Active control of flexural waves in a phononic crystal beam with staggered periodic properties

Journal Pre-proof Active control of flexural waves in a phononic crystal beam with staggered periodic properties Ping Chen, Yi-Ze Wang, Yue-Sheng Wang

PII: DOI: Reference:

S0165-2125(19)30171-4 https://doi.org/10.1016/j.wavemoti.2019.102481 WAMOT 102481

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Wave Motion

Received date : 10 May 2019 Revised date : 28 November 2019 Accepted date : 28 November 2019 Please cite this article as: P. Chen, Y.-Z. Wang and Y.-S. Wang, Active control of flexural waves in a phononic crystal beam with staggered periodic properties, Wave Motion (2019), doi: https://doi.org/10.1016/j.wavemoti.2019.102481. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

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Active control of flexural waves in a phononic crystal beam with staggered periodic properties Ping Chen 1, Yi-Ze Wang 2 , Yue-Sheng Wang 1, 2 1 Institute

of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China

2 Department

of Mechanics, Tianjin University, Tianjin, 300350, China

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Abstract

The band gaps of a phononic crystal beam with staggered periodic structure are investigated. The

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periodic system consists of a pure elastic (i.e. PMMA) matrix beam and some piezoelectric (i.e. PZT) patches with coupling between the mechanical-electrical components. The PZT patches connected by negative capacitance circuits are applied to function as the active control system. Based on the condition at the interface between adjacent unit cells, the transfer matrix and localization factor are derived. The influence of the degree of interlacing and negative capacitance circuits are discussed. The numerical results

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show that another band gap can be generated by the staggered periodic structure of PZT patches. The widths and locations of the band gaps can be changed by the degree of interlacing. Keywords: Active control; flexural wave; phononic crystal beam; staggered periodicity; band gap.

1. Introductions

Phononic crystals are composites consisting of two or more elastic materials in a periodic arrangement/structure. When the frequency falls within the band gap, elastic waves and vibration cannot propagate through the structure [1–7]. Phononic crystals have many applications, e.g. wave localization,

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filtering and isolation, focusing and cloaking, etc. Over the past decades, phononic crystals have received considerable attention. Chen et al. [8] studied the propagation of Lamb waves in a periodic plate with lower

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order. Wang et al. [9] discussed elastic wave localization in a randomly disordered layered phononic crystal with thermal effects. Ding et al. [10] investigated the wave propagation and band gaps in a periodic structure with piezoelectric patches bonded to the same surface of a beam. Phononic crystals often exhibit an ordered periodic structure. However, the staggered periodic structure can provide special band gap properties. With changes in periodic properties, the locations and widths of

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band gaps will present different behaviors. Sometimes, another band gap can be generated during the design of different phononic crystals [11]. Cheng et al. [12, 13] studied phononic crystal plates with staggered and sandwich-layered structures. It is known that when the filling ratio changes, the band gaps will exhibit different behaviors; therefore, we can tune the propagation of elastic waves. However, previous investigations have focused on fixed 

Corresponding author Email addresses: [email protected] 1

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systems in which material parameters are not easily controlled. On the other hand, it would be interesting to design tunable periodic structures [14]. Because of electrical and mechanical coupling, piezoelectric materials can achieve complex responses more easily. The piezoelectric effect is the ability of electrical and mechanical components to exchange energies [15, 16]. Then, piezoelectric composite materials can be applied to mechanical metamaterials with negative effective mass density and elastic modulus [17], which

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provide the ability to actively control their actions. Based on this superior characteristic, it has been shown that negative impedance circuits attached to piezoelectric patches can tune elastic waves [18–24]. In the present work, we propose a phononic crystal beam with an active control system attached to

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negative impedance shunting circuits. The staggered periodic structure of piezoelectric patches is considered and the band gaps can be tuned. Based on the transfer matrix and Lyapunov exponent method, the band gaps are presented. The theoretical results are verified by finite element calculations and experiments.

2. Basic equations and derivations

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The phononic crystal beam is shown in Fig.1, where the staggered periodic piezoelectric patches are considered. The periodic system consists of n unit cells with the piezoelectric layers and purely elastic (i.e. PMMA) beam. In a unit cell, four sub-cells are denoted as 1–4, their lengths are denoted as l1–l4, and the length of the base beam and piezoelectric layers are lb and lp, respectively. The thickness of the base beam in a unit cell is Hb. Two piezoelectric patches are bonded to the top and bottom of the base beam with the same thickness Hp and width Wb. Perfect continuity without slipping was applied at the interface between the base beam and piezoelectric layer. The transverse displacement, denoted as w(x, t), is the same in different layers.

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Based on the small deformation of the Euler beam model, the shear deformation and rotary inertia are negligible. The position of the neutral layer did not change during deformation [4, 10].

2. 1 Negative capacitance circuits

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The constitutive equation of the piezoelectric material can be expressed as [23]

 S1   s11E  =   D3   d 31

d 31   T1   , T  ε33   E3 

(1)

where 1 and 3 denote the x and z directions; S1 and T1 are the strain and stress along the x-axis; s11E is the

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compliance coefficient of the piezoelectric material; d31 and  3T3 are the piezoelectric and dielectric constants; and D3 and E3 are the electrical displacement and electrical field intensity along the z-axis, respectively.

Based on Eq. (1), the relation between the strain and stress of the piezoelectric patch is [24]

 sZd 312 As H p1  S1 =  s11E   T1 ,  1 + sZC p   2

(2)

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T

where Z is the complex impedance; As and C p = ε33 As H p are the electrode area and inherent

1 is the Laplace variable.

capacitance of the piezoelectric patch, and s = iω with i =

The elastic modules of the piezoelectric patch can be derived as

H p (1+ sZC p )

E p 11

H s

1+ sZC   sZd p

2 31

As

.

(3)

of

Ep =

In Fig.2, the negative capacitance circuits are applied to show the active control performance, with capacitance C and compensation resistance R0. The complex impedance of the operational amplifier can be

Z=

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defined as

1 , R2 Cs R1

(4)

where R1 and R2 are the fixed and slide rheostat resistances, respectively.

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2. 2 Transfer matrix of the periodic structure

Fig. 3.shows the deformation of a sub-cell with piezoelectric layers bonded to the top and bottom of the elastic base beam. Their lengths are l2, and we considered the same angle. In general speaking, the neutral axes are different, but according to previous investigations on multi-layered beams [10–13, 16, 18–22], this influence on the deformation is not obvious because the composite beam is thin. In order to obtain an explicit derivation for the numerical calculation, it is usually assumed that the neutral axis remains the same. The displacements of the neutral axes of the base beam and the top and bottom piezoelectric layers are u2b,

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u2pt and u2pb with the following expressions:

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u 2 pt = u 2 b 

u 2 pb = u 2 b +

H p + Hb 2 H p + Hb 2

θ2 ,

(5a)

θ2 ,

(5b)

where Hp and Hb are the thickness of the piezoelectric layers and base beam;

θ 2 = w2 / x

is the rotation

angle; and the subscripts b, pt, and pb represent the base beam, top and bottom of the piezoelectric layers, respectively.

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The strain energy of the three layer composite beam for sub-cell 2 can be expressed as

V2 =

2 2   2 w2   1 l2   u 2 b  E A + E I    dx b b b b  x  x 2   2 0     

2 2 2  u 2 pt   u 2pb    2 w2   1 l2  +   E p Apt   dx ,  + E p Apb   + 2E p I p  2  2 0   x    x   x  

3

(6a)

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where E, A and I are the Young’s modulus, cross-sectional area, and inertia moment, respectively. Similarly, the kinetic energy of sub-cell 2 is

 u2 pt 2  u2 pb 2  w 2    u2b 2  w2 2  1 l2  2 T2 =   ρb Ab   +  t   + ρp Ap  t  +  t  + 2  t    dx , (6b) t 2 0                

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where ρ is the mass density and t is the time. For sub-cells 1 and 3, where only the top or bottom piezoelectric layer is considered with the neutral axis assumption [10–13, 16, 18–22], the strain and kinetic energies are



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2 2   2 w1   1 l1   u1b  V1 =   Eb Ab   + Eb I b  2   d x 2 0  x   x   

2 2  u1 pt    2 w1   1 l1   E A + E I    dx , p pt  p p  x  x 2   2 0     

V3 =

(7b)

2 2   2 w3   1 l3   u 3b   E A E I +   dx b b  b b   x  x 2   2 0     

2 2   2 w3    u3 pb  1 l3   E A + E I   dx ,  p pb  p p  x  x 2   2 0     

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+

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 u1 pt 2  w 2    u1b 2  w1 2  1 l1  1 T1 =   ρb Ab   +  t   + ρp Ap  t  +  t    dx , t 2 0              

(7a)

(8b)

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 u3 pb 2  w 2    u3b 2  w3 2  1 l3  3 T3 =   ρb Ab   +  +   + ρp Ap     dx ,  t  t  t  t 2 0                 

(8a)

where u1b and u1pt are the axial displacements of the neutral axes of the base beam and the top piezoelectric layer in sub-cell 1; u3b and u3pb are the axial displacements of the neutral axes of the base beam and the bottom piezoelectric layer in sub-cell 3; w1 and w3 are the transverse displacements of the elastic beam in sub-cells 1 and 3; and l1 and l3 are the lengths of the base beam in sub-cells 1 and 3, respectively.

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Similarly, the strain and kinetic energies of the elastic beam for sub-cell 4 are 2 2   2 w4   1 l4   u 4 b  V4 =   Eb Ab   dx ,  + Eb I b  2  2 0   x   x   

 u4b 2  w4 2  1 l4 T4 =  ρb Ab    dx ,  + 2 0  t   t  

(9a)

(9b)

where u4b and w4 denote the axial and transverse displacements of the base beam in sub-cell 4 and l4 is the 4

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length. According to Eqs. (5a) and (5b), we can use ub instead of upt and upb in Eqs. (6a) and (6b). As shown in Fig.3, a beam element is considered with two nodes and three degrees-of-freedom (DOF) in each node. Therefore, the displacement vector is defined as ib

wi  x,t  θi  x,t  =  N iu  x  N iw  x  T

N'iw  x  

T

δ  t  ,

(i =1–4),

i

of

u  x,t 

(10)

where i is the number of the sub-cell; Niu(x) and Niw(x) are the shape functions, and δi(t) represents the nodal DOF vector expressed as follows:

2 3  x x N iw  x  =  0 1  3   + 2     li   li 

δ  t  = u

 0 0 , 

2

x x 0 3   2    li   li 

x 2 x3 x2 + 2 li li

wiL

iL

x li

θiL

uiR

wiR

θiR 

3



x 2 x3   2, li li  

T

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i

0 0

p ro

 x N iu  x  = 1   li

.

(11)

(12)

(13)

The strain and kinetic energies of different sub-cells are presented in Appendix A. We consider the flexural wave with angular frequency ω. As a result, the stiffness matrices of the periodic piezoelectric beam can be defined as

 K di  =  K i   ω 2  M i  ,

(i =1–4),

(14)

where Kd1–Kd4 denote the dynamic stiffness matrices of the corresponding sub-cell. According to previous studies on periodic layered structures [2, 4, 9], the localization factor can be

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calculated by the transfer matrix method and the governing equation of a sub-cell in the j-th unit cell is expressed as

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j)  K (diLL  ( j)  K diR L

j)   X iL( j )   FiL( j )  K (diLR =  ( j)  , j)   ( j)  K (diRR   X iR   FiR 

(i =1–4, j =1, 2, …, n),

(15)

(M, N=L,R) is the 3×3 partial matrix of Kdi in the j-th unit cell; and X(j) and F(j) are the where K (j) diMN vectors of the generalized displacements and forces, respectively.

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Then, Eq. (15) can be derived as

YiR( j ) = TiYiL( j ) ,

(i =1–4,

j =1, 2, …, n),

(16)

j j where YiR  and YiL  are the state vectors on the right and left sides, Ti is the 6×6 transfer matrix of the

four sub-cells and

( j) YiR( j ) =  X iR

5

T

FiR( j )  ,

(17a)

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YiL( j ) =  X iL( j ) 1 (j)   K (j) diLR K diLL Ti =  (j) (j) (j)  1 (j)  K diRL  K diRR K diLR K diLL

T

FiL( j )  ,

(17b)

1  K (j) diLR , (i =1–4, j =1, 2, …, n). (18) (j) (j)  1  K diRR K diLR 

According to the continuous condition at the interface, the displacement and stress state vectors are

On the other hand, Eqs. (19a) and (19b) can be written as (j) , YiR(j) = JY(i+ 1 )L

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(j) , (i =1–4, j =1, 2, …, n). FiR(j) =  F(i+ 1 )L

(j) , X (j) iR =X (i+1 )L

j =1, 2, …, n),

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(i =1–4,

where

1 0 0 I= 0 1 0 .    0 0 1 

I 0  J=  ,  0 I 

(19a, b)

(20)

(21a, b)

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The relation between the right side of sub-cell 4 and the left side of sub-cell 1 in the j-th unit cell is obtained as

Y4( Rj )  T Y1(Lj ) ,

(22)

where Tꞌ is the transfer matrix of the j-th unit cell with the following expression:

T  = T4J  1T3J 1T2J  1T1 .

(23)

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Then, the state vector of the left side in the j-th unit cell can be expressed as:

Y1(j)L = J 1Y4(jR-1 ) ,

(j =1, 2, …, n).

(24)

According to Eqs. (22) and (24), the state vectors of the (j-1)-th and j-th unit cells can satisfy the

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following relation:

Y4( Rj )  T ( j ) Y4( Rj -1) ,

(j =1, 2, …, n),

(25)

where T ( j )  T J 1 = T4J 1T3J 1T2J 1T1J 1 is the whole transfer matrix between two adjacent unit cells.

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2.3 Lyapunov exponent and localization factor The Lyapunov exponent, which is defined as the average exponential rate of convergence or divergence between two neighboring phase orbits, characterizes the temporal evolution of a dynamical system [25–27]. The localization factor, which denotes the spatial evolution of a nearly periodic system, similarly characterizes the average exponential rate of growth or decay of wave amplitudes. The localization factor can be calculated by the Lyapunov exponent of periodic systems like Eq.(25). The 6

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system is unstable with the positive Lyapunov exponent, while the negative Lyapunov exponent corresponds to the stable and dynamic one. The smallest positive Lyapunov exponent is the localization factor, and denotes the wave that propagates with the longest distance [28]. It has been proved that for the system with a 2m×2m transfer matrix, the m pairs of Lyapunov exponents have the following property [25, 29, 30]:

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1  2    m  0  m1 ( m )  m2 ( m-1 )    2m ( 1 ) .

(26)

The smallest positive Lyapunov exponent λm is defined as the localization factor because it carries the

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energy the farthest and has the least amount of decay. In the present work, the algorithm by Wolf et al. [31] was applied to calculate Lyapunov exponents of the continuous systems [4, 9, 28]. The d-th Lyapunov exponents (1≤d≤2m) are written as:

1 n ˆ  j . ln Y  4 R ,d n n j 1

ˆ  j is defined below. where vector Y 4R, d

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d  lim

(27)

To calculate the d-th Lyapunov exponent (1≤d≤2m), the d orthogonal unit vectors with the 2m-dimension ( u1(1) , u (21) , …, u(d1) ) are chosen as the initial state vectors. For the j-th iteration

Y4( Rj ),l  T ( j ) u (l j -1) ,

(i =1–4,

j =1, 2, …, n).

(28)

By the Gram-Schmidt orthonormalization procedure, d orthogonal unit vectors can be calculated as

ˆ  j   Y( j ) , Y 4 R,1 4 R,1

u1   j

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4 R ,1

ˆ  j   Y( j )  (Y( j ) , u j  )u j  , Y 4 R,2 4 R,2 4 R,2 1 1

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ˆ  j Y 4 R ,1 , j ˆ Y 

u 2   j

ˆ  j Y 4 R ,2 , j ˆ Y  4 R ,2



ˆ  j   Y( j )  (Y( j ) , u j  )u j    (Y( j ) , u j  )u j  , Y 4 R,d 4 R,d 4 R,d 4 R,d 1 1 d 1 d 1

u d   j

ˆ  j Y 4 R ,d , ˆ  j Y

(29)

4 R ,d

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where (ꞏ,ꞏ) denotes the dot-product.

As a result, the d pairs of contrary Lyapunov exponents can be calculated and the d-th one (λd) is the localization factor. In this study, the transfer matrix of the phononic crystal beam with the staggered periodic piezoelectric patches is 6×6 and the third Lyapunov exponent λ3 is the localization factor. The zero value of the localization factor denotes the pass band in which the elastic wave can propagate; the other frequency regions denote the band gaps in which the wave is forbidden. 7

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3. Results and discussions Table 1. Unit cell geometry lp (m)

Hb (m)

Hp (m)

lb (m)

0.01

0.05

0.003

0.001

0.12

of

Wb(m)

Table 2. Material parameters Young’s

Compliance

Piezoelectric

Dielectric

modulus

coefficient

strain constant

Constant

𝑠

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Materials

Mass densities

𝜀

E (N/m2)

(m3 N-1)

(C m-2)

(F m-1)

PMMA

1200

1.43×109

—

—

—

PZT-5H

7500

7.75×1010

1.29×10-11

-1.86×10-10

3.54×10-8

Table 3. Circuit parameters

d31

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ρ (kg/m3)

C(pF)

Cp (pF)

R2 (kΩ)

R1 (kΩ)

R0 (kΩ)

Operational amplifier

9.58×103

14.3×103

68.2

50.28

2000

LM324N

In this section, the results of theoretical derivation, finite element calculation and experiments are

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examined to show the band gaps of the flexural wave. The material constants, geometry, and circuit parameters are listed in Tables 1–3. Our attention will focus on the modulation of the wave band gaps by the

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staggered array and active control action.

Fig. 4 shows the relation between the wave frequency and localization factor for the phononic crystal beam with and without the staggered periodic properties. For the symmetric periodic structure, there are two stop bands in the frequency regions 346–3596 Hz and 5853–9584 Hz. For the staggered periodic structure, these two stop bands appear when the wave frequency is at 2513–3431Hz and 4755–8862 Hz. The stop band

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with low frequency becomes narrower while the other one becomes much wider. This means that the staggered periodic structure is more likely to generate wide band gaps in the high-frequency region. Fig.5 shows the influences of the degree of interlacing on band gap properties, where μ is the ratio of l2 and l1 with l1=0.01m. We can see that there are two band gaps for μ = 2 in the frequency regions 2889-3190 Hz and 4815-18790 Hz. It is worth noting that these band gaps have obvious differences, with the former being narrow and the latter being wide. There is another narrow band gap in the frequency regions 722Hz 8

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and 918 Hz when μ = 8. Fig. 6 illustrates the effects of the thickness ratio (η) on the band gaps of the staggered periodic structure. Different values of η = 1, 3, and 5 were considered with the piezoelectric patch width of 0.01m. We observed that the thickness of the base beam can change the location and width of the band gaps. As η increased, a stop band appeared in the high-frequency region with increasing width.

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We know that by adjusting the resistance R1, the negative capacitance circuit can change the Young’s modulus of the piezoelectric patch. Thus, we will discuss the effect the resistance and performance of the electric circuit on the band gaps which behaves as the active control action. In Fig. 7, γ represents the ratio of

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R2 and R1 with R1=50.28kΩ. We can see that as R2 increased, the band gaps moved toward the high-frequency regions and their widths continuously increased.

In order to show the influences of the electric circuits, the response of the staggered periodic beam is presented by finite element simulation. The parameters μ = 4 and l1 = 0.01m were used during the COMSOL

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simulation. To support the theoretical and finite element results, experiments were performed in Fig.8. The phononic crystal beam and corresponding instrument are shown in Figs. 8(a) and 8(b). Figs. 8(c) and (d) illustrate the negative capacitance circuits and their local magnification. Moreover, the beam was suspended by two vertical lines to maintain as a horizontal shape. Under this condition, the effects of gravity and the wires connected to the piezoelectric patches were reduced. During the experiment, the Fast Fourier Transform (FFT) results excited by a single frequency point are illustrated. As shown in Fig. 8(a), both the exciter and receiver are used at the left and right sides. According to these signals, the responses to frequencies located at the pass and stop bands are quite different.

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Seven unit cells were applied during the finite element simulation and experimental investigation with two frequencies (i.e. 5689Hz and 3446Hz). These frequencies represent the characteristics of stop and pass

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bands, respectively. Different experiments were performed to determine the influences of the negative capacitor circuits. Fig. 9(a) shows that for the excitation frequency corresponding to the band gap, the elastic wave propagates with sharp attenuation. However, if we change the electric circuits to generate a pass band for the same frequency in Fig. 9(b), the elastic wave can propagate from one side to the other without the obvious attenuation. Similarly, Fig.10(a) illustrates the response corresponding to the pass band before the

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electric circuits were used, which represents the same situation as in Fig.9(a). We can see in Fig. 10(a) that the elastic wave propagated freely to the right side of the phononic crystal beam. When the active control circuits were applied in Fig. 10(b), the previous pass band became a band gap and the wave response decayed rapidly.

The experimental results with different wave frequencies are shown in Figs. 11 and 12. Acceleration excitation was performed and the results from the beginning and end of the phononic crystal beam are 9

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presented [18, 32]. The exciting and receiving signals show the band gap properties which can be tuned by the external circuits. For example, because of the external circuit, the previous band gap in Fig. 11(a) without the piezoelectric circuit can behave as the pass band in Fig. 11(b). Similarly, the phenomenon of the pass band changing into the stop band can also be found in Figs. 12(a) and (b).

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4. Conclusions In this work, elastic wave propagation in phononic crystal beam was studied. An elastic base beam with staggered periodic arrangement of piezoelectric patches was considered, and negative capacitance circuits

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were applied as the active control system. The transfer matrix method was extended to the displacement of a unit cell, and the expression of the localization factor was derived. Calculation results show that the band gap was influenced by the thickness ratio and degree of interleaving. It is worth noting that the staggered periodic structure has an obvious modulation effect on the high-frequency waves. Furthermore, the negative

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capacitance circuits have superior ability to control band gaps.

Acknowledgements

The authors wish to express gratitude for the supports provided by the National Natural Science Foundation of China (Grant Nos. 11772039 and 11532001), the Joint Sino-German Research Project (Grant No. GZ 1355) and the German Research Foundation (DFG, Grant No. ZH 15/27-1).

Appendix A.

The strain and kinetic energies of sub-cell 1 can be derived from Eqs. (7a), (7b) and (11) with the nodal

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displacement as

T 1 δ1  t   K 1 δ1  t  ,  2

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V1 =

1  δ1  t  2



T1 

 K1  = 0

l1

T

1

1

(A.1b)

T

 Eb Ab + E p Ap   ddNx1u   ddNx1u  dx    

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where

 M δ  t  ,

(A.1a)

2 T   2 Hb + H p     d 2 N1w  N d w 1 dx +  Eb Ib + E p Ap + Ep I p   0    dx 2   dx 2  4   l1

  dN  T  d 2 N   d 2 N  T  dN   1 l1 1w 1w 1u   E p Ap  H b + H p    1u   + 2  2     dx , 2 0 d d d d x x x x           10

(A.2a)

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M1  = 0 ρb Ab  N1u   N1u + N1w   N1w  dx l1

T

T

H b + H p  d N1w    H b + H p  d N1w    ρ p A p  N 1 u     N 1 u      dx   dx 2 2  dx      

l1

0

+  ρ p A p  N 1w  l1

 N 1w  d x .

T

0

Similarly, the strain and kinetic energies of sub-cells 2–4 are

1  δ2 t  2



T2 

V3 =

(A.3b)

T 1 δ3  t   K 3 δ3  t  ,  2

(A.4a)

T 1 δ 4  t   K 4 δ 4  t  ,  2

(A.5a)



K 2  = 

l2

4

(A.5b)

4

T

 Eb Ab + 2E p Ap   dNdx2u   dNdx2u  dx    



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l2 Hb + H p  +  ρ p A p  N 2 u   0 2 

0

T

(A.6a)



ρb Ab  N2u   N2u  + N2w   N2w  dx

l2

+

3

2   H + H p  + 2E I   d2N2w T  d2N2w  dx ,  Eb I b + E p Ap b p p     dx 2   dx 2  2  

M2  = 0

l2

3

 M δ  t  ,

urn

0

T

al

0

2

(A.4b)

where

+

2

 M δ  t  ,

1  δ4 t  2

T4 

l2

T

Pr e-



V4 =

(A.3a)

 M δ  t  ,

1  δ3  t  2

T3 

p ro

T 1 δ 2  t   K 2 δ 2  t  ,  2

V2 =

(A.2b)

of

+

T

T

T

T

Hb + H p  dN 2 w     d x    N 2 u   2   

 dN 2 w    dx   dx  

T

H b + H p  dN 2 w    H b + H p  dN 2 w    ρ p A p  N 2 u  +   N 2 u  +    dx   dx 2 2  dx       +  2 ρ p A p N 2 w  l2

T

0

K 3  = 0

l3

N 2 w  d x , T

 Eb Ab + E p Ap   dNdx3u   dNdx3u  dx     11

(A.6b)

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+

l3

0

2 T   2 Hb + H p     d 2 N3w  N d w 3  Eb I b + E p Ap + Ep I p   dx    dx 2   dx 2  4  

  dN  T  d 2 N   d 2 N  T  dN   1 l3 3w 3w 3u +   E p Ap  H b + H p    3u   2  2     dx , x x x x 2 0 d d d d           l3

0

T

T



T

H b + H p  dN 3 w    H b + H p  dN 3 w    ρ p Ap  N 3u     N 3u      dx   dx 2 2  dx       +  ρ p Ap N 3w  l3

T

0

l4

N 3w  dx ,

T T   d2N 4w   dN 4 u   dN 4 u  E A + E I  b b b b  2      dx   dx   dx  

M4  = 0

l4

References



 d 2 N 4 w   dx ,  2   dx  

Pr e-

K 4  = 0

p ro

+

l3



ρb Ab  N3u   N3u  + N3w   N3w  dx

of

M3  = 0

(A.7a)



ρb Ab  N4u   N4u  + N4w   N4w  dx . T

T

(A.7b)

(A.8a)

(A.8b)

[1] Chen, Y.J., Huang, Y., Lu, C.F., Chen, W.Q., 2017. A two-way unidirectional narrow-band acoustic filter realized by a graded phononic crystal. ASME Journal of Applied Mechanics 84, 091003. [2] Barnwell, E.G., Parnell, W.J., Abrahams, I.D., 2016. Antiplane elastic wave propagation in pre-stressed

al

periodic structures; tuning, band gap switching and invariance. Wave Motion 63, 98–110 [3] Bibi, A., Liu, H., Xue, J.L., Fan, Y.X., Tao, Z.Y., 2019. Manipulation of the first stop band in periodically corrugated elastic layers via different profiles. Wave Motion 88, 205–213.

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[4] Li, F.M., Xu, M.Q., Wang, Y.S., 2007. Frequency-dependent localization length of sh-wave in randomly disordered piezoelectric phononic crystals. Solid State Communications 141, 296–301. [5] Sigalas, M., Economou, E.N., 1993. Band structure of elastic waves in two dimensional systems. Solid State Communications. 86, 141–143.

[6] Shi, P., Chen, C.Q., Zou, W.N., 2015. Propagation of shear elastic and electromagnetic waves in one

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dimensional piezoelectric and piezomagnetic composites. Ultrasonics 55, 42–47. [7] Wang, G., Wen, X.S., Wen, J.H., Shao, L.H., Liu, Y.Z., 2004. Two-dimensional locally resonant phononic crystals with binary structures. Physical Review Letters 93, 154302. [8] Chen, J.J., Zhang, K.W., Gao, J., Cheng, J.C., 2006. Stopbands for lower-order Lamb waves in one-dimensional composite thin plates. Physical Review B 73, 094307. [9] Wang, Y.Z., Li, F.M., Kishimoto, K., Wang, Y.S., Huang, W.H., 2010. Wave localization in randomly 12

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disordered layered three-component phononic crystals with thermal effects. Archive of Applied Mechanics 80, 629–640. [10] Ding, L., Zhu, H.P., Yin, T., 2013. Wave propagation in a periodic elastic-piezoelectric axial-bending coupled beam. Journal of Sound and Vibration 332, 6377–6388. [11] Ma, T., Chen, T.N., Wang, X.P., Li, Y.G., Wang, P., 2014. Band structures of bilayer radial phononic

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crystal plate with crystal gliding. Journal of Applied Physics 116, 104505.

[12] Cheng, Y., Liu, X.J., Wu, D.J., 2011. Band structure of a phononic crystal plate in the form of a staggered-layer structure. Journal of Applied Physics 109, 064904.

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[13] Cheng, Y., Liu, X.J., Wu, D.J., 2011. Band structures of phononic-crystal plates in the form of a sandwich-layered structure. Journal of the Acoustical Society of America 130, 2738–2745. [14] Chen, S., Fan, Y.C., Fu, Q.H., Wu, H.J., Jin, Y.B., Zheng, J.B., Zhang, F.L., 2018. A review of tunable acoustic metamaterials. Applied Sciences-Basel 8, 1480.

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[15] Tsai, M.S., Wang, K.W., 2002. A coupled robust control/optimization approach for active-passive hybrid piezoelectric networks. Smart Materials and Structures 11, 389–395. [16] Chen, Y.Y., Hu, G.K., Huang, G.L., 2016. An adaptive metamaterial beam with hybrid shunting circuits for extremely broadband control of flexural waves. Smart Materials and Structures 25, 105036. [17] Jin, Y.B., Bonello, B., Pan, Y.D., 2014. Acoustic metamaterials with piezoelectric resonant structures. Journal of Physics D-Applied Physics 47, 245301.

[18] Cardella, D., Celli, P., Gonella, S., 2016. Manipulating waves by distilling frequencies: a tunable shunt-enabled rainbow trap. Smart Material and Structures 25, 085017.

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[19] Chen, S.B., Wen, J.H., Yu, D.L., Wang, G., Wen ,X.S., 2011. Band gap control of phononic beam with negative capacitance piezoelectric shunt. Chinese Physics B 20, 014301.

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[20] Airoldi, L., Ruzzene, M., 2011. Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos. New Journal of Physics 13, 113010. [21] Dai, L.X., Jiang, S., Lian, Z.Y., Hu, H.P., Chen, X.D., 2015. Locally resonant band gaps achieved by equal frequency shunting circuits of piezoelectric rings in a periodic circular plate. Journal of Sound and Vibration 337, 150–160.

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[22] Rao, M., Narayanan, S., 2007. Active control of wave propagation in multi-span beams using distributed piezoelectric actuators and sensors. Smart Materials and Structures 16, 2577–2594. [23] Hagood, N.W., Vonflotow, A., 1991. Damping of structural vibrations with piezoelectric materials and passive electrical networks. Journal of Sound and Vibration 146, 243-268. [24] Wang, G., Chen, S.B., 2016. Large low-frequency vibration attenuation induced by arrays of piezoelectric patches shunted with amplifier–resonator feedback circuits. Smart Materials and 13

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Structures 25, 015004. [25] Castanier, M.P., Pierre, C., 1995. Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems. Journal of Sound and Vibration 183, 493–515. [26] Bouzit, D., Pierre, C., 2000. Wave localization and conversion phenomena in multi-coupled multi-span beams. Chaos Solitons and Fractals 11, 1575–1596.

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[27] Bendiksen, O.O., 2000. Localization phenomena in structural dynamics. Chaos Solitons and Fractals 11, 1621–1660.

[28] Li, F.M., Wang, Y.S., 2005. Study on wave localization in disordered periodic layered piezoelectric

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composite structures. International Journal of Solids and Structures 42, 6457–6474.

[29] Kissel, G.J., 1991. Localization factor for multichannel disordered-systems. Physical Review A 44, 1008–1014.

[30] Xie, W.C., 1998. Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners.

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Computers and Structures, 67, 175–189.

[31] Wolf, A., Swift, J.B., Swinney, H.L. Vastano, J.A., 1985. Determining Lyapunov exponents from a time-series. Physica D 16, 285–317.

[32] Celli, P., Zhang, W.T., Gonella, S., 2018. Pathway towards programmable wave anisotropy in cellular

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urn

al

metamaterials. Physical Review Applied 9, 014014.

14

…

lb

Hp

Pr e-

lp

p ro

Piezoelectric layers

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Sub-cell 1 Sub-cell 2

Sub-cell 3 Sub-cell 4

Active control system

Active control system

l2

l3

l4

al

l1

Jo

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Fig.1. Phononic crystal beam with staggered periodic piezoelectric patches.

1

Wb

Hb

Base beam

…

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R0 Cp

C0

of

_ +

R1

p ro

R2

z

u2pb u2b

o (a)

Hb

wiR

wiL

Hp

urn

u2pt

z

Hp

al

Piezoelectric layers

Pr e-

Fig. 2. Negative capacitance circuit.

θiL

uiR

uiL θiR

x

x (b)

Fig.3. (a) Deformation of the two layered elastic-piezoelectric beam and (b) the DOFs per

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node in a beam element.

2

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Staggered periodicities

p ro

of

Localization factors

Symmetric periodicities

(Hz)

Wave frequency

Pr e-

Fig.4. Localization factors of the phononic crystal beam with the symmetric or staggered periodicities.

μ=8 μ=4

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urn

Localization factors

al

μ=2

(Hz) Wave frequency

Fig.5. Localization factors of the phononic crystal beam with staggered periodicities for different length ratios.

3

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η=1 η=3

p ro

of

Localization factors

η=5

(Hz)

Wave frequency

Pr e-

Fig.6. Localization factors of the phononic crystal beam with different values of η for the staggered periodic patches.

R2=0Ω R2=40.0Ω

al urn

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Localization factors

R2=68.2Ω

(Hz) Wave frequency

Fig.7. Localization factors of the phononic crystal beam with different values of R2 for the staggered periodic patches.

4

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Exciter

Piezoelectric

Specimen

Sensor

Sensor

Circuits

Dynamic measuring system

Power amplifier (b)

Pr e-

DC power supply

p ro

(a)

of

patches

Signal generator

R1

R2

(c)

urn

al

R0 C

Operational amplifier

(d)

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Fig.8. Experimental instruments: (a) the phononic crystal beam with the external circuit, (b) the experimental platform, (c) the whole equivalent negative capacitance circuit and (d) the equivalent negative capacitance circuit in a unit cell.

5

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(m/s2)

(m/s2) 0.8

0.5

0.4

0

0

Excitation signal

of

1

-0.8

(a)

(b)

p ro

Excitation signal

-0.4

-0.5 -1

-1.5

Pr e-

Fig.9. Responses of the phononic crystal beam with 5689Hz of the excitation signal on the left side. (a) A stop band response without the active control by negative capacitance circuits. (b) a pass band response with the active control by negative capacitance circuits.

al

(m/s2) 0.8

0.8

0.4 0

(a)

urn

Excitation signal

(m/s2)

0.4 Excitation signal

0

-0.4

-0.4 -0.8

(b)

Fig.10. Responses of the phononic crystal beam with 3446Hz of the excitation signal is on the left

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side. (a) A pass band response without the active control by negative capacitance circuits. (b) a stop band response with the active control by negative capacitance circuits.

6

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1.6

Excitation signal Receiving signal

Excitation signal Receiving signal

1.2 Response

1.2 Response

m/s2

m/s2

0.8 0.4

of

1.6

0.8 0.4

(a)

(b

p ro

0 5550

0 ) 5850 (Hz) 5550

5650

5750 Wave frequency

5650

5750

5850(Hz)

Wave frequency

Fig.11. Response with the FFT form of the phononic crystal beam when f = 5689Hz for the input and output

m/s2

10

Excitation signal

Excitation signal

6

al

4 2

urn

(a)

0 3300

3400

m/s2

8

Receiving signal

3500

Receiving signal

6

4 2

(b)

0 3600 (Hz) 3300

Wave frequency

3400

3500

3600 (Hz)

Wave frequency

Fig.12. Response with the FFT form of the phononic crystal beam when f = 3446Hz for the input and output signals: (a) without the negative capacitance circuit and (b) with the negative capacitance circuit.

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Response

8

Response

10

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signals: (a) without the negative capacitance circuit and (b) with the negative capacitance

7

Journal Pre-proof 1. A new kind of active elastic wave metamaterial beam is presented by the staggered periodic characteristic. 2. Negative capacitance circuits connected by PZT patches are employed to design the active control system

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urn

al

Pr e-

p ro

of

3. The transfer matrix is derived and the localization factor is given to show the band gap properties.

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Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the

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urn

al

Pr e-

p ro

of

manuscript entitled,