Low-Frequency Vibration and Radiation Performance of a Locally Resonant Plate Attached with Periodic Multiple Resonators

Article Low-Frequency Vibration and Radiation Performance of a Locally Resonant Plate Attached with Periodic Multiple Resonators

Qi Qin, Meiping Sheng * and Zhiwei Guo

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China; [email protected] (Q.Q.); [email protected] (Z.G.) * Correspondence: [email protected]; Tel.: +86-029-8849-5861

Received: 16 March 2020; Accepted: 17 April 2020; Published: 20 April 2020

Abstract: The low-frequency vibration and radiation performance of a locally resonant (LR) plate with periodic multiple resonators is studied in this paper, with both infinite and finite structure properties examined. For the finite cases, taking the LR plate attached with two periodic arrays of resonators as an example, the forced vibration response and the radiation efficiency are theoretically derived by adopting a general model with elastic boundary conditions. Through a comparison with the band structures calculated by the plane-wave-expansion method, it shows that the band gaps in the infinite LR plate are in good agreement with the vibration-attenuation bands in the finite LR plate, no matter what boundary conditions are applied to the latter. In contrast to the vibration reduction in the band gaps, the radiation efficiency of the finite LR plate is sharply increased in the band-gap frequency ranges. Furthermore, the acoustic power radiated from the finite LR plate can be seriously affected by its boundary conditions. For the LR plate with greater constraints, the acoustic power is reduced in the band-gap frequency ranges, while that from the one with fully free boundary conditions is increased. When further considering the damping loss factors of the resonators, the attenuation performance can be improved for both the vibration and radiation of the LR plate.

Keywords: locally resonant plate; periodic multiple resonators; boundary condition; vibration attenuation; radiation eﬃciency

1. Introduction Owing to the band-gap property, periodic structures have attracted a great deal of interest over the past several decades. Periodic beams [1] and stiffened plates [2] are used widely in engineering, while the band gaps are mostly limited to mid-and-high frequency ranges due to the Bragg scattering mechanism [3,4], which requires the structure period to be of the same order as the wavelength of the band-gap frequency. Since the locally resonant (LR) mechanism was proposed by Liu et al. [5], the study of periodic structures based on the LR mechanism (e.g., phononic crystals [6,7], acoustic metamaterials [8,9], etc.) has been increased. They are usually manufactured by adding arrays of resonant microstructures periodically into/onto the structures to suppress flexural wave propagation. It is considered as an efficient method to achieve the low-frequency band gaps. Xiao et al. [10] studied the propagation of flexural waves in an LR thin plate made of a two-dimensional periodic array of one type of resonator attached on a thin homogeneous plate. Qian and Shi [11] extended the traditional plane-wave-expansion (PWE) method to calculate the band structures of the “spring-mass” simplified model and the “spring-torsional spring-mass” simplified model of LR plate systems, where the effect of the viscidity is also considered. Nouh et al. [12,13] presented the vibration characteristic of metamaterial beams and plates manufactured out of assemblies of periodic elements

Appl. Sci. 2020, 10, 2843; doi:10.3390/app10082843 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 2843 2 of 21 with built-in local resonances. Wang et al. [14] proposed a meta-plate model by periodically attaching high-static-low-dynamic-stiffness resonators onto the thin plate to attenuate very low-frequency flexural waves. To broaden the bandwidths, many researchers investigated the band gaps in periodic structures with multiple local resonators. Flexural wave propagation in locally resonant beams with multiple periodic arrays of attached spring-mass resonators was studied by Xiao et al. [15], showing that much broader band gaps can be achieved by applying damped resonators. Wang et al. [16] studied a homogeneous damped plate with parallel attached resonators on both surfaces, from which two band gaps were generated to improve the band-gap performance. Wang et al. [17] investigated wave propagation and power flow in an acoustic metamaterial plate with a lateral local resonance attachment, where two band gaps can be formed by tuning the parameters of the vertical spring-mass and the lateral spring-mass. A chiral-lattice-based elastic metamaterial beam with multiple resonators was numerically and experimentally studied for the broadband vibration suppression by utilizing their individual band gaps by Zhu et al. [18]. Zhou et al. [19] realized multi-low-frequency band gaps in beams by using the mechanism of multiple resonators containing negative-stiffness. Ma et al. [20] studied the band-gap properties of a periodic vibration suppressor composed of a bottom spring-mass oscillator with multiple secondary oscillators with an arbitrary degree of freedom. Qin and Sheng [21] developed the theory of the PWE method to calculate the band structures of the LR plate attached with multiple spring-mass resonators, which is applied in this paper. It was also concluded that the spring damping characteristic can assist in attenuating the vibration responses outside the band-gap frequency ranges. At present, most studies were focused on dynamic behavior based on band-gap properties. It is generally accepted that vibration attenuation leads to noise reduction. Since the band gaps affect the flexural wave propagation, it is expected that the vibro-acoustic radiation from the LR plate can also be suppressed, especially in the low-frequency range. Acoustic radiation from the LR plate-like structures has been investigated by several researchers. Claus et al. [22] investigated the acoustic radiation efficiency of local resonance-based band-gap materials in both infinite structures and corresponding finite structures, with a specific focus on the radiation efficiency variation affected by the correlation between the resonant frequency and the critical frequency of the plate. It was first found that the band-gap property may increase the radiation efficiency of the LR plate. Song et al. [23] investigated the suppression of vibration and acoustic radiation in a sandwich plate through the use of a periodic design by the finite-element method (FEM). Guo et al. [24] derived the average radiation efficiency of a sandwich plate with periodically inserted LR resonators theoretically. Recently, Jung et al. [25,26] used the LR band gaps to increase the efficiency of sound radiation employing defect modes on a phononic crystal. Although research has been conducted focusing on acoustic radiation in LR structures, more studies can be done to further obtain valuable acoustical characteristics of the LR structure in the low-frequency range. In this paper, the low-frequency vibration and radiation performance of the LR plate attached with periodic multiple resonators is studied. Following the introduction, the theoretical models are built for the forced response and the total radiation efficiency of the finite LR plate in Section2. A general model of the boundary conditions is adopted by using linear and torsional springs for the finite LR plate. In Section3, the band-gap property in the infinite LR plate and vibration attenuation in its finite counterpart are discussed. Furthermore, the acoustic radiation from the finite LR plate is analyzed, with a focus on the effect of the boundary conditions of the plate itself and the damping characteristic of the resonators on the acoustic radiation property. Finally, the conclusions are summarized in Section4. The theory built in this paper and the corresponding findings can guide the multi-band-gap design of the LR plate for noise control. Appl. Sci. 2020, 10, 2843 3 of 21

2. Model and formulations

2.1. Forced Response of the Finite LR Plate with Elastic Boundary Conditions Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 21 In this subsection, the displacement is derived by solving the forced response of the LR plate with elastic boundary conditions. Shown in Figure1 is a finite rectangular LR plate with multiple spring-mass resonators periodically attached on the base plate, of which all four edges are applied spring-mass resonators periodically attached on the base plate, of which all four edges are applied with elastic boundary conditions. The base plate has dimensions of thickness hp , length a , and with elastic boundary conditions. The base plate has dimensions of thickness hp, length a, and width b, ν ρ widthand material b , and parametersmaterial parameters of Young’s of modulusYoung’s modulusE, Possion’s E , ratio Possion’sν, and ratio mass density, and massρp. Adensity harmonic p jωt ω point force F0e is applied at thej t position of (x0, y0) on the base plate.() The differential equation of . A harmonic point force Fe0 is applied at the position of xy00, on the base plate. The motion for a forced vibration of the base plate is given by: differential equation of motion for a forced vibration of the base plate is given by: 4 2 D ∇−42w(()x, y) ρρωphpω w( ()x, y) == F, (1) D∇ wxy,,− pp h wxy F, (1) 3 2 4 4 4 4 2 2 4 4 where D ==−Eh32p/12()1 νν is the bending stiffness of the plate, =∇=∂∂444∂ /∂x + +∂2∂ 42244/∂ ∂x ∂ y +∂∂+ ∂ /∂y , where DEhp 12 1 − is the bending stiffness of the plate,∇ xxyy2 , and F includes both the external force and the forces from the resonators. and F includes both the external force and the forces from the resonators.

x y z

FigureFigure 1. 1. TheThe finite ﬁnite locally locally resonant resonant plate plate with with elastic elastic boundary boundary conditions. conditions.

In the case of the LR plate attached with two periodic resonators of (m , k ) and (m , k ) In the case of the LR plate attached with two periodic resonators of (,)mkR1 R1 and (,)mkR2 R2 along the x direction, F can be expressed as: RR11 RR22 along the x−- direction, F can be expressed as: XS STXT STXS XT FF=−δδ()() xx yy −+ F δ ( xx − )() δ yy −+ F δ ( xx − )() δ yy −, (2) F = F0δ(x 00x0)δ(y y0) + 0F1 stststδ(x x 11s)δ(y yt) +  2 ststF2stδ( 2x x2s)δ(y yt), (2) − − st==11− − st == 11 − − s = 1 t = 1 s = 1 t = 1 δδ()()−− where F0 is the amplitude of the external force and xx00 yy is the 2-D Dirac delta where F0 is the amplitude of the external× force and δ(x x0)δ(y y0) is the 2-D Dirac delta function. function. On the base plate 2ST resonators are attached.− −The terms F1st and F2st are the On the base plate 2S T resonators are attached. The terms F1st and F2st are the amplitudes of forces amplitudes of forces ×from the vibrating resonators of (,)mk and (,)mk, respectively. The from the vibrating resonators of (mR1, kR1) and (mR2, kR2), respectively.RR11 TheRR terms22(x1s, yt) and (x2s, yt) termsrepresent ()xy the, coordinates and ()xy, of the represent individuals the coordinates in both kinds, of the where individualss (s [in0, Sboth]) is kinds, an odd where number, s (t 1st 2st ∈ (t [0, T]) is an integer, and x s = x + aL. sS∈∈0, ) is an odd number, 2t (tT∈ 1s0, ) is an integer, and xxa=+. The transverse displacement of a thin rectangular plate can21ssL be expressed as [27,28]: The transverse displacement of a thin rectangular plate can be expressed as [27,28]: ! P ∞∞P 4P4 ∞P ∞P ( ) = ∞ ∞ ( ) ( ) + lll ( ) ∞ l ( ) + lll ( ) ∞ l ( ) w x,wxyy (), = AAmn coscos ()()λλλam xx coscos λbn yy ++ ξξ ()b yy c coscm cos ()() λξλ xamx ξ xa x d cosdn ()cos λ yλbn y. . (3)(3) m =0 n = 0 mn am bn l =1 b m = m0 am a n = n0 bn mn==00 l = 1 m = 0 n = 0 LetLet the the infinite inﬁnite series series be be truncated truncated to to mmM= M andand nnN= N inin the the following following calculation, calculation, and and then then EquationEquation (3) (3) can can be be written written as: as: MN4 M N ll ll ()M N ()()λλ++4 ξ ()M ()() λξN () λ! wxy, =PP Amn cos am x cos bn y P b y P c m cos am x a x  dP n cos bn y , (4) ( ) = ==( ) ( ) + =l ( ) =l ( ) + l ( =) l ( ) w x, y mn00Amn cos λamx cos λbn y l 1ξb y m 0cm cos λamx ξa x n 0 dn cos λbn y , (4) m = 0 n = 0 l = 1 m = 0 n = 0 λπ= λπ= l l where am ma and bn nb. The terms Amn , cm , and dn are unknown expansion coefficients that need to be determined by both boundary equations and governing differential ξ l () ξ l () equations. The terms a x and b y are given in the form of trigonometric functions and are defined as: T ξφ()==ξξξξ1234 () () () ()  aaaaaxxxxxΑ aa, , (5)  T ξφ()yyyyy==ξξξξ1234 () () () () A  bbbbb bb Appl. Sci. 2020, 10, 2843 4 of 21

l l where λam = mπ/a and λbn = nπ/b. The terms Amn, cm, and dn are unknown expansion coefficients that need to be determined by both boundary equations and governing differential equations. The l ( ) l ( ) terms ξa x and ξb y are given in the form of trigonometric functions and are defined as:

 h iT  1 2 3 4  ξa(x) = ξa (x) ξa (x) ξa (x) ξa (x) = Aaϕa,  h iT , (5)  ξ ( ) = 1( ) 2( ) 3( ) 4( ) =  b y ξb y ξb y ξb y ξb y Abϕb where Aa and A are two arbitrary 4 4 matrixes of full rank, and: b ×   h iT  ϕa = sin(πx/2a) cos(πx/2a) sin(3πx/2a) cos(3πx/2a) ,  (6)  h iT  ϕb = sin(πy/2b) cos(πy/2b) sin(3πy/2b) cos(3πy/2b)

The ﬁrst- to fourth-order derivatives used in the boundary equations and governing diﬀerential equations can be determined as:

( (1) (2) (3) (4) ξa (x) = AaBa1Ba0ϕa, ξa (x) = AaBa2ϕa, ξa (x) = AaBa3Ba0ϕa, ξa (x) = AaBa4ϕa, (1) (2) (3) (4) , (7) ξb (y) = AbBb1Bb0ϕb, ξb (y) = AbBb2ϕb, ξb (y) = AbBb3Bb0ϕb, ξb (y) = AbBb4ϕb where the matrixes Ba0, Ba1, Ba2, Ba3, Ba4, Bb0, Bb1, Bb2, Bb3, and Bb4 are deﬁned in Equation (A1) in AppendixA. To make the separation of variables in Equation (4) possible, ϕa and ϕb need to be expanded in the form of cosine series as:

XM XN ϕa = τm cos(λamx), ϕb = τn cos(λbn y), (8) m = 0 n = 0

T T h 1 2 3 4 i h 1 2 3 4 i where τm = τm τm τm τm and τn = τn τn τn τn , which are deﬁned in Equation (A3) in AppendixA. Hence, Equation (5) can be further expressed as:

XM XN (i) (i) ξa (x) = αim cos(λamx), ξb (y) = βin cos(λbn y) , (9) m = 0 n = 0 where the superscript of (i) means the ith derivative and i = 0, 1, 2, 3, 4. The terms h iT h iT α = α1 α2 α3 α4 and β = β1 β2 β3 β4 are 4 1 column vectors, given im im im im im in in in in in × by: ( α = A τ , α = A B B τ , α = A B τ , α = A B B τ , α = A B τ , 0m a m 1m a a1 a0 m 2m a a2 m 3m a a3 a0 m 4m a a4 m (10) β0n = Abτn, β1n = AbBb1Bb0τn, β2n = AbBb2τn, β3n = AbBb3Bb0τn, β4n = AbBb4τn

The base plate is elastically restrained on four edges bearing the shear forces and bending moments balanced by the linear and torsional springs. The boundary conditions of the base plate can be expressed by:   k w(0, y) = Q , K ∂w (0, y) = M , k w(a, y) = Q , K ∂w (a, y) = M ,  x0 x x = 0 x0 ∂x x x = 0 xa x x = a xa ∂x x x = a  | ∂w − | − | ∂w | (11)  ky0w(x, 0) = Qy , Ky0 (x, 0) = My , k w(x, b) = Qy , K (x, b) = My  y = 0 ∂y − y = 0 yb − y = 0 yb ∂y y = b where kx0, kxa, ky0, and kyb are the linear spring stiffnesses, and Kx0, Kxa, Ky0, and Kyb are the torsional spring stiffnesses. The bending moments Mx and My, and the so-called effective shear forces Qx and Appl. Sci. 2020, 10, 2843 5 of 21

Qy, are deﬁned in Equation (A4) in AppendixA. Substituting Equation (4) into Equation (11) and applying the orthogonality of trigonometric functions give:

 4 M 4 M  P P l P l P  g c + g d = g Amn (n = 0, 1, , N, i = 1, 2, 3, 4),  i,1 m i,2 n i,3 ···  l = 1 m = 0 l = 1 m = 0 , (12)  4 4 N N  P l P P l P  gi,1cm + gi,2dn = gi,3Amn (m = 0, 1, , M, i = 5, 6, 7, 8) l = 1 l = 1 n = 0 n = 0 ··· where gi,1, gi,2, and gi,3 are given by Equation (A5) in AppendixA. Equation (12) represents 4(M + 1) + 4(N + 1) boundary equations, and they can be rewritten in a matrix form as:

Hp = Qa, (13) where H and Q are coeﬃcient matrixes represented with the elements gi,1, gi,2, and gi,3. The terms p and a are deﬁned as: h = 1 1 1 2 2 2 3 3 3 4 4 4 p c0, c1, , cM, c0, c1, , cM, c0, c1, , cM, c0, c1, , cM, ··· ··· ··· ··· iT (14) d1, d1, , d1 , d2, d2, , d2 , d3, d3, , d3 , d4, d4, , d4 , 0 1 ··· N 0 1 ··· N 0 1 ··· N 0 1 ··· N T a = [A , A , , A N, A , A , , A , , Amn, , AM , A , , AMN] , (15) 00 01 ··· 0 10 11 ··· 1N ··· ··· 0 M1 ··· respectively. Thus, p can be expressed by a, namely:

1 p = H− Qa. (16)

In Equation (4), a total of (M + 1)(N + 1) + 4(M + 1) + 4(N + 1) unknown expansion coeﬃcients can be obtained. Except for 4(M + 1) + 4(N + 1) boundary equations supported by Equation (13), (M + 1)(N + 1) governing diﬀerential equations are provided in the following. Furthermore, express the force terms found in Equation (2) in Fourier series with a cosine form as:

XM XN F0δ(x x0)δ(y y0) = Fmn cos(λamx) cos(λ y), (17) − − bn m = 0 n = 0 where Fmn is the modal force. By using the orthogonality of trigonometric functions, multiplying cos(λamx) cos(λ y) on both sides of Equation (17), and integrating in the area of (x, y) ([0, a] [0, b]), bn ∈ × Fmn can be given by: F0 Fmn = cos(λamx0) cos(λbn y0), (18) LamLbn ( ( a, m = 0 b, n = 0 where Lam = and Lbn = . a/2, m , 0 b/2, n , 0 Hence, the external force in Equation (2) can be expressed as:

XM XN F δ(x x )δ(y y ) = Fmn cos(λamx) cos(λ y), (19) 0 − 0 − 0 0 bn m = 0 n = 0 where Fmn0 = (F0/LamLbn) cos(λamx0) cos(λbn y0). When considering the damping characteristic of the springs, the force equilibrium equation of both kinds of resonators can be written as:

.. m w + k (1 + jη )[w w(x , yt)] = 0, i = 1, 2, (20) Ri Rist Ri Ri Rist − is Appl. Sci. 2020, 10, 2843 6 of 21

where ηRi is the damping loss factor of the spring damper, and then the displacements of two kinds of resonators can be expressed by:

kRi(1 + jηRi) wRist = w(xis, yt), i = 1, 2. (21) k (1 + jη ) ω2m Ri Ri − Ri

Thus, Fist in Equation (2) are given by:

.. F = m w = Z (ω)w(x , yt), i = 1, 2, (22) Appl. Sci. 2020, 10, x FOR PEER REVIEWist − Ri Rist Ri is 6 o f 21 2 ω mRikRi(1+jηRi) where ZRi(ω) = 2 . The substitutionMN of Fistδ(x xis)δ(y yt) into Equation (17) gives: kRi(1+jηRi) ω mRi − − δδ−− −= λ λ= Fist ( xxis ) ( yyt ) ∑∑ Fmnistcos( am x) cos( bn y) , i 1, 2 , (23) mn=00 = XM XN where F= F( Fistδ( LLx xis))cosδ(y(λλyt x) )= cos( y). Fmnist cos(λamx) cos(λbn y), i = 1, 2, (23) mnist ist am− bn −am is bn t Substituting Equat ions (4), (9), (19)m, =and0 n =(23)0 into Equat ion (1) derives equations written in a matrix form as: where Fmnist = (Fist/LamLbn) cos(λamxis) cos(λbn yt). Substituting Equations (4), (9), (19), and (23) into Equation (1) derives equations written in a matrix (Ra+− Sp) ω2 ( Za +− Tp)  U( ωω) a + V( ) p = F , (24) form as:  2 (Ra + Sp) ω (Za +T Tp) [U(ω)a + V(ω)p] = F, (24) ST − − where F = ⋅⋅⋅, F + ( FF+), ⋅⋅⋅ . The coefficient matrixes R , S , Z , T , " mn0∑∑S mn 12T st mn st #T st=11 = P P (MN+1)( +× 11) where F = , Fmn0 + (Fmn1st + Fmn2st), . The coefficient matrixes R, S, Z, ··· s = 1 t = 1 ··· (M+1)(N+1) 1 U (ω) , and V (ω) are defined in Equat ions (A6) – (A10) in Appendix× A. T, U(ω), and V(ω) are defined in Equations (A6)–(A10) in AppendixA. The substitutionsubstitution ofof EquationEquat ion (16) makes EquationEquat ion (24) be expressed as:as: hK−−ωω2 MWi a = F, (25) K ω2M W((ω))a = F, (25) − − −1 −1 −1 where K= R+ SH Q1 , M= Z+ TH Q ,1 and W(ωω)= U( ) + V( ω) HQ. 1 where K = R + SH− Q, M = Z + TH− Q, and W(ω) = U(ω) + V(ω)H− Q. For a given frequency ω , by solving Equat ion (25), the coefficient vector can be achieved, For a given frequency ω, by solving Equation (25), the coefficient vector aa can be achieved, and then ppcan can be be determined determined by by Equation Equat ion (16). (16) The. The transverse transverse displacement displacement of theof the base base plate plate can can be furtherbe further obtained obtained by by Equation Equat ion (4). (4).

2.2.2.2 Wavenumber Wavenumber Domain Domain Approach Approach for for the the Total Total Radiation Radiation Efficiency Eﬃciency of the Finite LR PlatePlate

A finitefinite LRLR plate plate with with elastic elastic boundary boundary conditions conditions is embedded is embedded in an in infinite an infinite rigid barigidffle, asbaffle, shown as inshown Figure in2 Figure. As the 2. acoustic As the acoustic medium medium is air in is this air paper, in this the paper, e ffect the of effect the fluid of the load fluid is ignored.load is ignored.

z y x O Infinite rigid baffle

FigureFigure 2. 2.SchematicSchematic of of the the locally locally re resonant sonant plate embeddede mbe dde d in in an an inﬁnite infinite rigid rigid ba baffleﬄe. .

A h harmonicarmonic motion motion at at the the arbitrary arbitrary frequency frequency ωω is assumed. assumed. The The radiated radiated pressure pressure pxyzp((x,,, y, z)) satisﬁes the Helmholtz equation: satisfies the Helmholtz equation: 2 + k2 p(x, y, z) = 0, (26) ∇ (∇+22k) pxyz( ,,) = 0, (26)

2 2222 where ∇=∂∂+∂∂xy, kc= ω 0 , and c0 is the speed of sound. The propagation of acoustic pressure in a plane, harmonic, bending wave in 3-D space can be represented as:

−−−jk x jk y jk z pxyzt( ,,,) = pe xyz ejtω , (27)

222 2 where kkkkxyz++=. Substituting Equat ion (27) into the momentum equation: ∂p ∂v +=ρ 0 , (28) ∂∂zt0 Yields: Appl. Sci. 2020, 10, 2843 7 of 21 where 2 = ∂2/∂x2 + ∂2/∂y2, k = ω/c , and c is the speed of sound. The propagation of acoustic ∇ 0 0 pressure in a plane, harmonic, bending wave in 3-D space can be represented as:

jkxx jky y jkzz jωt p(x, y, z, t) = epe− − − e , (27)

2 2 2 2 where kx + ky + kz = k . Substituting Equation (27) into the momentum equation:

∂p ∂v + ρ = 0, (28) ∂z 0 ∂t Yields: ρ0c0k p(x, y, z) = v(x, y, z), (29) kz where v(x, y, z) is the particle velocity in air, and ρ0 is the air density. By using the spatial Fourier transform as: Z Z ∞ ∞ jkxx jky y P kx, ky, z = p(x, y, z)e− − dxdy, (30) −∞ −∞ Z Z ∞ ∞ jkxx jky y V kx, ky, z = v(x, y, z)e− − dxdy, (31) −∞ −∞ Equation (29) can be written as: P kx, ky, z = Za kx, ky V kx, ky, z , (32) where Za kx, ky is the acoustic impedance, expressed as:   ρ0c0k 2 2 2  q , kx + ky < k  k2 k2 k2  − x− y Za kx, ky = . (33)  ρ0c0k 2 2 2 ± j q , kx + ky > k  k2+k2 k2 x y− Considering the radiation of the air-plate interface, if V kx, ky, z = 0 is known, the upper surface pressure P kx, ky, z = 0 can be derived. The term V kx, ky, z = 0 can be determined by the base plate velocity v(x, y), which can be acquired by multiplying w(x, y) by jω. For the baﬄed LR plate under harmonic excitation, the total acoustic power can be calculated by integrating the acoustic intensity over the plate-baﬄe plane, i.e., z = 0, expressed as: (Z Z ) 1 ∞ ∞ W = Re p(x, y)v∗(x, y)dxdy , (34) 2 −∞ −∞ where the superscript of * means a complex conjugate. The inverse Fourier transforms of p(x, y) and v(x, y) are represented as:

Z Z 1 ∞ ∞ jkxx+jky y p(x, y) = P kx, ky e dkxdky, (35) 4π2 −∞ −∞ Z Z 1 ∞ ∞ jkxx+jky y v(x, y) = V kx, ky e dkxdky, (36) 4π2 −∞ −∞ respectively, and substituting them into Equation (34) gives: ( ) Z Z Z Z Z Z 1 ∞ ∞ ∞ ∞ jkxx+jky y ∞ ∞ jkx x jky y W = Re P kx, ky e dkxdky V∗ kx0, ky0 e− 0 − 0 dkx0dky0dxdy (37) 32π4 × −∞ −∞ −∞ −∞ −∞ −∞ Appl. Sci. 2020, 10, 2843 8 of 21

By changing the order of integration and ﬁrstly performing the integration over x and y, the Dirac delta function is obtained, namely:

Z Z ∞ ∞ j(kx kx )x j(ky ky )y 2 e − 0 e − 0 dxdy = 4π δ(kx kx0)δ ky ky0 . (38) − − −∞ −∞ The substitution of Equation (32) into Equation (37) derives:

(Z Z ) 1 ∞ ∞ 2 W = Re Za kx, ky V kx, ky dkxdky . (39) 8π2 × −∞ −∞ 2 2 2 It should be noted that only the wavenumber components kx and ky satisfying kx + ky < k contribute to an eﬀective acoustic radiation. The range of integration can be limited, and Equation (39) can be expressed as:

 q  2 2 Z k Z k ky  1  − 2  W = Re Z k , k V k , k dk dk . (40) 2  q a x y x y x y 8π  k k2 k2 ×   − − − y  The substitution of V kx, ky calculated by Equation (31) into Equation (40) can acquire W. Therefore, the radiation eﬃciency of the LR plate can be derived by:

W σrad = D E, (41) 1 ρ c ab v 2 2 0 0 | | D E where v 2 is the mean square velocity averaged over the plate surface. | | 3. Examples and Discussion In Section3, the band-gap properties in the inﬁnite LR plate are acquired by the theory in reference [21], and the vibration and radiation responses in the ﬁnite LR plate are obtained by the theory in Sections 2.1 and 2.2. Examples of the LR plate with two resonators in each periodic element (TRIEPE LR plate) are focused on, for a low-frequency vibro-acoustic performance. In the following 3 examples, the base plate is made of steel with a material density ρp = 7800 kg/m , Young’s modulus 11 E = 2.1 10 Pa, Poisson’s ratio ν = 0.3, and damping loss factor ηp = 0.001. The thickness of the × plate is set as hp = 0.003m and the lattice constant aL = 0.05 m.

3.1. Band-Gap Property and Vibration Attenuation It should be noted that introducing a new type of resonator will affect the original band-gap frequency range in the LR plate with one resonator in each periodic element (ORIEPE LR plate), which is related to the spring-mass parameters of the new resonators. Thus, the effect of the ratio between the resonant frequencies on the band-gap frequency range is studied first of all. Since the complete band gaps are closely related to the LR mechanism in the LR plate system, directional band gaps are not considered in this paper. Shown in Figure3 are the trends of two band-gap frequency ranges versus fR2/ fR1 in the TRIEPE LR plate with resonators (mR1, kR1) and (mR2, kR2), with the resonant frequencies of fR1 and fR2, respectively. The band-gap frequency ranges in the two corresponding ORIEPE LR plates are also shown in Figure3 for comparison. The parameters mR1 = 0.020 kg, 4 k = 3.2 10 N/m, f = 201Hz, and mR = 0.014 kg are kept unchanged, while fR / f is varied R1 × R1 2 2 R1 by changing kR2. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 21

22− 2 1 kkky WZkkVkkdkdk=×Re()() , , . (40) π 2 −−kkk22 − axy xy xy 8 y () The substitution of Vkxy, k calculated by Equation (31) into Equation (40) can acquire W . Therefore, the radiation efficiency of the LR plate can be derived by:

σ = W rad , (41) 1 2 ρ cabv 2 00

2 where v is the mean square velocity averaged over the plate surface.

3. Examples and Discussion

In Section 3, the band-gap properties in the infinite LR plate are acquired by the theory in reference [21], and the vibration and radiation responses in the finite LR plate are obtained by the theory in Sections 2.1 and 2.2. Examples of the LR plate with two resonators in each periodic element (TRIEPE LR plate) are focused on, for a low-frequency vibro-acoustic performance. In the following ρ = 3 examples, the base plate is made of steel with a material density p 7800 kg/m , Young’s modulus =×11 ν = η = E 2.1 10 Pa , Poisson’s ratio 0.3 , and damping loss factor p 0.001 . The thickness of the = plate is set as hp =0.003m and the lattice constant aL 0.05 m .

3.1 Band-Gap Property and Vibration Attenuation

It should be noted that introducing a new type of resonator will affect the original band-gap frequency range in the LR plate with one resonator in each periodic element (ORIEPE LR plate), which is related to the spring-mass parameters of the new resonators. Thus, the effect of the ratio between the resonant frequencies on the band-gap frequency range is studied first of all. Since the complete band gaps are closely related to the LR mechanism in the LR plate system, directional band gaps are not considered in this paper. Shown in Figure 3 are the trends of two band-gap frequency

ranges versus ffRR21 in the TRIEPE LR plate with resonators (,)mkRR11 and (,)mkRR22, with the

resonant frequencies of fR1 and fR2 , respectively. The band-gap frequency ranges in the two corresponding ORIEPE LR plates are also shown in Figure 3 for comparison. The parameters Appl. Sci. 2020, 10=, 2843 =×4 = = 9 of 21 mR1 0.020 kg , kR1 3.2 10 N/m , fR1 201Hz , and mR2 0.014 kg are kept unchanged, while

ffRR21 is varied by changing kR2 .

500

400

300 ORIEPE LR plate with (mR1, kR1)

ORIEPE LR plate with (mR2, kR2) 200 TRIEPE LR plate with (mR1, kR1) Frequency (Hz)

and (mR2, kR2) 100

0.5 1.0 1.5 2.0 2.5 fR2 / fR1

Figure 3. The effect of f f on the band-gap ranges in the locally resonant plate with two Figure 3. The eﬀect of fR2/ fR1 onRR21 the band-gap ranges in the locally resonant plate with two resonators resonators in each periodic element (TRIEPE LR plate) with (,)mkRR11 and (,)mkRR22, and two in each periodic element (TRIEPE LR plate) with (mR1, kR1) and (mR2, kR2), and two corresponding corresponding locally resonant plates with one resonator in each periodic element (ORIEPE LR locally resonant plates with one resonator in each periodic element (ORIEPE LR plates). plates). Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 21 It can be seen in Figure3 that the single band gap in the ORIEPE LR plate with (mR1, kR1) is

201–233 Hz,It can as shownbe seen inin theFigure dark 3 that grey the shaded single area. band The gap lightin the grey ORIEPE shaded LR areaplateshows with (,) that,mkRR11 within is the frequency201 – 233 range Hz, ofas 0–500shown Hz, in the with dark the grey increase shaded of area.fR2 /ThefR 1light, both grey the shaded lower area and shows upper that, frequencies within of the singlethe frequency band gap range in the of 0 ORIEPE – 500 Hz, LR with plate the withincrease(m Rof2,ffkRR21R2) keep, both increasing, the lower and and upper the lower frequencies frequency of the single band gap in the ORIEPE LR plate with (,)mk keep increasing, and the lower is very close to the resonant frequency fR2. When fR2/ fR1RR22= 1, the lower frequencies of the two frequency is very close to the resonant frequency f . When ff= 1 , the lower frequencies of single band gaps are totally the same, while the diﬀRerence2 betweenRR21mR1 and mR2 makes the upper the two single band gaps are totally the same, while the difference between m and m makes frequencies 233 Hz and 224 Hz, respectively, which means the resonator mass playsR1 an importantR2 part the upper frequencies 233 Hz and 224 Hz, respectively, which means the resonator mass plays an in the determination of the upper frequency. Furthermore, it can be seen that the lower frequencies of important part in the determination of the upper frequency. Furthermore, it can be seen that the lower the two single band gaps in both ORIEPE LR plates almost coincide with those of the two band gaps in frequencies of the two single band gaps in both ORIEPE LR plates almost coincide with those of the the TRIEPE LR plate, while the band gaps in the former cases cover those in the latter case. In the two band gaps in the TRIEPE LR plate, while the band gaps in the former cases cover those in the TRIEPE LR plate with both (mR1, kR1) and (mR2, kR2), when fR2/ fR1 < 1, the band gap< formed by the latter case. In the TRIEPE LR plate with both (,)mkRR11 and (,)mkRR22, when ffRR211 , the band resonance of (mR2, kR2) is the 1st band gap. The 1stst band gap is gettingst narrower as fR2 approaches gap formed by the resonance of (,)mkRR22 is the 1 band gap. The 1 band gap is getting narrower fR1, and meanwhile the 2nd band gap is gettingnd wider. When fR2/ fR1 = 1, the original 1st= band gap as fR2 approaches fR1 , and meanwhile the 2 band gap is getting wider. When ffRR211 , the vanishes andst only one band gap exists. As f / f continues increasing, two band gaps reappear, original 1 band gap vanishes and only one Rband2 R gap1 exists. As ffRR21continues increasing, two ( ) nd whileband the band gaps gapreappear, formed while by thethe resonanceband gap formed of mR2 by, k Rthe2 turnsresonance into theof (,) 2ndmkRR22 band turns gap. Theinto bandwidththe 2 of theband 1st band gap. The gap bandwidth is not changed of the after1st band a slight gap is increase, not changed and after the bandwidtha slight increase, of the and 2nd theband bandwidth gap keeps increasingof the after2nd band a decrease. gap keeps increasing after a decrease. ShownShown in Figure in Figure4 are 4 are the the total total bandwidths bandwidths of of thethe bandband gaps gaps in in three three corresponding corresponding LR LRplates plates when f is below 500 Hz. The bandwidth of the band gap in the ORIEPE LR plate with (,)mk when fR2 is belowR2 500 Hz. The bandwidth of the band gap in the ORIEPE LR plate withRR11(mR1, kR1) is 32 Hz. With the increase of ff, the bandwidth in the ORIEPE LR plate with (,)mk has a is 32 Hz. With the increase of fR2RR/21fR1, the bandwidth in the ORIEPE LR plate withRR22(mR2, kR2) has a nearlynearly linear linear increase, increase, which which makes makes itsits bandwidthbandwidth curve curve intersect intersect with with that that in the in theORIEPE ORIEPE LR plate LR plate with (,)mkRR11 when ffRR21 reaches a certain value. Moreover, the curve of the total bandwidth with (mR1, kR1) when fR2/ fR1 reaches a certain value. Moreover, the curve of the total bandwidth of of the two band gaps in the TRIEPE LR plate seems positively proportional to ffRR21 with a lower the two band gaps in the TRIEPE LR plate seems positively proportional to fR2/ fR1 with a lower slope slope than that in the ORIEPE LR plate with (,)mkRR22. Interestingly, three bandwidth curves than that in the ORIEPE≈ LR plate with (mR2, kR2). Interestingly, three bandwidth curves intersect at intersect at ffRR211.4 , where the total bandwidths are all 32 Hz. With the exception of this point, fR / f 1.4, where the total bandwidths are all 32 Hz. With the exception of this point, the bandwidth 2 theR1 ≈bandwidth in the TRIEPE LR plate stays between the bandwidths in both ORIEPE LR plates in thebelow TRIEPE 500 LRHz. plate stays between the bandwidths in both ORIEPE LR plates below 500 Hz.

40 ORIEPE LR plate with (mR1, kR1)

30 ORIEPE LR plate with (mR2, kR2)

TRIEPE LR plate with (mR1, kR1) 20 Frequency (Hz) and (mR2, kR2) 10

0.51.01.52.02.5 fR2 / fR1

FigureFigure 4. The 4. The total total bandwidths bandwidths of of the the band band gapsgaps in the three three locally locally resonant resonant plates plates in Figure in Figure 3. 3.

It can be seen that when aiming to achieve a new low-frequency band gap, the introduction of the new periodic resonators might make the total bandwidth decrease in the TRIEPE LR plate. Appropriate spring-mass parameters for both kinds of resonators could make the total bandwidth not be greatly affected. Hence, the chosen parameters of the periodic element of the TRIEPE LR plate in this paper are given in Table 1, together with the details of the finite base plate. In the following analysis, the parameters are kept unchanged unless otherwise stated.

Table 1. Parameters of the periodic element and the details of the base plate in the finite locally resonant plate with two resonators in each periodic element.

Symbol Description Value Appl. Sci. 2020, 10, 2843 10 of 21

It can be seen that when aiming to achieve a new low-frequency band gap, the introduction of the new periodic resonators might make the total bandwidth decrease in the TRIEPE LR plate. Appropriate spring-mass parameters for both kinds of resonators could make the total bandwidth not be greatly aﬀected. Hence, the chosen parameters of the periodic element of the TRIEPE LR plate in this paper are given in Table1, together with the details of the ﬁnite base plate. In the following analysis, the parameters are kept unchanged unless otherwise stated.

Table 1. Parameters of the periodic element and the details of the base plate in the ﬁnite locally resonant plateAppl. with Sci. 2020 two, 10 resonators, x FOR PEER in REVIEW each periodic element. 10 of 21

Symbol Description Value mR1 Mass of the resonator (,)mkRR11 0.020 kg m Mass of the resonator (m , k ) 0.020× 4 kg R1kR1 Spring stiffness of the resonatorR1 R(,)mk1 RR11 3.2 10 N/m 4 kR1 Spring stiffness of the resonator (mR1, kR1) 3.2 10 N/m mR2 Mass of the resonator (,)mkRR22 0.014× kg mR2 Mass of the resonator (mR2, kR2) 0.014 kg k Spring stiffness of the resonator (,)mk 510N/m× 44 kR2 R2 Spring stiffness of the resonator (mR2RR,22kR2) 5 10 N/m × aL aL LatticeLattice constant constant 0.05 0.05 m m h Thickness of the base plate 0.003 m p h Thickness of the base plate 0.003 m a p Length of the finite base plate 0.8 m 0.8 m b a WidthLength of of the the finite finite base base plate plate 0.5 m 0.5 m F0 b AmplitudeWidth of ofthe excitation finite base force plate 1 N

(x0, y0F0) Force positionAmplitude on of the excitation finite base force plate (0.1 m,1N 0.1 m) () () xy00, Force position on the finite base plate 0.1 m,0.1 m As illustrated in Section 2.1, the four boundaries of the plate are supported by a linear spring with stiffness k andAs illustrated a torsional in springSection with2.1, the sti fourffness boundariesK. The free, of the simply plate are supported, supported and by a clamped linear spring boundary conditionswith can stiffness be simulated k and a torsional by using spring specific with sti stiffnessffness valuesK . The offree,k and simplyK: supported, the free boundary and clamped condition can be expressedboundary conditions when k = can0 andbe simulatedK = 0; theby simplyusing specific supported stiffness boundary values of condition k and K can: the be free expressed = = when k boundarytends to infinitycondition and canK be= expressed0; and the when clamped k 0 boundary and K 0 condition; the simply can supported be expressed boundary when both condition can be expressed when k tends to infinity and K = 0 ; and the clamped boundary k and K tend to infinity. The variation of the boundary conditions can result in the difference in the condition can be expressed when both k and K tend to infinity. The variation of the boundary vibrationconditions response can and result acoustic in the difference performance in the [vibr29,30ation]. Inresponse this paper, and acoustic three kinds performance of diff erent[29,30]. boundary In conditionsthis paper, are applied three kinds to the of differen plate,t including boundary conditions fully free are boundary applied to conditions the plate, including (‘FFFF’), fully fully free simply supportedboundary boundary conditions conditions (‘FFFF’), (‘SSSS’), fully simply and fully suppo clampedrted boundary boundary conditions conditions (‘SSSS’), (‘CCCC’). and fully Beforeclamped the analysisboundary ofconditions the vibration (‘CCCC’). attenuation property of the LR plate, taking the SSSS boundary condition asBefore an example, the analysis the of theory the vibration to simulate attenuatio the vibrationn property response of the LR in plate, this papertaking isthe examined SSSS by using theboundary FEM softwarecondition ofas Ansys.an example, The the comparisons theory to simulate are shown the vibration in Figure response5a–c, respectivelyin this paper is for the bare plate,examined the mass-loaded by using the (ML) FEM plate, software and theof Ansys. LR plate. The Thecomparisons ML plate are indicates shown thein Figure plate periodically5a–c, respectively for the bare plate, the mass-loaded (ML) plate, and the LR plate. The ML plate indicates attached with point masses, for which the plate, the lattice constant, and the loaded masses are all the plate periodically attached with point masses, for which the plate, the lattice constant, and the identical to those of the LR plate. The above theory is also appropriate for the ML plate via replacing loaded masses are all identical to those of the LR plate. The above theory is also appropriate for the Equation (21) by wRist = w(xis, yt). Good agreements= () can be found between the root-mean-square ML plate via replacing Equation (21) by wwxyRist is, t . Good agreements can be found between velocitiesthe (RMSVs) root-mean-square averaged velocities over the (R wholeMSVs) surfaceaveraged calculated over the whole by the surface theory calculated and FEM, by the which theory validates the correctnessand FEM, of which the theoryvalidates in the this correctness paper. of the theory in this paper.

(a)10-1 (b)10-1 (c)10-1 10-2 10-2 10-2

10-3 10-3 10-3 RMSV (m/s) RMSV (m/s) RMSV 10-4 10-4 RMSV (m/s) 10-4 theory theory theory -5 -5 10 FEM 10 FEM 10-5 FEM 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (Hz) Frequency (Hz) Frequency (Hz)

Figure 5. The comparison between the root-mean-square velocities (RMSVs) averaged over the whole Figure 5. The comparison between the root-mean-square velocities (RMSVs) averaged over the whole surfacesurface calculated calculated by theby the theory theory and and finite ﬁnite element element method method (FEM) (FEM) for (a) the for bare (a) plate, the bare (b) the plate, mass- (b) the mass-loadedloaded plate,plate, and (c ()c )the the locally locally resonant resonant plate. plate.

Shown in Figure 6a are the band structure curves of the infinite bare plate, the infinite LR plate, and the infinite ML plate. The RMSVs averaged over the whole plate surface of the bare plate, the LR plate, and the ML plate, for the cases of various boundary conditions, are shown in Figure 6b–d. In the range of 0 – 500 Hz, the band gaps induced by the LR effect are labeled in Figure 6a, with band- Appl. Sci. 2020, 10, 2843 11 of 21

Shown in Figure6a are the band structure curves of the infinite bare plate, the infinite LR plate, and the infinite ML plate. The RMSVs averaged over the whole plate surface of the bare plate, the LR plate, and the ML plate, for the cases of various boundary conditions, are shown in Figure6b–d. In the range of 0–500 Hz, the band gaps induced by the LR e ffect are labeled in Figure6a, withAppl. band-gap Sci. 2020, frequency10, x FOR PEER ranges REVIEW of 201–215 Hz and 300–320 Hz, where no waves propagate.11 of 21 The two ranges correspond to the resonant frequencies of two kinds of resonators, i.e., fR1 = 201 Hz and gap frequency ranges of 201 – 215 Hz and 300 – 320 Hz, where no waves propagate. The two ranges fR2 = 300 Hz. It can also be seen that, for the bare plate and the ML plate, waves are allowed at correspond to the resonant frequencies of two kinds of resonators, i.e., f = 201 Hz and any frequency. The comparison between the band structures in Figure6a and theR1 vibration results in f = 300 Hz . It can also be seen that, for the bare plate and the ML plate, waves are allowed at any FigureR62b–d basically show a good coincidence between the band gaps of the infinite LR plate and the frequency. The comparison between the band structures in Figure 6a and the vibration results in vibrationFigure attenuation 6b–d basically bands show of a its good finite coincidence counterpart, between no matter the band what gaps boundary of the infinite conditions LR plate are and applied to thethe finite vibration plate. attenuation In the band-gap bands frequencyof its finite ranges,counterpart, the vibrationno matter ofwhat the boundary finite LR plateconditions is attenuated are whenapplied compared to the with finite those plate. of In the the bare band-gap plate andfrequency the ML ranges, plate. the vibration of the finite LR plate is Asattenuated illustrated when above, compared the with ML those plate of and the thebare LRplate plate and havethe ML the plate. same weight, heavier than the bare plate.As From illustrated Figure above,6b–d, the it ML is obvious plate and that the LR the pla firstte have few modalthe same frequencies weight, heavier of the than bare the platebare are a littleplate. larger From than Figure those 6b–d, of it the is obvious ML plate that andthe first the few LR modal plate. frequencies Except for of thethe ebareffect plate on are the a structural little vibration,largerthe than weight those of also the a MLffects plate the and acoustic the LR behavior plate. Except [22]. for Thus, the effect only theon the ML structural plate is consideredvibration, for comparisonthe weight below. also affects the acoustic behavior [22]. Thus, only the ML plate is considered for comparison below. (a) Band structure (b) FFFF (c) SSSS (d) CCCC 500 Bare plate Bare plate Bare plate ML plate ML plate ML plate 400 LR plate LR plate LR plate 2nd BG 300

st 200 1 BG Frequency (Hz) Frequency Bare plate 100 ML plate LR plate 0 Μ Γ X Μ Y Γ 10-1 10-2 10-3 10-4 10-1 10-2 10-3 10-4 10-5 10-1 10-2 10-3 10-4 10-5 RMSV (m/s) RMSV (m/s) RMSV (m/s)

FigureFigure 6. (a 6.) Band(a) Band structures structures of of the the inﬁnite infinite bare bare plate,plate, mass-loaded mass-loaded (ML) (ML) plate, plate, and and locally locally resonant resonant (LR)(LR) plate, plate, and and the the root-mean-square root-mean-square velocities velocities (RMSVs) averaged averaged over over the the whole whole surface surface of their of their ﬁnitefinite counterparts counterparts with with (b) fully (b) freefully (FFFF)free (FFFF) boundary boundary conditions, conditions, (c) fully (c) simplyfully simply supported supported boundary conditionsboundary (SSSS), conditions and (d (SSSS),) fully and clamped (d) fully (CCCC) clamped boundary (CCCC) boundary conditions. conditions. (BG: band (BG: gap). band gap).

It is notedIt is noted that that the the RMSV RMSV results results in in Figure Figure6 6b–db–d are all averaged averaged over over the the whole whole plate plate surface, surface, whichwhich makes makes the the vibration vibration attenuation attenuation in in the the LR LR plates plates not notvery veryobvious obvious at some at frequencies some frequencies in two in two bandband gaps, especially especially for for the the FFFF FFFF boundary boundary condit conditionsions in Figure in Figure 6b. In 6theb. Infinite the LR finite plates, LR the plates, vibration attenuation in the band-gap frequency ranges is usually evaluated by the vibration the vibration attenuation in the band-gap frequency ranges is usually evaluated by the vibration transmittance. With the band structure curves of the infinite ML and LR plates shown in Figure 7a transmittance. With the band structure curves of the infinite ML and LR plates shown in Figure7a for comparison, Figure 7b shows the vibration transmittances of the finite ML and LR plates with for comparison, Figure7b shows the vibration transmittances of the= finite ML and LR plates with FFFF boundary conditions. The vibration transmittance is given by Tvv20log10 res ext , where the FFFF boundary conditions. The vibration<< transmittance is given<< by T = 20 log 10 vres/vext , where the RMSVs averaged over the area of 00.1mx and 0.5 mx 0.6 m within the base| plate surface| RMSVs averaged over the area of 0 < x < 0.1 m and 0.5 m < x < 0.6 m within the base plate surface are chosen as vext and vres , respectively. From the comparison between the vibration transmittances are chosenof the ML as v andext and LR plates,vres, respectively. it can be seen From that the the flexural comparison waves betweenin the LR theplate vibration can be dramatically transmittances of thesuppressed ML and LRin the plates, band-gap it can frequency be seen thatranges, the be flexuraltter revealing waves its in band-gap the LR plate property. can beFigure dramatically 8a,b suppressedshows the in displacement the band-gap profiles frequency of the ranges,LR plate betterat 202 Hz revealing and 310 itsHz, band-gap respectively property. within two Figure band 8a,b showsgaps. the It displacement can be seen profilesthat, with of the the exception LR plate atof 202the Hzarea and around 310 Hz,the respectivelyforce position, within the excited two band gaps.responses It can be seenover almost that, with the thewhole exception surface ofof thethe areaLR plate around are thesignificantly force position, suppressed the excited at the responses two overfrequencies. almost the whole surface of the LR plate are significantly suppressed at the two frequencies. Appl. Sci. 2020, 10, 2843 12 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 21

(a) 500 (b)

400

nd 300 2 BG

st 200 1 BG Frequency (Hz) 100 ML plate ML plate LR plate LR plate 0 Μ Γ X Μ Y Γ 10 0 -10 -20 -30 -40 -50 Vibration transmittance (dB) .

FigureFigure 7. 7.( (aa)) BandBand structuresstructures ofof thethe inﬁniteinfinite mass-loadedmass-loaded (ML)(ML) plateplate andand locallylocally resonantresonant (LR)(LR) plate,plate, andand ( b(b)) the the vibration vibration transmittances transmittances of of their their ﬁnite finite counterparts counterparts with with fully fully free free boundary boundary conditions. conditions. Displacement (m) Displacement (m)

(a) (b) Figure 8. Displacement profiles of the finite locally resonant plate with fully free boundary conditions 8 atFigure (a) 202 8. HzDisplacement within the 1stprofiles band of gap the and finite (b) locally 310 Hz reso withinnant the plate 2nd with band fully gap. free boundary conditions at (a) 202 Hz within the 1st band gap and (b) 310 Hz within the 2nd band gap. For all of the above results of the LR plates, the damping characteristic of the resonators is not considered, while it could broaden the vibration attenuation band outside the band-gap frequency For all of the above results of the LR plates, the damping characteristic of the resonators is not ranges [21]. Shown in Figure9a–c are the RMSVs averaged over the whole surface of the FFFF ML considered, while it could broaden the vibration attenuation band outside the band-gap frequency plate and the FFFF LR plate with the η and η values of the resonators varying separately and ranges [21]. Shown in Figure 9a–c are theR1 RMSVsR2 averaged over the whole surface of the FFFF ML simultaneously. When applying the dampingη lossη factors to the resonator springs, it has little effect on plate and the FFFF LR plate with the R1 and R 2 values of the resonators varying separately and the RMSV in a lower frequency range under 150 Hz in this case. With ηR2 kept as zero and ηR1 varied simultaneously. When applying the damping loss factors to the resonator springs, it has little effect by 0, 0.05, and 0.1, the attenuation band covering the 1st band-gap frequency range becomes wider, on the RMSV in a lower frequency range under 150 Hz in this case. With η kept as zero and η though the RMSV in the original 1st band-gap frequency range is increased.R Meanwhile,2 the RMSVR1 withinvaried theby 2nd0, 0.05, band-gap and 0.1, frequency the attenuation range changesband covering little, as the shown 1st band-gap in Figure frequency9a. Figure range9b shows becomes an 13 st analogouswider, though effect the in theRMSV case in where the originalηR1 is zero 1 band-gap and ηR2 is frequency varied by range 0, 0.05, is and increased. 0.1. With Meanwhile, the damping the lossRMSV factor within increasing the 2nd simultaneously,band-gap frequency as shown range in changes Figure 9little,c, the as RMSV shown is in decreased Figure 9a. between Figure 9b the shows two η η band-gapan analogous frequency effect ranges,in the case and where a much widerR1 is zero attenuation and R band2 is varied appears by covering 0, 0.05, and them. 0.1. Moreover, With the thedamping RMSV curvesloss factor become increasing smoother simultaneously, in the higher frequency as shown range in withFigure the 9c, application the RMSV of theis decreased resonator damping.between the It istwo revealed band-gap that frequency a better attenuation ranges, and performance a much wider can attenuation be achieved band in the appears LR plate covering with multiplethem. Moreover, resonators the when RMSV further curves considering become their smoot dampingher in characteristic.the higher frequency range with the application of the resonator damping. It is revealed that a better attenuation performance can be achieved in the LR plate with multiple resonators when further considering their damping characteristic.

Appl. Sci. 2020, 10, 2843 13 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 21

(a) 10-1

10-2 ML plate η 10-3 LR plate with R1=0 LR plate with η =0.05 RMSV (m/s) R1 η 10-4 LR plate with R1=0.1

10-5 0 100 200 300 400 500 Frequency (Hz) (b) 10-1

10-2 ML plate η 10-3 LR plate with R2=0 LR plate with η =0.05 RMSV (m/s) R2 η 10-4 LR plate with R2=0.1

10-5 0 100 200 300 400 500 Frequency (Hz) (c) 10-1 ML plate -2 η 10 LR plate with R1,2=0 η LR plate with R1,2=0.05 -3 η 10 LR plate with R1,2=0.1

RMSV (m/s)RMSV η LR plate with R1,2=0.2 10-4

10-5 0 100 200 300 400 500 Frequency (Hz) Figure 9. The root-mean-square velocities (RMSVs) averaged over the whole surface of the mass-loaded Figure 9. The root-mean-square velocities (RMSVs) averaged over the whole surface of the mass- (ML) plate and the locally resonant (LR) plate with the resonator damping loss factors of (a) ηR1 varied loaded (ML) plate and the locally resonant (LR) plate with the resonator damping loss factors of (a) by 0, 0.05, 0.1, and η kept as 0, (b) η kept as 0 and η varied by 0, 0.05, and 0.1, and (c) η and η η R2 η R1 η R2 η R1 R2 simultaneouslyR1 varied by varied0, 0.05, by0.1, 0, and 0.05, 0.1,R2 and kept0.2. as 0, (b) R1 kept as 0 and R2 varied by 0, 0.05, and 0.1, and (c) η and η simultaneously varied by 0, 0.05, 0.1, and 0.2. 3.2. Acoustic RadiationR1 PropertyR2 of the Finite LR Plate 3.2 AcousticAs derived Radiation in Sections Property 2.1 ofand the 2.2 Finite, the LR theory Plate in this paper can calculate the radiation efficiencies of the finite LR plate with various boundary conditions. Figure 10a–c gives the radiation efficiencies of As derived in Sections 2.1 and 2.2, the theory in this paper can calculate the radiation efficiencies the FFFF, SSSS, and CCCC plates, including the results of the ML plate, the LR plates with undamped of the finite LR plate with various boundary conditions. Figure 10a–c gives the radiation efficiencies resonators, and the LR plate with damped resonators of ηR = 0.05. It is not difficult to observe that of the FFFF, SSSS, and CCCC plates, including the results1,2 of the ML plate, the LR plates with more peaks and dips appear in the curves of the RMSV and the radiation efficiency of the FFFF plates undamped resonators, and the LR plate with damped resonators of η = 0.05 . It is not difficult to in 0–500 Hz, than in those in the case of greater constraints, such as theR1,2 SSSS and CCCC boundary observe that more peaks and dips appear in the curves of the RMSV and the radiation efficiency of conditions simulated in this paper. Moreover, the modal frequencies shift to higher frequencies when morethe FFFF constraints plates in are 0 applied– 500 Hz, to than the plate in those boundaries; in the ca forse instance,of greater the constraints, 1st modal frequenciessuch as the ofSSSS the and LR platesCCCC with boundary FFFF, SSSS,conditions and CCCC simulated boundary in this conditionspaper. Moreover, are respectively the modal 22 frequencies Hz, 36 Hz, shift and to 69 higher Hz in st thisfrequencies paper. As when for the more radiation constraints efficiency, are applied it can be to easily the seenplate from boundaries; Figure 10 fora–c instance, that, in contrast the 1 modal to the vibrationfrequencies attenuation of the LR in plates the two with band-gap FFFF, SSSS, frequency and CCCC ranges, boundary the radiation conditions efficiency are hasrespectively been obviously 22 Hz, increased36 Hz, and in 69 both Hz bands in this compared paper. As with for the radiation results of theefficiency, ML plates. it can When be easily the resonators seen from are Figure damped, 10a– c that, in contrast to the vibration attenuation in the two band-gap frequency ranges, the radiation efficiency has been obviously increased in both bands compared with the results of the ML plates. Appl.Appl. Sci.Sci. 20202020,, 1010,, 2843x FOR PEER REVIEW 1414 ofof 2121

When the resonators are damped, the curves become smoother than those in the undamped case from thebelow curves the becomelower frequency smoother of than the those1st band inthe gap, undamped while the caseacoustic from radiation below the is lower still high frequency in the ofband- the 1stgap band frequency gap, while ranges the in acoustic each case. radiation is still high in the band-gap frequency ranges in each case.

(a) 100 10-1 10-2 10-3 10-4 ML plate 10-5 LR plate with η =0 -6 R1,2

Radiation efficiency 10 η LR plate with R1,2=0.05 10-7 0 100 200 300 400 500 Frequency (Hz) (b) 100 10-1 10-2 10-3 10-4 ML plate 10-5 LR plate with η =0 -6 R1,2

Radiation efficiency 10 η LR plate with R1,2=0.05 10-7 0 100 200 300 400 500 Frequency (Hz) (c) 100 10-1 10-2 10-3 10-4 ML plate 10-5 LR plate with η =0 -6 R1,2

Radiation efficiency 10 η LR plate with R1,2=0.05 10-7 0 100 200 300 400 500 Frequency (Hz)

FigureFigure 10.10. TheThe radiation radiation e ffiefficienciesciencies of of the the mass-loaded mass-loaded (ML) (ML) plate, plate, the locallythe locally resonant resonant (LR) plate(LR) withplate undamped resonators (ηR =η 0),= and the LR plate with damped resonators (ηR η= 0.05= ), with (a) with undamped resonators1,2 ( R1,2 0 ), and the LR plate with damped resonators1,2 ( R1,2 0.05 ), with fully free boundary conditions, (b) fully simply supported boundary conditions, and (c) fully clamped (a) fully free boundary conditions, (b) fully simply supported boundary conditions, and (c) fully boundary conditions. clamped boundary conditions. It is known that the acoustic radiation from the infinite bare plate is effective only above the criticalIt frequencyis known that [31] wherethe acoustic the acoustic radiation wave from number the infinite exceeds bare the plate structural is effective wave number.only above For the its finitecritical counterpart, frequency [31] though where its the acoustic acoustic power wave can nu emberffectively exceeds radiate the structural in the whole wave frequency number. range,For its thefinite radiation counterpart, efficiency though reaches its acoustic the highest power at the can critical effectively frequency radiate and in tendsthe whole to unity frequency above the range, critical the frequency.radiation efficiency Based on reaches this relationship, the highest the at the structural critical andfrequency acoustic and wave tends numbers to unity ofabove the infinitethe critical LR frequency. Based on this relationship, the structural and acoustic wave numbers of the infinite LR plate are then calculated. In an infinite LR plate with the wave vector k = kx, ky , the displacement = () functionplate are canthen be calculated. assumed as: In an infinite LR plate with the wave vector k kkxy, , the displacement jkxx jky y function can be assumed as: w(x, y) = We− e− (42)

− − The substitution of Equation (42) into the governingjk x equationjkyy of the inﬁnite LR plate: wxy(), = Wex e (42)

X2 XS XT The substitution of Equation4 2 (42) into the governing equation of the infinite LR plate: D ω ρphp w(r) = Fistδ(x xis)δ(y yt), (43) ∇ − 2 ST − − ()∇−42ωρ()i == 1 s = 1 t = 1 δ ( − ) δ ( − ) DhwFxxyyppr  ist is t , (43) === Yields: ist111  2 2  Yields: ω mR1 ω mR2    Dk4 ω2ρ h ejkxaL  W  p p p 2a2 2a2    − − L − L    2  wR1  = 0, (44)  kR1 kR1 ω mR1 0    − jkxa − 2  w  kR e L 0 kR ω mR R2 − 2 − 2 − 2 Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 21

ωω22 mmjk a 42−−−ωρ RR12xL Dkppp h22 e 22aaLLW  −−kkmω2 0 w =0 , (44) RRR111 R 1 Appl. Sci. 2020, 10, 2843 − jk a 15 of 21 −−xL ω 2 w keRRR2220 k m R2  q  2 2 where kP ==+k22x + ky is the structural wave number. To obtain the non-zero displacement responses of where kkkPxy is the structural wave number. To obtain the non-zero displacement responses theof the base base plate plate and and the resonators,the resonators, the matrixthe matrix determinant determinant of Equation of Equation (44) should(44) should equal equal zero, zero, and then and thethen structural the structural and acoustic and acoustic wave wave numbers numbers can be can obtained. be obtained. TheThe structuralstructural andand acousticacoustic wavewave numbersnumbers of of thethe infiniteinfinite LRLR plateplate areare comparedcompared inin 0–5000–500 HzHz andand 3–63–6 kHz,kHz, wherewhere thethe eeffectiveffective radiation radiation frequency frequency ranges ranges are are shaded, shaded, as as shown shown in in Figure Figure 11 11a.a. ItIt cancan bebe seenseen that,that, besidesbesides thethe rangerange wherewhere thethe infiniteinfinite LRLR plateplate radiatesradiates soundsound eeffectivelyffectively aboveabove thethe criticalcritical frequencyfrequency ofof aboutabout 44 kHz,kHz, thethe acousticacoustic wavewave numbernumber alsoalso exceedsexceeds thethe structuralstructural wavewave numbernumber inin thethe twotwo additional additional bands bands of of 201–215 201–215 Hz Hz and and 300–320 300–320 Hz. Hz The. The two two bands bands coincide coincide well withwell thewith band the gaps,band whichgaps, meanswhich thatmeans the that acoustic the acoustic radiation radiation from the from infinite the LR infinite plate isLR eff plateective is in effective the band-gap in the frequencyband-gap ranges.frequency Therefore, ranges. theTherefore, acoustic the radiation acoustic from radiation the finite from LR the plate finite of identicalLR plate resonatorof identical parameters resonator canparameters reach a highcan reach effectiveness a high effectiveness in the band-gap in the frequency band-gap ranges, frequency as expressly ranges, as shown expressly in Figure shown 11 b.in ItFigure can be 11b. seen It thatcan be the seen radiation that the effi radiationciencies ofefficiencies the finite FFFF,of the SSSS,finite andFFFF, CCCC SSSS, LR and plates CCCC are LR sharply plates increasedare sharply in twoincreased band-gap in two frequency band-gap ranges. frequency Furthermore, ranges. byFurthermore, comparing by the comparing results of three the results different of LRthree plates, different the results LR plates, behave the almost results the behave same andalmost closer the tosame unity and in thecloser band-gap to unity frequency in the band-gap ranges. However,frequency atranges. most frequenciesHowever, at in most 0–500 frequencies Hz, the greater in 0 – the500 boundaryHz, the greater conditions the boundary are, the higherconditions the radiationare, the higher efficiency the radiat of theion LR efficiency plate is in of this the case. LR plate is in this case.

(a) (b) 120 Structural wave number 100 FFFF SSSS CCCC 120

) ) -1

Acoustic wave number -1 100 10-1 100 -2 80 10 80 10-3 60 60 10-4 40 critical frequency 40 10-5

20 -6 20 (m number Wave Wave number(m Radiation efficiency 10 0 10-7 0 0 200 400 3k 4k 5k 6k 0 100 200 300 400 500 Frequency (Hz) Frequency (Hz)

FigureFigure 11.11. ((aa)) TheThe structuralstructural andand acousticacoustic wavewave numbersnumbers ofof thethe infiniteinfinite locallylocally resonantresonant (LR)(LR) plate,plate, and (b) the radiation efficiencies of the finite LR plates with undamped resonators (ηR1,2 =η 0),= with fully and (b) the radiation efficiencies of the finite LR plates with undamped resonators ( R1,2 0 ), with free (FFFF) boundary conditions, fully simply supported (SSSS) boundary conditions, and fully clamped fully free (FFFF) boundary conditions, fully simply supported (SSSS) boundary conditions, and fully (CCCC) boundary conditions, together with the wave numbers for comparison. clamped (CCCC) boundary conditions, together with the wave numbers for comparison. Figure 12a–c shows the acoustic power radiated from the finite ML and LR plates with FFFF, SSSS, and CCCCFigure boundary 12a–c shows conditions. the acoustic It can bepower observed radiated from from the comparisonthe finite ML between and LR Figures plates 12 witha and FFFF, 12b SSSS, and CCCC boundary conditions. It can be observed from the comparison between Figure 12a (or Figure12c) that the acoustic behavior behaves quite di fferent when no constraints are applied to and Figure 12b (or Figure 12c) that the acoustic behavior behaves quite different when no constraints the plate. It has been explained in reference [29] that the acoustic power in the low-frequency range are applied to the plate. It has been explained in reference [29] that the acoustic power in the low- radiated from the FFFF bare plate is mainly dominated by three rigid body modes. The rigid body frequency range radiated from the FFFF bare plate is mainly dominated by three rigid body modes. motion and rotation of the FFFF plate make the first few flexural modes inefficiently radiating ones, The rigid body motion and rotation of the FFFF plate make the first few flexural modes inefficiently and then the acoustic power at these modal frequencies is not highlighted, unlike the results of the radiating ones, and then the acoustic power at these modal frequencies is not highlighted, unlike the finite plates with higher constraints. As shown in Figure 12a, the acoustic power radiated from the ML results of the finite plates with higher constraints. As shown in Figure 12a, the acoustic power plate is almost constant below 200 Hz in this case. For the LR plate, the radiated power is approximate radiated from the ML plate is almost constant below 200 Hz in this case. For the LR plate, the radiated to that from the ML plate in 0–100 Hz and is decreased below the lower frequency of the 1st band gap, power is approximate to that from the ML plate in 0 – 100 Hz and is decreased below the lower while it turns out to be increased at a certain frequency within the band gap. A similar phenomenon is frequency of the 1st band gap, while it turns out to be increased at a certain frequency within the band found around the 2nd band-gap frequency range. gap. A similar phenomenon is found around the 2nd band-gap frequency range. Appl. Sci. 2020, 10, 2843 16 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 21

(a) -3 10 304 Hz 10-5 10-7 10-9 10-11 ML plate LR plate with η =0 10-13 R1,2 Acoustic power (W) η LR plate with R1,2=0.05 10-15 0 100 200 300 400 500 Frequency (Hz) (b) 10-3 10-5 10-7 10-9 10-11 ML plate η -13 LR plate with R1,2=0

Acoustic power (W) 10 η LR plate with R1,2=0.05 10-15 0 100 200 300 400 500 Frequency (Hz) (c) 10-3 10-5 10-7 10-9 10-11 ML plate η -13 LR plate with R1,2=0

Acoustic power (W) 10 η LR plate with R1,2=0.05 10-15 0 100 200 300 400 500 Frequency (Hz)

FigureFigure 12.12. The acoustic power radiated from the mass-loaded (ML) plate, the locally resonant (LR) plate with undamped resonators (ηRη1,2 == 0), and the LR plate with damped resonators (ηRη1,2 == 0.05), plate with undamped resonators ( R1,2 0 ), and the LR plate with damped resonators ( R1,2 0.05 ), with (a) fully free boundary conditions, (b) fully simply supported boundary conditions, and (c) fully with (a) fully free boundary conditions, (b) fully simply supported boundary conditions, and (c) fully clamped boundary conditions. clamped boundary conditions. For the LR plates with greater constraints, such as the SSSS and CCCC boundary conditions For the LR plates with greater constraints, such as the SSSS and CCCC boundary conditions considered in this paper, it is expected that the acoustic power is decreased in the band-gap frequency considered in this paper, it is expected that the acoustic power is decreased in the band-gap frequency ranges, as shown in Figure 12b,c. It can be concluded that, although the radiation eﬃciency of the ranges, as shown in Figure 12b,c. It can be concluded that, although the radiation efficiency of the LR LR plate with greater constraints, is increased in the band-gap frequency ranges, due to the good plate with greater constraints, is increased in the band-gap frequency ranges, due to the good performance of the vibration attenuation the acoustic power is reduced much in the same bands. performance of the vibration attenuation the acoustic power is reduced much in the same bands. Moreover, when considering the damping characteristic of the resonators, the acoustic power curve Moreover, when considering the damping characteristic of the resonators, the acoustic power curve becomes smooth from below the lower frequency of the 1st band gap. Although the degree of acoustic becomes smooth from below the lower frequency of the 1st band gap. Although the degree of acoustic power reduction in the band-gap frequency ranges is weakened, the attenuation band of acoustic power reduction in the band-gap frequency ranges is weakened, the attenuation band of acoustic radiation is broadened. radiation is broadened. It can be seen that the application of the periodic resonators on the FFFF base plate cannot be It can be seen that the application of the periodic resonators on the FFFF base plate cannot be a a good noise-control method in this case, especially in and above the band gap frequency ranges. In good noise-control method in this case, especially in and above the band gap frequency ranges. In fact, it can be understood from the vibration proﬁle shown in Figure 13a that the vigorously vibrating fact, it can be understood from the vibration profile shown in Figure 13a that the vigorously vibrating FFFF ML plate at the two frequencies may not radiate much sound power due to radiation cancellation FFFF ML plate at the two frequencies may not radiate much sound power due to radiation from the uniform distribution of anti-phase regions. Conversely, though the vibration over the whole cancellation from the uniform distribution of anti-phase regions. Conversely, though the vibration surface of the FFFF LR plate is not strong, as shown in Figure 13b, the acoustic power radiated from over the whole surface of the FFFF LR plate is not strong, as shown in Figure 13b, the acoustic power the vibrating area near the excitation can be very high when no cancellation exists. radiated from the vibrating area near the excitation can be very high when no cancellation exists.

Appl. Sci. 2020, 10, 2843 17 of 21 8

Figure 13. The displacement proﬁles of (a) the mass-loaded plate and (b) the locally resonant plate 13 with fully free boundary conditions at 304 Hz.

However, when considering the damping loss factors of the resonators, the high acoustic power radiated from the FFFF LR plate can be decreased. The vibration proﬁle of the FFFF LR plate with damped resonators is shown in Figure 14. It can be seen that the LR plate with undamped resonators in Figure 13b vibrates vigorously near the excitation and hardly over the other surface. When replaced by damped resonators, the response of the FFFF LR plate is suppressed in the region where it is originally vibrating strongly, as shown in Figure 14, and thus the radiated power is reduced. This is why the acoustic power radiated from the LR plate with undamped resonators is peculiarly high and decreased greatly after considering13 the damping characteristic of the resonators at 304 Hz, as shown in Figure 12a.

Figure 14. The displacement proﬁle of the locally resonant plates with damped resonators with fully free boundary conditions at 304 Hz.

Overall, the acoustic power radiated from the plates is related to their boundary conditions. For the FFFF ML plate, the radiation cancellation could appear more easily in a wide frequency range, and the plate radiates sound weakly. However, for the FFFF LR plate without good radiation cancellation, despite a lower RMSV in the band-gap frequency ranges, the higher radiation eﬃciency will translate to a higher acoustic power. In other cases where the base plate has greater constraints, such as the SSSS and CCCC LR plates in the manuscript, even though the radiation eﬃciency is higher in the band-gap frequency ranges, a lower RMSV will lead to a lower acoustic power.

4. Conclusions In this paper, the low-frequency vibration and1 radiation performance of an LR plate is analyzed. The theoretical model is established for the band structures of the infinite LR plate with periodic multiple resonators. Taking the finite LR plate attached with two periodic arrays of resonators as an example, the forced vibration response and the radiation efficiency of the finite LR plate are derived, by adopting a general model with elastic boundary conditions. First, we discuss the impact of the parameter of the ratio between the resonant frequencies of two resonators on the band-gap frequency ranges. The band-gap width in the TRIEPE LR plate stays between those in both corresponding ORIEPE LR plates. By using the proper spring-mass parameters, the band structures of the infinite bare, ML, and LR plates and the RMSVs of their finite FFFF, SSSS, and CCCC counterparts are calculated. The comparison shows a good coincidence1 between the band

Appl. Sci. 2020, 10, 2843 18 of 21 gaps in the infinite LR plate and the vibration attenuation bands in the finite LR plate, no matter what boundary conditions are applied to the latter. A better vibration attenuation performance can be achieved when further considering the damping characteristic of the resonators. In contrast to the obvious decrease of the RMSV, the radiation efficiency of the finite LR plate has been sharply increased in the band-gap frequency ranges compared with the ML plate. By comparing the structural and acoustic wavenumbers, it can be seen that the acoustic radiation from the infinite LR plate is effective in the band gaps, and correspondingly that its finite counterpart can radiate sound highly effectively in the band-gap frequency ranges. Moreover, the acoustic power radiated from the finite LR plate around the band-gap frequencies may be seriously affected by its boundary conditions. The acoustic power radiated from the FFFF LR plate could be increased in the band-gap frequency ranges due to no effective cancellation of radiation from the strongly vibrating area near the excitation. However, for the LR plate with greater constraints, such as the SSSS and CCCC boundary conditions considered in this paper, the acoustic power is decreased in the band-gap frequency ranges. When further considering the damping characteristic of the resonators, a better vibration attenuation performance can result in the improvement of a radiation reduction around and above the band-gap frequencies.

Author Contributions: Conceptualization, Q.Q.; Data curation, M.S.; Formal analysis, Q.Q.; Investigation, M.S. and Z.G.; Methodology, Q.Q.; Resources, M.S.; Software, M.S. and Z.G.; Supervision, M.S.; Validation, Q.Q., M.S. and Z.G.; Visualization, Z.G.; Writing—original draft, Q.Q.; Writing—review & editing, M.S. and Z.G. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A

The terms Ba0, Ba1, Ba2, Ba3, Ba4, Bb0, Bb1, Bb2, Bb3, and Bb4 in Equation (7) can be expressed as

 0 1 0 0       1 0 0 0  Bs0 =  , (A1)  0 0 0 1     0 0 1 0    B = diag(1, 1, 3, 3) π/2/s,  s1  − − · 2 2  Bs2 = diag( 1, 1, 9, 9) π /4/s ,  − − − − · 3 3 , (A2)  Bs3 = diag( 1, 1, 27, 27) π /8/s ,  − − ·  B = diag(1, 1, 81, 81) π4/16/s4 s4 · where s in the above equations represents correspondingly a or b. The terms τm and τn in Equation (8) are given by

   2   2 , r = 0  , r = 0  1  π 2  π r  τr =  4 , τr =  4( 1) ,   2 , r , 0  − , r 0   (1 4r )π  (1 4r2)π ,   −  − , (A3)  2 2 =   , r = 0  3π , r 0  3  3π 4  − r+1  τr =  12 , τr =  12( 1)   2 , r , 0  − , r 0   (9 4r )π  (9 4r2)π , − − where r in the above equation represents correspondingly m or n. The terms Mx, My, Qx, and Qy in Equation (11) are given by

 2 2 3 3  M = D ∂ w + ∂ w , Q = D ∂ w + (2 ) ∂ w ,  x ∂x2 ν ∂y2 x ∂x3 ν ∂x∂y2  − − − . (A4)  ∂2w ∂2w ∂3w ∂3w  My = D + ν , Qy = D + (2 ν) − ∂y2 ∂x2 − ∂y3 − ∂x2∂y Appl. Sci. 2020, 10, 2843 19 of 21

The coeﬃcients gi,1, gi,2, and gi,3 in Equation (12) are given by

 l l l (3) l (1) 2  g = kx β , g = kx ξ (0) + Dξ (0) D(2 ν)ξ (0)λ , g = kx ,  1,1 0 0n 1,2 0 a a − − a bn 1,3 − 0  l 2 l l(1) l(2) l 2 2 2  g2,1 = D β λam νβ , g2,2 = Kx0ξ (0) Dξ (0) + Dνξ (0)λbn , g2,3 = D λam + νλbn ,  0n − 2n a − a a −  m l l l(3) l(1) 2 m+1  g3,1 = ( 1) kxaβ , g3,2 = kxaξ (a ) Dξ (a ) + D(2 ν)ξ (a )λbn , g3,3 = ( 1) kxa,  − 0n a − a − a −  m l l 2 l(1) l(2) l 2  g4,1 = D( 1) νβ β λam , g4,2 = Kxaξa (a ) + Dξa (a ) Dνξa(a )λbn ,  − 2n − 0n −  m 2 2 l l(3) l(1) 2 l g = D( 1) λam + νλ , g = ky ξ (0) + Dξ (0) D(2 ν)ξ (0)λam , g = ky α , (A5)  4,3 − bn 5,1 0 b b − − b 5,2 0 0m  l(1) l(2) l 2 l 2 l  g = ky , g = Ky ξ (0) Dξ (0) + Dνξ (0)λam , g = D α λ να ,  5,3 − 0 6,1 0 b − b b 6,2 0m bn − 2m  2 2 l l(3) l(1) 2 n l  g = D λ + νλam , g = k ξ (b) Dξ (b) + D(2 ν)ξ (b)λam , g = ( 1) k α ,  6,3 − bn 7,1 yb b − b − b 7,2 − yb 0m  n+1 l(1) l(2) l 2 n l l 2  g7,3 = ( 1) k , g8,1 = K ξ (b) + Dξ (b) Dνξ (b)λam , g8,2 = D( 1) να α λ ,  − yb yb b b − b − 2m − 0m bn  n 2 2  g = D( 1) λ + νλam 8,3 − bn The terms R, S, Z, and T in Equation (24) are all diagonal matrixes. The elements of R and Z are deﬁned respectively as   4 2 2 4  Rs,s = D λam + 2λam λbn + λbn  , s = mN + n, (A6)  Zs,s = ρh with dimensions of [(M + 1) (N + 1)] [(M + 1) (N + 1)]. The terms S and T can be expressed as × × ×  h 1 2 3 4 1 2 3 4 i  S = S S S S Sα Sα Sα Sα  h β β β β i , (A7)  1 2 3 4 1 2 3 4  T = Tβ Tβ Tβ Tβ Tα Tα Tα Tα of which Sl and Tl have the dimensions of [(M + 1) (N + 1)] (M + 1), and Sl and Sl have the α α × × β β dimensions of [(M + 1) (N + 1)] (N + 1), expressed as × ×   Sl = D βl λ 4 2βl λ 2 + βl  βs,m 0n am 2n am 4n  −  Sl = D αl λ 4 2αl λ 2 + αl  αs,n 0m bn 2m bn 4m , s = mN + n. (A8)  l l −  T = ρhβ  βs,m 0n  l = l Tαs,n ρhα0m

The terms U(ω) and V(ω) in Equation (24) are given ] by \      X2 XS XT   h i   ···κistmn   U(ω) = ZRi(ω)  L L  κistmn , (A9)  am bn  × ··· ··· 1 (M+1)(N+1) i = 1 s = 1 t = 1  ×  (M+1)(N+1) 1 ··· ×     T         ···l   2 S T   ξ (yt) cos(λamx )   X X X    b is    ···κistmn     V(ω) = ZRi(ω)     , (A10)  LamLbn       ×  ···l   i = 1 s = 1 t = 1  ξ (x ) cos(λ yt)    (M+1)(N+1) 1  a is bn    ··· ×     1 4(M+1+N+1) ··· × where κistmn = cos(λamxis) cos(λbn yt).

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