Numerical and experimental investigations on the band-gap characteristics of metamaterial multi-span beams

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Physics Letters A ••• (••••) ••••••

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Numerical and experimental investigations on the band-gap characteristics of metamaterial multi-span beams Shuaimin Hao a , Zhijing Wu a,b,∗ , Fengming Li a,∗ , Chuanzeng Zhang b a b

College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany

a r t i c l e

i n f o

Article history: Received 29 July 2019 Received in revised form 27 September 2019 Accepted 27 September 2019 Available online xxxx Communicated by R. Wu Keywords: Metamaterial multi-span beams Periodic boundary conditions Band-gap coupling Spectral element method Experiments

a b s t r a c t A novel metamaterial multi-span beam with periodic simple supports and local resonators is designed and investigated. The frequency responses of the proposed metamaterial multi-span beam are computed by the spectral element method (SEM). The accuracy and feasibility of the SEM are verified by the finite element method (FEM) and the vibration experiments. The results show that the metamaterial multi-span beam could generate both the local resonance band-gaps in the low-frequency ranges and the Bragg band-gaps in the medium and high frequency regions. By adjusting the natural frequencies of the local resonators, the thickness of the base beam and the length of the unit-cell, the local resonance and the Bragg band-gaps can be controlled, respectively. The coupling effects of these two kinds of band-gaps are investigated by the parametrical design, which broadens the band-gaps and consequently improves the vibration reduction performance. © 2019 Elsevier B.V. All rights reserved.

1. Introduction “Phononic crystals” and “acoustic metamaterials” are artificial periodic structures which can suppress the elastic wave propagation in the band-gap (or stop-band) frequency ranges and control the wave propagation by the artificial design of elastic periodic structures [1–7]. The generation mechanisms of band-gaps can be divided into the Bragg scattering and the local resonance mechanisms. Since the Bragg band-gap frequency is limited by the lattice constant (i.e. the elastic wavelength corresponding to the center frequency of the lowest band-gap is about twice long of the lattice constant), it is difficult to obtain the band-gaps in the lowfrequency ranges for a small-sized unit-cell [8–13]. Liu et al. [14] originally proposed the local resonance mechanism in 2000. The position of the local resonance band-gap corresponds to the natural frequency of a single resonator, which breaks the limitation of the Bragg scattering mechanism [15–20]. The band-gap characteristics of the phononic crystals and acoustic metamaterials have important application prospects in the structural vibration and sound suppression. Wang and Wang [21] used the piezoelectric spring model to control the nonlin-

*

Corresponding authors at: College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China. E-mail addresses: [email protected] (Z. Wu), [email protected] (F. Li). https://doi.org/10.1016/j.physleta.2019.126029 0375-9601/© 2019 Elsevier B.V. All rights reserved.

ear phononic crystals. Based on the Lindstedt–Poincaré method, the approximate solution was given and the band-gap properties were studied. Ning et al. [22] proposed a novel strategy of active elastic metamaterials and a concealed design of double Helmholtz cavities. The structure has a controllable effective bulk modulus which may have a negative value in a wide frequency range. Li et al. [23,24] studied the non-reciprocity in linear and nonlinear acoustic metamaterials and discussed the band-gap and propagation characteristics of elastic waves. Xiao et al. [25] studied the acoustic transmission loss of the unbounded uniform thin plate consisting of several sub-wavelength arrays of spring-mass resonators. The results show that the acoustic transmission loss of the metamaterial-based plate is much higher than that of the bare plate. Shu et al. [26] studied the band-gap formation mechanism of a phononic crystal plate with tubular pillars, which was used in the design of structural stability. Su et al. [27] proposed an acoustic metamaterial model for the low-frequency vibration suppression. The Bloch phase of a unit-cell was calculated using the transfer matrix method and the Bloch-Floquet theorem. The spectral element method (SEM) has been widely applied in the wave propagation and vibration analysis. Bahrami et al. [28] analyzed the vibration characteristics of thin shallow shell under the impact load by the SEM and discussed the influence of boundary conditions of the thin shallow shell. Shirmohammadi and Bahrami [29] presented the application of the SEM in the vibration analysis of circular and annular circular plates under impact load. Wu et al. [30–32] analyzed the band-gap characteris-

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Fig. 1. (a) A metamaterial multi-span beam with N unit-cells, (b) the unit-cell, (c) the local resonator and (d) the cross-section of the beam.

tics of a novel lattice with a hierarchical periodicity, piezoelectric square lattice structures and three-dimensional Kagome lattices by the SEM. Compared with the finite element method (FEM) results, the accuracy and validity of the SEM are verified. Although numerous investigations on the frequency band-gap characteristics of almost all kinds of phononic crystals and acoustic metamaterial structures have been conducted, few results on the combination of the Bragg scattering and local resonance mechanisms to design special periodic structures for generating more band-gaps from low to high frequency ranges have been reported so far, which motivates the present work. In this paper, a novel metamaterial multi-span beam is designed by combining the Bragg scattering and local resonance mechanisms. The multi-span beams are commonly found in the bridge and frame structures in the engineering applications. The band-gap properties in broad frequency ranges are studied by the SEM. The accuracy and feasibility of the SEM are verified by the FEM and experimental results. The effects of the structural parameters on the coupling of the two kinds of band-gaps are further analyzed, which provides a reference for the vibration attenuation design in multi-frequency bands of periodic multi-span beam structures. 2. Problem description

Based on the Euler-Bernoulli beam theory, the equation of motion of a uniform beam is given by

(1)

where v(x, t) is the transverse displacement, ρ is the mass density, E is the Young’s modulus, A is the cross-sectional area of the beam, and I is the area moment of inertia. The transverse displacement in a spectral expression has the following form: N 0 −1 1 

N0

n =0

V n (x, ωn ) ei ωn t ,

V (x, ωn ) = A 1 e−i k F x + A 2 e−k F x + A 3 e−i k F (a−x) + A 4 e−k F (a−x) , (3)



3. Spectral element equation

v (x, t ) =

where V n (x, ωn ) is the spectral component of v (x, t), N 0 is the number of samples in the time domain and ωn is the circular frequency. Substituting Eq. (2) into Eq. (1), the solution of the equation of motion can be assumed to be

where A 1 , A 2 , A 3 and A 4 are the coefficients related to the fre-

Fig. 1(a) shows a metamaterial multi-span beam on the periodically distributed simple supports. The metamaterial multi-span beam includes N unit-cells. The unit-cell is illustrated in Fig. 1(b), and the beam of each unit-cell can be divided into two spectral beam elements with length a by three nodes for the SEM. The cross-section of the beam is rectangular with the width b and the height h. The local resonator is composed of a cylindrical rubber rod and a cylindrical steel mass. The geometrical parameters of the resonator are shown in Fig. 1(c).

∂ 4 v (x, t ) ∂ 2 v (x, t ) EI + ρ A = 0, ∂ x4 ∂t2

Fig. 2. Symbolic convention of Euler-Bernoulli beam.

(2)

 ω 2 ρ A 1/ 4

n is the wave-number of the flexural quency, and k F = EI wave. The rotation angle Θ is obtained from the derivation of the transverse displacement to the coordinate x

Θ(x, ωn ) =

∂V = A 1 (−i k F ) e−i k F x + A 2 (−k F ) e−i k F x ∂x + A 3 (i k F ) e−i k F (a−x) + A 4 (k F ) e−i k F (a−x) . (4)

For an Euler-Bernoulli beam, the bending moment and the shear force can be expressed as

∂2V , ∂ x2 ∂3V . Q (x) = − E I ∂ x3

M (x) = E I

(5a) (5b)

The symbolic convention of the Euler-Bernoulli beam is shown in Fig. 2 from which the nodal forces can be expressed as

⎧ Q ⎪ ⎪ ⎨ 1

⎫ ⎪ ⎪ ⎬

⎤ − Q (0) ⎢ − M (0) ⎥ M1 ⎥ =⎢ ⎣ Q (a) ⎦ . Q2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ M2 M (a) ⎡

(6)

The displacements and rotation angles of the nodes are written as

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Fig. 4. Spectral beam element.

Fig. 3. Two beam elements with a mass-spring system.

⎧ ⎫ ⎡ ⎤ V ⎪ V (0) ⎪ ⎪ ⎨ 1⎪ ⎬ ⎢ Θ1 Θ(0) ⎥ ⎥. =⎢ ⎣ V V (a) ⎦ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ Θ2 Θ(a)

(7)

⎧ ⎫ Q ⎪ ⎪ ⎪ ⎨ 1⎪ ⎬

⎧ ⎫ V ⎪ ⎪ ⎪ ⎨ 1⎪ ⎬ M1 Θ1 =S , Q2 ⎪ V2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ M2 Θ2

(8)

where S is the spectral stiffness matrix of the beam element which can be expressed as

α EI ⎢ γa S= 3 ⎢ a ⎣ −α −γ a

−α −γ a

γa βa

2

βa

γa

⎤

βa ⎥ ⎥, 2

(9)

α −γ a ⎦ −γ a β a 2

γa 2

where

α = [sin(k F a) + sinh(k F a)] (k F a) / , β = [− cos(k F a) sinh(k F a) + sin(k F a) cosh(k F a)] (k F a)/ , β = [− sin(k F a) + sinh(k F a)] (k F a)/ ,

γ = [− cos(k F a) + cosh(k F a)] (k F a)2 / , γ = sin(k F a) sinh(k F a)(k F a)2 / , = 1 − cos(k F a) cosh(k F a). Fig. 3 shows two beam elements with an oscillator. In this SEM model, the local resonator in Fig. 1 is simplified as a mass-spring system, which is composed of the spring with the stiffness k1 and the mass m2 . In the case of a time-harmonic vibration, the discrete vibration equation of the mass-spring system can be expressed as

(K − ω

= F,

where



K=

 F=

−k1

k1 −k1 F1 0

k1



(10)



 M=

,

0 0 0 m2



 ,

V =

vb vs

 ,

,

in which v b is the transverse displacement of the beam at the connection point B, v s is the displacement of the mass m2 along the y-axis, and F 1 is the force acting on the oscillator at point B. Eq. (10) can be further expressed as

F 1 = kr v b ,

(11)

where kr is the additional dynamic stiffness of the oscillator to the beam structure, and it has the following form:

kr = k 1 −

k21 k1 − ωn2 m2

.

k1 = E R

A L mx h1

(12)

(14)

,

where E R is the Young’s modulus of the rubber rod, and mx = 1 + 1.65

A 2L

in which A L =

A 2f

π d21 4

and A f = π d1 h1 .

Each node of the spectral beam element has two degrees of freedom as shown in Fig. 4. The degree of freedom of the oscillator is coupled with the transverse displacement of the beam. The dynamic stiffness matrix of the beam with the mass-spring system can be obtained by assembling those of the two spectral beam elements and the mass-spring element [30–35], and it has the following form:

S 1,1 S R = ⎣ S 1,3

3

(13)

In this paper, the rubber rod with the diameter d1 and height h1 is used as the spring. The stiffness of the cylindrical rubber rod can be expressed as [36]

⎡

α = [cos(k F a) sinh(k F a) + sin(k F a) cosh(k F a)] (k F a)3 / ,

2 n M )V

The mass m1 of the spring is also considered in the modeling, especially when it is large. In this situation, the additional dynamic stiffness can be modified as

kr = −ωn2 m1 + kr .

From Eqs. (3)–(7), the relationship between the nodal forces and the nodal displacements can be obtained as

⎡

3

S 1,2 S 1,4 + S 2,1 + D S 2,3

⎤

S 2,2 ⎦ , S 2,4

(15)

where S i , j (i = 1, 2; j = 1, 2, 3, 4) is a 2 × 2 sub-matrix which represents the dynamic stiffness matrices of the beam element, and D is the additional dynamic stiffness matrix of the oscillator and it has the following form:



D=

kr 0



0 . 0

(16)

Thus, the dynamic equation of the beam with the mass-spring system can be expressed as

SRVn = f ,

(17)

where V n and f are the nodal displacement and force vectors. Considering the boundary conditions in Fig. 1, the transverse displacements at the simply supported points are zero. So the rows and columns of the dynamic stiffness matrix corresponding to the displacements at the simple supports should be taken as zero. Then, by solving Eq. (17), the frequency response curves of the metamaterial multi-span beam can be obtained and the band-gap properties can be analyzed. 4. Results and discussions 4.1. Finite element comparison and experimental validation The vibration characteristics of a metamaterial multi-span beam are calculated by the SEM and compared with the FEM and the experimental results to verify the validity of the analytical model and the correctness of the numerical results. Fig. 5 shows the model of the metamaterial multi-span beam and the unit-cell for the FEM. In Fig. 5(a), the simply supported boundary condition is applied to the beam at the middle positions

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of two adjacent oscillators. The number of the unit-cells is N = 5. The material and structural parameters used in the calculation are listed in Table 1. Fig. 6 shows the comparison of the frequency response curves calculated by the SEM and the FEM (ANSYS software). For the FEM models, the element sizes are set to be 0.1 mm and 0.5 mm, respectively. As the element size becomes finer (smaller), the results of the FEM converge to those of the SEM. However, increasing the number of the FEM element will inevitably increase the time consumption. In the SEM, each unit-cell is divided into two spectral elements. The SEM is able to obtain more accurate results by using fewer elements. Consequently, compared with the FEM, the SEM is more efficient for the band-gap analysis of the metamaterial multispan beam structure. Fig. 7 shows a prototype in the experiment. In Fig. 7, a smooth ring is bonded to the support position of the beam and the screw

Fig. 5. (a) The finite element model of the metamaterial multi-span beam and (b) the unit-cell.

on the supporting steel sheet passes through the ring to realize the simple supports. Fig. 8 shows the comparisons of the frequency responses obtained by the SEM, the FEM and the experiments. It is seen that the band-gaps measured by the experiments are in fairly good agreement with those obtained by the SEM and the FEM. The experimental errors are mainly caused by the manufacturing inaccuracy, the measurement error and the structural damping. 4.2. Band-gap properties of the metamaterial multi-span beam The band-gap properties of the proposed metamaterial multispan beam are discussed in this sub-section. The structural and material parameters are the same as those in the sub-section 4.1, unless otherwise stated. Fig. 9 shows the frequency response curves of a local resonance beam (without the periodic simple supports), a multi-span beam (without the local resonators) and the proposed metamaterial multi-span beam calculated by the SEM. The figure presents two types of band-gaps: (a) the local resonance band-gaps caused by the periodic oscillators on the base beam and (b) the Bragg band-gaps caused by the periodic simple supports. It is clearly seen from Fig. 9 that for the local resonance and the multi-span beams, only the local resonance and the Bragg band-gaps occur, respectively. Compared with the other two kinds of beam structures, the metamaterial multi-span beam can produce both types of the

Fig. 6. Comparison of the frequency response curves between the SEM and the FEM.

Fig. 7. A prototype of the proposed metamaterial multi-span beam.

Table 1 Material and structural parameters of the metamaterial multi-span beam. Material

E (MPa)

μ

ρ (kg/m3 )

Length (m)

Width (m)

Height (m)

Diameter (m)

Aluminum beam Rubber rod Steel rod

70000 0.1175 200000

0.34 0.47 0.33

2700 1300 7850

1.00 -

0.01 -

0.002 0.003 0.004

0.008 0.01

Fig. 8. (a) Frequency responses calculated by the SEM and FEM and (b) the experimentally measured frequency responses at the free end of the beam.

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band-gaps, which increases the frequency band-gap ranges and improves the vibration reduction capacity of the beamstructure.

5

Fig. 9. Frequency response curves of a local resonance beam, a multi-span beam and the proposed metamaterial multi-span beam. In the calculations, N = 9 is used.

tant parameter to be considered in the design of the metamaterial multi-span beam. Fig. 10 shows the frequency responses of the metamaterial multi-span beam with the different natural frequencies of the oscillators. The parameters are listed in Table 2. The natural frequency of the oscillator determines the location of the local resonance band-gap. The local resonance band-gap appears in the high-frequency range with the increase of the natural frequency of the oscillator. Fig. 11 shows the frequency response curves of the metamaterial multi-span beam when the unit-cell number N is 5, 7 and 9, respectively. The displacements within the band-gaps decrease with the unit-cell number increasing. The unit-cell number has no effect on the start frequencies of the band-gaps, but the cut-off frequencies of the band-gaps will move to the lower frequencies with the increase of the unit-cell number. Fig. 12 shows the frequency responses of the metamaterial multi-span beams with different base beam thicknesses and unitcell lengths. These two parameters have little effects on the local resonance band-gaps but large influences on the Bragg band-gaps. The Bragg band-gaps move to the higher frequency ranges with the increase of the base beam thickness and the decrease of the unitcell length. Therefore, the positions of the Bragg band-gaps can be adjusted by these two structural parameters. Furthermore, the coupling characteristic of the Bragg and the local resonance band-gaps will be investigated. From Figs. 10 and 12 it is observed that the natural frequency of the oscilla-

Fig. 10. Frequency response curves of the metamaterial multi-span beam with different natural frequencies of the oscillators.

Fig. 11. Comparisons of the frequency responses for metamaterial multi-span beams with different unit-cell numbers.

k1 The natural frequency of the oscillator is f = 21π m = 186 Hz. 2 In Fig. 9, for the metamaterial multi-span beam, the frequency interval of the local resonance band-gap is about 160-236 Hz. The natural frequency of the oscillator has a corresponding relation with the position of the local resonance band-gap, and the natural frequency is always within the band-gap frequency range. Therefore, the natural frequency of the local resonator is an impor-

Table 2 Parameters of the oscillators. Natural frequencies of the oscillators (Hz)

d1 (mm)

h1 (mm)

d2 (mm)

h2 (mm)

Start frequencies (Hz)

Cut-off frequencies (Hz)

151 186 237

8.0 8.0 8.0

4.0 3.0 2.3

10.0 10.0 10.0

3.7 4.0 4.2

142 160 178

196 236 293

Fig. 12. Frequency responses of the metamaterial multi-span beam with different (a) base beam thicknesses and (b) unit-cell lengths.

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Table 3 Parameters of the oscillators and the beam thickness. Natural frequencies of the oscillators (Hz)

d1 (mm)

h1 (mm)

d2 (mm)

h2 (mm)

h (mm)

Start frequency of second band-gap (Hz)

Cut-off frequency of second band-gap (Hz)

186 365

8.0 10.0

3.0 2.0

10.0 10.0

4.0 5.1

2.0 1.9

160 179

236 495

implies that the vibration reduction capacity of the metamaterial multi-span beam can be increased by the combination of the Bragg scattering and the local resonance mechanisms. (2) The natural frequency of the oscillators is a key parameter to obtain the target local resonance band-gap position. (3) Increasing the base beam thickness and decreasing the unitcell length will lead to a shift of the Bragg band-gaps to the higher frequency ranges. But they have only little effects on the local resonance band-gaps. (4) The band-gap coupling effect can be achieved by properly choosing the parameters of the local resonators and the base beam so as to broaden the band-gap frequency range and enhance the vibration reduction capacity of the metamaterial multi-span beams. Fig. 13. Comparison of the band-gap positions of the metamaterial multi-span beam with different natural frequencies of the oscillators.

tor determines the position of the local resonance band-gap, while the positions of the Bragg band-gaps are mainly controlled by the structural parameters of the base beam. So if we select the structural parameters of the oscillators and the beam as shown in Table 3, a very interesting phenomenon can be observed. For this case, the frequency response curves of the metamaterial multispan beam are shown in Fig. 13. It is seen here that when the natural frequency of the oscillator is increased from 186 Hz to 365 Hz and the thickness of the beam is chosen to be 1.9 mm, the local resonance band-gap will move to higher frequency region and couple with the Bragg band-gap as observed from the dashed line. Especially, the coupled band-gap is very broad from 179 to 495 Hz, which implies that the coupling effect of the two kinds of band-gaps can efficiently broaden the band-gap frequency range and improve the vibration reduction capacity of the beam structure. It can be concluded from the above analyses that the positions of the local resonance and the Bragg band-gaps can be adjusted by changing the natural frequency of the oscillators and the unitcell length or the thickness of the base beam for the design of the band-gap properties of the metamaterial multi-span beams. The local resonance and the Bragg band-gaps can be combined and even coupled to increase the band-gap widths and improve the vibration reduction performance of the metamaterial multi-span beams. 5. Conclusions A novel metamaterial multi-span beam with more band-gaps in broad frequency ranges is designed and investigated by combining the Bragg scattering and local resonance mechanisms. The SEM is used to calculate the vibration characteristics of the metamaterial multi-span beam, and the validity of the numerical method is verified by the FEM and the experimental results. The influences of the natural frequency of the oscillators, and the structural parameters of the base beam and the unit-cell on the band-gap characteristics are analyzed. From the present investigations, the following main conclusions can be drawn: (1) Compared with the local resonance beam and the multi-span beam, the proposed metamaterial multi-span beam can generate both the local resonance and the Bragg band-gaps, which

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research is supported by the National Natural Science Foundation of China (Nos. 11761131006, 11602065 and 11572007), the German Research Foundation (DFG, Project-No. ZH 15/30-1), the China Postdoctoral Science Foundation (Nos. 2018M630338 and 2019T120253) and the Heilongjiang Postdoctoral Science Foundation (LBH–Z18056). Zhijing Wu is grateful to the financial support from the China Scholarship Council (201806685004) for her scientific visit at the Chair of Structural Mechanics, University of Siegen, Germany. References [1] Y. Pennec, B. Djafari-Rouhani, H. Larabi, J.O. Vasseur, Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate, Phys. Rev. B 78 (10) (2008) 104105. [2] L. Fan, H. Ge, S.Y. Zhang, H.F. Gao, Y.H. Liu, H. Zhang, Nonlinear acoustic fields in acoustic metamaterial based on a cylindrical pipe with periodically arranged side holes, J. Acoust. Soc. Am. 133 (6) (2013) 3846–3852. [3] Z.W. Li, C. Wang, X.D. Wang, Modelling of elastic metamaterials with negative mass and modulus based on translational resonance, Int. J. Solids Struct. 162 (2019) 271–284. [4] M.H. Lu, L. Feng, Y.F. Chen, Phononic crystals and acoustic metamaterials, Mater. Today 12 (12) (2009) 34–42. [5] D. Beli, J.R.F. Arruda, M. Ruzzene, Wave propagation in elastic metamaterial beams and plates with interconnected resonators, Int. J. Solids Struct. (2018) 139–140, 105–120. [6] R.X. Feng, K.X. Liu, Tuning the band-gap of phononic crystals with an initial stress, Physica B, Condens. Matter 407 (12) (2012) 2032–2036. [7] M.L. Wu, L.Y. Wu, P.W. Yang, L.W. Chen, Elastic wave band gaps of onedimensional phononic crystals with functionally graded materials, Smart Mater. Struct. 18 (11) (2009) 115013. [8] H.J. Zhao, H.W. Guo, M.X. Gao, R.Q. Liu, Z.Q. Deng, Vibration band gaps in double-vibrator pillared phononic crystal plate, J. Appl. Phys. 119 (1) (2016) 014903. [9] M. Oudich, Y. Li, B.M. Assouar, Z.L. Hou, A sonic band gap based on the locally resonant phononic plates with stubs, New J. Phys. 12 (8) (2010) 083049. [10] Y. Chen, L. Wang, Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps, Appl. Phys. Lett. 105 (19) (2014) 191907. [11] Y.K. Dong, H. Yao, J. Du, J.B. Zhao, J.L. Jiang, Research on local resonance and Bragg scattering coexistence in phononic crystal, Mod. Phys. Lett. B 31 (11) (2017) 1750127.

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