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Bandgap properties in simpliﬁed model of composite locally resonant phononic crystal plate

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State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street No. 29, Nanjing, Jiangsu, 210016, China

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Denghui Qian, Zhiyu Shi ∗

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a r t i c l e

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Article history: Received 5 July 2017 Received in revised form 20 August 2017 Accepted 28 August 2017 Available online xxxx Communicated by M. Wu Keywords: Bandgap property Simpliﬁed model Phononic crystal plate Plane wave expansion

a b s t r a c t

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This paper extends the traditional plane wave expansion (PWE) method to calculate the band structure of the proposed simpliﬁed model of composite locally resonant phononic crystal (LRPC) plate. Explicit matrix formulations are developed for the calculation of band structure. In order to illustrate the accuracy of the results, the band structure calculated by PWE method is compared to that calculated by ﬁnite element (FE) method. In addition, in order to reveal the bandgap properties, band structures of the “spring–mass” simpliﬁed model of stubbed-on LRPC plate, “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate and “spring–torsional spring–mass” simpliﬁed model of composite LRPC plate with and without the viscidity considered are presented and investigated in detail. © 2017 Elsevier B.V. All rights reserved.

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1. Introduction

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As is well-known, plates are extensively used as the containment structures in many areas such as marine, transport, aerospace engineering and civil construction projects [1,2]. Vibrations are mostly propagated along the containment structures from the vibration sources and structural noises are produced by the radiation of the vibrations. The proposition of PC concept provides a new idea for the investigation on the theory of vibration insulation and noise reduction. Over the past two decades, the propagation of elastic waves in PCs has attracted a lot of attentions which mainly focus on the calculation methods and bandgap properties, but the application researches particularly in the ﬁeld of vibration insulation and noise reduction are still immature. Bragg scattering [3–6] and locally resonant (LR) [7–10] are considered as the two main mechanisms for the creation of acoustic band gaps, which the frequency range of band gaps based on the ﬁrst mechanism is almost two orders of magnitude higher than that based on the second mechanism [7]. Hence, studies on plates with the design idea of LRPC introduced will provide a new idea for restraining the structure vibration and reducing the noise in the unmanageable low frequency range [11–14] of some industrial products. In recent years, bandgap properties of plates with the design idea of LRPC introduced have been researched. By etching holes

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*

Corresponding author. E-mail addresses: [email protected] (D. Qian), [email protected] (Z. Shi).

http://dx.doi.org/10.1016/j.physleta.2017.08.058 0375-9601/© 2017 Elsevier B.V. All rights reserved.

periodically in a solid matrix plate and then ﬁlling them with scatters, the so-called ﬁlled-in system is formed; by stubbing resonant units periodically onto the free surfaces of the plate, the stubbedon system is formed [15]. Hsu et al. [16] and Xiao et al. [17] investigated the vibration band gaps of epoxy base plates with ﬁlled-in rubber resonant units and ﬁlled-in rubber-coated heavy mass resonant units by using PWE method, respectively. Similarly, the three-component and two-component stubbed-on systems constructed by periodically depositing rubber stubs with and without Pb capped on the surface of the base plate was studied by using FE method by Oudich et al. [18]. Besides, Xiao et al. [13] researched the ﬂexural wave propagation and vibration transmission in an LR thin plate with a two-dimensional periodic array of attached spring–mass resonators. Zhao et al. [19] proposed a double-vibrator (rubber–steel–rubber–steel layers) threecomponent pillared PC plate on the basis of the traditional univibrator (rubber–steel layers) three-component pillared PC plate and studied the propagation characteristics of band gaps of ﬂexural vibration and longitudinal vibration in the two-layer stubbed-on system. By revisiting the ﬁlled-in and stubbed-on structures, Ma et al. [15] proposed a new structure with the three-layered spherical resonant units, which opens a large sub-wavelength full acoustic band gap. By combining the ﬁlled-in and stubbed-on units, Li et al. [20] investigated the propagation characteristics of Lamb waves in an LRPC plate with the combined resonant unit. Based on this, Li et al. [21] further researched the expansion of LR complete acoustic band gaps in two-dimensional PCs using a double-sided stubbed composite PC plate with composite stubs. Recently, Qian et al. [22]

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Table 1 Parameters used in calculations.

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μ1

E 1 (GPa)

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77.6

0.35

68

ρ1 (kg m−3 )

E 2 (GPa)

μ2

ρ1 (kg m−3 )

2730

1.175 × 10−4

0.469

1300

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kF (N m−1 )

mR (kg)

kM (N m2 )

J R (kg m2 )

a (m)

h (m)

R s (m)

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7

4 × 105

0.1

100

0.01

0.1

0.002

4 × 10−6

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Fig. 1. (a) Simpliﬁed model of composite LRPC plate, and (b) simpliﬁed model of resonant unit attached on the plate in a unit cell.

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investigated the propagation characteristics of ﬂexural waves in the LRPC double panel structure made of a two-dimensional periodic array of a spring–mass resonator surrounded by n springs connected between the upper and lower plates. In this paper, we investigate the propagation characteristics of ﬂexural vibration in the proposed “spring–mass” simpliﬁed model of stubbed-on LRPC plate, “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate and “spring–torsional spring– mass” simpliﬁed model of composite LRPC plate with and without the viscidity considered. The traditional PWE method is extended and formulized to treat such PC plates. All the results are expected to be of theoretic signiﬁcances and engineering application prospects in the ﬁeld of vibration and noise reduction.

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2. Model and method

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The simpliﬁed model of composite LRPC plate is composed by periodically attaching the simpliﬁed model of resonant unit onto the composite plate, as sketched in Fig. 1(a). The simpliﬁed model of resonant unit is composed of spring, torsional spring and mass. The composite plate is formed by etching holes periodically in the matrix plate and then ﬁlling them with another soft material. Here, the viscidity of the soft material in the composite plate is not considered at ﬁrst, the plate and mass are connected by the spring and torsional spring in each unit cell, as shown in Fig. 1(b). In this paper, the thickness of the plate is assumed to meet the requirement of thin plate. Besides, only the x-axis is considered as the direction that the mass twists around in order to simplify the derivation. According to the bending theory of thin plate, the governing equation of the model shown in Fig. 1(a) can be written as:

2 ⎧ 2 ∂ 2 W 1 (r ) ∂ W 1 (r ) ∂ ⎪ ⎪ D (r ) + μ(r ) ⎪ ⎪ ∂ x2 ∂ x2 ∂ y2 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪

∂ W 1 (r ) ∂ ⎪ ⎪ 2 D ( r ) 1 − μ ( r ) + ⎪ ⎪ ⎪ ∂ x∂ y ∂ x∂ y ⎪ ⎪ 2 ⎪ 2 ⎪ ∂ 2 W 1 (r ) W 1 (r ) ∂ ∂ ⎪ ⎨ + μ(r ) + 2 D (r ) ∂y ∂ y2 ∂ x2 ⎪ 2 ⎪ ω ρ ( r ) h ( r ) W ( r ) − ⎪ 1 ⎪ ⎪ ⎪ ∂ ⎪ ⎪ ⎪ Q 1( R) + M1( R ) = ⎪ ⎪ ∂x ⎪ ⎪ R R ⎪ ⎪ ⎪ −ω2m W ( R ) = Q ( R ) ⎪ ⎪ R R R ⎪ ⎩ −ω2 J R θR ( R ) = M R ( R ),

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where,

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(3)

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In the equations, W 1 (r ) is the transverse displacement of the plate, ρ (r ) is density of the plate, h(r ) is the thickness of the plate, D (r ) = E (r )h(r )3 /12(1 − μ(r )2 ) is the bending stiffness of the plate, E (r ) and μ(r ) are the elasticity modulus and Poisson ratio of the plate. Because the composite plate is composed by two different materials, the density, bending stiffness, elasticity modulus and Poisson ratio of the hard and soft materials are represented by (ρ1 , D 1 , E 1 , μ1 ) and (ρ2 , D 2 , E 2 , μ2 ), respectively. For the simpliﬁed model of resonant unit, all parameters are denoted by spring stiffness kF , mass mR , torsional spring stiffness kM and rotational inertia J R . Moreover, the radius of the soft material is R s , and the lattice constant is a. By means of a series of spatial Fourier expansions, ﬁnally equation (1) can be expressed by a matrix formulation (see Appendix A for details) as

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⎛ ⎡

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⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎤ ⎞

[U G ] + kF [ Q G ] − kM [ S G ] −kF [ P G ] kM [ R G ] ⎦ ⎣ kF 0 −kF [ P G ]T T kM [ R G ]⎡ 0 ⎤ −kM S[F G] 0 0 mR 0 ⎦ −ω2 ⎣ 0 ⎤

⎡

0

0

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⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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JR

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[ W 1 (G )] × ⎣ W R (0) ⎦ = 0. θR (0)

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(4)

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Equation (4) represents a generalized eigenvalue problem for ω2 . Finally, the band structure of the proposed simpliﬁed model of composite LRPC plate can be obtained by solving the equation for each Bloch wave vector limited in the irreducible ﬁrst Brillouin zone (1BZ). Besides, all the parameters used in calculations are displayed in Table 1.

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3. Numerical results and analyses

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3.1. “Spring–mass” simpliﬁed model of stubbed-on LRPC plate

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(1)

If the plate is composed of only one material and the torsional springs are ignored in the simpliﬁed model of composite LRPC plate shown in Fig. 1(a), the model can be further reduced to “spring–mass” simpliﬁed model of stubbed-on LRPC plate. Because the plate is composed of one material, equation (A.3) can be simpliﬁed as

ξ( G ) =

(2)

[ AG ]i j =

D 1 (k + G i )4x , 0,

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ξ1 , G = 0 0, G = 0.

(5)

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Substituting equation (5) into equations (A.10)–(A.15), we obtain that

Q 1 ( R ) = −kF W 1 ( R ) − W R ( R ) δ(r − R ) Q R ( R ) = kF W 1 ( R ) − W R ( R ) ,

⎧ ∂ ⎪ ⎪ M ( R ) = k ( R )δ( r − R ) − θ ( R )δ( r − R ) W M 1 R ⎨ 1 ∂x ⎪ ∂ ⎪ ⎩ M R ( R ) = −kM W 1 ( R ) − θR ( R ) . ∂x

i= j i = j ,

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(6)

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Fig. 2. Band structures of “spring–mass” simpliﬁed model of stubbed-on LRPC plate.

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2γ1 (k + 0,

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(12)

In addition, the items relevant to the torsional spring are eliminated in equation (4), which can be reduced as

[U G ] + kF [ Q G ] −kF [ P G ] − ω2 kF −kF [ P G ]T [ W 1 (G )] = 0. × W R (0)

S[F G] 0 0 mR

(13)

The band structure of the “spring–mass” simpliﬁed model of stubbed-on LRPC plate calculated by equation (13) is displayed in Fig. 2. The parameters used in the calculation are shown in Table 1, and the number of the plane waves is chosen as N = (5 ∗ 2 + 1)2 . As shown in the ﬁgure, a wide acoustic band gap is opened in the low frequency range (almost from 236 Hz to 472 Hz). Besides, the band structure calculated by FEM is also displayed to verify the accuracy. Here, the thin plate in the unit cell is divided by the four node quadrilateral plate element, the spring and mass are represented by the spring element and mass element. Moreover, the boundaries of the unit cell are applied by the periodic boundary conditions. By comparing the band structures calculated by the two methods, they agree very well. Hence, the extended PWE method can be eﬃciently applied to calculate the band structure of the proposed simpliﬁed model. Further, in order to illustrate the effect of the “spring–mass” resonator on the simpliﬁed model, the parameters of the resonator used in the example of Fig. 2 are changed to kF = 6 × 105 N m−1 and mR = 0.05 kg, and the band structure calculated is shown in Fig. 3. By comparing Figs. 2 and 3, the bands labeled in Fig. 2 are affected by the resonator except for band g 3 . The results reveal

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that the vibration modes corresponding to bands g 1 , g 2 and g 4 are the coupling between the vibrations of resonator and plate, but the vibration mode corresponding to bands g 3 is just relevant to the plate. In Ref. [13], band g 3 is called as Bragg scattering band, and the eigenfrequencies corresponding to the band can be written as

(9)

By substituting equations (6)–(11) to equation (A.9), it can be reduced as

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(7)

0,

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β1 (k + G i )2x (k + G i )2y , i = j i= j, 0,

Fig. 3. Band structure of “spring–mass” simpliﬁed model of stubbed-on LRPC plate with the parameters of the resonator kF = 6 × 105 N m−1 and mR = 0.05 kg.

f g3 (k y ) =

1 2π

π a · cos{atan[k y /(π /a)]}

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D

ρh

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.

(14)

As shown in equation (14), no relations are occurred between band g 3 and the resonator. In addition, the critical frequency f 2 of the acoustic band gap is smaller than f 3 in Fig. 2, but f 2 is bigger than f 3 in Fig. 3. Hence, the width of the acoustic band gap f w is decided by the starting frequency f 1 and the ending frequency, which is chosen as the smaller one between f 2 and f 3 . Particularly, the acoustic band gap cannot be opened if f 1 is bigger than f 2 or f 3 . The effects of spring stiffness kF on critical frequencies f 1 , f 2 , f 3 and band width f w are displayed in Fig. 4. Here, all the parameters used in the calculation except for kF are shown in Table 1, and the range of kF is from 0 to 2 × 106 N m−1 . As shown in the ﬁgure, with the increase of kF , both f 1 and f 2 increase, and f 3 keeps still, which leads f w to increase at ﬁrst and then decrease. Besides, kF = 4.8 × 105 N/m is the critical stiffness where f w reaches the maximum. Hence, the widest acoustic band gap can be obtained by adjusting the spring stiffness to the critical value, and the low starting frequency can be obtained by decreasing the stiffness. The inﬂuences of mass mR on critical frequencies f 1 , f 2 , f 3 and band width f w are shown in Fig. 5. Similarly, all the parameters used in the calculation except for mR are shown in Table 1, and the range of mR is from 0.002 to 0.2 kg. As shown in the ﬁgure, with the increase of mR , both f 1 and f 2 decrease, and f 3 keeps still, which leads f w to increase constantly. Besides, two critical points are existed. When mR < 0.02 kg, the acoustic band gap cannot be opened because the starting frequency f 1 is bigger than f 3 ; when mR > 0.066 kg, f 2 < f 3 , so the ending frequency changes from f 3 to f 2 , which leads the growth rate to decrease. In consequence, the wider acoustic band gap and lower starting frequency can be obtained by increasing the mass.

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3.2. “Spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate

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For the simpliﬁed model studied in the above section, if the torsional spring is considered, the “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate is formed. The researches in this section are beginning with the comparison of the

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Fig. 4. The inﬂuences of spring stiffness kF on critical frequencies f 1 , f 2 , f 3 and band width f w .

Fig. 6. Band structure of “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate.

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Fig. 5. The inﬂuences of mass mR on critical frequencies f 1 , f 2 , f 3 and band width f w .

Fig. 7. The inﬂuences of torsional spring stiffness kM on critical frequencies f 1 , f 2 , f 3 and band width f w .

band structures of the simpliﬁed models with and without torsional spring. Then the effects of the parameters of the torsional spring on the acoustic band gap are investigated. Here, the eigenvalue equation (4) is applied to calculate the band structure, and the matrix [U G ] is also determined by equation (12) as only one material in the plate. Fig. 6 displays the band structure of the “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate. All the parameters used in the calculation are shown in Table 1, and the number of plane waves is also chosen as N = (5 ∗ 2 + 1)2 . By comparing Figs. 6 and 2, the torsional spring has effects on bands g 1 , g 2 and g 4 labeled in Fig. 2 without g 3 , which illustrates again that there are no relations between band g 3 and the resonator. In addition, the torsional spring makes band g 1 move down and bands g 2 , g 4 move up as shown in the ﬁgure. The inﬂuences of torsional spring stiffness kM on critical frequencies f 1 , f 2 , f 3 and band width f w are displayed in Fig. 7. Here, all the parameters used in the calculation except for kM are shown in Table 1, and the range of kM is from 0 to 200 N m2 . As shown in the ﬁgure, with the increase of kM , f 1 decreases, f 2 increases and f 3 keeps still, which leads f w to increase constantly. Besides, kM = 105 N m2 is the critical stiffness, after which the ending frequency changes from f 2 to f 3 , and the growth rate decreases. Hence, by increasing the torsional spring stiffness, the width of the acoustic band gap will get bigger and the starting frequency will get lower.

In order to reveal the inﬂuences of rotational inertia J R on acoustic band gap, we calculate the acoustic band gap of the proposed “spring–torsional spring–mass” simpliﬁed model with J R = 0.1 kg m2 , as shown in Fig. 8. All the other parameters are the same as the example in Fig. 6. By comparing Figs. 8 and 6, the two band structures agree very well, which illustrates that there are no effects between the rotational inertia and the band structure of the “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate.

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3.3. Simpliﬁed model of composite LRPC plate

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If the homogeneous plate in the “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate is replaced by the composite plate, the simpliﬁed model of composite LRPC plate is formed, as shown in Fig. 1(a). When calculating the band structure, the matrix [U G ] in equation (4) is determined by equations (A.10)–(A.14). In addition, the bandgap properties of the simpliﬁed model are investigated with and without the consideration of viscidity of the soft material, respectively. Here, in order to improve the calculation accuracy, the improved PWE method [23] is applied. If the viscidity of soft material is not considered, the band structure of simpliﬁed model of composite LRPC plate is displayed in Fig. 9. Here, all the parameters used in the calculation are shown in Table 1. In addition, the number of plane waves is chosen as

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Fig. 8. Band structure of “spring–torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate with the parameters of the resonator kM = 100 N m2 and J R = 0.1 kg m2 .

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Fig. 10. The inﬂuences of soft material radius R s in composite plate on critical frequencies f 1 , f 2 , f 3 and band width f w .

G ∞ + G 0 τG2 ω2 1 + τG2 ω2

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Fig. 9. Band structure of simpliﬁed model of composite LRPC plate with the viscidity of soft material not considered.

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N = (10 ∗ 2 + 1)2 to increase the accuracy on account of the existence of two different materials in the composite plate. By comparing Fig. 2 with Fig. 9, what can be seen is that the effect of soft material is similar to the torsional spring, which makes band g 1 move down and bands g 2 , g 4 move up. Hence, replacing the homogeneous plate with composite plate has a role in promoting the acoustic band gap with wider band width and lower starting frequency. The inﬂuences of soft material radius R s in composite plate on critical frequencies f 1 , f 2 , f 3 and band width f w are displayed in Fig. 10. Here, all the parameters used in the calculation except for R s are shown in Table 1, and the range of R s is from 1 to 5 × 10−6 m. As shown in the ﬁgure, with the increase of R s , f 1 decreases, f 2 increases and f 3 keeps still. Besides, the increasing constantly f w is determined by the starting frequency f 1 and ending frequency f 3 because f 2 is always bigger than f 3 in the relation curves. So by increasing the soft material radius in composite plate, the acoustic band gap will get wider and the starting frequency will get lower. If the viscidity of soft material is considered, the elasticity modulus E and shear modulus G are dependent on the frequency in the frequency domain, and can be represented as

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(16)

E (ω) =

E ∞ + E 0 τ E2 ω2 1+τ

2 E

ω

2

( E 0 − E ∞ )τ E ω + i, 1 + τ E2 ω2

(15)

E − 2G

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(17)

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Substituting the real parts of complex modulus in equations (15) and (16) to (17) yields

104

μ=

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(G 0 − G ∞ )τG ω + i, 1 + τG2 ω2

where E 0 and E ∞ are the initial and ﬁnal state values of elasticity modulus. Similarly, G 0 and G ∞ are the initial and ﬁnal state values of shear modulus. τ E and τG represent the relax time of elasticity modulus and shear modulus, respectively. Here, the relax time of elasticity modulus and shear modulus are assumed to be the same, that is τ E = τG = τ . Besides, deﬁne initial–ﬁnal value ratios α E = E 0 / E ∞ and αG = G 0 /G ∞ , and assume α E = αG = α . For the investigation of wave acoustic band gaps, only real parts of complex modulus need to be considered [24]. As is well known, the relation between Poisson ratio μ and modulus is

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G (ω) =

22

84

μ=

2G

.

E 0 − 2G 0 2G 0

(18)

.

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Equation (18) illustrates that μ is just determined by the initial state values of modulus but not relevant to the frequency. Hence, the bending stiffness of plate D = Eh3 /12(1 − μ2 ) is also frequency dependent and can be obtained by the real parts of complex modulus. Because of the frequency-dependence of modulus, the eigenvalue problem of equation (4) should be solved by iterative process, and the main steps are as follows:

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1. For a given Bloch wave vector k limited in the irreducible ﬁrst Brillouin zone, let the elasticity modulus of soft material E = E 0 and obtain the bending stiffness D (ω), insert D (ω) into equation (A.2) and compute the eigenvalues from eigenvalue equation (4). The eigenvalues are corresponding to those of the simpliﬁed model of stubbed-on LRPC plate with viscidity p of soft material not considered. Call them as ωn ( p = 1, 2, . . .), here p is the order of eigenvalue, and n = 0. p 2. Insert ωn into equation (15) to obtain the real part of complex elasticity modulus E (ω) and further obtain the bending stiffness D (ω). Insert D (ω) into equation (A.2) and compute the eigenvalues from eigenvalue equation (4). Call the new p eigenvalues as ωn+1 , which can be treated as a revision of old eigenvalues

ωnp .

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Fig. 11. The relations between the real part of complex elasticity modulus and frequency, (a) relax time (b) initial–ﬁnal value ratio α = 2.0, relax time τ takes different values.

τ = 3 × 10−5 s, initial–ﬁnal value ratio α takes different values,

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Fig. 12. Band structures of simpliﬁed model of composite LRPC plate with the viscidity of soft material considered when relax time τ = 3 × 10−5 s, initial–ﬁnal value ratio (a) α = 1.5, (b) α = 2.0, (c) α = 3.0, and (d) the inﬂuences of initial–ﬁnal value ratio α on critical frequencies f 1 , f 2 , f 3 and band width f w .

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p

p

3. Repeat the second process until |ωn+1 − ωn | < δ , where δ is the prescribed error tolerance.

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124

Fig. 11 shows the relations between the real part of complex elasticity modulus and frequency when relax time τ and initial–

ﬁnal value ratio α take different values. Consider that G and E have similar dependence on frequency, only dependence curves of E are given here. The initial state value of elasticity modulus is shown in Table 1. Fig. 11(a) and (b) display the relation curves when τ = 3 × 10−5 s, α = 1.0, 1.5, 2.0 and α = 2.0, τ = 3 × 10−5 s,

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Fig. 13. Band structures of simpliﬁed model of composite LRPC plate with the viscidity of soft material considered when initial–ﬁnal value ratio τ = 3 × 10−5 s, (b) τ = 1 × 10−4 s, (c) τ = 5 × 10−4 s, and (d) the inﬂuences of relax time τ on critical frequencies f 1 , f 2 , f 3 and band width f w .

α = 2.0, relax time (a)

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10−4 s, 5

10−4 s, respectively. As shown in the ﬁgure,

1× × with the increase of frequency, the real part of elasticity modulus reaches a deﬁnite value, which is the initial state value of elasticity modulus. Hence, the frequency domain is divided into the frequency dependence domain and frequency independence domain at low and medium frequency and high frequency, respectively. Fig. 9 shows that the acoustic band gap is under 300 Hz, so it is completely affected by the viscidity of soft material. In addition, Fig. 11(a) and (b) display two ﬂat bands corresponding to α = 1.0 and τ = 0 s separately. When α = 1.0, that is E ∞ = E 0 , then E (ω) = E 0 is obtained by substituting it into equation (15). Hence, the ﬂat band corresponding to α = 1.0 represents the initial state value of elasticity modulus. When τ = 0 s, E (ω) = E ∞ is obtained by substituting it into equation (15). Hence, the ﬂat band corresponding to τ = 0 s represents the ﬁnal state value of elasticity modulus. Moreover, what can be seen in the ﬁgure is that the starting values of the relation curves are different but the frequency domains to reach the deﬁnite value are almost the same when α takes different values, and the frequency domains to reach the deﬁnite value are different but the starting values of the relation curves are the same when τ takes different values. In consequence, the initial–ﬁnal value ratio and relax time determine the change amplitude of modulus insider the frequency dependence domain and the frequency dependence domain.

Fig. 12(a), (b) and (c) display the band structures of simpliﬁed model of composite LRPC plate with the viscidity of soft material considered when relax time τ = 3 × 10−5 s, initial–ﬁnal value ratio α = 1.5, 2.0 and 3.0, respectively. Here, all the parameters used in the calculation are shown in Table 1. As shown in the ﬁgure, bands g 1 , g 2 and g 4 are all affected by α . In order to further illustrate the inﬂuence rule, the inﬂuences of α on critical frequencies f 1 , f 2 , f 3 and band width f w are displayed in Fig. 12(d). As sketched in the ﬁgure, with the increase of α , starting frequency f 1 decreases, f 2 increases and f 3 keeps still. Because f 2 is always bigger than f 3 in the relation curves, the constantly increasing f w is determined by f 1 and f 3 . In consequence, increasing the initial–ﬁnal value ratio of soft material plays a role in widening the acoustic band gap and lowering the starting frequency. Fig. 13(a), (b) and (c) display the band structures of simpliﬁed model of composite LRPC plate with the viscidity of soft material considered when initial–ﬁnal value ratio α = 2.0, relax time τ = 3 × 10−5 s, 1 × 10−4 s and 5 × 10−4 s, respectively. Here, all the parameters used in the calculation are shown in Table 1. As shown in the ﬁgure, the band structure is inﬂuenced by τ slightly. Besides, the effects of τ on critical frequencies f 1 , f 2 , f 3 and band width f w are displayed in Fig. 13(d). As shown in the ﬁgure, with the increase of τ , starting frequency f 1 increases and f 2 decreases, which is opposite to the inﬂuences of α on the band structure. The phenomenon can be attributed to that the real part of modulus

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decreases with the increase of α but increases with the increase of τ as shown in Fig. 11. Because f 2 is always bigger than f 3 in the relation curves, the constantly decreasing f w is determined by f 1 and f 3 . In general, the frequency domain affected by τ is smaller than that inﬂuenced by α , but decreasing the relax time of soft material plays a certain role in widening the acoustic band gap and lowering the starting frequency.

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4. Conclusions

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In this paper, the traditional PWE method is extended and formulized to calculate the band structure of the proposed simpliﬁed model of composite LRPC plate. For the further simpliﬁed “spring–mass” simpliﬁed model of stubbed-on LRPC plate, “spring– torsional spring–mass” simpliﬁed model of stubbed-on LRPC plate and “spring–torsional spring–mass” simpliﬁed model of composite LRPC plate with and without the viscidity considered, the bandgap properties are investigated in detail. The main conclusions are as follows:

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1. For the band structure of the “spring–mass” simpliﬁed model of stubbed-on LRPC plate, the accuracy is veriﬁed by comparing the band structure calculated by PWE and that calculated by FEM. Numerical results demonstrate that a wide acoustic band gap is opened in the low frequency domain. With the increase of spring stiffness in the resonator, the bandgap width increases ﬁrst and then decreases, and the starting frequency increases. With the increase of mass, the bandgap width increases and the starting frequency decreases, but this will lead the total quality to increase. 2. By comparing the band structures of the “spring–mass” simpliﬁed model of stubbed-on LRPC plate with and without the torsional spring added, the added torsional spring have obvious inﬂuences on the band structure. With the increase of torsional spring stiffness, the bandgap width increases and the starting frequency decreases. But the rotational inertia has nothing to do with the band structure. 3. For the simpliﬁed model of composite LRPC plate, with the increase of soft material radius, the bandgap width increases and the starting frequency decreases. If the viscidity of soft material is considered, increasing the initial–ﬁnal value ratio or decreasing the relax time of soft material plays a role in widening the acoustic band gap and lowering the starting frequency.

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Acknowledgements

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This research is supported by the National Natural Science Foundation of China through Grant No. 11172131 and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant No. 0515G01).

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Appendix A

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In this appendix, we present the derivation process of equation (4) from (1) in detail. Deﬁne α (r ) = ρ (r )h(r ), β(r ) = D (r )μ(r ) and γ (r ) = D (r )(1 − μ(r )), equation (1) is simpliﬁed as:

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(A.1)

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On account of the periodicity of the model in x and y directions, α (r ), β(r ), γ (r ) and D (r ) can be expressed in spatial Fourier series as

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⎧ ⎪ D (r ) = D ( G )e iGr , ⎪ ⎨ G ⎪ β( r ) = β( G )e iGr , ⎪ ⎩

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α (r ) = γ (r ) =

G

G

α ( G )e

iGr

γ ( G )e iG r ,

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(A.2)

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G

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with

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ξ( G ) =

ξ2 f + ξ1 (1 − f ), G = 0 (ξ2 − ξ1 )ψ( G ), G= 0,

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(A.3)

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where ξ( G ) is used to represent the parameters α (r ), β(r ), γ (r ) and D (r ) uniformly, ξ1 and ξ2 denote the corresponding parameters of hard and soft materials, respectively. f = π R 2s /a2 represents the ﬁlling ratio of soft material, and ψ( G ) = 2 f J 1 (G R S )/(G R S ). J 1 is the Bessel function of the ﬁrst kind of order one, and G is the modulus of reciprocal-lattice vector G . According to the Bloch theory and the periodicity of the model, W 1 (r ) can be written as

W 1 (r ) =

W 1 G e i (k+G )r ,

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(A.4)

G

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where k = (kx , k y ) denotes the Bloch wave vector limited in the irreducible ﬁrst Brillouin zone (1BZ) and G is the reciprocal-lattice vector. In addition, note that the Bloch theory and the periodic condition imply that

e ik R δ(r − R ) = e ikr 1 S

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(A.5)

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δ(r − R ),

(A.6)

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and the delta function suggests the following relation:

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⎧ ik R ⎪ ⎨ W 1 ( R ) = W 1 (0)e W R ( R ) = W R (0)e ik R ⎪ ⎩ θR ( R ) = θR (0)e ik R ,

R

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⎧ 2 ∂ 2 W 1 (r ) ∂ 2 W 1 (r ) ∂ ⎪ ⎪ ⎪ D ( r ) + β( r ) ⎪ ⎪ ∂ x2 ∂ x2 ∂ y2 ⎪ ⎪ ⎪ 2 2 ⎪ ∂ W 1 (r ) ∂ ⎪ ⎪ γ (r ) +2 ⎪ ⎪ ∂ x ∂ y ∂ x∂ y ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∂ ∂ ∂ 2 W 1 (r ) W ⎪ 1 (r ) ⎪ ⎨ + 2 D (r ) + β( r ) ∂y ∂ y2 ∂ x2 2 ⎪ ⎪ − ω α (r ) W 1 (r ) ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ Q 1( R) + M1( R ) = ⎪ ⎪ ⎪ ∂x ⎪ R R ⎪ ⎪ ⎪ ⎪ −ω2m W ( R ) = Q ( R ) ⎪ ⎪ R R R ⎪ ⎪ ⎩ 2 −ω J R θR ( R ) = M R ( R ).

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where . R δ(r − R ) = Ge Besides, equation (A.4) suggests the following relation:

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(A.7)

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By substituting equations (A.2)–(A.7) to equation (A.1) gives

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G ⎪ ⎪ + 2 k + G k + G y ⎪ ⎪ x ⎪

⎪ ⎪ ⎪ × γ G − G k + G x k + G y W 1 G ⎪ ⎪ ⎪ ⎪ G ⎪ ⎪ 2

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⎪ ⎪ ⎪ + k + G y D G − G k + G y W1 G ⎪ ⎪ ⎪ ⎪ G ⎪ ⎪ 2

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⎪ ⎪ ⎪ + k+G y β G − G k + G x W1 G ⎪ ⎪ ⎨ G (A.8) 2 ⎪ −ω α G − G W1 G ⎪ ⎪ ⎪ ⎪ G ⎪ ⎪ ⎪ ⎪

k ⎪ F ⎪ ⎪ = − W 1 G − W R (0) ⎪ ⎪ S ⎪ ⎪ G ⎪ ⎪ ⎪

⎪ kM ⎪ 2 ⎪ − + k + G W G k + G θ ( 0 ) ⎪ 1 x x R ⎪ S ⎪ ⎪ G ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ −ω2mR W R (0) = kF W 1 G − W R (0) ⎪ ⎪ ⎪ ⎪ ⎪ G ⎪ ⎪

⎪ ⎪ ⎪ ω2 J R θR (0) = kM − θ k + G W G ( 0 ) . ⎪ 1 R x ⎩

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If the number of reciprocal-lattice vectors is chosen as N, equation (4) can be obtained from (A.8). In the equation,

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[U G ] = S [ AG ] + S [ B G ] + S [C G ] + S [ D G ] + S [ E G ], [ AG ]i j = (k +

G i )2x D ( G i

[ B G ]i j = (k +

G i )2x β( G i

(A.9)

− G j )(k + G

2 j )x ,

(A.10)

− G j )(k + G

2 j)y,

(A.11)

[C G ]i j = 2(k + G i )x (k + G i ) y × γ ( G i − G j )(k + G j )x (k + G j ) y , [ D G ]i j = (k +

G i )2y D ( G i

− G j )(k + G

2 j)y,

(A.12) (A.13)

[ E G ]i j = (k + G i )2y β( G i − G j )(k + G j )2x ,

(A.14)

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(A.15)

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(A.16)

1 ⎢1⎥ ⎢ ⎥ 1

, N 2 ×1 T

[ Q G ] = [ P G ][ P G ] , ⎡ ⎤ (k + G 1 )x ⎢ (k + G 2 )x ⎥ ⎢ ⎥ [R G] = ⎢ ⎥, .. ⎣ ⎦ . (k + G N × N )x

(A.17)

(A.18)

⎡

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(k + G 1 )2x (k + G 1 )2x ··· (k + G 1 )2x ··· (k + G 2 )2x ⎥ ⎢ (k + G 2 )2x (k + G 2 )2x ⎥. [S G] = ⎢ . . . .. ⎦ ⎣ .. .. .. . 2 2 2 ··· (k + G N × N )x (k + G N × N )x (k + G N × N )x

(A.19) References

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