Active vibration control of CNT-reinforced composite cylindrical shells via piezoelectric patches

Accepted Manuscript Active vibration control of CNT-reinforced composite cylindrical shells via piezoelectric patches Z.G. Song, L.W. Zhang, K.M. Liew PII: DOI: Reference:

S0263-8223(16)31633-6 http://dx.doi.org/10.1016/j.compstruct.2016.09.031 COST 7757

To appear in:

Composite Structures

Received Date: Accepted Date:

23 August 2016 13 September 2016

Please cite this article as: Song, Z.G., Zhang, L.W., Liew, K.M., Active vibration control of CNT-reinforced composite cylindrical shells via piezoelectric patches, Composite Structures (2016), doi: http://dx.doi.org/10.1016/ j.compstruct.2016.09.031

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Active vibration control of CNT-reinforced composite cylindrical shells via piezoelectric patches Z.G. Song1 , L.W. Zhang2,* K.M.Liew1, 3,* 1

Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China 2 College of Information Science and Technology, Shanghai Ocean University, Shanghai 201306, China 3 City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, China

Abstract The active vibration control of carbon nanotube (CNT) reinforced functionally graded composite cylindrical shell is studied in this investigation using piezoelectric materials. Piezoelectric patches are bonded onto the outer and inner surfaces of the cylindrical shell to act as the actuator and sensor, respectively. Thermal effects are taken into account. Reddy’s high-order shear deformation theory is used in the structural modeling. The displacement fields of the piezoelectric actuator and sensor are given, according to the geometrical deformation relationship. The equation of motion of the CNT reinforced composite cylindrical shell is formulated by way of Hamilton’s principle, the solution of which is derived using the assumed mode method. In the research surrounding active vibration control, the controller is designed using velocity feedback and LQR methods. Influences of thickness on the vibration control effects of the cylindrical shell are analyzed. The control results gained by way of different control methods are compared. The active control effects of cylindrical shells with different placements of piezoelectric patches are also researched.

Keywords: CNT-reinforced cylindrical shell; functionally graded; higher-order shear deformation theory; velocity feedback; LQR; piezoelectric patches

*Corresponding authors. E-mail addresses: [email protected] (L.W. Zhang), [email protected] (K.M. Liew) 1

1 Introduction In recent years, carbon nanotubes (CNTs) have attracted the increasing attention of researchers for their excellent mechanical properties. It is well known that the addition of CNTs to a matrix can significantly improve both thermal and physical behaviors. Typically, the CNTs in a polymeric matrix are functionally graded, and are known as FG-CNT reinforced composite materials. A large number of studies have researched the buckling, vibration and bending behaviors of FG-CNT reinforced composite structures [1]. Shen and Xiang [2] investigated the large amplitude vibration behavior of nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes in thermal environments. Lin and Xiang investigated the linear [3] and nonlinear [4] free vibration characteristics of FG-CNT beams based on the first- and third-order shear deformation theories. The postbuckling behaviors of FG-CNT reinforced cylindrical shells subjected to either axial compression or lateral pressure in thermal environments were studied by Shen [5]. Using the element-free IMLS-Ritz method [6, 7], Zhang et al. [8-12] studied the flexural strength, free vibration, buckling and nonlinear bending behaviors of FG-CNT reinforced composite plates and cylindrical panels of various planforms with different boundary conditions. The first-order shear deformation theory (FSDT) was applied to the structural modeling. Zhang et al. [13] and Lei et al. [14] investigated the static, dynamic and buckling properties of FG-CNT reinforced composite plates using the element-free kp-Ritz method. They also conducted free vibration analyses of FG-CNT reinforced composite plates, considering thermal effects [15]. Based on the state-space Levy solution, Zhang et al. [16] studied the vibration and buckling behaviors of high-order FG-CNT composite plates. Vibration is typically harmful to engineering structures. Therefore, it is necessary to suppress any such structural vibration. Many studies have focused on the active vibration control of structures using piezoelectric materials. Zhang et al. [17] studied the shape control of CNT reinforced composite plate using the genetic algorithm. Song and Li [18] presented a method for the active flutter control of beams in supersonic airflow using piezoelectric materials, in which displacement and velocity feedback control methods were used. Barboni et al. [19] investigated the dynamic behaviors of flexural beam actuated by induced strain surface bonded actuators. The bending moment produced by the single actuator was evaluated using the pin-forced model. Based on the genetic algorithm, Han and Lee [20] studied the optimal placement of piezoelectric actuators and sensors for the active vibration control of composite plates. Song et al. [21] proposed a discrete LQG controller by which to study the active nonlinear flutter and aerothermal post buckling control of composite laminated panels in supersonic airflow. Shooshtari et al. [22] investigated the

2

nonlinear vibrations of viscoelastic micro cantilevers with a piezoelectric actuator layer on the top surface. The Hamilton principle and Galerkin methods were used to formulate the equation of motion. Ray and Batra [23] investigated the performance of a vertically reinforced 1-3 piezoelectric composite distributed actuator in the ACLD system bonded to functionally graded FG plates. From the literature review, it is noted that a large number of studies have studied the static, buckling and vibration behaviors of FG-CNT reinforced composite plate. However, the active vibration control of high-order CNT reinforced composite cylindrical shell has not been fully studied. In recent work, we have studied the free and forced vibration behaviors of FG-CNT reinforced plate [24]. Active vibration control has also been carried out. In this paper, the active vibration control of FG-CNT reinforced cylindrical shell will be further investigated. Reddy’s high-order shear deformation theory will be used in the structural modeling. The equation of motion of the CNT reinforced composite cylindrical shell will be formulated by way of the Hamilton’s principle. The active controller will be designed using the velocity feedback and LQR methods. The active control effects of cylindrical shell with different placements of piezoelectric patches will be researched. The effects of thickness on the active vibration control effects will also be studied.

2 Formulation of the equation of motion The FG-CNT reinforced composite cylindrical shell with piezoelectric patches is shown in Fig. 1, in which R, h and L denote radius, thickness and length of the host cylindrical shell. In this study, piezoelectric actuators are bonded onto the outer surface of the host CNT reinforced cylindrical shell, and sensors are located on the corresponding positions on the inner surface of the shell. Based on Reddy’s third-order shear deformation theory, the displacement fields of the host cylindrical shell have the following form: u = u 0 + ζφ1 −

4 3 ∂w  4 1 ∂w0    ζ  ζ  φ1 + 0  , v = 1 + v0 + ζφ 2 − 2 ζ 3  φ 2 +  , w = w0 , 2 3h ∂x  3h R ∂θ   R  

(1)

where u, v and w are the displacements at any point (x, θ, ζ) of the cylindrical shell, u0, v0 and w0 are the in-plane and transverse displacements in the neutral plane, ϕ1 and ϕ2 are the rotations of the transverse normal about the θ and x axes, and ζ is the transverse coordinate. According to the geometrical relationship of deformation [25, 26], the displacement fields of the piezoelectric actuator and sensor are given as follows: h 2  ∂w h ζ  h h ∂w0  h  1 ∂w0   , wa = w0 , u a = u 0 + φ1 +  ζ − h  0 , va = 1 + + v 0 + φ 2 − + ζ −  3 3  ∂x 3 6 R∂θ  2  Rˆ ∂θ   2 R Rˆ 

3

(2a)

h 2  ∂w h ζ  h h ∂w0  h  1 ∂w0   , ws = w0 , u s = u 0 − φ1 +  ζ + h  0 , v s = 1 − +  v0 − φ 2 + + ζ +  3 3  ∂x 3 6 R∂θ  2  Rˆ ∂θ   2 R Rˆ 

where the subscripts “a” and “s” denote the actuator and sensor, and Rˆ

(2b)

= R + h/2. The

strain-displacement relationships between the host cylindrical shell, actuator and sensor can be expressed as: ε = ε 0 + ζκ 1 + ζ 3κ 3 , ε a = ε 0 a − zκ , ε s = ε 0 s − z κ ,  2 γ = γ 0 + ζ γ 2

(3)

where  ∂φ1 ∂ 2 w0  ∂u 0 ∂φ1     +   2     ∂ x ∂ x ∂x ∂x   2  1 ∂v    w  1 ∂ w0    1 ∂φ 2   1 ∂φ 2 0 ε0 =  +  , κ1 =  + 2  , κ 3 = −c1  , 2  R ∂θ R   R ∂θ   R ∂θ R ∂θ 2   ∂v 0 + 1 ∂u 0   ∂φ 2 + 1 ∂φ1   ∂φ 2 + 1 ∂φ1 + 2 ∂ w0   ∂x R ∂θ   ∂x R ∂θ   ∂x R ∂θ R ∂x∂θ 

1 ∂w0  1 ∂w0    φ + φ +  2 R ∂θ   2 R ∂θ  γ0 =  , γ 2 = −c2  , ∂w  ∂w   φ1 + 0   φ1 + 0  ∂x  ∂x   

(4a)

  ∂u 0 h ∂φ1 2h ∂ 2 w0 + −   2 ∂x 3 ∂x 3 ∂x   2 h  ∂v0 h ∂φ 2 h  1 3  ∂ w0 w0   , ε0a =  + − + 1 +   +   ˆ 2 R  Rˆ ∂θ 3 Rˆ ∂θ 6 Rˆ  R Rˆ  ∂θ 2 R    2  ∂u 0 + 1 + h  ∂v0 + h  ∂φ1 + ∂φ 2  − h  7 + 1  ∂ w0   Rˆ ∂θ  2 R  ∂x 3  Rˆ ∂θ ∂x  6  Rˆ R  ∂x∂θ    ∂u 0 h ∂φ1 2 h ∂ 2 w0 − +   2 ∂x 3 ∂x 3 ∂x   2 h  ∂v0 h ∂φ2 h  1 3  ∂ w0 w0   , ε0s =  − + + 1 −   +   ˆ 2 R  Rˆ ∂θ 3 Rˆ ∂θ 6 Rˆ  R Rˆ  ∂θ 2 R    2  ∂u 0 + 1 − h  ∂v0 − h  ∂φ1 + ∂φ2  + h  7 + 1  ∂ w0   Rˆ ∂θ  2 R  ∂x 3  Rˆ ∂θ ∂x  6  Rˆ R  ∂x∂θ  T

 ∂ 2 w0 1  ∂ 2 w0 ∂v0  2 ∂ 2 w0 1 ∂v0  ,− κ = − ,− 2  + −  ' 2 2 ∂θ  Rˆ ∂x∂θ Rˆ ∂x  Rˆ  ∂θ  ∂x

(4b)

where c1 = 4/3h2 and c2 = 3c1 . The constitutive equation of the host cylindrical shell and piezoelectric patches are described by the following equations:  N0    M 0  = P   0 

A 0 B  0  E 0

B0 D0 F0

E 0  ε 0   N T      Q 0  A1 F0   κ 1  +  M T  ,   =  R 0  D1 H 0  κ 3   PT 

σ p = Q p ( ε p − α p ∆ T ) − e T E p , D = eε + ΞE p

4

p

D1  γ 0   , F1  γ 2 

(5a)

.

(5b)

p

In Eq. (5a), the thermal stress resultants and coefficient matrices A0, A1, B0, D0, D1, E0, F0, F1 and H0 are calculated by: h/ 2

(A 0 , B 0 , D0 , E0 , F0 , H 0 ) =

∫

ˆ (ζ )(1, ζ , ζ 2 , ζ 3 , ζ 4 , ζ 6 )dζ , (A , D , F ) = Q b 1 1 1

−h / 2

(NT , MT , PT ) =

∫

h/ 2

h/ 2

∫

ˆ (ζ )(1, ζ 2 , ζ 4 )dζ , Q s

−h / 2

ˆ (ζ )α (ζ )(1, ζ , ζ 3 )∆Tdζ , Q b 0

(6)

−h / 2

in which ∆T is the temperature change, and:  E11 1 − υ υ 12 21  υ12 E 22  ˆ Q b (ζ ) = 1 − υ12υ 21  0  

υ 21 E11 1 − υ12υ 21 E 22 1 − υ12υ 21 0

 0   G 0  , Qˆ s (ζ ) =  23   0 G12   

α 11  0  , α 0 (ζ ) = α 22  , G13   0   

(7)

where α11 and α22 are the effective thermal expansion coefficients, and E11, E22, G12, G23, G13, υ12 and υ12 are the effective Young’s moduli and Poisson’s ratio, and are calculated by [23, 27, 28]: α11 =

VCNT (ζ ) E11CNT α11CNY + Vm (ζ ) E mα m , α 22 = (1 + υ12CNT )VCNT (ζ )α 22CNY + (1 + υ m )Vm (ζ )α m −υ12α11 , VCNT (ζ ) E11CNT + Vm (ζ ) E m

E11 = η1VCNT (ζ ) E11CNT + Vm (ζ ) E m ,

η2 E22

=

VCNT (ζ ) Vm (ζ ) η 3 VCNT (ζ ) Vm (ζ ) + , = + , E22CNT Em G12 G12CNT Gm

υ12 = V ∗υ12CNT + Vmυ m , υ21 = υ12E22 E11 ,

(8)

where ECNT, GCNT and αCNT are the Young’s moduli and thermal expansion coefficient of CNTs, Em, Gm and αm are the corresponding parameters of the isotropic matrix, υCNT and υm are the Poisson’s ratios of the CNTs and matrix, V* is the total volume fraction of the CNTs, η1, η2 and η3 are the CNT efficiency parameters, and VCNT and Vm are the volume fractions of the CNT and matrix. In this study, three types (FGX, FGO and UD) [2] of CNT distributions for the FG-CNT reinforced composite cylindrical shells are considered, the volume fractions of which are expressed as follows: V CNT ( ζ ) = V

∗

 2ζ VCNT (ζ ) = 21 −  h 

VCNT (ζ ) =

4ζ h

, (UD)  ∗ V , (FGO)  

V ∗ . (FGX)

(9a) (9b)

(9c)

In Eq. (5b), σp is the stress vector, Ep = [0, 0, V0(t)/hp]T is the electric field intensity vector where V0 (t) is the external voltage [29], Dp = [0, 0, Dz ]T is the electric displacement vector and αp , Qp, e and Ξ are the thermal expansion coefficient, stiffness coefficient, piezoelectric constant and dielectric constant matrices of the piezoelectric material, respectively, given as: 5

α p  C11   α p = α p  , Q p = C12 0  0  

0 0 0 0  , e =  0 0 e31 e32 C66 

C12 C11 0

0 Ξ11 0 0 , Ξ =  0 Ξ 22  0 0 0

0 0  , Ξ33 

(10)

in which αp is the thermal expansion coefficient, e31 and e32 are the piezoelectric constants, Ξ11 , Ξ22 and Ξ33 are the dielectric constants, and C11 = E/(1–υ2), C12 = υE/(1–υ2 ) and C66 = E/[2(1+υ)] are the stiffness coefficients where E and υ are the elastic modulus and Poisson’s ratio of the piezoelectric material. The equation of motion of the CNT reinforced structural system is formulated using the following Hamilton’s principle: t

∫ [δ(T − U ) + δW ]dt = 0 ,

(11)

0

where T, U and W are the kinetic energy, potential energy and work done by the external loads, respectively; detailed expressions are given as follows:

T=

U=

1 1 ρ (ζ )(u
2 + v
2 + w
2 )dV + 2V 2

∫

1 1 (N T0 ε 0 + M T0 κ1 + P0T κ 3 + Q T0 γ 0 + R T0 γ 2 )dA + 2A 2

∫

W =−

np

∑ ∫ ρ (u


2

p

+ v
2 + w
2 )dV ,

(12a)

i =1 V +V ai si

np

∑∫

(ε p ) T σ p dV −

i =1 V +V ai si

1 2

np

∑ ∫ (D ) E dV , (12b) p

T

p

i =1 V ai

2 2 R   ∂w  ∂w ∂w   ∂w   + FTθ   + 2 FTxθ  FTx   dxdθ + k1 λ (t ) w ( x0 ,θ0 ) + k 2 λ (t ) wdA , (12c) 2 A   ∂x  ∂x R∂θ   R ∂θ  A

∫

∫

where np is the number of pairs of piezoelectric actuators and sensors, FTx, FTθ and FTxθ are the in-plane thermal loads, A and V are the surface area and volume of the cylindrical shell, Va and Vs are the volumes of piezoelectric actuator and sensor, respectively, k1 and k2 are the coefficients, λ(t) is the external load, ρp is the mass density of the piezoelectric material, and ρ(z) is the effective mass density of the FG-CNT reinforced cylindrical shell, which can be calculated by the following equation:

ρ (ζ ) = VCNT (ζ ) ρ CNT + Vm (ζ ) ρ m .

(13)

In order to formulate the equation of motion for the cylindrical shell, the assumed mode method (AMM) is applied. In the AMM, the displacement fields of the structural system are expressed as (for convenience, the subscript “0” is omitted): u ( x,θ , t ) =

v( x,θ , t ) =

m

n

i =1

j =1

∑∑ ϕ ( x,θ ) p (t ) = φ ij

m

n

i =1

j =1

∑∑ψ

ij

ij

T

( x, θ ) p ( t ) ,

( x , θ ) qij (t ) = ψ T ( x, θ )q (t ) ,

6

(14a)

(14b)

w ( x, θ , t ) =

φ x ( x, θ , t ) =

φ y ( x,θ , t ) =

m

n

i =1

j =1

m

n

∑∑ ϖ

ij

( x , θ ) rij (t ) = ϖ T ( x , θ )r (t ) ,

∑∑ϑ ( x,θ ) f (t ) = ϑ

( x, θ )f (t ) ,

(14d)

( x , θ ) g ij (t ) = ξ T ( x , θ ) g (t ) ,

(14e)

ij

i =1

j =1

m

n

i =1

j =1

∑∑ ξ

ij

(14c)

ij

T

where m and n are the mode numbers in the x and θ directions, φ, ψ, ϖ, ϑ and ξ are the column vectors of the assumed mode, and p, q, r, f and g are the generalized coordinate vectors. Then Eqs. (1)-(3), (5), (13) and (14) are substituted into Eqs. (12a)-(12c), by means of Eqs. (4), (6)-(9) and (10), and the results are substituted into Eq. (11); after conducting the variation calculations, the equation of motion of the structural system can be obtained as:



(t ) 2 + (K + K )X(t) + K V (t ) = k λ (t)F + k λ(t )F , MX ∆T a p 1 c 2 d

(15)

where X(t) = [p(t)T, q(t)T, r(t)T, f(t)T, g(t)T]T is the generalized coordinate vector, Vp(t) = [V01(t),

V02(t), … , V0np (t)]T is the external voltage vector, M and K are the modal mass and stiffness matrices, K∆T is the thermal stiffness matrix, Ka is the electromechanical coupling matrix, and Fc and Fd are the force vectors. According to the positive piezoelectric effect of piezoelectric materials, charges are produced on the piezoelectric sensor when it is subjected to external loads. The value of the charge in the ith sensor can be computed by [30]: Qsi (t ) =

∫

Asi ( ζ =ζ

ˆ X(t ) , D z dA =K si

(16)

m)

where Asi is the surface area of the ith sensor, zm is the transverse coordinate of the mid-plane of the sensor from the neutral plane of the host FG-CNT reinforced cylindrical shell and Kˆ si is the coefficient row vector. The sensing voltage Vsi (t) of the ith sensor is calculated by:

Vsi (t ) =

hp Ξ33 Asi

Qsi (t ) .

(17)

The substitution of Eq. (16) into Eq. (17) yields the following sensor voltage: Vsi (t ) =

hp Ξ 33 Asi

ˆ X(t ) = K X(t ) , K si si

(18)

where K si =

hp Ξ 33 Asi

7

ˆ . K si

(19)

Then the sensor voltage vector of the whole control system can be expressed as: Vs (t ) = K s X(t ) ,

(20)

where Ks is a matrix, and the ith row of Ks is Ksi . In order to determine the control voltage Vp(t), firstly the velocity feedback control algorithm is used because this control method is relatively robust. The sensor signal is differentiated with respect to time, which is then amplified and fed back to the actuators. Therefore, the velocity feedback control voltage signal can be obtained as: V p (t ) = G v

dVs (t )
(t ) , = G vK s X dt

(21)

where Gv is the velocity feedback control gain matrix expressed by:

Gv1 0  0 G v2 Gv =   0   0  0

0   0  ,      Gvnp 



(22)

in which Gvi (i = 1, 2, … , np) is the velocity feedback control gain for the ith actuator. Then Eq. (21) is substituted into Eq. (15) and the controlled equation of motion can be obtained as:



(t ) 2 + K G K X
(t) + (K + K )X(t) = k λ(t )F + k λ(t )F . MX a v s ∆T 1 c 2 d

(23)

It can be seen from Eq. (23) that the velocity feedback controller can provide active damping for the structural system. Therefore, it can effectively suppress the vibration amplitude of the structure to zero, which means the closed-loop control system is stable. In order to calculate the free vibration responses (k1 = 1 and k2 = 0) of the structural system, a numerical method is used which can be realized through the function ode45 or impulse of MATLAB software. As for the forced vibration responses (k1 = 0 and k2 = 1) of the CNT reinforced structural system, the solution procedure is set out below. The distributed external load in the transverse direction is expressed as:

λ(x,θ , t) = λ1 sinΩt + λ2 cosΩt ,

(24)

where λ1, λ2 and Ω are the amplitude and frequency of the dynamic load. Under the application of the external dynamic load, the steady-state solution of Eq. (23) can be written as:

X(t) = Π1 sinΩt + Π2 cosΩt , where Π1 and Π2 are the amplitude values, to be determined by the following equations:

8

(25)

K + K ∆T − Ω 2 M − K a G v K s Π1  λ1F     =   , K + K ∆T − Ω 2 MΠ 2  λ2 F  K aG vK s

(26)

from which the amplitude Π1 and Π2 of the forced vibration can be obtained. The LQR method is also used in this investigation to design the controller. Based on the equation of motion Eq. (15), a standard state-space model for controller design and numerical simulation can be formulated as: Z
= AZ + BU , Y = CZ ,

(27)

where 

I

0



0




]T , A = Z = [X, X   , B = − M −1K  , C = [K s −1 a − M (K + K ∆T ) 0 

0] , U = Vp ,

(28)

and Y is the sensor output. By using the feedback, the control voltage is defined as: U = −G l Z ,

(29)

where Gl is the feedback control gain. U minimizes a quadratic performance index that is a cost function of the system states and control input:

∫

∞

J = [Z T QZ + U T RU]dt ,

(30)

0

where Q and R are the semi-positive-definite and positive-definite weighting matrices on the states and control inputs. By solving the functional extremum problem, the feedback control gain Gl can be obtained as Gl = R–1BTP, where P satisfies the following Riccati equation: PA + AT P − PBR−1BT P + Q = 0 .

(31)

3 Numerical results and discussion Numerical examples are presented in this section to illustrate the above solution techniques. The material properties of the matrix are assumed to be ρm = 1150kg/m3, υm = 0.34, and Em = (3.52 − 0.0034T) Gpa, in which T = 300 K [2]. The Young’s moduli and thermal expansion of the CNTs are listed in Table 1. The Poisson’s ratio and mass density of the CNTs are υ12CNT = 0.175 and ρCNT = 1400 kg/m3, respectively. The CNT efficiency parameter ηi is as follows: η1 = 0.137, η2 = 1.022 and η3 = 0.715 for the case of V* = 0.12; η1 = 0.142, η2 = 1.626 and η3 = 1.138 for the case of V* = 0.17; and η1 = 0.141, η2 = 1.585 and η3 = 1.109 for the case of V* = 0.28 [2], and the shear moduli are assumed as: G13 = G12 and G23 = 1.2G12. The thickness of the cylindrical shell is h = 5 mm. The length of the shell is equal to L = 10(Rh)1/2, where R = 10h is the radius of the shell. The material properties

9

of the piezoelectric patches are: ρp = 7600 kg m–3, E = 63 × 109 N m–2, υ = 0.3, d31 = d32 = 245 × 10−12 m V–1, Ξ33 = 1.5 × 10−8 F m–1, and hp = 0.0002m. The velocity feedback control gain is set to Gvi = 0.003. Firstly, the active vibration control of thin FG-CNT reinforced cylindrical shells is studied. The radius-to-thickness ratio is R/h = 100. Fig. 2 displays a comparison of controlled and uncontrolled time histories of FGX-CNT cylindrical shell. It is evident from the figure that, when the velocity feedback controller is on, the vibration response of the cylindrical shell becomes convergent. In other words, the velocity feedback controller is effective in the vibration suppression of the thin FG-CNT reinforced cylindrical shell. This control effect can also be noted from the frequency-domain responses. Fig. 3 presents a frequency-amplitude curve for the forced vibration of FGX-CNT reinforced cylindrical shell. It is evident that both the low-frequency and high-frequency amplitudes are suppressed by the velocity feedback controller. The controlled forced vibration response of FGO-CNT reinforced cylindrical shell is computed and shown in Fig. 4, which shows that the velocity feedback controller is also effective in the active vibration control of the FGO-CNT cylindrical shell. In the above examples, the placements of piezoelectric patches are shown in Fig. 5(a). In order to determine how the placements of piezoelectric patches affect vibration control effects, four different types of placement of the piezoelectric patches are studied, schematic diagrams of which are presented in Fig. 5. For the first three cases in Fig. 5, the areas and locations in the circumferential direction of the piezoelectric patches are the same, but the number and width of the piezoelectric patches are different. In the final case, more piezoelectric patches are used in the axial direction, but the total area of piezoelectric materials is the same as for Case I. The control effects of the four different structural systems are shown in Fig. 6, where it can be seen that the control effects of structural system Case III are superior to those of Case II and Case I. That is to say that, in order to improve the control effect of the velocity feedback controller for the vibration of CNT reinforced cylindrical shell, more information on displacement and velocity in the circumferential direction is needed. On the other hand, the control effect of the structural system of Case I is almost identical to that of Case IV, which indicates that the state details in the axial direction of the cylindrical shell are not significant. Fig. 7 displays the control results for the forced vibration of the FG-CNT reinforced cylindrical shell. The same conclusion can be drawn for the active control as for the free vibration. Next, the active vibration of the thick FG-CNT reinforced cylindrical shell is carried out. The radius-to-thickness ratio is R/h = 10. Uncontrolled and controlled free and forced vibration responses are shown in Fig. 8, in which the piezoelectric patch placements of Case III in Fig. 5 are utilized. Both 10

the controlled free and forced vibration responses indicate that the velocity feedback controller is also effective in the active vibration control of thick CNT reinforced cylindrical shells. Fig. 9 displays the active vibration control effects of FGX-CNT reinforced cylindrical shell using different piezoelectric patch locations. It is evident from the figure that the control result of Case III is better than that of Case I, which is the same conclusion as the vibration control for thin shells. However, in comparing the control results of thick and thin cylindrical shell, shown in Figs. 6(b) and 8(a), it is noted that the vibration control effects of the velocity feedback controller for the thin cylindrical shell are superior to those of the thick shell. In order to suppress the vibration of thick CNT reinforced cylindrical shell, another control method, LQR, is applied. Fig. 10 shows the controlled time histories of the FGX-CNT reinforced cylindrical shells using different control methods under different temperature changes. It is obvious that the control effect of LQR is significantly superior to that of the velocity feedback controller, especially in a thermal environment. Comparisons of control results for FGO-CNT reinforced cylindrical shells using different control methods under different temperature conditions are displayed in Fig. 11; these comparisons further verify that the vibration control results of LQR are far superior to those of the velocity feedback controller.

4 Conclusions The active vibration control of FG-CNT reinforced cylindrical shell was studied using piezoelectric materials in this investigation. Piezoelectric patches were bonded onto the outer and inner surfaces of the cylindrical shell to act as the actuator and sensor, respectively. Thermal effects were considered. Reddy’s high-order shear deformation theory was used in the structural modeling. The displacement fields of the piezoelectric actuator and sensor were provided through the geometrical deformation relationship. The equation of motion of the CNT reinforced composite cylindrical shell was formulated using Hamilton’s principle. The active controller was designed by way of the velocity feedback and LQR methods. From the numerical results, the following conclusions can be drawn: (a) the velocity feedback controller is more effective in the vibration suppression of thin CNT reinforced cylindrical shells than thick ones, and both the low and high frequency amplitudes can be reduced; (b) the vibration control effects of LQR are much better than those of the velocity feedback control method, especially in a thermal environment; (c) LQR is effective in the vibration control of thick CNT reinforced cylindrical shells; and (d) additional details of displacement and

11

velocity in the circumferential direction are required in order to improve vibration control effects.

Acknowledgements The work described in this paper was fully supported by grants from the National Natural Science Foundation of China (Grant No. 11402142 and Grant No. 51378448) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9042047, CityU 11208914).

References [1] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Compos Struct 2015;120:90-97 [2] Shen HS, Xiang Y. Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput Methods Appl Mech Engrg 2012;213-216:195-205 [3] Lin F, Xiang Y. Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Appl Math Modell 2014;38:3741-3754 [4] Lin F, Xiang Y. Numerical analysis on nonlinear free vibration of carbon nanotube reinforcement composite beams. Int J Struct Stab Nyn 2014;14:1350056 [5] Shen HS. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments. Part I: axially-loaded shells. Compos Struct 2011;93:2496-2503 [6] Zhang LW, Liew KM. An improved moving least-squares Ritz method for two-dimensional elasticity problems. Appl Math Comput 2014;246:268-282 [7] Zhang LW, Li DM, Liew KM. An element-free computational framework for elastodynamic problems based on the IMLS-Ritz method. Eng Anal Bound Ele 2015;54:39-46 [8] Zhang LW, Lei ZX, Liew KM. Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Compos Struct 2015;120:189-199 [9] Zhang LW, Lei ZX, Liew KM. Vibration characteristic of moderately thick functionally graded carbon nanotube reinforced composite skew plates. Compos Struct 2015;122:172-183 [10] Zhang LW, Lei ZX, Liew KM. Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach. Compos Part B 2015;75:36-46 [11] Zhang LW, Lei ZX, Liew KM. Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLS-Ritz method. Appl Math Comput 2015;256:488-504

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[12] Zhang LW, Song ZG, Liew KM. Nonlinear bending analysis of FG-CNT reinforced composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method. Compos Struct 2015;128:165-175 [13] Zhang LW, Lei ZX, Liew KM, Yu JL. Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels. Compos Struct 2014;111:205-212 [14] Lei ZX, Liew KM, Yu JL. Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Compos Struct 2013;98:160-168 [15] Lei ZX, Liew KM, Yu JL. Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment. Compos Struct 2013;106:128-138 [16] Zhang LW, Song ZG, Liew KM. State-space levy method for vibration analysis of FG-CNT composite plates subjected to in-plane loads based on higher-order shear deformation theory. Compos Struct 2015;134:989-1003 [17] Zhang LW, Song ZG, Liew KM. Optimal shape control of CNT reinforced functionally graded composite plates using piezoelectric patches. Compos Part B 2016;85:140–149 [18] Song ZG, Li F M. Active aeroelastic flutter analysis and vibration control of supersonic beams using the piezoelectric actuator/sensor pairs. Smart Mater Struct 2011;20:055013 [19] Barboni R, Mannini A, Fantini E, Gaudenzi P. Optimal placement of PZT actuators for the control of beam dynamics. Smart Mater Struct 2000;9:110-120 [20] Han J H, Lee I. Optimal placement of piezoelectric sensors and actuators for vibration control of a composite plate using genetic algorithms. Smart Mater Struct 1999;8:257-267 [21] Song ZG, Li FM, Zhang W. Active flutter and aerothermal postbuckling control for nonlinear composite laminated panels in supersonic airflow. J Intel Mat Syst Str 2015;26(7):840-857 [22] Shooshtari A, Hoseini SM, Mahmoodi SN, Kalhori H. Analytical solution for nonlinear free vibration of viscoelastic microcantilevers covered with a piezoelectric layer. Smart Mater Struct 2012;21:075015 [23] Ray MC, Batra RC. Vertically reinforced 1-3 piezoelectric composites for active damping of functionally graded plates. AIAA J 2007;45(7):1779-1783 [24] Song ZG, Zhang LW, Liew KM. Active vibration control of CNT reinforced functionally graded plates based on a higher-order shear deformation theory. Int J Mech Sci 2015;105:90-101 [25] Zhang LW, Song ZG, Liew KM. Computation of aerothermoelastic properties and active flutter control of CNT reinforced functionally graded composite panels in supersonic airflow. Comput Methods Appl Mech Engrg 2016;300:427-441 [26] Song ZG, Li FM. Aeroelastic analysis and active flutter control of nonlinear lattice sandwich beams. 13

Nonlinear Dynam 2014;76:57-68. [27] Ray MC, Pradhan AK. The performance of vertically reinforced 1-3 piezoelectric composites in active damping of smart structures. Smart Mater Struct 2006;15:631-641 [28] Kim HW, Kim JH. Effect of piezoelectric damping layers on the dynamic stability of plate under a thrust. J Sound Vib 2005;284:597–612 [29] Gao JX, Liao WH. Vibration analysis of simply supported beams with enhanced self-sensing active constrained layer damping treatments. J Sound Vib 2005;280:329–357 [30] Reddy JN. On laminated composite plates with integrated sensors and actuators. Eng Struct 1999;21:568–593

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List of tables

Table 1 Temperature-dependent material property for CNTs Temperature

α 11CNT

α 22CNT

(×10−6/K)

(×10−6/K)

1.9445

3.4584

5.1682

6.9348

1.9643

4.5361

5.0189

6.8641

1.9644

4.6677

4.8943

E11CNT (TPa)

E 22CNT (TPa)

G12CNT (TPa)

300

5.6466

7.0800

500

5.5308

700

5.4744

(K)

15

List of figures

Piezoelectric patch

ζ θ L x

(m

R

h

Fig. 1 Schematic diagram of the cylindrical shell

16

-3

1.5

x 10

1

w (m)

0.5 0 -0.5 -1

Controlled -1.5

0

0.2

0.4

Uncontrolled 0.6

0.8

1

t (s)

Fig. 2 Controlled time history of thin CNT reinforced cylindrical shell by velocity feedback

control

17

-190 -200 -210 -220 -230 -240 -250

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

-210

w (dB)

-220 -230 -240 -250 -260 -270 600

-210 Controlled

-220

Uncontrolled

-230 -240 -250 -260 -270 1200

1400

1600

1800

2000

2200

2400

Ω (Hz)

Fig. 3 Controlled and uncontrolled frequency-amplitude curves of FGX-CNT reinforced cylindrical shell

18

-190 -200 -210 -220 -230 -240 -250

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1600

1700

1800

w (dB)

-200 -220

-240

-260 600 -220 -230

Uncontrolled

Controlled

-240 -250 -260 -270 1200

1300

1400

1500 Ω (Hz)

Fig. 4 Controlled and uncontrolled frequency-amplitude curves of FGO-CNT reinforced cylindrical shell

19

1

8 7

2

6

3 4

5

3

2

6

7

(a) Case I

15

16

1 2 3

14 13

4

12

5

11

6 10

9

8

7 3 4 5 6

11 12 13 14

(b) Case II 10

11

2

3 (d) Case IV

(c) Case III

Fig. 5 Four cases of placements of piezoelectric patches

20

14

15

6

7

(a)

-4

8

x 10 Case I

6

Case II

4 w (m)

2 0 -2 -4 -6 -8

0

0.05

0.1

0.15

0.2

t (s) -4

(b)

x 10

3

Case III Case II

2

w (m)

1 0 -1 -2 -3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t (s)

(c)

-4

8

x 10 Case I

6

Case IV

4 w (m)

2 0 -2 -4 -6 -8

0

0.05

0.1

0.15

0.2

t (s)

Fig. 6 Controlled time history of thin CNT reinforced cylindrical shell with different placements of piezoelectric patches

21

-200

w (dB)

-220 -240 -260 -280

Case I -300

0

1000

2000

3000

Case II 4000

5000

Case III 6000

7000

8000

Ω (Hz)

Fig. 7 Controlled forced vibration responses of thin CNT reinforced cylindrical shell with different placements of piezoelectric patches

22

-3

(a)

3

x 10

2

w (m)

1 0 -1 -2 -3 -4

0

0.002

0.004

0.006

0.008

0.01

t (ms)

(b)

-200 Uncontrolled

Controlled

w (dB)

-220 -240 -260 -280 -300

0

1000

2000

3000

4000

5000

6000

7000

8000

Ω (Hz)

Fig. 8 Controlled and uncontrolled (a) free and (b) forced vibration responses of thick (R/h = 10) CNT reinforced cylindrical shell

23

-3

2

x 10 Case I Case III

w (m)

1

0

-1

-2

0

0.002

0.004

0.006

0.008

0.01

t

Fig. 9 Controlled time histories of thick (R/h = 10) CNT reinforced cylindrical shell with different placements of piezoelectric patches

24

-3

(a)

1

x 10 LQR

Velocity feedback control

0.5

w (m)

0 -0.5 -1 -1.5

0

1

2

3

4

5

t (ms) -3

(b)

1.5

x 10 LQR

1

Velocity feedback control

0.5 w (m)

0 -0.5 -1 -1.5 -2 0

1

2

3

4

5

t (ms)

Fig. 10 Controlled time histories of thick (R/h = 10) FGX-CNT reinforced cylindrical shell under (a) ∆T = 0K and (b) ∆T = 100K by different control methods

25

(a)

-3

3

x 10

2

w (m)

1 0 -1 -2 Velocity feedback control -3

0

0.5

1

1.5

2 2.5 t (ms)

LQR 3

Uncontrolled 3.5

4

4.5 -3

-3

(b)

6

x 10 Uncontrolled

Velocity feedback control

LQR

4

w (m)

2 0 -2 -4 -6 0

1

2

3

4

5

6

t (ms)

Fig. 11 Controlled time histories of thick (R/h = 10) FGO-CNT reinforced cylindrical shell under (a) ∆T = 0K and (b) ∆T = 200K by different control methods

26