Optimal locations of discontinuous piezoelectric laminated cylindrical shell with point supported elastic boundary conditions for vibration control

Journal Pre-proofs Optimal locations of discontinuous piezoelectric laminated cylindrical shell with point supported elastic boundary conditions for vibration control Chaofeng Li, Peiyong Li, Zixuan Zhang, Bangchun Wen PII: DOI: Reference:

S0263-8223(19)32607-8 https://doi.org/10.1016/j.compstruct.2019.111575 COST 111575

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Composite Structures

Received Date: Revised Date: Accepted Date:

10 July 2019 9 October 2019 21 October 2019

Please cite this article as: Li, C., Li, P., Zhang, Z., Wen, B., Optimal locations of discontinuous piezoelectric laminated cylindrical shell with point supported elastic boundary conditions for vibration control, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111575

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Optimal locations of discontinuous piezoelectric laminated cylindrical shell with point supported elastic boundary conditions for vibration control Chaofeng Lia,b,*, Peiyong Lia, Zixuan Zhanga, Bangchun Wena a. School of Mechanical Engineering and Automation, Northeastern University, 110819, Shenyang, China b. Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, Shenyang, Liaoning, China, 110819. Abstract: In this paper, the vibration control of discontinuous piezoelectric laminated shell with point supported elastic boundary conditions are investigated, and the location of piezoelectric layer are optimized. The point supported boundary condition are simulated by using artificial springs. The position with the piezoelectric layer are considered to be a laminated shell, and the position without the piezoelectric layer are regarded as a thin-walled cylindrical shell. The strain and curvature expressions of the cylindrical shell are obtained by the first-order shear shell theory, and the Chebyshev polynomial is used as the admissible displacement functions. Then, the differential equation of coupling motion of piezoelectric laminated cylindrical shells is established by using Lagrange equation, and the negative velocity feedback control is used as the control strategy. The Newmark method is used to obtain the response curves. The accuracy of the model are verified by comparing with the ANSYS results. For better vibration reduction, the optimal locations of the piezoelectric layer are obtained by using the Multi-Objective Particle Swarm Optimization algorithm based on the crowding distance. Finally, the optimization results and the vibration reduction of the piezoelectric layer are verified by analyzing the radial displacement response of the cylindrical shell.

*

Corresponding author. Tel.: +86 13514215459 E-mail address: [email protected] (C. Li).

Keyword: Smart materials; Point supported elastic boundary condition; Active vibration control; Position optimization. 1. Introduction In the aerospace, petrochemical, mechanical engineering, energy and other fields, the cylindrical shell is an important component of many machines, and its vibration will seriously affect the overall performance of the machine. The vibration control of cylindrical shell has always been one of the most important topics that scholars pay attention to. In recent years, with the development of piezoelectric materials, the piezoelectric intelligent structure can transform the energy through the positive and negative piezoelectric effects. It can effectively assume the functions of the inductor and the actuator, and can make the intelligent structure integrate structure, induction, drive and control. It has broad application prospects. In the past few decades, the piezoelectric laminated cylindrical shell has been the research hotspot and difficulty of scholars. For commonly used shell theory and modeling methods, Leissa, Kapuria, Qatu et al. made some detailed introductions [1-3]. The common methods for modeling piezoelectric laminated cylindrical shells are as follow: the finite element method (FEM), the Hamilton’s principle, the Rayleigh–Ritz method, the differential quadrature method (DQM). For FEM, Correia et al [4] introduced a new mixed finite element model of the frusta conical shell, in which the piezoelectric actuators and sensors were considered. The finite element model combined with the high-order shear shell theory was used to analyze the influence of the amplification factor on the time domain response. Balamurugan and Narayanan [5] used a finite element model of nine-node shell element to study the vibration control of piezolaminated plate and curvilinear shell, and the inner and outer layers of the plate and shell are PZT layers, and the middle layer is the composite laminated in this model. The influence of the length and position of piezoelectric layer on the vibration response were studied. For the Hamilton’s principle, Sheng and Wang [6, 7] studied the force vibration and nonlinear vibration of simply supported Sensor-FGM-Actuator cylindrical shell based on the first order shear shell theory, beam function and Hamilton’s principle, and the effects of piezoelectric and temperature on the shell were considered in the model. Arefi [8] used the first shear nonlinear shell theory to research the force vibration of graded function shell with PZT layer,

and the influence of L/R and H/R on frequency were given. The Rayleigh–Ritz method is also a common method for studying piezoelectric laminated cylindrical shells. Kwak et al.[9] conducted the dynamic modelling, vibration control and experience for cylindrical shell equipped with piezoelectric layer by using the Donnell’s shell theory and Rayleigh–Ritz method. Kwak presented a multiple input and output positive position feedback controller based on the block-inverse theory. Ramirez et al. [10] presented a new approximate approach to solve the free vibration of magneto-electro-elastic laminated plates by using the discrete layer approach and the Ritz method, and the nature characteristic of shell with different piezoelectric layer were compared. In addition to the commonly used FEM, Rayleigh–Ritz method and the Hamilton’s principle, some scholars also used the differential quadrature methods. Such as, Zhang et al.[11] used the differential quadrature method (DQM) to analyze the free vibration of multilayered piezoelectric composite plates. Alibeigloo and Kani [12] investigated the 3D free vibration of laminated hybrid shell by using the state space method and DQM. Banks et al.[13]used the Fourier polynomials, the Lagrange equation and the POD reduced basis method to establish a model of reduced order thin-wall shell. Compared with the above several modeling methods, the combination method of polynomial and Lagrange equation is less used, and the method has the advantages of simple use, less freedom and the like. At present, the research on piezoelectric cylindrical shells mostly focuses on the classical boundaries [7, 12, 14, 15], but the working environment of the shell structure is mostly complicated in practical engineering. It is also the focus of scholars to establish elastic boundaries and discontinuous elastic boundaries. Wang et al. [16] investigated the graded function carbon nanotube reinforced composite spherical panels and shells with elastic boundary condition. Song et al. [17, 18] established an laminated cylindrical shell model with arbitrary boundary conditions by using a series of artificial spring, and the frequency of composite laminated cylindrical shell are studied. Sofiyev et al.[19, 20] studied the vibration of laminated and functionally graded (FG) orthotropic cylindrical shell with different boundary conditions. Qin et al. [21] studied the rotating shell-plate vibration by using Sanders shell theory and Mindlin plate theory, Chebyshev polynomials and Rayleigh-Ritz method. Qin et al. [22] presented an analysis approach for rotating functionally graded CNT reinforced composite cylindrical shells. Safaei et al. [23] analyzed the influence of excitation frequency on the vibration of nanocomposite sandwich panels under thermo-mechanical loading. For

the elastic boundary condition of the whole circumference constraint, the connection such as bolting and local welding appearing in the engineering project are not simulated. In recent years, some scholars have focused on the complex boundary such as discontinuous elastic boundary conditions and nonlinear boundary conditions [24-28]. Tang et al. [26, 27] presented a bolted joined cylindrical shell model to analyze the vibration. Li et al. [28] established a composite laminated cylindrical shell with point supported and arcs supported boundary condition to study the geometrically nonlinear vibration. The establishment of a discontinuous elastic boundary model is closer to the actual engineering situation and has greater engineering significance. For the parameter optimization of discontinuous piezoelectric layers, more optimization criteria have been proposed. Zhang et al. [29] investigated the active vibration control of piezoelectric laminated cylindrical shell with clamped boundary condition, and the number of layers of PVDF on the inhibitory effect were verified. Sohn et al. [30] used the Donnell’s shell theory and Lagrange equation to establish a simply supported hull structure. the genetic algorithm was used to optimize the location of piezoelectric layer, and the effectiveness of the piezoelectric layer was verified by experiments. Zhai et al. [31] established a piezoelectric curved shell model by using FEM and a performance index based on the locations and thicknesses of piezoelectric layer. The simulated annealing algorithm was used in the paper. Hu and Li [32] presented a novel technique to optimize the position and angle of piezoelectric actuators. The linear quadratic energy index was used as optimization criterion and Genetic Algorithm was used to optimize. From the preparation process and cost of the piezoelectric material (PVDF), it is very difficult to completely cover the piezoelectric layer on the surface of the cylindrical shell in engineering. Most of PVDF are pasted or embedded in a small part of the surface of the shell. This paper established a differential equation of discontinuous piezoelectric laminated cylindrical shells with point supported elastic boundary conditions based on the first-order shear shell theory, Chebyshev polynomial and Lagrange equation. The different boundary condition are simulated by using artificial spring in five directions. The accuracy of the model are validated by comparing with the result of ANSYS. To find the optimal placement of piezoelectric sensors and actuators, the optimal performance index are established based on controllability Gramian matrix. The Multi-Objective Particle Swarm Optimization algorithm based on the crowding distance is used to find the optimal solution.

Nomenclature

A

Stretching stiffness matrix

Ax

System matrix

B

Coupling stiffness matrix

B

Control matrix

Bf

Disturbance matrix

CR , C A

Damping matrix

Cx

Output matrix

D

Bending stiffness matrix

Da , Ds

Electric displacements

Df

Location matrix of external excitation

Eb , Ea , Es

Yong’s modulus of the based layer, actuator layer and sensor layer

E

Electric fields

F

Point pulse excitation

GF

Amplification factors

J

Secondary performance index of the system

K qq , K spr , K q , K

Generalized stiffness matrix, spring stiffness matrix, electrical mechanical coupled stiffness, electrical stiffness matrix

L

Length of the shell

M qq

Generalized mass matrix

M x , M  , M x

Moments of the in-plane stresses

N

Number of terms for circumferential wave

N x , N  , N x

Force of the in-plane stresses

NA

Number of supported point

NT

Number of terms for Chevbyshev polynomials

Qa , Qs , Q b

Plane stresses-strain matrix

Q , Qx

Transverse shear force

P

Applied electrical charge for piezoelectric layers

R

Radius of the shell

T

Kinetic energy

Tm*  

Admissible displacement functions

U , U spr

Strain energy, potential energy

U , V , W , x , 

Mode vector satisfying the boundary condition

Wc , Q

Gramian matrix

amn , bmn , cmn , d mn , emn , f mn , g mn

Unknown corresponding coefficients

e

Piezoelectric constants

f0

Amplitude of pulse excitation

h , ha , hs

Thickness of the shell of the based layer, actuator layer and sensor layer

ku , kv , k w , k x , k

Stiffness of axial, circumferential, radial, torsional spring

kc

Shear correction factor

n

Circumferential wave number

q

Global coordinates

u , v, w, x , 

Displacement in the x, θ, z directions and rotations of the transverse normal respect to the x and θ axes

a ,s

Distribution of electric potential along thickness coordinate



Mass normalized mode shape matrix

 x ,  ,  x ,   z ,  xz

Strains of the shell



Dielectric constants

 x ,  ,  x

Curvature of the shell

b

Poisson’s ratios of based layer



Modal coordinates

ψ

Values of electric potential of actuator and sensor layer



Non-dimensional axial coordinate

( s ,  s ) , ( s,  s )

Starting and ending coordinates of the sth piezoelectric layer

b , b , b

Mass density of the based layer, actuator layer and sensor layer

 x ,   , xy ,  y , xz

Stresses of the shell



Natural angular frequency of the shell

2. Theoretical formulations 2.1. Description of laminated cylindrical shells mode As shown in Fig. 1, the piezoelectric laminated cylindrical shell with radius R and length L is established. The thickness of the base layer is denoted as h, and the thickness of the piezoelectric sensor and actuator layers embedded on inner and outer surfaces are denoted as hs and ha. An orthogonal coordinate system (x, θ, z) is considered along the axial, circumferential and radial directions of the middle surface of the shell, respectively. The displacements of a point on the middle surface are denoted as u, v and w in the x, y, z directions, and x and  are the rotations of the transverse normal respect to the x and θ axes. To simplify the calculation, the non-dimensional coordinate ξ is defined as ξ=x/L. There are locally distributed piezoelectric sensor and actuator layers on the shell, and the sensor layer and actuator layer are the symmetric distributed about the middle surfaces. The number of piezoelectric layers is defined as NP. The starting and ending coordinates of the sth piezoelectric layer are ( s ,  s ) and ( s,  s ) , respectively. In Fig. 1(b), for the point elastic supported laminated cylindrical shells, the stiffness of artificial spring restraining u, v, w, x and  of supported points are denoted by ku0 , kv0 , kw0 , k x0 , k0 and ku1 , kv1 , kw1 , k 1x , k1 (N/m). The number of evenly distributed point is defined as NA.

(a)

(b) Base Layer

Actuator Layer

kv ,

k , kw, ku ,

k x ,

s

O

R Sensor Layer

z, w x, u

(c)

sth piezoelectric layer ξ=ξs

z,w

(d)

ξ=ξ's

θ ,v

ha

ξ=1 θs

ξ=0

hs

θ's

R h

L sth piezoelectric layer

Fig.1 Schematic diagram of discontinuous piezoelectric laminated cylindrical shell with point supported elastic boundary conditions: (a) diagram of piezoelectric laminated cylindrical shell; (b) shell with point supported boundary condition; (c) partial cross-sectional view of the shell in the axial and radial

directions; (d) partial cross-sectional view of the shell in the circumferential and radial directions; 2.2. Stress-strain relations and stress resultants Based on the first shear deformation theory, the displacement of a point  ,   in the piezoelectric laminated shell are expressed as

u  , , z, t  =u0  , , t  +zx  , , t   v  , , z, t  =v0  , , t  +z  , , t    w  , , z, t  =w0  , , t  where u0 , v0 , and w0 are the displacements of the middle surface in the axial, circumferential and radial directions, respectively, x and  are the rotations of the transverse normal respect to the x and θ axes. The strains of a point  ,   in the shell are given by

(1)

x  x  x                   x     x   z   x          θz    θz  0      0     xz   xz (0)

(2)

where the subscript (0) is used to represent the middle surface of the shell. The strain component in the middle surface and curvature are defined as u0   x 0  L   1 v0 w0     0  R  R  u  v 1  0  0   x  0  R  L   1 w0 v0    z  0    R  R   1 w0  xz  0  x  L  

(3)

x   x = L   1     R    1  1 x  x  L   R  

(4)

The stress-strain relationship for the base layer in the laminated shell is given according to the general Hooke’s law

 xb  Q b  b   11    Q12b  b    x    b     z    b    xz 

Q12b Q22b Q66b b 44

Q

b   x    b       xb   b     z  b  Q55   b   xz 

(5)

where the plane stress-strain relation in the material coordinates Q are defined by

Q11b  Q22b 

Eb E Eb b , Q12b  Q21  b b2 , Q66b  , Q44b  Q66b , Q55b  Q66b 2 1  b 1  b 2 1  b 

where Eb , b is the material Yong’s modulus and Poisson’s ratio of the base layer, respectively. The stress-strain relationship for the piezoelectric sensor and actuator layer in the laminated shell is given by

(6)

 xi   Q i  i   11e    Q12i e  i   x    0  i   0   z    i   0  xz 

Q12i e i Q22 e 0 0 0

 Dxi  0  i   D   0  i   ei  Dz   31e

0 0 i e32 e

0 0 Q66i e 0 0

0 0 0

0 0 0 i Q44 e 0

0 i e24 e 0

i i  0   x   0 0 e31 e   i   i  Ei  0 e32e  x 0     0  i  i    0  x   0 0 0   E     i 0   i z   0 e24 0   Ezi  e   Q55i e   i  e15i e 0 0   xz  ,  xi   i  i e15i e     11i e 0 0   Ex        0   xi    0  22i e 0   Ei  0   i z   0 0  33i e   Ezi       i   xz 

 i =a, s 

(7)

where Di and Ei is the electric displacements and electric fields of the piezoelectric layer, respectively,

Qi , ei ,  i is the elastic constants, piezoelectric constants and dielectric constants, respectively. The subscript (i) is used to represent the sensor layer (i=s) and actuator layer (i=a). Due to the poling direction of the piezoelectric layer is coincident with the thickness direction, the relation of the electric fields and the electric potential can be given by

Ex i  

 i  x, , z , t  L

, E i  

 i  x, , z , t  R

, Ez i  

 i  x, , z , t  z

,

 i =a, s 

(8)

The potential of the piezoelectric layer due to elastic deformation can be expressed as

 2  ha 2    x, , z , t    za     a  x, , t   2    2  h    s  x, , z , t    zs 2   s   s  x, , t   2    a

where a  x, , t  is the distribution of the electric potential of the actuator layer along the thickness coordinate, za is the coordinate relative to the middle surface of the actuator layer, za  z  (h  ha ) 2 ;

 s  x, , t  is the distribution of the sensor layer, za is the coordinate of the sensor layer, zs  z  (h  hs ) 2 . In this model, the position at which the piezoelectric layer is considered as a laminated shell, and the position at which the piezoelectric layer is not disposed is regarded as a thin-walled cylindrical shell. In the location of base-piezoelectric layers, the matrix form of the force and moment resultant relations to the strains in the middle surface and curvature changes are defined in Eq.(10), in which

(9)

N xE , NE , N xE , M xE , M E , M xE is the piezoelectric resultants of the piezoelectric layer.  N x   A11     N   A12  N x   0    M x   B11  M    B12     M x  0 Q   A44    kc  Q  0  x

A12

0

B11

A22 0

0 A66

B12 0

B12

0

D11

B22 0

0

D12 0

B66

E   x (0)   N x    E     (0)   N    B66   x (0)   N xE     0   x   M xE  0     M E      D66   x   M E      x 

0 0

B12 B22 0 D12 D22 0

(10)

0    z (0)  QE     A55   xz (0)  QxE 

where Aij , Bij , Dij (i, j  1, 2,6) are the stretching, coupling and bending stiffness coefficients of laminated cylindrical shell, respectively. It is noted that all the Bij terms become zero for shells laminated symmetrically with respect to their middle surfaces. kc is the shear correction factor, generally taken as 5/6.

Aij = 

h / 2  ha

Bij = 

h / 2  ha

Dij = 

h / 2  ha

h/2

h/2

h/2

Qija dz  

h/2

h/2

Qija zdz  

Qijb dz  

h/2

h/2

Qija z 2 dz  

h/2

 h / 2  hs

Qijb zdz  

h/2

h/2

Qijs dz

h/2

 h / 2  hs

Qijb z 2 dz  

Qijs zdz

h/2

 h / 2  hs

(11)

Qijs z 2 dz

The force and moment resultant produced by the piezoelectric layer can be given by 2ea h / 2  ha z  dz  2es  h / 2 z  dz  a a 31e  h / 2  h s s  31e h / 2  s N    h / 2  ha h/ 2 s  E   2e a za a dz  2e32 32 e h / 2 e  h / 2  h zs s dz  N     s   N E  0   x   E =   h / 2  ha h/ 2 a s M  x  2e31e  zza a dz  2e31e  zzs s dz  h/2  h / 2  hs M E    h/ 2     a h / 2  ha  s 2 d e zz z  2 e zz  d z  a a 32 e  h / 2  h s s  M xE   32 e h / 2  s   0  E x

2 2   zs 2   hs 2 2    h / 2  ha za   ha 2   h/ 2 a s a s   d e z e   dz 24 e h / 2 24 e  h / 2  h  E s  Rz R z    Q         E   Qx   a h / 2  ha 2 h/ 2    2  a 2 s s 2  z   hs 2   dz dz  e15e  e15e h / 2  za   ha 2      h / 2  hs  s L   L    

In the location of base layers, the matrix form of the force and moment resultant relations to the strains in the middle surface and curvature changes are defined as

(12)

 N x   A    11  N    A12    N x   0     M x   B11     B  M    12    0  M x   Q   A 44     kc   0 Qx 

A12 A

0 0 A

22

0 B12 B

66

0 0 B66

22

0

B11 B 12

B12 B

0 D11 D

0 D12 D

0

0

12

  x (0)      (0)      66   x (0)  0   x   0      D 66   x 

0 0 B

22

22

(13)

0    z (0)     A55   xz (0) 

where Aij , Bij , D ij (i, j  1, 2, 6) is the stretching, coupling and bending stiffness coefficients of thin-wall cylindrical shell. h/2 Aij =  Qijb dz h/2

h/2 Bij =  Qijb zdz

(14)

h/2

h/2 D ij =  Qijb z 2 dz h/2

2.3. Energy equations The kinetic energy of the piezoelectric laminated shell can be given as

T=

LR NP  s  s  2 u  v 2  w 2  I 0  2 ux  v I1  x2  2 I 2 d d         s s 2 s 1









LR NP r r  2   u  v2  w 2  I 0  2 ux  v I 1  x2  2 I 2 d d  2 r 1 r r 









(15)

where NP is the number of non-piezoelectric layers.  and  are the starting coordinates of non-piezoelectric position in axial and circumferential directions.  and  are the ending coordinates. h / 2  ha

 I 0 , I1 , I 2   h / 2

a 1, z , z 2  dz  

h/2

h/2

 b 1, z , z 2  dz  

h/2

 h / 2  hs

s 1, z , z 2  dz

 I 0 , I 1 , I 2     b 1, z , z 2  dz   h/2 h/2

where  is the density of the kth layer The strain energy of the piezoelectric laminated shell can be given as

U=



h / 2  ha LR NP  s  s T N x  x  NT   N xT  x  M xT x  M T  M xT  x  QTz   z  QxzT  xz   Da T Ea dz    h/2 2 s 1  s  s



h/2

 h / 2  hs



Ds T Es dz d d 



Q xzT  xz d d

LR NP r r  T N x  x  N T   N xT  x  M xT x  M T  M xT  x  QTz   z      r r 2 r 1



(16)

(17) As shown in Fig.1, the artificial springs are used to simulate different boundary condition. The energy produced by the springs is considered in the Lagrange equation. For the piezoelectric laminated cylindrical shell with point supported elastic boundary condition, the potential energy point U springs can be given as

U spr 



2 2 2 2 1 NA 0 ku , u (0,  t  , t )   kv0, v(0,  t  , t )  + kw0 ,  w(0,  t  , t )   k x0, x (0,  t  , t )   2  =1

 k0,  (0,  t  , t ) 

2





2 2 2 2 1 NA   ku1, u (1,  t  , t )   kv1, v(1,  t  , t )  + kw1 ,  w(1,  t  , t )   k 1x , x (1,  t  , t )  2  =1

 k1 ,  (1,  t  , t ) 

2

(18)



2.4. Admissible displacement functions When analyzing the vibration characteristics of the cylindrical shell, it is very important to choose the appropriate displacement admissible functions for the high accuracy and convergency of the calculation results. In the research [33], Qin studied the free vibration of thin-walled cylindrical shells under different displacement admissible functions (the modified Fourier series, the Orthogonal polynomials, and the Chebyshev polynomials), and compared the accuracy, convergency and computational efficiency. The comparison results show that the three sets of admissible functions have good precision. The convergence rate and the computational efficiency of the Orthogonal polynomial and Chebyshev polynomial are higher than the modified Fourier series, and the Chebyshev polynomial has the highest computational efficiency. Therefore, the Chebyshev polynomials is used in this paper. The displacement of shell and the distribution of the electric potential can be written as[34, 35] NT N  u t amnTm*  e jt cos n =U T qu   , = ,     m 1 n 1  NT N  *  j t T  v  , , t  =  bmnTm  e sin n  V qv m  1 n  1  NT N  *  j t T  w  , , t  =  cmnTm  e cos n  W qw m 1 n 1  NT N  *  j t  x  , , t    d mnTm  e cos n  x qx m n  1  1  NT N  *  j t    , , t     emnTm  e sin n   q m 1 n 1  NT N   s  , , t    f mnTm*  e jt cos n   s q s m 1 n 1  NT N   a  , , t    g mnTm*  e jt cos n   a q a  m 1 n 1

(19)

where amn , bmn , cmn , d mn , emn , f mn , g mn are the unknown corresponding coefficients.  is the natural frequency of shell. NT is the number of polynomial terms in the calculation. n is the number of circumferential wave. U( ) , V ( ) , W ( ) x ( ) and  ( ) are the mode vector satisfying boundary condition. Tm*   is the displacement component of the Chebyshev polynomials. Tm*    Tm  2  1 .

Tm () is the first kind of the Chebyshev polynomial, and the recurrence expressions are written as[28] T0    1, T1     , Tm 1    2 Tm    Tm 1   ,  m  2 

(20)

2.5. Equations of motion The admissible displacement functions from Eq.(19) is substituted into Eqs (15), (17) and (18).They can be written the quadratic form, and the expressions are given by 1 1 1 1 1 1 1 T = quT M uu qu  qvT M vv qv  qwT M ww qw + qTx M xx qx + qT M   q + quT M ux qx + qvT M v q 2 2 2 2 2 2 2

(21) 1 1 1 1 1 1 1 U   quT K uu qu  quT K uv qv  quT K uw qw  quT K ux qx  quT K u q  qvT K vv qv  qvT K vw qw 2 2 2 2 2 2 2 1 1 T  x x 1 T w 1 T wx 1 T ww 1 T v 1 T vx  qv K qx  qv K q  qw K qw  qw K qx  qw K q  qx K qx  qTx K x q 2 2 2 2 2 2 2 1 T u s 1 T v a 1 T v s 1 T w a 1 T   1 T u a  q K q  qu K q a  qu K q s  qv K q a  qv K q s  qw K q a 2 2 2 2 2 2 1 T w s 1 T x  a 1 T x s 1 T   a 1 T  s 1  qw K q s  qx K q a  qx K q s  q K q a  q K q s  qT a K  a  a q a 2 2 2 2 2 2 1 T a s 1 T  s s  q a K q s  q s K q s 2 2

(22) 1 1 1 1 1  x x   uu vv ww U spr  quT K spr qu  qvT K spr qv  qwT K spr qw  qTx K spr qx  qT K spr q 2 2 2 2 2

(23)

The Lagrange equation with the dissipative function D is defined as

d  T  T  U   U spr  D    F t    q q dt  q  q

(24)

The differential equations of coupling motion of laminated cylindrical shells with the elastic displacements q and voltage vector ψ can be re-written as [31, 36]

M qq q  C R q   K qq  K spr  q  K q ψ =F K qT q  K ψ =P where M qq , K qq and K spr are the generalized mass matrix, stiffness matrix and spring stiffness matrix.

(25)

K q is the electrical mechanical coupled stiffness. K is the electrical stiffness matrix. The dimensions of the stiffness and mass matrix Mqq, Kqq, Kspr and CR are 5(NT+1)×5(NT+1), the dimensions of Kqψ is 5(NT+1)×2(NT+1), and the dimensions of K

ψψ

is 2(NT+1)×2(NT+1). P is

applied electrical charge for piezoelectric layers, and F is the radial excitation on the position (x, θ).

q  qu

qv

qw

qx

q  , ψ   a  s  T

T

(26)

In order to facilitate the decoupling of the equation, the equation of motion is arranged in the following form

M qq q  C R q +  K qq  K spr  q  K qS ψ S  K qA ψ A =F A K qS T q  K ψ A =P

K

S T q

(27)

q  K ψ =0 S

S

where ψA and ψS are the values of the actuator and sensor layer electric potential. The superscript A and S are used to represent the sensor layer and actuator layer. The differential equation of motion is decoupled, in which the converse piezoelectric effect in the sensor layer is negligible, so it can be written as [37] S 1 M qq q  C R q   K q q  K spr + K qS K K qS T  q =F  K qA ψ A S 1 ψ S  K K qS T q

(28)

The control is consider as a negative velocity feedback control. The input potential of the actuator layer can be obtained by S 1 ψ A  GGcψ S  GF ψ S  GF K K qS T q

(29)

where Gc is the constant gain of the charge amplifier, G is the gain of the amplifier. The differential equation of motion can be written as follows S 1 M qq q +  C A  C R  q   K q q  K spr  K qS K K qS T  q =F

(30)

where CR is the Rayleigh damping matrix. CA is the damping matrix caused by the control potential. S 1 C A =  GF K qA K K qS T

(31)

CR = M qq   K q q

(32)

where α and β is the Rayleigh damping parameters.

 2

 =2 



2 2



1   1 1  2 2  2  2  ,  =2  22  11  2  1  2  1   2 1 

(33)

where 1 , 2 are the First order and second order natural frequencies of laminated cylindrical shell, and 1 ,  2 are the damp coefficient. The differential equation is solved by Newmark method in this paper. 3. Model Validation and comparison In order to verify the accuracy of the proposed model in Section 2, the finite element software ANSYS was used to establish a piezoelectric laminated cylindrical shell model for comparison verification. As shown in Fig. 2, the base layer of laminated cylindrical shell uses SOLID 45 element, and the piezoelectric layer uses SOLID 5 element. The boundary condition is simulated by using COMBIN14 element. The structural parameters of the shell are shown in Table 1. The material parameters of the base layer and the piezoelectric layer are shown in Table 2. The piezoelectric layer is made of PVDF material. In Table 3, the natural frequencies of different piezoelectric layer sizes are compared, and the piezoelectric layer ranges are   [0,1] and   [0, 2] ,

  [0,0.5] and

  [0, 2] ,   [0,1] and   [0,  , respectively. Then, the different number of terms for Chevbyshev polynomials NT are calculated. According to the results, as NT greater than 5, the natural frequency of piezoelectric laminated cylindrical shell gradually stabilized, therefore, NT is set as 5 in next section. In the Table3, compared with the results obtained by ANSYS, the error is small, and the model proposed in this paper is considered to be accurate. Table 1 Geometric parameters of cylindrical shell hs

ha

h

L

R

0.001

0.001

0.002

0.1

0.1

Units: hi(m), L(m), R(m). Table 2 Material properties of base layers and piezoelectric layers

Base layer

PVDF

PVDF

Eb

ρb

μ

200

7850

0.26

Q11

Q12

Q22

Q44

Q55

Q66

ρe

238.24

3.98

23.6

2.15

4.4

6.43

1800

e31

e32

e24

e15

ζ11

ζ22

ζ33

-0.13

-0.14

-0.01

-0.01

0.885

0.885

10.6

Units: Ei(GPa), ρ(kg·m-3), eij(C·m-2), Qij(GPa), ζij(10-11F·m-1), ρ(kg·m-3).

Fig.2 Finite element model of the piezoelectric laminated cylindrical shell with point supported boundary condition in ANSYS Table 3 The verification of natural frequency (HZ) of piezoelectric laminated cylindrical shell Piezoelectric layer distribution

   0,1,    0, 2

   0,0.5,    0, 2

   0,1,    0, 

NT=4

Matlab NT=5

NT=6

(1, 2)

154.4

154.3

(1, 3)

436.4

436.1

Mode

ANSYS

Error (%)

154.3

157

1.7

436.1

437

0.2

(1, 4)

835.9

835.3

835.3

844

1.0

(1, 5)

1350.2

1349.3

1349.3

1346

0.2

(1, 6)

1977.6

1976.5

1976.5

1970

0.3

(1, 2)

142.8

142.7

142.7

146

2.2

(1, 3)

402.3

401.9

401.8

405

0.8

(1, 4)

766.8

765.9

765.6

774

1.0

(1, 5)

1230.6

1229.0

1228.3

1234

0.4

(1, 6)

1789.5

1786.8

1785.4

1792

0.3

(1, 2)

143.4

143.2

143.2

141

1.5

(1, 3)

405.5

404.8

404.8

400

1.2

(1, 4)

776.8

775.8

775.8

757

2.4

(1, 5)

1254.9

1253.4

1253.4

1233

1.6

(1, 6)

1838.5

1836.5

1836.5

1784

2.9

4. Position optimization of piezoelectric layer In order to optimize the position of the piezoelectric sensor and the actuator, the differential equation of motion is organized into S 1 M qq q  C R q   K q q  K spr + K qS K K qS T  q =F  K qA ψ A

In the equation of motion, there are multiple orders of frequency and mode shapes. In the process of the actual vibration control, all modes are not displayed. So it is necessary to establish a

(34)

reduced-order dynamic model based on the modal characteristics. The natural frequency  and the mass normalized mode shape matrix 

of the shell can be obtained by solving the differential

equation of motion(30). The global coordinates q are converted to modal coordinates  as follow q = 

(35)

Taking Eq.(35) into (34) and multiply both sides of the equation by  T at the same time. Then, the differential equation can be expressed as S 1  T M qq   T C R   T  K q q  K spr + K qS K K qS T  = T F   T K qA ψ A

(36)

Because the mode matrix is orthogonal, the following formulas can be obtained

 T M qq =  T C R  diag  2 ii   Z 

T

K

qq

S 1

 K spr + K K K S q

(37) S T q

  diag     2

i

Here, the state space variables X   ,  , X   ,  are introduced, and the state space of the shell are written as

X  Ax X  B f f + B

(38)

Y = Cx X where Ax is the system matrix. B is the control matrix . B f is the disturbance matrix. Cx is the output matrix. D f is the location matrix of external excitation.  0 Ax     F  Df f

0    0  I  , B f   T  , B   T A  , C x    Z   D f    K q 

0

(39)

Taking the retained energy as the optimization goal, the secondary performance index of the system is established as follow. J  J1  J 2 

1  1  T T   t    t dt   Y  t  Y  t dt  0 0 2 2

where J1, J2 are the vibration energy and control energy, respectively. In order to achieve the effective vibration control of the piezoelectric laminated shell under the minimum control force, it can be accomplished by the following two conditions

(40)

tf

minmize

J1     t    t dt

maxmize

J 2   Y  t  Y  t dt

T

0

tf

(41)

T

0

4.1 Optimal position of actuators The Pontryagin’s minimum principle are used to solve the J1 in the Eq.(41), and the control energy can be given as [38] J1  e Ax t f x  0   x  t f   Wc  t f T



1

e Ax t f x  0   x  t f    

(42)

where Wc (t f ) is the controllability Gramian matrix at the tf, and it can be expressed as

Wc  t f  =  e Ax t B B T e Ax t dt tf

T

(43)

0

With the increasing of Gramian matrix, the control energy is decreasing. When time approaches infinity, Wc (t f ) is related with the Gramian matrix of stable system Wc () .

Wc  t f  =Wc     e Ax t f Wc    e Ax

T

tf

(44)

where Wc () can be obtained by solving the Lyapunov stability equation. AxWc     Wc    Ax T =  B B T

(45)

The performance index can be defined as the form of Eq.(46), in which the first n order modes of the controlled system are considered.[39]

PI ac 

 2 n ac  2 2 n ac  i n  i   iac   i 1  i 1 1

which i is the eigenvalues of the controllability Gramian matrix, and   i  is the standard deviation of i .The larger value of PI in the optimized design of the actuator, the better the control performance of the system. In the optimal design of actuator, the higher PI value represent the higher control performance of the system. 4.2 Optimal position of sensors The initial displacement of the system is X (0)  X 0 , then the displacement at time t can be expressed as X (t )  e Ax t X 0 . The vibration energy of the system is expressed as

(46)

tf

J 2 max   Y  t  Y  t dt =X 0T QX 0 T

0

(47)

where Q is the controllability Gramian matrix. tf

Q =  e Ax t C x T C x e Ax t dt T

0

(48)

For the stable system, it satisfies the Lyapunov stability equation.

Ax T Q  QAx =  C x T C x

(49)

As the performance index of actuator, the following expression can be used as the performance index of the sensor.

PI se 

 2 n se  2 n se  i 2n  i    ise   i 1  i 1 1

where i is the eigenvalues of Q. 4.3 Implementation of MOPSO-CD for optimization process The Multi-Objective Particle Swarm Optimization (MOPSO-CD) algorithm based on the crowding distance is an optimization algorithm to simulate the foraging of birds and fish. In this method, the crowding distance calculation method is introduced into the Particle Swarm Optimization algorithm with global optimal selection and the deletion method of external archives of non-dominant solutions[40]. Compared with Genetic algorithms, the PSO uses simple and no complicated parameters, and these parameters seriously affect the optimization results, which are mostly guided by experience. Moreover, the information sharing mechanism for finding the optimal solution is different with the genetic algorithm. There is an information sharing mechanism between the particles in the POS, which can make the algorithm find the optimal solution faster. In this paper, the MOPSO-CD algorithm is used in the position optimization of piezoelectric layer.

(50)

1

Axial coordinate

N 

 s s

Ns

 s  s

N 1

N1

0

1 1

Ns

N 2

1 1

0

N

Circumferential coordinate

2

Fig.3 the distribution of the piezoelectric layer In order to achieve the optimal solution of the piezoelectric layer position, the optimization index (PIse, PIac) in the Sect. 4.1 and 4.2 are taken as the objective function, and the starting to coordinate (ξs, θs) of the piezoelectric layer are used as the optimization variable. During the optimization process, the position of the piezoelectric layer is continuously changing. In Fig. 3, the distribution of the piezoelectric layer is shown, in which 2 / N and 1 / N are the size of piezoelectric layer in the circumferential and axial directions of shell. As shown in Fig. 4, the flowchart of MOPSO-CD is given. Before the process, some parameter are defined, such as the number of particles m, the range of optimization variable, the maximum number of iterations T, learning factor c and inertia weight w . (1) At the first iteration, the position and speed of each particle are randomly initialize. (2) After solving the performance index ( PI se , PI ac ) of each particle, the solution of each particle are saved to an external archive A, and the crowded distance of each particle are calculated to select the global optimal solution. (3) The new position and speed of each particle are updated by the next formulas. Vi  t  1  wVi  t   c1r1  Pi  t   X i  t    c2 r2  Gi  t   X i  t   X i  t  1  Vi  t  1  X i  t 

where r1 and r2 are random numbers in [0,1]. w  wmax 

( wmax  wmin )(i  1) m 1

where i represent the ith particle. (4) After solving the new performance index ( PI se , PI ac ) of each new particle, the external

archive A are updated and the global optimal solution are updated. (5) Determine whether the number of iterations is greater than the maximum number of iterations. If it  T , end for; if it < T , repeat the second step (2). 开始 Start Defining parameters

Initialize the position X i  t  and speed Vi  t  of each particle randomly, local optimal position Pi  t  =X i  t  global optimal position Gi  t  =X i  t 

Objective function calculation

Save the solution to archive A, and calculate the 将粒子群的解存入外部档案A中，并计算A中 crowded distance of each particle in A, and select 每个粒子的拥挤距离，选择全局最优位置 the global optimal position.

Put X i  t   i  t  , i  t  into T, Uε , Uspr 得到运动微分方程: Get the differential   of motionSA A Mequation qq q  C R q  Kq =F  K q ψ

M qq q  C R q  Kq =F  K qSA ψ A

Update the 更新粒子的位置和速度 position and speed of particles No

Vi  t  1  wVi  t   c1r1  Pi  t   X i  t    c2 r2  Gi  t   X i  t   X i  t  1  Vi  t  1  X i  t 

得到状态方程: Get the state space

XX AAx xXXBBf f ff++BB YY ==CCx xXX

Update archive A, update the local optimal 更新外部档案A，更新粒子i局部最优位置Pi(t) position Pi  t 

Solve the performance 求解性能指标 PI se,PI ac se index 1/PI ,1/PI ac Reach the maximum number of iterations iterations＞T Yes End

Fig.4 the flowchart of MOPSO-CD 5. Application and discussion In order to verify the optimization method proposed in this paper, this section optimizes the position of piezoelectric layer of laminated shell with clamped supported in one edge while the other edge freed. In this case, the number of piezoelectric layer is set as 1, and the axial and circumferential size of piezoelectric layer is chosen as N =5 , N =20 . The parameters of MOPSO-CD are as follows: m=20, T=200, c1=2, c2=2, wmax=0.9, wmin=0.4. The excitation position is (0.1, 0), the response position is (0.1, π/16), and the excitation amplitude f is 50N. The parameters of piezoelectric laminated cylindrical shells are shown in Table 1 and Table 2. One end is free

boundary( kv0 =kw0 =k x0 =k0  0 N/m (N/rad) ),while the other end is clamped point supported ( ku1 =kv1 =kw1 =k 1x =k1  1012 N/m(N/rad) ), and the number of supported point NA is 16. The frequency of first seven order modes (n=1~7) of cylindrical shell without piezoelectric layer are as follow: 4741，2937，1964，1535，1555，1903，2450，3131. There are five frequencies less than 3000 HZ. Therefore, the control mode is considered as 2~6 in the optimization process. In Fig. 5, the Pareto frontier of MOPSO-CD for the optimal position of piezoelectric layer of laminated shell with free-clamped supported condition are given. It is clear showed that the present method can great converge to the true Pareto frontier. In this, the performance index of actuator and sensors are interacting. For free-clamped supported piezoelectric laminated cylindrical shell, when the piezoelectric layer is on the free end (ξ=0), the performance index of sensor layer is better. When the piezoelectric layer is on the clamped end (ξ=1), the performance index of actuator layer is better. Then, the evolution of the global optimal position and optimal performance indicators are given in Fig. 6. As the number of iterations increases, the position and performance of the piezoelectric layer tend to be stable, and eventually remain unchanged as the number of iterations increases. By optimizing results, the optimal position (ξ, θ) of the piezoelectric layer distribution of the free-clamped supported piezoelectric laminated cylindrical shell can be obtained, as (0, 3.8) and (0.8, 4.2).

Fig.5 Pareto frontier of MOPSO-CD

Fig.6 For free-clamped supported piezoelectric laminated cylindrical shell, optimization process of global optimal position and optimal performance index 5.1 Pulse excitation This section will analyze the influence of the piezoelectric layer on vibration response when the cylindrical shell is subjected to pulse excitation. In Eq. (30), F is the radial pulse excitation on the position (x, θ) as follow F  f  x,    x  x0      0  t  t0  f  W f  x,     0 0 t  t0 0 The time domain of the piezoelectric laminated cylindrical shell subjected to pulse excitation are

plotted in Fig. 7, in which the piezoelectric layer at optimized locations and at random locations are considered. In Fig. 7(a), the time domain response of piezoelectric layer at optimized locations of the free end (0, 3.8) are calculated, and the amplification factors GF are 0, 0.05 and 0.1 respectively. It can be seen that when the amplification factor is not 0, the piezoelectric layer can reduce the amplitude of the time domain response of the cylindrical shell, and the response amplitude has a significant reduction with the increase of amplification factor. Then, the time domain response of piezoelectric layer at optimized locations of the clamped end (0.8, 4.2) is given in Fig. 7(b). As GF increases, the response amplitude also has an obvious decrease. Next, the time domain response of piezoelectric layer at random locations are given in Fig. 7(b). As GF increases, the response amplitude has a small decrease. By comparing Fig. 7, it can be seen that the piezoelectric layer at the optimized position has a greater inhibitory effect on the vibration response amplitude of the cylindrical shell, and when the piezoelectric layer is at the free end have a better inhibitory effect.

(a)

(b)

(c)

Fig.7 The effect of GF for time domain response subjected to pulse excitation: (a) optimal position 1(0, 3.8); (b) optimal position 2(0.8, 4.2); (c) random position (0.05, 4) 5.2 Harmonic excitation This section will analyze the influence of the piezoelectric layer on vibration response when the cylindrical shell is subjected to harmonic excitation. In Eq. (30), F is the radial harmonic excitation on the position (x, θ) as follow

F  f  x,    x  x0      0  f  x,   fcosωt  W0 The time domain of the piezoelectric laminated cylindrical shell subjected to harmonic excitation are showed in Fig. 8. In Fig. 8(a), the time domain response of piezoelectric layer at optimized locations of the free end (0, 3.8) are calculated, and the amplification factors GF are 0, 0.05 and 0.1 respectively; in Fig. 8(b), the time domain response of piezoelectric layer at optimized locations of the clamped end (0.8, 4.2) is given; in Fig. 8(c), the time domain response of piezoelectric layer at random locations is shown. In Fig. 7, the piezoelectric layer have obvious influence on the time domain response. With the increase of GF, the radial displacement of piezoelectric laminated shell decrease. By comparing Fig. 7, it can be seen that the piezoelectric layer at the optimized position has a greater inhibitory effect on the vibration response amplitude, and when the piezoelectric layer is at the free end have a better inhibitory effect. This result is mutually verified with the conclusion drawn by Section 5.1. (a)

(b)

(c)

Fig.8 The effect of GF for time domain response subjected to harmonic excitation: (a) optimal position 1(0, 3.8); (b) optimal position 2(0.8, 4.2); (c) random position (0.05, 4) 5.3 Discussion The result of optimization can be explained through the stress and the model shape. At the free end, the amplitude of model shape is the largest, and the amplitude is the smallest at the clamped end. In addition, the largest stress of cylindrical shell is at the clamped end. Therefore, the performance index of sensor layer is better when the piezoelectric layer is on the free end, and the performance index of actuator layer is better when the piezoelectric layer is on the clamped end. For the phenomenon in Figs. 7 and 10, it can be explained by Eqs. (29), (30) and (31). As GF increase (the constant gain of the charge amplifier or the gain of the amplifier G increase in Eq. (29)), the damping matrix caused by the control potential increase in Eq. (31). Then, when the mass matrix, stiffness matrix, Rayleigh damping matrix and the excitation matrix keep unchanged in the Eq. (30), the damping of system increase and the radial displacement decrease. S 1 ψ A  GGcψ S  GF ψ S  GF K K qS T q

(29)

S 1 C A =  GF K qA K K qS T

(30)

S 1 M qq q +  C A  C R  q   K q q  K spr  K qS K K qS T  q =F

(31)

6. Conclusion In this paper, a piezoelectric laminated cylindrical shell with point supports elastic boundary condition model is presented, in which the artificial springs are used to simulate different boundary conditions, and the position optimization of the piezoelectric layer and the vibration control of shell are studied. The conclusions can be obtained as

(1) As shown in the time domain response, the piezoelectric layers have great effect on the vibration control. (2) The optimization results show that when the piezoelectric layer is at the free end, the performance index of actuator is the largest; the performance index of sensor is great at the clamped end. (3) Through comparing the time domain response, the piezoelectric layer at the optimal position have great inhibitory effect than the piezoelectric layer at other positions. Especially, the piezoelectric layer have better inhibitory effect at the free end than at the clamped end. Generally speaking, the modeling method of piezoelectric structure presented in this paper can well handle the vibration control and optimization of piezoelectric structure. Acknowledgments This project is supported by the National Natural Science Foundation of China (No.51575093) and the Fundamental Research Funds for the Central Universities (Nos. N180313008 and N170308028). Conflict of interests The authors declare that there is no conflict of interests regarding the publication of this article.

Appendix A Expressions for mass matrix The generalized mass matrix of piezoelectric laminated shell is expressed as

M qq

 uu M  0    0  1 ux T  M 2  0

where

0

0

1 ux M 2

M vv

0

0

0

M ww

0

0

0

M  x x

1 v T M 2

0

0

   1 v  M  2  0   0      M  0

(A.52)

NP

M uu = LR   s 1

 s

 s



s

NP

M uu = LR   s 1

 s

s

NP

M ux = 2LR   s 1

M vv = LR   s 1

s

NP

M v  2LR   s 1

M ww = LR   M xx  LR   NP

 s

s

M    LR   s 1

 s

s

NP

s 1

 s

 s

s



s

s

 s



s

 s



s

s

UU T d d  I 0

r

r

NP

r

Ux T d d  I1  2 LR   





r

r 1

r

r





r

r 1

 s

r 

VV T d d  I 0  LR   



UU T d d  I 0

r



r

NP

 s



r

r 1

s



r

NP

 s



r



UU T d d  I 0  LR   

s

 s

s

NP

s 1

 s

 s

r 1

 s



s

NP

s

r

NP

UU T d d  I 0  LR   

VV T d d  I 0 r

U T d d  I1  2 LR    r

WW T d d  I 0  LR    r 1 NP

r r

xx T d d  I 2  LR    r 1 NP

r r

  T d d  I 2  LR    r 1





r

r 1

NP

r

Ux T d d  I 1

r

r

NP

r

r





U T d d  I 1

WW T d d  I 0

r

r



r



r

r





r

xx T d d  I 2   T d d  I 2

Appendix B Expressions for stiffness matrix The generalized stiffness matrix of piezoelectric laminated shell is expressed as  uu K   1 K uv T 2  1 K qq =  K uwT 2 1  K ux T 2  1 u T  K  2

where

1 uv K 2 K vv 1 vwT K 2 1 vx T K 2 1 v T K 2

1 uw K 2 1 vw K 2 K ww 1 wx T K 2 1 w T K 2

1 ux K 2 1 vx K 2 1 wx K 2 K  x x 1 x T K 2

1 u K 2 1 v K 2 1 w K 2 1 x K 2 K  

             

(B.54)

 A11 U U T A66 U U T   2  2 d d  s  s R    s 1  L   NP       A U U T A66 U U T  r r  LR       112  2 d d r r R    r 1  L   NP     U V T 2 A66 U V T  s  2A s K uv =LR     12  d d s s L R    s 1  LR   NP

K uu =LR  

 s

 s

 2 A12 U V T 2 A66 U V T    d d  r r LR    r 1  LR   NP     NP      2 A U T  U T  s s  2A r r K uw =LR     12 W d d  LR       12 W d d s s r r s 1 r 1   LR   LR   T T NP     2 B U x  U x s s  2B K ux =LR     211  266 d d s s R    s 1  L   NP

r

 LR   

r

 2 B11 U x T 2 B66 U x T   2  d d 2  r r R    r 1  L   NP     U  T 2 B66 U  T  s s  2B K u =LR     12  d d s s LR    s 1  LR   NP

r

 LR   

r

 2 B12   r r r 1  LR NP     V s s  A K vv =LR     222 s s R  s 1  NP

r

 LR   

r

U  T 2 B66 U  T   d d   LR    A V V T V T kc A44  2 VV T  662 R L   

 d d 

 A 22 V V T kc A 44 A66 V V T  T   VV  d d 2  r r R2 L2    r 1  R   NP     V T 2kc A44 W T  s  2A s K vw =LR     222 W  V d d s s R2   s 1  R  NP

r

 LR   

r

 2 A 22 V T 2kc A 44 W T  W  V  d d 2  r r R2   r 1  R  NP     V x T 2 B66 V x T  s s  2B K vx =LR     12  d d s s LR    s 1  LR   NP

r

 LR   

r

 2 B12 V x T 2 B66 V x T    d d  r r LR    r 1  LR   NP      V  T 2 B66 V  T 2kc A44 s s  2B K v =LR     222 V  T d d  2  s s L   R s 1  R    NP

r

 LR   

r

 2 B 22 V  2 r r r 1  R  NP     W s s  k A K ww =LR     c 244 s s s 1  R  NP

r

 LR   

NP

r

 LR    r 1

r

r





r





r

  T 2 B66 V  T 2kc A 44  2  V  T d d  L   R   W T kc A55 W W T A22  2  2 WW T d d  L   R 

 kc A 44 W W T kc A55 W W T A 22   2  2 WW T d d  2 L   R  R   

 2kc A55 W T 2 B12 x T  x  W  d d s s LR   s 1  L  NP      2k A W T 2 B12 x T  r r  LR       c 55 x  W d d  r LR   r 1 r  L  NP

K wx =LR  

 s

 s



 2kc A44 W T 2 B22  T    2 W  d d  s  s R   s 1  R  NP      2k A W T 2 B 22  T  r r  LR       c 44   2 W d d  r   R r 1 r  R  NP

K w =LR  

 s

 s

 D11 x x T D66 x x T   2  kc A55xx T d d  2  s  s R   s 1  L    T T NP        D x x x x r r  D  LR       11  662  kc A55xx T d d 2 r r R   r 1  L    NP

K xx =LR  

 s

 s

 2 D12 x  T 2 D66 x  T    d d  s  s L R    s 1  LR   NP        T 2 D   T  r r  2D x  x  66   LR       12 d d  r       LR LR   r 1 r   T NP      D   T D   s s K   =LR     222  kc A44  T  66 s s L2   s 1  R   NP

K x =LR  

 s

 s

 d d  T T NP         D    r r  D  LR       222  kc A 44  T  66 d d r r L2    r 1  R  

The electrical mechanical coupled stiffness and the electrical stiffness matrix of piezoelectric laminated shell is expressed as

K q

 0   1 K v a 2  1 =  K w a 2 1  K x  a 2 1    K a 2

K a a K =  0

   1 v s  K  2  1 w s  K  2 1 x  s   K 2  1   s  K   2

0

 0 s s  K 

where  e24e a ha 3  a T  V  d d 2   s    s 1 s  3R s 3 NP      e  s T  s s 24 e hs v s K =LR     V d d   s  3R 2   s 1 s  NP

K v a =LR  

 s

 s

(B.55)

(B.56)

a 3 T e24e a ha 3 W  a T   e15e ha W  a   d d  2  s  s   3R 2    s 1  3L T NP      e s h 3 e24e s hs 3 W  s T  s  s 15e s W  s K w s =LR      d d 2 s s 3R 2    s 1  3L   NP

 s

 s

NP

 s

 s

K w a =LR  

K x a =LR  

s

s 1



s

a 3  e15e a ha 3  a T e31e ha x T   a d d     x 3L   3L  

s 3 s 3  s T e31e hs x T   e15e hs  s d d     x  s  s  3L 3L  s 1   a a 3 3 NP      e  a T   T e24e ha s  32 e ha s a   K  a =LR     d d s s   3R s 1  3R  s 3 s 3 NP      e  s T   T e24e hs s  32 e hs s  s s   K =LR     d d s s   3R s 1  3R  a 5 T NP        22e a ha 5  a  a T  33e a ha 3 s s   11e ha  a  a  a a T d d K  a a =LR       2 2 s s 30 R   3 s 1   30 L   NP

 s

 s

NP

 s

 s

K x s =LR  

K  s s =LR  

s

s 1



s

   11e s hs 5  s  s T  22e s hs 5  s  s T  33e s hs 3  s s T d d    2 2 3 30 R     30 L  

Appendix C Expressions for spring stiffness matrix The spring stiffness matrix of piezoelectric laminated shell is expressed as

K spr

uu  K spr  0  0  0 0 

0 vv K spr 0 0 0

0 0 ww K spr 0 0

 0  0   0  0     K spr 

0 0 0  x x K spr 0

(C.58)

where

     k V  0, V  0,   k V 1, V 1,      k W  0, W  0,   k' W 1, W 1,      k   0,   0,   k  1,  1,      k   0,   0,   k  1,  1,  

uu   ku,o pU  0, p  U T  0, p   ku1, pU  0, p  U T  0, p  K spr NA

p 1

vv K spr ww K spr

 x x K spr   K spr

NA

p 1

NA

p 1

NA

p 1 NA

p 1

0 v, p

T

p

0 w, p

0 x, p 0

,p

1 v, p

p

T

p

p

1 w, p

p

1 x, p

p

,p

T

x

p

x



p



T

p

T

1

p

T

p

p

T

x



p

p

x



p

T

p

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