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7/27/2019 Active Control of Acoustic Radiation From Laminated Cylindrical Shells Integrated With a Piezoelectric Layer

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Active control of acoustic radiation from laminated cylindrical shells integrated with a

piezoelectric layer

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2013 Smart Mater. Struct. 22 065003

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 22 (2013) 065003 (16pp) doi:10.1088/0964-1726/22/6/065003

Active control of acoustic radiation from

laminated cylindrical shells integratedwith a piezoelectric layer

Xiongtao Cao, Lei Shi, Xusheng Zhang and Guohe Jiang

Laboratory of Marine Power Cabins, Shanghai Maritime University, Haigang Avenue 1550,

Peoples Republic of China

E-mail: [email protected]

Received 19 October 2012, in final form 4 March 2013

Published 25 April 2013Online at stacks.iop.org/SMS/22/065003

Abstract

Active control of sound radiation from piezoelectric laminated cylindrical shells is

theoretically investigated in the wavenumber domain. The governing equations of the smart

cylindrical shells are derived by using first-order shear deformation theory. The smart layer is

divided into lots of actuator patches, each of which is coated with two very thin electrodes at

its inner and outer surfaces. Proportional derivative negative feedback control is applied to the

actuator patches and the stiffness of the controlled layer is derived in the wavenumber domain.

The equivalent driving forces and moments generated by the piezoelectric layer can produce

distinct sound radiation. Large actuator patches cause strong wavenumber conversion and

fluctuation of the far-field sound pressure, and do not make any contribution to sound

reduction. Nevertheless, suitable small actuator patches induce weak wavenumber conversion

and play an important role in the suppression of vibration and acoustic power. The derivative

gain of the active control can effectively suppress sound radiation from smart cylindrical

shells. The effects of small proportional gain on the sound field can be neglected, but large

proportional gain has a great impact on the acoustic radiation of cylindrical shells. The

influence of different piezoelectric materials on the acoustic power is described in the

numerical results.

(Some figures may appear in colour only in the online journal)

1. Introduction

The discovery of piezoelectric materials has initiated an era

in which electroacoustic transducers are playing an important

role in distinguishing the threat of submarines and warships.

A great many piezoelectric patches are closely distributed in

planar, cylindrical, or spherical sonar arrays mounted on the

submarines and ships. An external control voltage excites the

piezoelectric patches and causes the sonar arrays to radiate

sound into the far field, which is the fundamental principle

of active sonar. Therefore, laminated cylindrical shells coated

with a piezoelectric layer which is divided into lots of

piezoelectric patches can be taken as simple sound projectors

when an applied control voltage drives the smart patches. If anexternal load is applied on the piezoelectric cylindrical shells,

the active control method can be used to suppress acoustic

radiation from the smart cylindrical shells by applying a

feedback control voltage on the actuator patches.

Since piezoelectric plates and shells have wide applica-

tions in the vibration control of structures, vibrational analysis

of smart structures has been extensively investigated. Tzou

et al [1] studied the spatial actuation and control effectiveness

of distributed segmented actuator patches affixed to a

laminated cylindrical shell. A feedback control voltage is

applied to densely distributed piezoelectric actuator patches

in the radial direction. Zhang et al explored [2, 3] active

vibration control of a cylindrical shell partially covered by

a laminated piezoelectric actuator which was composed of

several piezoelectric patches and bonding layers. The controlforces of the actuator can be significantly enhanced by

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Smart Mater. Struct. 22 (2013) 065003 X Cao et al

increasing the piezoelectric layer number while the driving

voltage is kept unchanged. An analytical method for vibration

optimal control of simply supported thin laminated shells

integrated with two piezoelectric layers has been presented

by Ray [4]. The electric potential variable in the actuator

is assumed to be a linear variation across its thickness

and is taken as an independent variable. Yuan et al [5]derived the first-order differential governing equation for

thin circular cylindrical shells partially covered with active

constrained layer damping (ACLD) in the axial direction and

analyzed the effects of the circumferential dominant modal

control strategy on the damping of ACLD circular cylindrical

shells by using a single-point feedback control method.

In [15], classical thin shell theory was used to describe

the equations of motion for cylindrical shells, restricting

the applicable scope of these models, especially for the

vibration of laminated cylindrical shells in the medium and

high frequency range. The previous studies [13, 5] make

use of the uncoupled classical shell theory and the electric

potential is not established as an independent equation. Dube

et al [6] examined modal sensitivity factors and the controlled

damping ratio for simply supported cross-plied composite

cylindrical shells with segmented distributed piezoelectric

sensor and actuator layers on the basis of first-order shear

deformation theory. Nevertheless, the electric potential is still

not treated as an independent variable.

For thin piezoelectric layers, the electric potential

approximately satisfies the linear potential theory. However,

the electric potential in a moderately thick piezoelectric layer

follows a nearly quadratic variation across its wall [79]

and the electric potential variables should be considered

as independent field variables. Wang et al [7] proposeda quadratic function to describe the electric potential

distribution across the thickness of the piezoelectric layers

in piezoelectric coupled circular plates and verified it by

using a finite element method. Recently, Larbi and Deu [9]

also showed using a 3D state-space solution that an electric

potential with a quadratic evolution in the radial direction

acted on the piezoelectric cylindrical shell. An analytical

model of free vibration for piezoelectric coupled moderately

thick circular plates has been given by Liu et al [10] based on

first-order shear deformation plate theory in the case where the

electrodes on the piezoelectric layers are short circuited. The

sinusoidal function employed in their work has a similar shapeto that of the quadratic function adopted by Wang et al [7].

Sheng and Wang [11, 12] investigated the dynamic response

of functionally graded cylindrical shells with surface-bonded

piezoelectric layers by means of first-order shear deformation

shell theory. Similarly, a layerwise quadratic distribution of

the electric potential was considered. Free vibration and the

dynamic response of simply supported functionally graded

piezoelectric cylindrical panels impacted by time-dependent

blast pulses were analytically explored by Bodaghi and

Shakeri [13]. The governing equations of the smart structure

based on first-order shear deformation theory and a quadratic

distribution of the electric potential were derived.

Studies on vibration of piezoelectric laminated platesand shells have been done by several researchers on

the basis of higher-order shear deformation theory. Torres

and Mendonca [14] derived the equations of motion

in terms of generalized displacements for rectangular

piezoelectric laminated plates using Levinsons higher-order

shear deformation theory. A layerwise discretization of

the electric potential is used in the piezoelectric lamina.

Hosseini-Hashemi et al [15] examined free vibration ofannular moderately thick plates integrated with two surface-

bond piezoelectric layers on the basis of Levinsons plate

theory [16], which serves as a compromise between Mindlins

plate theory and Reddys plate theory. Levinsons plate theory

neglects the higher-order moments and higher-order shear

forces shown in the variational formulation of Reddys plate

theory [17]. By using Reddys third-order shear deformation

plate theory, they [18] also provided analytical solutions for

free vibration of thick circular or annular piezoelectric plates.

The distribution of the electric potential through the thickness

of the piezoelectric layer is assumed as a sinusoidal function.

An efficient coupled zigzag theory has been developed by

Kapuria and Achary for hybrid piezoelectric plates under

thermoelectromechanical loading [19]. The thermal and

potential fields through the sublayers are assumed to be

piecewise linear. The transverse displacement of the plate is

described by a combination of a global uniform term across

the thickness and local piecewise quadratic variations across

the sublayers. A new improved third-order theory has been

presented for hybrid piezoelectric angle-plied plates under

thermal loading by Kumari et al [20]. The authors made

comparisons with the results from the third-order coupled

zigzag theory presented in [19]. An improved third-order

zigzag theory and its smeared counterpart were recently

developed by Nath and Kapuria [21] for the vibration ofpiezoelectric laminated cylindrical shells. The zigzag theory

takes the layerwise variation of in-plane displacements into

account and satisfies the conditions of continual transverse

shear stresses at the layer interfaces and free tractions at the

inner and outer surfaces of the shell.

Reported studies on acoustic radiation from piezoelectric

plates and shells with active control are not enough. Laplante

et al [22] analyzed the vibrational and acoustic performance

of submerged cylindrical shells with ACLD patches using a

finite element method. An underwater hydrophone generates

the sensor output voltage and proportional derivative feedback

control is applied to the ACLD patches. Wang andVaicaitis [23] investigated active control of noise transmission

into double wall composite cylindrical shells under random

pressure and point loadings by means of pairs of spatially

discrete piezoelectric actuators. Velocity feedback and sound

pressure rate feedback control procedures were developed.

An approach to the design of a fluid-loaded lightweight

structure with surface-mounted piezoelectric actuators and

sensors capable of actively reducing vibration and sound

radiation was presented by Ringwelski and Gabbert [24].

The finite element method was employed to model the shell

structure and fluid domains which are partially or totally

bounded by the structure. A boundary element method was

used to characterize the unbounded acoustic pressure field.Testa et al [25] studied tonal noise control in an aircraft

2


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cabin by using a fuselage skin embedded with piezoelectric

actuators. An optimal control approach was applied to the

actuation of the piezoelectric patches in order to suppress

cabin noise. Ray et al [26] developed a finite element model

of active structural acoustic control for a thin homogeneous

isotropic plate coupled with an acoustic cavity by using a

patch of ACLD affixed to the plate. The ACLD treatmentperforms excellently in controlling sound radiation from

the vibrating flexible wall of the cavity. Li and Zhao [27]

explored active control of vibration and acoustic radiation

for a fluid-loaded laminated plate with piezoelectric layers.

Control of acoustic radiation from a baffled fluid-loaded

laminated plate with ACLD was formulated on the basis of the

finite element method. Active control of sound radiation from

a composite plate with anisotropic polyvinylidene fluoride

(PVDF) actuators has been researched by Kim and Yoon [28]

based on the coupled finite element and boundary element

methods. Active control of the sound field is performed

through minimization of the radiated sound power. Recently,

Li [29] examined a model of active modal control for the

vibroacoustic response of plates with piezoelectric actuators

and sensors. The active modal damping is added to the

coupled system via velocity negative feedback. Kim et al

[30] demonstrated active control of sound radiation from

plates in subsonic flow using a piezoelectric sensor and

actuator. Vibration and sound radiation were analyzed using

the finite element and the boundary element methods. A

linear quadratic Gaussian controller was designed to estimate

the controlled modes from sensor output and minimize the

performance index.

A literature survey reveals that the issue of active

control of acoustic radiation from piezoelectric cylindricalshells by means of piezoelectric actuators has not been

addressed in depth and is only concerned with active control

of sound radiation based on the finite element method and

assumed modal method [2230]. In the present study, active

control of acoustic radiation from laminated cylindrical shells

integrated with a piezoelectric actuator layer is presented

in the wavenumber domain using the Fourier transform. In

order to achieve distributed control of the smart cylindrical

shells, the piezoelectric actuator layer must be segmented

into a great many cells with electrodes [31, 32]. Though this

model is difficult to analytically establish in the modal space,

a novel analytical method can be found in the wavenumberdomain. The electrical potential and displacement equations

of the piezoelectric cylindrical shell are derived on the basis of

first-order shear deformation theory and a quadratic variation

of the electrical potential across the thickness of the actuator

layer. The elastic and electric properties of the piezoelectric

layer are assumed to be orthotropic. Control stiffness of

the distributed segmented actuator patches is obtained by

using the periodic Fourier transform and a Poissons sum

formulation. The radial displacement of the cylindrical shell in

the wavenumber domain is found by solving the simultaneous

governing equations. Vibrational and acoustic characteristics

of the cylindrical shell with or without active control are

compared and the effects of the piezoelectric patches onsound reduction performance are explored. For the harmonic

Figure 1. An infinite laminated cylindrical shell integrated with apiezoelectric layer.

vibration, a time-dependent factor eit will be suppressed

throughout.

2. The physical model

An infinite laminated cylindrical shell integrated with a

piezoelectric layer is shown in figure 1. The piezoelectric layer

is bonded to the outer surface of the laminated shell. Each

layer of the smart cylindrical shell is made of orthotropic

material and the poling direction of the piezoelectric layer

is coincident with the radial direction. ha and h are the

thicknesses of the piezoelectric layer and composite laminas.

a is the radius of the middle surface of the cylindrical shell.

The sensor points sense the radial displacements of the smart

cylindrical shells and the actuator layer is segmented into a

great many patches. In order to sense or inject the voltage

signals of the piezoelectric layer, a feasible method is that the

smart layer is divided into many segments. Recently, Hu et al

[33] studied the vibration of parabolic cylindrical shells with a

piezoelectric sensor layer using a similar method. The effects

of the temperature due to piezoelectric hysteresis losses as

heat in the piezoelectric layer on the dynamic characteristics

of the cylindrical shells are neglected in the present studies.

2.1. Governing equations of laminated cylindrical shells with

a piezoelectric layer

The displacement field of the smart composite cylindrical

shell can be described by

U1(1, 2, 3, t) = u1(1, 2, t) + 31(1, 2, t), (1)

U2(1, 2, 3, t) = u2(1, 2, t) + 32(1, 2, t), (2)

U3(1, 2, t) = u3(1, 2, t), (3)

where U1, U2 and U3 are the displacements of the composite

cylindrical shell in the 1, 2 and 3 directions. u1, u2 and u3are the displacements of a point at the middle surface. 1 and

2 are the rotations of a transverse normal about the 2 and1 axes, respectively. The straindisplacement relations of the

3


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circular cylindrical shell are

11

22

12

13

23

=

11

22

12

13

23

+ 3

11

22

12

0

0

, (4)

with

11 =u1

1, 22 =

u3

a+

1

a

u2

2,

12 =1

a

u1

2+

u2

1, 13 = 1 +

u3

1,

23 =1

a

u3

2+ 2

u2

a, 11 =

1

1,

22 =1

a

2

2, 12 =

1

a

1

2+

2

1.

(5)

The constitutive equations for the jth orthotropic lamina witha piezoelectric effect are given by

(j)

11

(j)

22

(j)

12

(j)

23

(j)

13

=

Q(j)

11 Q(j)

12 Q(j)

16 0 0

Q(j)

12 Q(j)

22 Q(j)

26 0 0

Q(j)

16 Q(j)

26 Q(j)

66 0 0

0 0 0 Q(j)

44 Q(j)

45

0 0 0 Q(j)

45 Q(j)

55

11

22

12

23

13

0 0 e(j)

31

0 0 e(j)

32

0 0 e(j)

36

e(j)

14 e(j)

24 0

e(j)

15 e(j)

25 0

E(j)

1

E(j)

2

E(j)

3

, (6)

D(j)

1

D(j)

2

D(j)

3

=

0 0 0 e(j)

14 e(j)

15

0 0 0 e(j)

24 e(j)

25

e(j)

31 e(j)

32 e(j)

36 0 0

11

22

12

23

13

+

(j)

11 (j)

12 0

(j)

12

(j)

22

0

0 0 (j)

33

E

(j)

1

E(j)

2E

(j)

3

. (7)Equations (6) and (7) can be denoted by

j = Qj ejEj,

Dj = eTj + jEj,

(8)

where the matrices are

j =

(j)

11 (j)

22 (j)

12 (j)

23 (j)

13

T,

= 11 22 12 23 13T

,

Dj =D

(j)

1 D(j)

2 D(j)

3

T, Ej =

E

(j)

1 E(j)

2 E(j)

3

T.

(9)

j and are the stress and strain vectors. Ej and Dj are the

electric field and the electric displacement vectors. Qj, ejand j are the reduced stiffness matrix, piezoelectric matrix

and dielectric matrix, respectively. The relations between the

electric field Ej and electric potential j in the piezoelectric

layer are described by

Ej =

j

1

1

3

j

2

j

3

T. (10)

Q(j)im are the reduced stiffness coefficients of the jth layer,

expressed by

Q(j)

11 = Q(j)

11 cos4 + 2(Q

(j)

12 + 2Q(j)

66)

sin2 cos2 + Q(j)

22 sin4,

Q(j)

12 = Q(j)

12 + (Q(j)

11 + Q(j)

22 2Q(j)

12 4Q(j)

66)

sin2 cos2,

Q(j)

22 = Q(j)

22 cos4 + 2(Q

(j)

12 + 2Q(j)

66)

sin2 cos2 + Q(j)

11 sin4,

Q(j)

66 = Q(j)

66 + (Q(j)

11 + Q(j)

22 2Q(j)

12 4Q(j)

66)

sin2 cos2,

Q(j)

16 = (Q(j)

11 Q(j)

12 2Q(j)

66) sin cos3

(Q(j)

22 Q(j)

12 2Q(j)

66) sin3 cos ,

Q(j)

26 = (Q(j)

11 Q(j)

12 2Q(j)

66) sin3 cos

(Q(j)

22 Q(j)

12 2Q(j)

66) sin cos3,

Q(j)

44 = Q(j)

44 cos2 + Q

(j)

55 sin2,

Q(j)

45 = (Q(j)

55 Q(j)

44) cos sin ,

Q(j)

55 = Q(j)

55 cos2 + Q

(j)

44 sin2, (11)

where is the fiber orientation (the anti-clockwise direction

is assumed to be positive). Unless the orthotropic layer

is a piezoelectric layer, only the reduced stiffnesses Q(j)im

should be kept unchanged and the parameters associated

with piezoelectric effect vanish in equation (6). By using

the assumption of zero normal stress in the first-order shear

deformation shell theory and the method shown in [10], the

resultant material parameters Q(j)im, e

(j)im and

(j)im for the jth

lamina are derived as

Q(j)

11 = C(j)

11 C(j)13C

(j)13

C(j)

33

, Q(j)

12 = C(j)

12 C(j)13C

(j)23

C(j)

33

,

Q(j)

22 = C(j)

22 C

(j)

23C(j)

23

C(j)

33

, Q(j)

44 = C(j)

44,

Q(j)

55 = C(j)

55, Q(j)

66 = C(j)

66,

e(j)

31 = e(j)

p31 C

(j)

13

C(j)

33

e(j)

p33, e(j)

32 = e(j)

p32 C

(j)

23

C(j)

33

e(j)

p33,

(j)

33 = (j)

p33 +e

(j)

p33e(j)

p33

C

(j)

33

,

e(j)

14 = e(j)

p14, e(j)

25 = e(j)

p25,

4


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(j)

11 = (j)

p11, (j)

22 = (j)

p22, (12)

in which C(j)im , e

(j)pim and

(j)pim are the elastic, piezoelectric and

dielectric material constants. is the shear correction factor,

taking into account the non-uniformity of the shear strain

distribution through the thickness of the shell and is given

by 2/12, as presented by Mindlin [34]. e(j)im and (j)im are thereduced piezoelectric moduli and dielectric coefficients of the

piezoelectric layer, given by

e(j)

31 = e(j)

31 cos2 + e

(j)

32 sin2,

e(j)

32 = e(j)

31 sin2 + e

(j)

32 cos2,

e(j)

36 = (e(j)

31 e(j)

32) sin cos ,

e(j)

14 = (e(j)

15 e(j)

24) sin cos ,

e(j)

24 = e(j)

24 cos2 + e

(j)

15 sin2,

e(j)

15 = e(j)

15 cos2 + e

(j)

24 sin2,

e

(j)

25 = (e

(j)

15 e

(j)

24) sin cos ,

(j)

11 = (j)

11 cos2 +

(j)

22 sin2,

(j)

22 = (j)

11 sin2 +

(j)

22 cos2,

(j)

12 = ((j)

11 (j)

22 ) sin cos , (j)

33 = (j)

33 .

(13)

In this context, the upper symbol j can be replaced with a

to denote the piezoelectric and dielectric coefficients of the

actuator layer. For a actuator layer, due to both the direct

piezoelectric effect and the inverse piezoelectric effect, the

layerwise quadratic distribution of the electric potential (e)a

is given by [12, 35]

(e)a = 23a

haV(1, 2) +

23a

ha2

2 a(1, 2),

3a = 3 (h + ha)/2. (14)

The extended Hamilton principle can be used to show

t10

N

j=1

1

2

3

(T(Qj ejEj)

Ej(eTj + jEj) u

TMju)a d1 d2 d3

1

2

uTFa d1 d2 dt = 0, (15)where N is the number of orthotropic layers. Therefore, the

governing equations of the smart laminated cylindrical shells

are derived as

L11 L12 L13 L14 L15 L16 L17

L21 L22 L23 L24 L25 L26 L27

L31 L32 L33 L34 L35 L36 L37

L41 L42 L43 L44 L45 L46 L47

L51 L52 L53 L54 L55 L56 L57

L61 L62 L63 L64 L65 L66 L67

u1

u2

u3

1

2

a

V

=

f1

f2

pe pa

m1

m2

0

,

(16)

where the differential operators Lij are defined as

L11 = A11 2

2

1

A66

a2

2

2

2

2A16

a

2

12+ I1

2

t2,

L12 = L21 = A12 + A66

a

2

12

A26

a2

2

22

A162

21

,

L13 = L31 = A12

a

1

A26

a2

2,

L14 = L41 = B112

21

B66

a2

2

22

2B16

a

2

12+ I2

2

t2,

L15 = L51 = B12 + B66a

2

12 B26

a2

2

22 B16

2

21,

L16 = L61 =

ha+h/2h/2

2e(a)31 3a d3

1

ha+h/2h/2

2e(a)36

a3a d3

2,

L17 =

ha+h/2h/2

2e(a)31

had3

1

ha+h/2h/2

2e(a)36

ahad3

2,

L22 = A66

2

21

A22

a2

2

22 +

A44

a2

2A26

a

2

12+ I1

2

t2,

L23 = L32 = A22 + A44

a2

2

A45 + A26

a

1,

L24 = L42 = B66 + B12

a

2

12

A45

a

B26

a2

2

22

B16 2

21

,

L25 = L52 = B662

21

B22

a2

2

22

A44

a

2B26

a

2

12+ I2

2

t2,

L26 = L62 =

ha+h/2h/2

23ae(a)32

ad3

+

ha+h/2h/2

e(a)24

a (3 + a)

23a

ha

2

2d3

2

ha+h/2

h/2e

(a)36 23a d3

5


7/17


+

ha+h/2h/2

e(a)14

a

23a

ha

2

2d3

1,

L27 =

ha+h/2h/2

e(a)32

a

2

had3

+

ha+h/2

h/2

e(a)24

a (3 + a)23a

had3

2

ha+h/2h/2

2e(a)36

had3

+

ha+h/2h/2

2e(a)14 3a

haad3

1,

L33 = A552

21

A44

a2

2

22

2A45a

2

12+ A22

a2+ I1

2

t2,

L34 = L43 =

B12

a A55

1+

B26

a2

A45

a

2,

L35 = L53 =

B22

a2

A44

a

2+

B26

a A45

1,

L36 = L63 =

ha+h/2h/2

e(a)24

a (3 + a)

23a ha

2

2

d3 2

22

ha+h/2h/2

e(a)15

23a

ha

2

2d3

2

21

+

ha+h/2h/2

2e(a)32 3a

ad3

ha+h/2h/2

e

(a)25

(a + 3)+

e(a)14

a

23a ha2

2

d3 2

12

,

L37 =

ha+h/2h/2

e(a)24

a (3 + a)

23a

had3

2

22

ha+h/2h/2

2e(a)15 3a

had3

2

21

+

ha+h/2h/2

2e(a)32

haad3

ha+h/2

h/2 e

(a)25

(a + 3)

+e

(a)14

a

23a

had3

2

12,

L44 = D11 2

21

D66

a2

2

22

2D16

a

2

12 + A55 + I3

2

t2 ,

L45 = L54 = D12 + D66

a

2

12

+ A45 D26

a2

2

22

D16 2

21

,

L46 = L64 =

ha+h/2h/2

e(a)15

23a

ha

2

2d3

ha+h/2

h/2

2e(a)31 33a d3

1

+

ha+h/2h/2

e

(a)25

(a + 3)

23a

ha

2

2

2e

(a)36

a3a3

d3

2,

L47 =

ha+h/2h/2

2e(a)15 3a

had3

ha+h/2

h/2

2e(a)31 3

ha

d3 1

+

ha+h/2h/2

e(a)25

(a + 3)

23a

had3

ha+h/2h/2

e(a)36

a

23

had3

2,

L55 = D22

a2

2

22

D662

21

2D26

a

2

12

+ A44 + I3 2

t

2,

L56 = L65 =

ha+h/2h/2

e(a)24

(3 + a)

23a

ha

2

2d3

ha+h/2h/2

2e(a)32 33a

ad3

2

+

ha+h/2h/2

e

(a)14

23a

ha

2

2

e(a)36 23a3 d3

1

,

6


8/17


L57 =

ha+h/2h/2

2e(a)24 3a

ha(3 + a)d3

ha+h/2h/2

2e(a)32 3

haad3

2

+

ha+h/2

h/2

e

(a)14

23a

ha

2e(a)36ha

3

d3

1,

L66 =

ha+h/2h/2

(a)11

23a

ha

2

22d3

2

21

+

ha+h/2h/2

(a)22

(a + 3)2

23a

ha

2

22d3

2

22

ha+h/2h/2

423a(a)33 d3 +

ha+h/2h/2

23a

ha

2

22 2(a)12(a + 3)

d32

12,

L67 =

ha+h/2h/2

(a)11

23a

ha

23a

ha

2

2d3

2

21

+

ha+h/2h/2

(a)22

(a + 3)223a

ha

23a

ha

2

2d3

2

22

ha+h/2h/2

4(a)33 3a

had3

+

ha+h/2h/2

23a

ha

2

2

2

(a)12

(a + 3)

23a

had3

2

12, (17)

in which the reduced stiffnesses Aim, Bim and Dim are given by

Aim =

Nj=1

Q(j)

im(hj hj1),

Bim =12

Nj=1

Q(j)im(h

2j h

2j1),

Dim =13

Nj=1

Q(j)im(h

3j h

3j1), (i, m = 1, 2, 6),

Aim =N

j=1

Q(j)im(hj hj1) = (i, m = 4, 5),

(18)

and the general inertial moments I1, I2 and I3 are given by

(I1,I2,I3) =h

a+h/2

h/2

Nj=1

(1, 3, 23 )(j) d3. (19)

The Fourier transform and the inverse Fourier transform are

defined by

f(k) =1

2

+

f(1)eik1 dx, (20)

f(1) =+

f(k)e

ik1 dk. (21)

The displacements of the cylindrical shell, electric potential,

external control voltage, external general forces and fluid

loadings should be decomposed into the following series

g(1, 2) =

n=

gn(1)ein2 . (22)

Before equation (16) is solved, the function g(1, 2) will be

replaced with u1, u2, u3, 1, 2, f1, f2, pe, m1, m2, pa, a and

V in the following derivation.

2.2. External point force

The external radial point force pe(1, 2) with an amplitude f3at the point (1j, 2j) acting on the laminated shell is described

by

pe(1, 2) =f3

a(1 1j)(2 2j). (23)

Taking the Fourier transform with respect to 1 and the

periodic Fourier transform with respect to 2, one obtains

pe(k, 2) =

n=

f3ei(k1j+n2j)

a(2 )2 e

in2

. (24)

Therefore, the force pe(1, 2) can be decomposed into the

circumferential components in the following form

pen(k) =f3e

i(k1j+n2j)

a(2 )2. (25)

2.3. The fluid loadings

The acoustic pressure p in the fluid satisfies the Helmholtz

equation in cylindrical coordinates.

2p + k20p = 0, (26)

where the wavenumber k0 is /c with c being the speed of

sound in the fluid. The Laplace operator 2 is given by

2 =1

23

2

22

+1

3

3

3

3

+

2

21

, (27)

where 1, 2 and 3 denote the coordinates of the axial,

circumferential and radial directions in circular cylindrical

coordinates, respectively. The boundary condition at the

interface of the fluid and the piezoelectric layer is

p

3

3=ac

= 2u3(1, 2). (28)

7


9/17


Taking the Fourier transform of equations (26) and (28) with

respect to 1, one obtains the sound pressure satisfying the

Sommerfeld radiation conditions at infinity

p(k, 2, 3)

=

n=

2u3n(k)H(1)n (k2

0

k23)k20 k

2H(1)n (

k20 k

2ac)ein2 , (29)

where H(1)n (z) is the nth-order Hankel function of the first

kind. If k20 is less than k2, H

(1)n (z) is replaced with Kn(z),

which is the nth-order modified Bessel function of the second

kind, and

k20 k

2 is replaced with

k2 k20. Therefore, the

circumferential decompositions pan of fluid loadings acting on

the smart layer can be written as

pan(k, a) =2au3n(k)H

(1)n (

k20 k

2a)

k

2

0 k2

H

(1)

n (

k

2

0 k2

a)

= Zn(k)u3n(k),

(30)

where Zn(k) are the impedances of the fluid loadings.

3. Solutions in the wavenumber domain

Taking the Fourier transform of equation (16) with respect to

1, one obtains

L11 L12 L13 L14 L15 L16L21 L22 L23 L24 L25 L26

L31 L32 L33 L34 L35 L36L41 L42 L43 L44 L45 L46L51 L52 L53 L54 L55 L56L61 L62 L63 L64 L65 L66

u1n

u2n

u3n1n

2n

an

=

f1n L17Vnf2n L27Vn

pen L37Vnm1n L47Vn

m2n L57Vn

L67Vn

,

(31)

where the elements Lij are the transformed operators and are

given by

L11 = A11k2 +

A66n2

a2+

2A16kn

a I1

2,

L12 = L21 =

(A12 + A66)kn

a +

A26n2

a2 + A16k2

,

L13 = L31 = ikA12

a

inA26

a2,

L14 = L41 = B11k2 +

B66n2

a2+

2B16kn

a I2

2,

L15 = L51 =(B12 + B66) kn

a+

B26n2

a2+ B16k

2,

L16 = L61 = ik

ha+h/2h/2

2e(a)31 3a d3

in

ha+h/2

h/2

2e(a)36a

3a d3,

L17 = ik

ha+h/2h/2

2e(a)31

had3 in

ha+h/2h/2

2e(a)36

ahad3,

L22 = A66k2 +

A22n2

a2+

A44

a2+

2A26kn

a I1

2,

L23 =

L32 =

in(A22 + A44)

a2

ik(A45 + A26)

a ,

L24 = L42 =kn(B66 + B12)

a

A45

a+

B26n2

a2+ B16k

2,

L25 = L52 = B66k2 +

B22n2

a2

A44

a+

2knB26

a I2

2,

L26 = L62 = in

ha+h/2h/2

23ae(a)32

ad3

+

ha+h/2h/2

e(a)24

a (3 + a)

23a

ha

2

2d3

ik

ha+h/2h/2

e(a)36 23a d3

+

ha+h/2h/2

e(a)14

a

23a

ha

2

2d3

,

L27 = in

ha+h/2h/2

e(a)32

a

2

had3

+ ha+h/2

h/2

e(a)24

a (3 + a)

23a

had3

ik

ha+h/2h/2

2e(a)36

had3+

ha+h/2h/2

2e(a)14 3a

haad3

,

L33 = A55k2 +

A44n2

a2+

2A45kn

a+

A22

a2+ Zn (k) I1

2,

L34 = L43 = ik

B12

a A55

+ in

B26

a2

A45

a

,

L35 = L53 = in

B22

a2

A44

a

+ ik

B26

a A45

,

L36 = L63 = n2

ha+h/2

h/2

e(a)24

a(3 + a)

23a

ha

2

2d3

+ k2ha+h/2

h/2e

(a)15

23a

ha

2

2d3

+

ha+h/2h/2

2e(a)32 3a

ad3

+ nk

ha+h/

2

h/2

e

(a)

25

(a + 3)+ e

(a)

14

a

8


10/17


23a

ha

2

2d3,

L37 = n2

ha+h/2h/2

e(a)24

a(3 + a)

23a

had3

+ k2

ha+h/2

h/2

2e(a)15 3a

had3 +

ha+h/2

h/2

2e(a)32

haad3

+ nk

ha+h/2h/2

e

(a)25

(a + 3)+

e(a)14

a

23a

had3,

L44 = D11k2 +

D66n2

a2+

2D16kn

a+ A55 I3

2,

L45 = L54 =(D12 + D66) kn

a

+ A45 +D26n

2

a2

+ D16k2,

L46 = L64 = ik

ha+h/2h/2

e(a)15

23a

ha

2

2d3

ha+h/2h/2

2e(a)31 33a d3

+ in

ha+h/2h/2

e

(a)25

(a + 3)

23a

ha

2

2

2e

(a)36

a

3a3 d3,L47 = ik

ha+h/2h/2

2e(a)15 3a

had3

ha+h/2h/2

2e(a)31 3

had3

+ in

ha+h/2h/2

e(a)25

(a + 3)

23a

had3

ha+h/2

h/2

e(a)36

a

23

had3 ,

L55 =D22n

2

a2+ D66k

2 +2D26kn

a+ A44 I3

2,

L56 = L65 = in

ha+h/2h/2

e(a)24

(3 + a)

23a

ha

2

2d3

ha+h/2

h/2

2e(a)32 33a

a

d3

+ ik

ha+h/2h/2

e

(a)14

23a

ha

2

2

e(a)36 23a3

d3,

L57 = in

ha+h/2

h/2

2e(a)24 3a

ha(3 + a)d3

ha+h/2h/2

2e(a)32 3

haad3

+ ik

ha+h/2h/2

e

(a)14

23a

ha

2e(a)36

ha3

d3,

L66 = k2

ha+h/2h/2

(a)11

23a

ha

2

22d3

n2ha+h/2

h/2

(a)22

(a + 3)2

23a

ha

2

22d3

ha+h/2h/2

423a(a)33 d3

nk

ha+h/2h/2

23a

ha

2

22 2(a)12(a + 3)

d3,

L67 = k2

ha+h/2h/2

(a)11

23a

ha

23a

ha

2

2d3

n2ha+h/2

h/2

(a)22

(a + 3)2

23a

ha

23a

ha

2

2d3

ha+h/2h/2

4(a)33 3a

had3

nk

ha+h/2h/2

23a

ha

2

2 2(a)12(a + 3)

23a

had3.

(32)

3.1. Sound projector induced by the external control voltage

The external control voltage applied to the piezoelectric layer

is described by

V(1, 2) = VaA(1)C(2), (33)

where A(1) and C(2) are the decompositions of the external

control voltage in the axial and circumferential directions,

as illustrated in figure 2. The applied control voltage V is

composed of constant voltage patches with an amplitude Va.

The periodic square waves A(1) and C(2) can be easily

expanded as the Fourier series

A(1) =

m=1,3,5

2i m

(ei2 m1/s1 ei2 m1/s1 ), (34)

9


11/17


Figure 2. External axial and circumferential square wave voltage functions A(1) and C(2).

C(2) =

n=1,3,5

2i

(ein2 ein2 ). (35)

Substituting equations (34) and (35) into (33), one obtains

V(1

, 2

) =

n=1,3,5

m=1,3,5

4Va

2m(ei2 m1/s1 ei2 m1/s1 )

(ein2 ein2 ). (36)

Taking the Fourier transform ofV(1, 2) with respect to 1,

one obtains

V(k, 2) =

n=1,3,5

m=1,3,5

4VaVc

2m((k 2m/s1)

(k+ 2 m/s1))(ein2 ein2 ). (37)

Since the moduli of L6j, Li6 and Li7 caused by the

piezoelectric and dielectric parameters are much smallerthan those of Lij (i,j = 15 ) generated by elastic material

constants, equation (31) is ill-conditioned. Eliminating a in

equation (31), one obtains a simplified equation with good

numerical characteristics

L11 L12 L13 L14 L15L21 L22 L23 L24 L25L31 L32 L33 L34 L35L41 L42 L43 L44 L45L51 L52 L53 L54 L55

u1n

u2n

u3n

1n

2n

=

f1f2

pe

m1

m2

, (38)

where

f1 = f1n ( L17 L16 L67/ L66)Vn,

f2 = f2n ( L27 L26 L67/ L66)Vn,

pe = pen ( L37 L36 L67/ L66)Vn,

m1 = m1n ( L47 L46 L67/ L66)Vn,

m2 = m2n ( L57 L56 L67/L66)Vn,

Lij = Lij Li6 L6i/ L66, i = 15.

(39)

Distinctly, f1, f2, pe, m1 and m2 can be taken as the equivalent

general forces induced by the actuator. Substituting the

circumferential harmonics Vn into equation (38) and taking

the inverse Fourier transform of equation (29), one obtains thesound pressure.

3.2. Active control method for acoustic radiation frompiezoelectric cylindrical shells

The proportional derivative negative feedback control strategy

is adopted when an external radial point force is applied on the

smart cylindrical shell. It is assumed that the sensor sampling

points for the radial displacements are located at the centers

of the segmented patches, as shown in figure 1. The feedback

control of the external voltage in a segment is given by

V(1, 2) = Gvu3(ml,j ), j = 0, 1, 2, . . . , c 1 (40)

where is the circumferential angle between two adjacent

sampling points and l is the axial length of a patch. Note that

ml and j are chosen according to 1 and 2, respectively. Gvis the gain, expressed by

Gv = (Gp iGd), (41)

where Gp and Gd are the proportional and derivative gain

coefficients respectively. Taking the Fourier transform of V

with respect to 1, one obtains

V(2) =1

2

m=

ml+l/2mll/2

Gvu3(ml,j )eik1 d1

=1

2

m=

ieikl/2 eikl/2

keikml

Gvu3(ml,j ), (42)

where u3(ml,j ) can be expressed by

u3(ml,j ) =

n=

einj

u3n(k1)eik1ml dk1. (43)

Substituting equation (43) into (42), one obtains

V(2) =1

2

n=

ieikl/2 eikl/2

kGve

inj

m=

u3n(k1)ei(k1k)ml dk1. (44)

Poissons summation formula can be used to show

m=

eimlk1 = 2

m=

(lk1 2m ). (45)

10


12/17


Therefore,

m=

u3n (k1) ei(k1k)ml dk1

= 2

m=

1

l

u3n k 2ml . (46)

Thus, substituting equation (46) into (44), one obtains

V(2) =

s=

ieikl/2 eikl/2

klGve

isj

m=

u3s

k

2m

l

. (47)

By using the definition in equation (22), one has the

decomposition

Vn(k) =1

22

0V(2)ein2 d2. (48)

Substituting equation (47) into (48) one obtains

Vn(k) =1

2

c1j=0

Gv

s=

j+/2j /2

eisj

eikl/2 eikl/2

ik

1

l

m=

u3s

k

2m

l

ein2 d2

=1

2

s=Gv

c1

j=0ei(sn)j

eikl/2 eikl/2

ik

(ein /2 ein/2)

in

1

l

m=

u3s

k

2m

l

.

(49)

The following relations hold,

=2

c, r1 = s n = bc,

q = eir1 = 1,c1j=0

eir1j =1 qc

1 q= 0.

(50)

Therefore, equation (49) can be simplified as

Vn(k) = 1

2

cGv

l

eikl/2 eikl/2

k

ein/2 ein/2

n

b=

m=

u3(n+bc)

k

2m

l

. (51)

A special case can be taken in equation (51). When the

electrode patches are very small (l 0, c ), one obtains

a simplified expression in equation (51)

Vn(k) = Gvu3n(k). (52)

It is understandable that this control strategy approaches anideal surface control method. Substituting k = k 2r/l into

equation (51), one obtains

Vn(k 2r/l) = eir

2

cGv

l

eikl/2 eikl/2

k 2r/l

ein/2 ein /2

n

b=

m=

u3(n+bc)

k

2m

l

. (53)

The masses of the sensor points are taken into account in the

active control model and assumed to be placed at the locations

of the sensors. Therefore, the uniform distributed inertia point

forces fi in the radial direction can be expressed as

fi(1, 2) = Ii2

m=

c1j=0

u3(1, 2)

(1 ml)(2 j ), (54)

where Ii is the mass at the sensor point. Taking the

continuous Fourier transform and periodic Fourier transform

of equation (54) with respect to 1 and 2, respectively, and

simplifying the equation by using the above method, one

obtains

fin =Ii

2c

2 al

b=

m=

u3(n+bc)

k

2m

l

. (55)

Substituting equations (53) and (55) and k = k 2r/l into

equation (38), one will obtain a new control equation that

can be used to form linear simultaneous equations for the

combination with rand n. The displacement field of the smart

cylindrical shell in the wavenumber domain can be found

by solving the linear simultaneous equations. The far-field

sound pressure of the piezoelectric cylindrical shells can be

described using the stationary phase method [36]

p(R, , ) = 2i2eik0R

k0R sin

n=

u3n(k0 cos )

H(1)n (k0a sin )

ein(i)n, (56)

where and are the polar and azimuthal angles in spherical

coordinates. The acoustic power radiated by the cylindrical

shell can be expressed by

() = 4 3

k0k0

k0an=k0a

1

k20 k2

u3n(k)

H(1)n (a

k20 k

2)

2

dk, (57)

where k0a stands for the largest integer approximating k0a.

Note that this expression is approximately established when

the exciting frequency is above the ring frequency of the largecylindrical shell [36]. The sound pressure level (SPL) and

11


13/17


Table 1. Material parameters of piezoelectric layers and laminas.

Elastic constantBaTiO3[38, 39]

PVDF[40]

Compositelamina

C11 (GPa) 166.0 238.24 226.51C22 166.0 23.6 9.64C33 162.0 10.64 9.64

C13 78.0 2.19 3.02C23 78.0 1.92 2.44C12 77.0 3.98 3.02C44 43.0 2.15 54C55 43.0 4.4 28C66 44.5 6.43 54ep31 (C m

2) 4.4 0.13ep32 4.4 0.145ep33 18.6 0.276ep24 11.6 0.009ep15 11.6 0.135

p11 (109C2N1m2) 11.2 0.111

p22 11.2 0.106p33 12.6 0.106 (kg m3) 5800.0 1000.0 2600.0

acoustic power level (APL) are defined by

SPL = 20 log

|p|

p0

, APL = 10 log

W0

, (58)

where p0 is the reference sound pressure 1 106 Pa and W0

is the reference acoustic power 1 1012 W.

4. Numerical results

In the numerical calculation, the vibrational and acousticcharacteristics of smart cylindrical shells are investigated

by using the foregoing theory. Attention is paid to

the active control of sound radiation from composite

laminated piezoelectric shells. The material parameters of

the piezoelectric layers and composite laminas are listed in

table 1. The sound field point is located at Q1(R = 50 m, =

/4, = /4) in spherical coordinates. The external fluid

is water. The sound speed c in the fluid is 1500 m s1 and

the mass density of the fluid is 1000 kg m3. Mass Iiof each sensor point is taken as zero and these parameters

are kept unchanged unless specially noted. The present

cylindrical shells are assumed to be infinite, which means

that elastic waves are dissipated through a finite distance. Afinite cylindrical shell can be modeled as an infinite cylindrical

shell as long as the finite cylindrical shell is sufficiently long.

For long cylindrical shells, reflection of the elastic waves due

to the end boundary conditions can be neglected. Wang and

Lai [37] have explored the acoustic characteristics of finite

cylindrical shells and infinite cylindrical shells in detail.

4.1. Free vibration of piezoelectric cylindrical shells

In order to verify the governing equations for the cylindrical

shells with a piezoelectric layer, the natural frequency of the

homogeneous BaTiO3 piezoelectric cylindrical shell given byBhangale and Ganesan [41] is compared with that obtained by

Figure 3. Comparisons of natural frequency for the BaTiO3piezoelectric cylindrical shell using the present theory and 3D finiteelement method.

the present method. The geometric parameters of two simply

supported BaTiO3 piezoelectric cylindrical shells [41] are

given as follows. The length ls and radius a are 4 m and 1 m.

The wall thicknesses ha of the two smart cylindrical shellsare 0.02 m and 0.005 m, respectively. BaTiO3 piezoelectric

material shown in table 1. The wavenumber of the first

axial natural frequency of the two smart cylindrical shells

given by the present shear deformable shell theory and the

semi-analytical 3D finite element method is shown in figure 3.

It can be observed that the results presented by these twomethods are in good agreement.

4.2. Sound induced by the external control voltage

We use an external control voltage applied on the piezoelectric

layer, as has been described in the section 3.1. The parameters

of the square waves are given as follows, Va = 1, s1 =

4 m. The material parameters of the infinite BaTiO3cylindrical shell have already been described. The thickness

and structural damping of the piezoelectric cylindrical shellare 0.02 m and 0.02, respectively. SPL values at Q1 induced

by all the equivalent forces or just the equivalent axial force

are shown in figure 4. SPL at Q1 caused by the equivalent

circumferential force or the equivalent radial force is shownin figure 5. By all the equivalent forces we mean that all

the right-hand items in equation (38) are considered. Theequivalent axial, circumferential and radial forces correspond

to f1, f2 and pe, respectively. It can be observed that the

effects of the general forces f1, f2 and pe on the sound field

are significant in figures 4 and 5. The axial, circumferential

and radial forces generated by the piezoelectric layer can

cause strong sound radiation. s1 can be used to control the

acoustic radiation behavior of some harmonics. Cao et al [36]have elaborated the ability for sound radiation of the axial

wavenumber and circumferential wavenumber harmonics for

large cylindrical shells, which can also be used to judge the

sound radiation characteristics of harmonics in the context ofsmart cylindrical shells.

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Figure 4. SPL induced by all the equivalent forces or just theequivalent axial force, Va = 1.

Figure 5. SPL induced by the equivalent circumferential force orthe equivalent radial force, Va = 1.

4.3. Active control of sound radiation from piezoelectriclaminated shells

A smart laminated cylindrical shell is composed of five

laminas and the structural damping of the smart laminated

cylindrical shell is still assumed to be 0.02. The outer

layer is a BaTiO3 piezoelectric lamina with a thickness

ha of 0.01 m. The other four layers are the composite

laminas depicted in table 1. The thickness of each composite

lamina is 0.005 m and angle plies of the smart shell are

(15/ 45/60/75/0). An external radial point force is

located at (0.1, 0). The SPL of the piezoelectric laminated

cylindrical shell with or without active control is shown

in figure 6. The circumferential sensor sampling points c

and the spacing l between adjacent axial sensing sampling

points are 15 and 0.1 m, respectively. Gp and Gd are both

1 105 for the smart cylindrical shell with active control.The axial and circumferential wavenumber conversion is

Figure 6. SPL of a BaTiO3 piezoelectric cylindrical shell with orwithout active control, c = 15.

Figure 7. Effects ofGd on SPL of the BaTiO3 piezoelectriccylindrical shell with or without active control, c = 60.

given by equation (51), which has a similar mechanism of

wavenumber conversion to that of cylindrical shells reinforcedby periodic orthogonal stiffeners [42]. As has been point out inprevious studies on acoustic radiation from cylindrical shells

with periodic stiffeners [36, 42], the larger l is, the strongerfluctuation of the far-field sound pressure. Therefore, large lis not suited to suppress sound radiation from piezoelectriccylindrical shells in the entire frequency range. Small l =0.1 m makes wavenumber conversion weak and the soundpressure slightly fluctuates. In figure 6, since a small value

of c induces strong circumferential wavenumber conversion,the far-field sound pressure fluctuates. Therefore, c = 15 is

not suitable for active control of sound radiation and a largervalue ofc has advantages in reducing acoustic radiation.

Turning out attention to the derivative gain Gd, the SPLof the piezoelectric laminated cylindrical shell with or without

active control is presented in figure 7. It is shown that active

control can be used to reduce the sound pressure in the entirefrequency range for c = 60. A special case for l 0 and

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Figure 8. Effects ofGp on SPL of the BaTiO3 piezoelectriccylindrical shell with or without active control, c = 60.

Figure 9. Effects of the sensor point masses on SPL of the BaTiO 3piezoelectric cylindrical shell with active control.

c 0 has been derived in equation (52). In fact, the ideal

control method is not feasible, but the control method can be

approached by using a small value of l and a large value ofc.

Thus, active control of acoustic radiation works. l and c should

be chosen to have suitable values, such as l = 0.1 m, c = 60.

The larger Gd is, the lower SPL at Q1 will be. Moreover, Gdcan be taken as active damping gain. The effects ofGp on SPL

of the piezoelectric laminated cylindrical shell with or without

active control are shown in figure 8. It can be seen that the

largest Gp has the best sound reduction performance in the low

and medium frequency range. Comparisons are made between

figures 7 and 8 for Gp = 1 105 and Gp = 1 10

8, and show

that the influence of Gp less than 1 108 on the sound field

can be neglected for Gd = 1 105. Since Gp corresponds to

dynamic stiffness gain, only large proportional gain plays a

role in sound suppression.

The effects of the sensor point masses on SPL of theBaTiO3 piezoelectric cylindrical shell with active control are

Figure 10. Helical wave spectra of BaTiO3 piezoelectriccylindrical shell without active control, Gp = 0, Gd = 0, 4 kHz.

Figure 11. Helical wave spectra of BaTiO3 piezoelectric

cylindrical shell with active control, c = 60, Gp = 1 105,

Gd = 1 105, 4 kHz.

shown in figure 9. The mass Ii of each sensor point is 0.05 kg.

It can be observed that the far-field sound pressure in figure 9

remains almost unchanged for Ii = 0, 0.05 when compared

with the cases in figures 6 and 7, respectively. This indicates

that the influence of the masses of the sensor points on the

vibrational and acoustic characteristics of the cylindrical shell

can be neglected.

The helical wave spectra of cylindrical shell have been

defined in the previous studies [36]. The helical wave spectra

of radial displacement for piezoelectric cylindrical shells

without or with active control are shown in figures 10 and

11, respectively. By comparison with the patterns of the radial

displacement helical wave spectra presented in figures 10

and 11, it is found that active control has a significant

impact on the radial displacement of the piezoelectric shell.The mean square radial displacement of the smart laminated

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Figure 12. APL of the BaTiO3 piezoelectric cylindrical shell withor without active control, c = 60.

cylindrical shell without active control is larger than that ofthe cylindrical shell with active control. APL of piezoelectriclaminated cylindrical shell with or without active control ispresented in figure 12. APL of the smart cylindrical shellwith active control is distinctly decreased for Gd = 5 10

5.Due to the weak wavenumber conversion caused by l and c,the acoustic power of cylindrical shells with active controlfluctuates slightly. APL of PVDF piezoelectric laminatedcylindrical shells with or without active control is illustrated infigure 13. The outer piezoelectric layer is composed of PVDFand the two lamination schemes (15/45/60/75/0) and

(15

/ 45

/60

/75

/45

) as calculated in the study. The plyangles of PVDF have minor effects on the acoustic powerof the laminated cylindrical shells without active control infigure 13. This is due to the fact that the four compositelaminas dominate the dynamic characteristics of the structure.It can be observed that the PVDF piezoelectric layer does noteffectively reduce the acoustic power of the smart cylindricalshell with active control. Comparisons made between figures12 and 13 show that the sound suppression performanceof PVDF is not as good as that of BaTiO3. Since thepiezoelectric parameters of PVDF listed in table 1 are muchsmaller than those of BaTiO3, the equivalent dynamic controlforces of PVDF are much smaller than those of BaTiO3

in equation (38). They cause a poorer sound suppressionperformance of PVDF in comparison with that of BaTiO3.

5. Conclusions

The governing equations of laminated cylindrical shells with apiezoelectric layer are derived on the basis of first-order sheardeformation theory and a quadratic distribution of the electricpotential. Active control of acoustic radiation from laminatedcylindrical shells with a piezoelectric layer is investigatedin the wavenumber domain. Some principal conclusions aredrawn as follows.

(1) An external control voltage with a tailored distribution canbe devised to induce acoustic radiation from piezoelectric

Figure 13. APL of PVDF piezoelectric laminated cylindrical shellswith or without active control, c = 60.

cylindrical shells, generating a great many harmonics

owing to different acoustic characteristics. The equivalent

axial, circumferential and radial forces produce distinct

sound radiation from piezoelectric cylindrical shells.

(2) Feedback control adopting dense piezoelectric patches

approaches an ideal surface feedback control when the

patches are small enough. Large piezoelectric patches

induce strong axial and circumferential wavenumber

conversion, making the far-field sound pressure fluctuate,

which causes active control of the acoustic radiation to be

ineffective. Small piezoelectric patches with active control

can be employed to suppress sound radiation from smartcylindrical shells in a wide frequency range. The effects

of the masses of light sensor points on the vibrational and

acoustic characteristics of smart cylindrical shells can be

neglected.

(3) The derivative gain Gd and proportional gain Gp of active

control play important roles in the reduction of vibration

and acoustic power for smart cylindrical shells. Even

small vales ofGd can also control acoustic radiation from

cylindrical shells. Small values of Gp have only a minor

impact on the vibrational and acoustic characteristics of

smart cylindrical shells. Nevertheless, large Gp can be

used to suppress sound radiation from the cylindricalshells. Gp and Gd influence the stiffness and damping of

the piezoelectric cylindrical shells, respectively.

(4) Because the piezoelectric parameters of PVDF are much

smaller than those of BaTiO3, the equivalent dynamic

control forces of PVDF are much smaller than those of

BaTiO3. PVDF piezoelectric layers have a much poorer

sound suppression performance in comparison with thatof BaTiO3 piezoelectric layers.

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