567.93 574.00 611.22

S S F S S F S F F S S F F F S S

283.62

2

354.74

3 4

425.68 454.18

5 6 7

Mode

511.52 554.12 567.00

Mode

1

6 7

Cavity alone

1

283.62

2

354.74

4

454.18

Mode

Plate alone

Mode

Coupled system

Type

1 2 3 4

113.40 173.25 268.06 268.69

5

313.11

285

449.89

286

491.35

1 2 3 4 5 6 7 288 289 290

114.04 172.11 266.95 267.81 284.08 312.26 355.50 449.89 454.06 491.14

S S S S F S F S F S

Mode

Plate alone

Mode

Coupled system

Type

1 2 3 4

113.60 174.06 272.41 273.08

5 285

322.19 449.89

286

491.35

1 2 3 4 5 6 290 291 292 293

114.24 172.89 271.18 272.10 284.13 321.14 549.79 554.04 568.78 599.12

S S S S F S S F F S

283.62

554.12 567.00

we consider that the plate alone (Fig. 1a) is subjected to a harmonic force excitation with a magnitude of 1 N which is applied at point A with coordinates [𝑥A = 300, 𝑦A = 25] mm which is shown graphically in Fig. 6a. The transverse displacement of the plate is evaluated at point C with coordinates [𝑥C = 300, 𝑦C = 375] mm for the PAP vs. MMP using the two previously considered designs as shown in Figs. 7a and 8a. Next we move the location of the harmonic force excitation (but retain the same magnitude) to be applied at point B with coordinates [𝑥B = 30, 𝑦B = 200] mm which is shown graphically in Fig. 6b and the transverse displacement is evaluated at point D with coordinates [𝑥D = 570, 𝑦D = 200] mm to obtain similar plots as shown in Figs. 7b and 8b. Figs. 7 and 8 represent FRFs for the uncoupled structural system, in the absence of fluid coupling and they are created for comparison purposes as will be discussed later. Secondly, we consider the fluid-structure coupled system (Fig. 2). Here, the sound pressure level in decibel (dB) can be expressed as

Table 3 Eigenfrequencies (Hz) of the coupled system (with MMP Design #1). Mode

Cavity alone

156

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

ing Fig. 10a, an increase in the displacement amplitude is observed for the MMP-cavity system within the predicted frequency bandgap which occurs at 𝜔 ≃ 454 Hz. This resonant frequency corresponds to the fluid predominant coupled mode #289 listed in Table 3. On the other hand, the pressure FRF (Fig. 10b) produces a relatively large amplitude for the MMP-cavity system at the same resonant frequency. Nevertheless, this pressure amplitude is smaller when compared to its counterpart corresponding to the PAP-cavity system. In fact, the latter produces two resonant frequencies for the pressure (and displacement) FRF within the predicted bandgap. The first resonant frequency occurs at 𝜔 ≃ 450 Hz (fluid predominant coupled mode #9) whereas the second resonant frequency occurs at 𝜔 ≃ 464 Hz (structure predominant coupled mode #10) as listed in Table 2. It is worth noting that the slight difference in the fluid predominant resonant frequencies in both systems (PAP-cavity vs. MMPcavity) is attributed to the difference in their fluid-structure coupling nature. Nevertheless, the shift in frequency values is modest in this case (less than 1%) and therefore, the transverse displacement and sound pressure levels at these two resonant frequencies can be reasonably compared. Furthermore, Fig. 10b reveals that the pressure is attenuated over the majority of the bandgap range when the PAP is replaced by the MMP with a significant attenuation in the vicinity of the bandgap tuning frequency. Another advantage of using the MMP is the absence of structural predominant modes within the designed frequency bandgap which is clearly captured in Fig. 10 for the pressure (and displacement) FRF and, as a consequence, we can reduce the number of resonant frequencies within a predesigned frequency range of interest. In this specific case, the second resonant peak in the pressure (and displacement) FRF for the PAP-cavity system, within the designed LR bandgap, can be avoided when the PAP is replaced by the MMP. This is a significant advantage of using LR MMPs over conventional plates for many of the existing noise control applications such as vehicle and helicopter cabins which can be modeled as 3-D acoustic cavities bounded by flexible structures. By comparing Figs. 9b and 10b, we observe that the fluid predominant coupled mode #289 for the MMP-cavity system is not excited when the force is applied at point A. This is attributed to the fact that point A is a displacement node for this particular mode. It is straightforward to prove this observation by looking at the corresponding coupled system mode shape shown in Fig. 4b. For the second design (𝜔𝑛 = 550 Hz), we obtain similar FRFs as shown in Figs. 11 and 12. Starting with Fig. 11a, we see that when the harmonic force is applied at point A (with displacement sensor at point C), the MMP-cavity system produces an increase in the displacement amplitudes within the predicted frequency bandgap at 𝜔1 ≃ 554 Hz and 𝜔2 ≃ 569 Hz. These two resonant frequencies correspond to the fluid pre-

Fig. 6. FE mesh used for the MMP along with the location of points A, B, C and D.

( ) Sound Pressure Level dB = 20 log10

(

𝑝 𝑝𝑟𝑒𝑓

) ,

(26)

where 𝑝𝑟𝑒𝑓 = 20 𝜇Pa is the reference sound pressure. Starting with the first design (𝜔𝑛 = 450 Hz), FRFs are obtained for the mechanical transverse displacement of the plate at point C and the fluid spatial average pressure inside the acoustic cavity when the harmonic force (with magnitude of 1 N) is applied at point A as shown in Fig. 9. In Fig. 9, a significant drop can be observed in the displacement and pressure amplitudes for the MMP-cavity system within the LR bandgap predicted earlier (see Fig. 5c) with maximum attenuation of the sound pressure levels attained in the vicinity of the tuning frequency. Next, the harmonic force is applied at point B and similar FRFs are shown in Fig. 10 for the transverse displacement of the plate at point D and the fluid spatial average pressure inside the acoustic cavity. Upon inspect-

Fig. 7. Displacement FRFs of the PAP (red) vs. MMP Design #1 (black): (a) Harmonic force applied at point A and (b) Harmonic force applied at point B. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

157

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 8. Displacement FRFs of the PAP (red) vs. MMP Design #2 (black): (a) Harmonic force applied at point A and (b) Harmonic force applied at point B. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. (a) Displacement and (b) Spatial average pressure FRFs of the coupled fluid-structure system using PAP (red) vs. MMP Design #1 (black) when the harmonic force is applied at point A. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. (a) Displacement and (b) Spatial average pressure FRFs of the coupled fluid-structure system using PAP (red) vs. MMP Design #1 (black) when the harmonic force is applied at point B. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

158

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 11. (a) Displacement and (b) Spatial average pressure FRFs of the coupled fluid-structure system using PAP (red) vs. MMP Design #2 (black) when the harmonic force is applied at point A. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. (a) Displacement and (b) Spatial average pressure FRFs of the coupled fluid-structure system using PAP (red) vs. MMP Design #2 (black) when the harmonic force is applied at point B. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

dominant coupled modes #291 and #292 as listed in Table 4. In addition, the pressure FRF (Fig. 11b) produces relatively large amplitudes for the MMP-cavity system at the same resonant frequencies. However, these pressure (and displacement) amplitudes are smaller when compared to their counterparts which correspond to the PAP-cavity system (fluid predominant modes #13 and #14). In this specific case, the two considered resonant frequencies and their counterparts are shifted from each other by less than 1 Hz which is equivalent to < 0.2% (see Tables 2 and 4). Furthermore, one additional resonant peak appears in the pressure (and displacement) FRF at 𝜔 ≃ 574 Hz for the PAP-cavity system. This peak, which corresponds to the structural predominant coupled mode #15 (see Table 2) is absent from the FRFs of the MMP-cavity system. Similar observations can be drawn from Fig. 12 for the vibroacoustic characteristics of the coupled systems under consideration where FRFs are obtained for the transverse displacement of the plate at point D and the fluid spatial average pressure inside the acoustic cavity when the force is applied at point B. We note that the fluid predominant mode #291 is not excited when the harmonic force is applied at point B which is a displacement node for this particular mode (see Fig. 4c). Tables 5 and 6 summarize the spatial average pressure attenuations within the predicted bandgaps for the first and second designs, respectively. Pressure amplitudes are collected by sweeping at constant frequency steps throughout the entire bandgap for each design. The pressure attenuations obtained by re-

placing the PAPs with MMPs confirm the significance of the proposed concept. 5.3. Acoustic indicators The vibroacoustic performance of the PAP-cavity and MMP-cavity systems can be further investigated by considering the total radiated sound power and the mean quadratic normal velocity (MQNV) as acoustic indicators in this work [42]. The sound intensity I at each point r on the vibrating surface of the plate is equivalent to the sound power radiated per unit area of the transmission. It can be expressed as [ ] [ ( ) ] 1 1 𝐼 = 𝐑𝐞 𝑝(𝑟)𝑣∗𝑛 (𝑟) = 𝐑𝐞 𝑝(𝑟) 𝑗𝜔𝐮(𝑟).𝐧𝑆 ∗ , (27) 2 2 where vn denotes the normal velocity at the fluid-structure interface Σ, the superscript ∗ denotes the complex conjugate and 𝐑𝐞[ ] is the real part of the expression in parentheses. The total radiated sound power Π can be calculated by integrating the intensity over Σ as follows [ ] [ ] 𝜔 ̃ 𝑢𝑝 𝐏Σ = 𝜔 𝐈𝐦 𝐡† 𝚽𝑇 𝐂 ̃ Π= 𝐼𝑑𝑠 = 𝐈𝐦 𝐔† 𝐂 (28) 𝑢 𝑢 𝑢𝑝 𝐏Σ , ∫Σ 2 2 where U† denotes the Hermitian transpose of the mechanical degrees of ̃ 𝑢𝑝 are the fluid degrees of freedom vector and freedom vector, PΣ and 𝐂 the fluid-structure coupling matrix, respectively, reduced to the degrees 159

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Table 5 Spatial average pressure attenuations within the bandgap using MMP Design #1. Frequency (Hz)

Force at point A PAP (dB) MMP (dB) Attenuation (dB) Force at point B PAP (dB) MMP (dB) Attenuation (dB)

451

455

459

463

467

471

475

479

483

487

79.6 42.5 37.1

80.7 51.2 29.5

83.3 54.1 29.2

89.2 56.1 33.1

101.9 57.7 44.2

84.8 59.4 25.4

79.1 61.1 18.0

75.5 63.1 12.4

73.0 65.5 7.5

71.2 69.2 2.0

121.8 63.9 57.9

106.9 84.6 22.3

106.7 74.3 32.4

116.6 72.0 44.6

106.6 71.1 35.5

97.4 71.0 26.4

92.1 71.3 20.8

88.2 72.0 16.2

85.3 73.3 12.0

82.8 75.4 7.4

Table 6 Spatial average pressure attenuations within the bandgap using MMP Design #2. Frequency (Hz) 551

556

561

566

571

576

581

586

591

596

Force at point A PAP (dB) MMP (dB) Attenuation (dB)

81.8 60.3 21.5

92.1 75.4 16.7

91.4 69.8 21.6

106.5 77.3 29.2

111.1 81.4 29.7

107.9 73.7 34.2

92.5 71.8 20.7

85.4 71.8 13.6

81.0 73.0 8.0

77.9 77.1 0.8

Force at point B PAP (dB) MMP (dB) Attenuation (dB)

73.2 43.9 29.3

77.3 58.4 18.9

83.9 66.6 17.3

97.9 78.1 19.8

101.0 82.1 18.9

95.3 73.8 21.5

76.8 71.1 5.7

69.5 70.0 −0.5

68.8 69.8 −1.0

69.8 71.1 −1.3

Fig. 13. Acoustic indicators of the coupled fluid-structure system using PAP (red) vs. MMP Design #1 (black) when the harmonic force is applied at point A: (a) Mean quadratic normal velocity level and (b) Total radiated sound power. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of freedom corresponding to the interface Σ. The sound power level in decibel (dB) can be expressed as ( ) Π , Sound Power Level (dB) = 10 log10 (29) Π𝑟𝑒𝑓

confirm the effectiveness of using LR MMPs within the designed frequency bandgaps. Tables 7 and 8 summarize the sound power attenuations within the predicted bandgaps for the first and second designs, respectively. Here, we can draw conclusions similar to those obtained from Tables 5 and 6. Finally, Figs. 17–18 and 19–20 show the pressure distribution inside the acoustic cavity and the transverse displacement distribution over the MMP using both designs when the harmonic force is applied at points A and B, respectively, evaluated at a single frequency of oscillation inside and outside the bounds of its corresponding LR bandgap. These plots provide useful information for the study of noise localization inside the acoustic cavity. The colorbars shown in these figures represent nodal pressure values (in Pa) and nodal displacement values (in m). It is very clear that the sound pressure levels are minimal when the frequency of oscillation is very close to the LR bandgap tuning frequency which was also predicted by the sound pressure FRFs discussed in this section of the paper.

where Π𝑟𝑒𝑓 = 1 pW is the reference sound power. On the other hand, the mean quadratic normal velocity ⟨V2 ⟩ is given by ⟨𝑉 2 ⟩ =

1 𝜔2 ̃ 𝑢𝑢 𝐰, 𝑣2𝑛 𝑑𝑠 = 𝐰† 𝐌 2Σ ∫Σ 2𝜌𝑆 ℎ𝐿𝑥 𝐿𝑦

(30)

where w is the vector associated with the nodal normal displacements ̃ 𝑢𝑢 is the mechanical mass matrix reduced to the of the structure and 𝐌 degrees of freedom of interest. Figs. 13 and 14 show the MQNV and the total radiated sound power levels of the coupled system using the first design of the MMP when the harmonic force is applied at points A and B, respectively, and similar plots are obtained for the second design as shown in Figs. 15 and 16. The obtained acoustic indicators 160

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 14. Acoustic indicators of the coupled fluid-structure system using PAP (red) vs. MMP Design #1 (black) when the harmonic force is applied at point B: (a) Mean quadratic normal velocity level and (b) Total radiated sound power. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Acoustic indicators of the coupled fluid-structure system using PAP (red) vs. MMP Design #2 (black) when the harmonic force is applied at point A: (a) Mean quadratic normal velocity level and (b) Total radiated sound power. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 16. Acoustic indicators of the coupled fluid-structure system using PAP (red) vs. MMP Design #2 (black) when the harmonic force is applied at point B: (a) Mean quadratic normal velocity level and (b) Total radiated sound power. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

161

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 17. Pressure distribution inside the acoustic cavity and transverse displacement distribution over the metamaterial plate using Design #1 evaluated at: (a) 𝜔 = 440 Hz, (b) 𝜔 = 451 Hz, (c) 𝜔 = 470 Hz, and (d) 𝜔 = 500 Hz. The harmonic force is applied at point A.

Fig. 18. Pressure distribution inside the acoustic cavity and transverse displacement distribution over the metamaterial plate using Design #1 evaluated at: (a) 𝜔 = 440 Hz, (b) 𝜔 = 451 Hz, (c) 𝜔 = 470 Hz, and (d) 𝜔 = 500 Hz. The harmonic force is applied at point B.

Fig. 19. Pressure distribution inside the acoustic cavity and transverse displacement distribution over the metamaterial plate using Design #2 evaluated at: (a) 𝜔 = 540 Hz, (b) 𝜔 = 551 Hz, (c) 𝜔 = 580 Hz, and (d) 𝜔 = 620 Hz. The harmonic force is applied at point A.

162

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Table 7 Sound power attenuations within the bandgap using MMP Design #1. Frequency (Hz)

Force at point A PAP (dB) MMP (dB) Attenuation (dB) Force at point B PAP (dB) MMP (dB) Attenuation (dB)

451

455

459

463

467

471

475

479

483

487

69.5 34.1 35.4

72.7 43.2 29.5

76.9 46.7 30.2

84.1 49.2 34.9

97.9 51.2 46.7

81.6 53.1 28.5

76.5 54.9 21.6

73.4 56.9 16.5

71.4 59.2 12.2

69.8 62.8 7.0

102.5 44.4 58.1

81.2 58.5 22.7

88.2 55.2 33.0

100.2 54.9 45.3

91.3 55.2 36.1

82.6 55.8 26.8

77.5 56.8 20.7

73.7 58.2 15.5

70.5 60.3 10.2

67.4 63.4 4.0

Table 8 Sound power attenuations within the bandgap using MMP Design #2. Frequency (Hz)

Force at point A PAP (dB) MMP (dB) Attenuation (dB) Force at point B PAP (dB) MMP (dB) Attenuation (dB)

551

556

561

566

571

576

581

586

591

596

73.5 41.3 32.2

75.0 51.8 23.2

79.9 36.9 43.0

88.5 56.1 32.4

86.0 58.8 27.2

89.7 55.2 34.5

80.8 53.6 27.2

77.0 51.8 25.2

74.6 43.2 31.4

72.9 60.6 12.3

66.7 35.9 30.8

68.5 47.9 20.6

71.6 52.7 18.9

79.5 58.4 21.1

76.8 58.2 18.6

79.4 49.3 30.1

71.7 51.9 19.8

69.5 56.5 13.0

68.8 60.1 8.7

69.0 65.5 3.5

Fig. 20. Pressure distribution inside the acoustic cavity and transverse displacement distribution over the metamaterial plate using Design #2 evaluated at: (a) 𝜔 = 540 Hz, (b) 𝜔 = 551 Hz, (c) 𝜔 = 580 Hz, and (d) 𝜔 = 620 Hz. The harmonic force is applied at point B.

5.4. Impact on human audible hearing range

In the process of searching for the operational limits of the proposed MMPs, it was found that tuning the vibration absorbers to frequencies above 8 kHz becomes more challenging. The evolution of the MMP dispersion behavior is shown in Fig. 22 as the tuning frequency is increased from 8000 to 8830 Hz. Consistent with the previous results, Fig. 22a and b are produced by considering that the absorber mass is 𝑚 = 0.18𝑀𝑐 where Mc is the mass of the plate substructure for a single unit cell. The predictions show that the bandgap width increases to reach 632 Hz at 𝜔𝑛 = 8 kHz. However, bandgaps can be maintained only until we reach a tuning frequency of about 8830 Hz where it starts to disappear (see Fig. 22b) unless the mass ratio is increased. In Fig. 22c, the absorber mass is increased to 𝑚 = 0.24𝑀𝑐 which leads to the re-formation of a bandgap that is unreachable with the previous MMP designs.

The working frequency range of a locally resonant metamaterial is a designer’s choice. In the previous sections, we demonstrated two different MMP designs, each spanning a specific frequency range of interest. Given the motivation to attenuate undesirable noise in acoustic environments, it is worth investigating the operational limits of the coupled MMP-cavity system and how it compares to audible sounds and the human hearing range, typically considered to be capped at 20 kHz [43]. Fig. 21 shows the band structures and their corresponding FRFs (for the mechanical transverse displacement of the plate at point C and the fluid spatial average pressure when the harmonic force is applied at point A) as the tuning frequency increases from 1 kHz to 3 kHz. It can be observed that the effectiveness of the locally resonant MMPs becomes more prominent at higher frequencies. For 𝜔𝑛 = 3 kHz, four resonant peaks in the pressure FRF are avoided within the bandgap when the PAP is replaced by the MMP, as opposed to two resonant peaks for 𝜔𝑛 = 1 kHz. In both cases, significant attenuation is noticeable across the shaded region corresponding to the bandgap frequency range.

5.5. Full order model vs. reduced order model Thus far, the frequency response functions (FRFs) were obtained using reduced order models of the coupled fluid-structure systems. The process of reducing the size of the system from a large set of 163

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

Fig. 21. MMP band structures and corresponding displacement and spatial average pressure FRFs of the coupled fluid-structure system. The operating frequency of the MMP is tuned to: (a-c) 𝜔𝑛 = 1 kHz and (d-f) 𝜔𝑛 = 3 kHz.

Fig. 22. Evolution of the MMP’s bandgap at higher tuning frequencies: (a) 𝜔𝑛 = 8 kHz, 𝑚 = 0.18𝑀𝑐 , (b) 𝜔𝑛 = 8.83 kHz, 𝑚 = 0.18𝑀𝑐 (Bandgap disappears), and (c) 𝜔𝑛 = 8.83 kHz, 𝑚 = 0.24𝑀𝑐 .

Fig. 23. Displacement FRFs of the reduced order model (red) vs. full order model (black) for the: (a) PAP and (b) MMP. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

164

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

physical equations into a smaller set of modal equations depends on the frequency range of main interest. Typically, the number of retained modal equations is chosen such that the frequency of the highest mode retained should sufficiently exceed the frequency of the applied load by a predetermined margin. Since the coupled fluid-structure system consists of the global vectors U and P of mechanical and fluid degrees of freedom, two coordinate transformations are needed. However, after testing a sufficient number of reduced order models (with different numbers of retained modal equations) for the MMP alone as well as the MMP-cavity system, it has been observed that undesirable inaccuracies can be present in the resulting structural response and hence, the equations of motion of the coupled system were solved by retaining the physical coordinates of the structural domain and only apply the coordinate transformations to the fluid domain. In order to illustrate the differences between the structural responses obtained using the full order model vs. reduced order model, we revisit Fig. 7a and reproduce the same results using both models. The previously considered mesh produces a total of 855 DOF for the PAP vs. a total of 1140 DOF for the MMP. Fig. 23a and 23b show a comparison between the two models for the transverse displacement of the PAP vs. MMP, respectively. The reduced order models used to obtain the structural response consider 400 DOF for the PAP vs. 750 DOF for the MMP. It can be observed that the reduced order model using standard modal reduction approach can be used effectively to capture the structural response of the PAP as shown in Fig. 23a. On the other hand, Fig. 23b shows that some characteristic features in the structural response of MMPs within the bandgap frequency range can be suppressed when reduced order models are used in the numerical simulations. It is worthy to mention that there exist other reduced order model techniques that can be used to minimize the errors introduced by the standard modal reduction approach. The modal truncation augmentation and the mode acceleration method are two such techniques. However, the study of such strategies is beyond the scope of this paper and will be the subject of further investigations.

to 2880 rpm of operating range. This corresponds to a 16% increase in mass which is as a consequence of using locally resonant metamaterials. It should be noted that even the use of constrained or unconstrained layer damping will inevitably also add mass to the structure. In conclusion, it becomes a choice of evaluating the desired reduction in interior noise and balancing that with design considerations including the total weight of the metastructure. References [1] Pai F., Huang G. Theory and design of acoustic metamaterials. Proceedings of the international society for optics and photonics (SPIE); 2015. [2] Deymier PA. Acoustic metamaterials and phononic crystals, 173. Springer Science; 2013. [3] Zhu X, Liang B, Kan W, Peng Y, Cheng J. Deep-subwavelength-scale directional sensing based on highly localized dipolar mie resonances. Phys Rev Appl 2016;5(5):054015. [4] Zhu X, Ramezani H, Shi C, Zhu J, Zhang X. P t-symmetric acoustics. Phys Rev X 2014;4(3):031042. [5] Süsstrunk R, Huber SD. Observation of phononic helical edge states in a mechanical topological insulator. Science 2015;349(6243):47–50. [6] Peng Y-G, Qin C-Z, Zhao D-G, Shen Y-X, Xu X-Y, Bao M, et al. Experimental demonstration of anomalous floquet topological insulator for sound. Nat Commun 2016;7:13368. [7] Zhu X, Li K, Zhang P, Zhu J, Zhang J, Tian C, et al. Implementation of dispersion-free slow acoustic wave propagation and phase engineering with helical-structured metamaterials. Nat Commun 2016;7:11731. [8] Huang GL, Sun CT. Band gaps in a multiresonator acoustic metamaterial. J Vib Acoust 2010;132(3):031003. doi:10.1115/1.4000784. [9] Achaoui Y, Khelif A, Benchabane S, Robert L, Laude V. Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars. Phys Rev B 2011;83(10):104201. [10] Al Ba’ba’a H, Nouh M. An investigation of vibrational power flow in one-dimensional dissipative phononic structures. J Vib Acoust 2017;139(2). 021003–10 [11] Zhu X, Xu T, Liu S, Cheng J. Study of acoustic wave behavior in silicon-based one-dimensional phononic-crystal plates using harmony response analysis. J Appl Phys 2009;106(10):104901. [12] Xue-Feng Z, Sheng-Chun L, Tao X, Tie-Hai W, Jian-Chun C. Investigation of a silicon-based one-dimensional phononic crystal plate via the super-cell plane wave expansion method. Chin Phys B 2010;19(4):044301. [13] Huang HH, Sun CT, Huang GL. On the negative effective mass density in acoustic metamaterials. Int J Eng Sci 2009;47(4):610–17. [14] Pai PF. Metamaterial-based broadband elastic wave absorber. J Intell Mater Syst Struct 2010;21(5):517–28. doi:10.1177/1045389X09359436. [15] Nouh M, Aldraihem O, Baz A. Vibration characteristics of metamaterial beams with periodic local resonances. J Vib Acoust 2014;136(6):61012. doi:10.1115/1.4028453. [16] Al Ba’ba’a HB, Attarzadeh MA, Nouh M. Experimental evaluation of structural intensity in 2D plate-type locally resonant elastic metamaterials. J Appl Mech 2018. [17] Al Ba’ba’a H, Nouh M, Singh T. Pole distribution in finite phononic crystals: understanding Bragg-effects through closed-form system dynamics. J Acoust Soc Am 2017;142(3):1399–412. [18] Al Ba’ba’a H, Nouh M, Singh T. Formation of local resonance band gaps in finite acoustic metamaterials: a closed-form transfer function model. Sound Vib J 2017;410:429–46. [19] Chen Y, Huang G, Sun C. Band gap control in an active elastic metamaterial with negative capacitance piezoelectric shunting. J Vib Acoust 2014;136(6):061008. [20] Nouh M, Aldraihem O, Baz A. Periodic metamaterial plates with smart tunable local resonators. J Intell Mater Syst Struct 2016;27(13):1829–45. [21] Yamamoto T. Acoustic metamaterial plate embedded with helmholtz resonators for extraordinary sound transmission loss. J Appl Phys 2018;123(21):215110. [22] Zhu X, Zou X, Liang B, Cheng J. One-way mode transmission in one-dimensional phononic crystal plates. J Appl Phys 2010;108(12):124909. [23] Islam MT, Newaz G. Metamaterial with mass-stem array in acoustic cavity. Appl Phys Lett 2012;100(1):011904. [24] Meng H, Wen J, Zhao H, Wen X. Optimization of locally resonant acoustic metamaterials on underwater sound absorption characteristics. J Sound Vib 2012;331(20):4406–16. [25] Lee H, Jung M, Kim M, Shin R, Kang S, Ohm W-S, et al. Acoustically sticky topographic metasurfaces for underwater sound absorption. J Acoust Soc Am 2018;143(3):1534–47. [26] Song K, Lee S-H, Kim K, Hur S, Kim J. Emission enhancement of sound emitters using an acoustic metamaterial cavity. Sci Rep 2014;4:4165. [27] Song K, Kim K, Hur S, Kwak J-H, Park J, Yoon JR, et al. Sound pressure level gain in an acoustic metamaterial cavity. Sci Rep 2014;4:7421. [28] Al Ba’ba’a HB, Nouh M. Mechanics of longitudinal and flexural locally resonant elastic metamaterials using a structural power flow approach. Int J Mech Sci 2017;122:341–54. [29] Pai PF, Peng H, Jiang S. Acoustic metamaterial beams based on multi-frequency vibration absorbers. Int J Mech Sci 2014;79:195–205. [30] Peng H, Pai PF. Acoustic metamaterial plates for elastic wave absorption and structural vibration suppression. Int J Mech Sci 2014;89:350–61. [31] Nouh M, Aldraihem O, Baz A. Wave propagation in metamaterial plates with periodic local resonances. J Sound Vib 2015;341:53–73. doi:10.1016/j.jsv.2014.12.030.

6. Concluding remarks In this paper, the fluid-structure interaction between a 2-D locally resonant metamaterial plate and an adjacent 3-D fluid cavity was studied. Through a generic variational formulation, a finite element model was used to extract the vibroacoustic characteristics of the coupled system depicted by the displacement and pressure frequency response functions (FRFs) for two different metamaterial designs. Numerical results reveal that metamaterial plates possess several advantages over conventional plates for the attenuation of sound radiation inside acoustic cavities which can be summarized in three main points: (1) Despite that pressure levels can be relatively high at the fluid predominant resonant frequencies within the desired frequency bandgap, these levels are smaller when compared to their counterparts which result from using conventional plates, (2) The pressure can be attenuated over the majority of the bandgap frequency spectrum with an assured significant attenuation in the proximity of the design tuning frequency and (3) Structural predominant resonant frequencies can be avoided within a predesigned frequency range of interest which can be reflected onto the total number of resonant peaks that occur in the pressure FRFs within the same frequency range. Materials with strong damping properties are often soft and exhibit low stiffness properties. As a result, they are highly incompatible with applications which require vibroacoustic mitigation without a stiffness trade-off. Although local resonance bandgaps are shown to only span a given frequency range, they provide a mechanism by which both vibrations and noise can be attenuated without compromising on the structural strength and load-bearing capability of the mechanical system; a key criteria in both vehicles and helicopter cabins. From a design standpoint, the locally resonant bandgap predicted from the second design, for example, spans around 48 Hz of frequency range which is equivalent 165

A. Aladwani, A. Almandeel and M. Nouh

International Journal of Mechanical Sciences 152 (2019) 151–166

[32] Morand HJ-P, Ohayon R. Fluid structure interaction-applied numerical methods. Wiley; 1995. [33] Olson LG, Bathe K-J. Analysis of fluid-structure interactions. a direct symmetric coupled formulation based on the fluid velocity potential. Comput Struct 1985;21(1–2):21–32. [34] Sandberg G, Goransson P. A symmetric finite element formulation for acoustic fluid-structure interaction analysis. J Sound Vib 1988;123:507–15. [35] Larbi W, Deü J-F, Ohayon R. A new finite element formulation for internal acoustic problems with dissipative walls. Int J Numer Methods Eng 2006;68(3):381–99. [36] Aladwani A, Aldraihem O, Baz A. Piezoelectric vibration energy harvesting from a two-dimensional coupled acoustic-structure system with a dynamic magnifier. J Vib Acoust 2015;137(3):031002. [37] Deü J-F, Larbi W, Ohayon R. Piezoelectric structural acoustic problems: symmetric variational formulations and finite element results. Comput Methods Appl Mech Eng 2008;197(19–20):1715–24.

[38] Krattiger D, Hussein MI. Generalized bloch mode synthesis for accelerated calculation of elastic band structures. J Comput Phys 2018;357:183–205. [39] Farzbod F, Leamy MJ. The treatment of forces in bloch analysis. J Sound Vib 2009;325(3):545–51. [40] Sun H, Du X, Pai PF. Theory of metamaterial beams for broadband vibration absorption. J Intell Mater Syst Struct 2010;21(11):1085–101. [41] Peng H, Pai PF, Deng H. Acoustic multi-stopband metamaterial plates design for broadband elastic wave absorption and vibration suppression. Int J Mech Sci 2015;103:104–14. [42] Larbi W. Numerical modeling of sound and vibration reduction using viscoelastic materials and shunted piezoelectric patches. Comput Struct 2017. [43] Dunn F, Hartmann W, Campbell D, Fletcher NH. Springer handbook of acoustics. Springer; 2015.

166