Finite element method for the feedback control of FGM shells in the frequency domain via piezoelectric sensors and actuators

Comput. Methods Appl. Mech. Engrg. 193 (2004) 257–273

www.elsevier.com/locate/cma

Finite element method for the feedback control of FGMshells in the frequency domain via piezoelectric

sensors and actuators

K.M. Liew a,*, X.Q. He a, S. Kitipornchai b

a Nanyang Centre for Supercomputing and Visualisation, School of Mechanical and Production Engineering,

Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singaporeb Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Received 30 December 2002; received in revised form 7 April 2003; accepted 23 September 2003

Abstract

An attempt on the dynamic control of functionally graded material (FGM) shells in the frequency domain is carried

out by using self-monitoring sensors and self-controlling actuators. Based on the first-order shear deformation theory

(FSDT), a generic finite element formulation is developed to account for the coupled mechanical and electrical re-

sponses of FGM shells with piezoelectric sensors and actuator layers. The properties of the FGM shells are graded in

the thickness direction according to a volume fraction power-law distribution. A constant displacement and velocity

feedback control algorithm is applied in a closed loop system to provide feedback control of the integrated FGM shell

structure. Numerical simulations are presented to show that mode shapes and resonance frequencies can be controlled

by adjusting the displacement control gain, and that the resonance amplitude peaks can be very significantly reduced by

providing an appropriate velocity feedback control.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Feedback control; FGM shells; Frequency domain piezoelectric; Sensors and actuators

1. Introduction

Recently, many researchers have investigated the application of piezoelectric materials as sensors and

actuators for the purpose of monitoring and controlling in active structural systems. Numerous approaches

have been introduced into the analysis of plates with bonded piezoelectric sensors and actuators. Among

such researches on piezoelectric plates, Tzou and Tseng [1] proposed a ‘‘thin’’ piezoelectric solid element by

adding three internal degrees of freedom for the vibration control of structures with piezoelectric materials.Based on the classical theory, the first-order theory, and the Reddy third-order theory, Reddy [2] developed

* Corresponding author. Tel.: +65-6790-4076; fax: +65-6793-6763.

E-mail address: [email protected] (K.M. Liew).

0045-7825/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.09.009

mail to: [email protected]

258 K.M. Liew et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 257–273

theoretical formulations of laminated plates with piezoelectric layers. The Navier solutions and finite ele-ment models were also presented for the analysis of laminated composite plates with integrated sensors and

actuators. Liew et al. [3] developed a finite element model based on the first-order shear deformation theory

for the active control of functionally gradient material plates with piezoelectric sensor/actuator layers that

were subjected to a thermal gradient. Numerical results for the control of both bending and torsional

vibrations were presented for an FGM plate that was comprised of zirconia and aluminum. Based on a

simple higher-order plate theory, Fukunaga et al. [4] presented a finite element model for analyzing

moderately thick composite laminates that contained the piezoelectric layers statistically or dynamically. A

C0-type FEM scheme was contracted through the introduction of two artificial variables in the displace-ment field and the penalty function method.

A generalized finite element formulation for active vibration control of a laminated plate integrated with

piezoelectric polymer layers acting as distributed sensors and actuators is presented by Samanta et al. [5]. In

their work, an eight-noded two-dimensional quadratic quadrilateral isoparametric element is derived for

modeling the global coupled electroelastic behavior of the overall structure using higher-order shear

deformable displacement theory. Other finite element models for the analysis of laminated plates that

contained active and passive piezoelectric layers were reported by Allik and Hughes [6], Lammering [7],

Hwang et al. [8] and He et al. [9].A number of FEM models were also proposed for shells with bonded piezoelectric layers. Tzou and

Zhong [10] established a linear piezoelectric thin shell vibration theory using Hamilton�s principle and

linear piezoelasticity theory. Tzou and Ye [11] subsequently developed a laminated piezoelectric triangular

shell element by using a layerwise constant shear angle theory. Natural frequencies and distributed control

effects of the ring shell with piezoelectric actuators of various lengths were studied. Heyliger et al. [12]

proposed a shell element for laminated piezoelectric shells. A discrete-layer theory that allowed for dis-

continuous shear strain through the shell thickness was employed. Chandrashekhara et al. [13] developed a

modal dynamic model for the active vibration control of laminated shells with piezoelectric sensors andactuators. A neural network controller was designed and trained to emulate the performance of the linear

quadratic gaussian with loop transfer recovery (LQG/LTR) controller. Lee and Saravanos [14] proposed a

mixed multi-field finite element formulation for thermopiezoelectric composite shells. They developed a

new mixed multi-field laminate theory that combined ‘‘single layer’’ assumptions for the displacements

along with layerwise fields for the electric potential and temperature. Correia et al. [15] developed a semi-

analytical axisymmetric shell finite element model with embedded and/or surface bonded piezoelectric ring

actuators and/or sensors for active damping vibration control of structures. A higher order shear defor-

mation theory was used in the displacement field with the condition of zero transverse shear stresses at thebottom and top surface of the shell for avoiding the need of shear correction factors. Other noteworthy

studies of piezoelectric shells include those of Koconis et al. [16], Tzou and Howard [17], Saravanos [18], Ye

and Tzou [19], and He et al. [20].

Moreover, a new class of composite materials known as ‘‘functionally graded materials’’ (FGMs) has

drawn considerable attention. Typically, FGMs are made from a mixture of ceramics and metals and are

further characterized by a smooth and continuous change of the mechanical properties from one charac-

teristic surface to the other. The ceramic constituents of the FGMs are able to withstand high temperature

environments due to their better thermal resistance, while the metal constituents provide strongermechanical performance and reduce the possibility of catastrophic fracture. Yamanouchi et al. [21] pub-

lished the first study of FGMs. After that, researchers such as Obata and Noda [22], Praveen and Reddy

[23], Reddy [24], Reddy and Chin [25], Ng et al. [26] and Wu et al. [27] have also carried out substantial

work in this topic.

In author�s previous work, finite element formulations were developed for shape and vibration control of

piezoelectric FGM plates [3,9,28] and piezoelectric FGM shells based on the Kirchhoff–Love hypothesis

[20,29]. In these work, an efficient finite element model is developed for static and dynamic control of

K.M. Liew et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 257–273 259

functionally graded material (FGM) plates and shells under temperature gradient environments. By usingelectromechanical coupling effect of piezoelectric inserts in smart structures, active control for shape and

vibration of structures is provided in a self-monitoring and self-controlling system. This paper presents a

finite element model based on FSDT for the dynamic control of FGM shells by using self-monitoring

sensors and self-controlling actuators. Finite element analysis is performed on the feedback control of

FGM shells in the frequency domain to show the effects of the volume fraction of different materials and

control gains on the natural frequencies, mode shapes, and resonance amplitudes of an FGM shell with

piezoelectric sensor and actuator layers. This work is an extension of the authors� previous work

[3,9,20,28,29] on the active control of FGM plates and shells.

2. Theoretical formulation based on FSDT

Consider an FGM shell with integrated piezoelectric sensors and actuators, as shown in Fig. 1. The shell

occupies a two-dimensional arbitrary shaped region X, and is bounded by its boundary C ¼Cu [ Cp ¼ C/ [ Cq. Here, the displacement u, the surfaces force p, potential /, and charge q are specified on

sub-surfaces Cu, Cp, C/, and Cq, respectively. An orthogonal curvilinear system ðn1; n2; n3Þ is chosen so thatn1 and n2 curves are lines of curvature of neutral surface, and n3 defines the normal. The top surface of the

FGM shell is ceramic rich, and the bottom surface is metal rich. The region between the two surfaces is

made of the combined ceramic–metal material with continuously varying mix-ratios. As the functionally

graded material is a mixture of ceramic and metal materials, the effective properties can be expressed as

Peffðn3Þ ¼ PcVc þ PmVm; ð1Þ

Vc þ Vm ¼ 1; ð2Þ

Actuator Input Feedback Voltage Gain Control

Structure Sensor Output

PiezoelectricSensing Layer

PiezoelectricActuator Layer

FGM Shell

1ξ

2ξ

3ξ

(a)

(b)

Fig. 1. (a) A laminated FGM shell with surface bonded piezoelectric sensor/actuator layers and (b) schematic illustration of the

configuration of the feedback scheme in the FGM shell with integrated piezoelectric sensor/actuator layers.


where Peff is the effective material property of the functionally graded material, Pc and Pm are the properties

of the ceramic and metal constituent materials respectively, and Vc and Vm are the volume fractions of the

constituent materials within the functionally graded material. The volume fraction of ceramic is assumed to

be

Vc ¼2n3 þ h

2h

� �n

� h=26 n3 6 h=2; ð3Þ

where n is the volume fraction exponent ð06 n61Þ, and h is the thickness of the plate. In the present

study, the functionally graded material consists of two constituent material types, i.e., ceramic and metal.

According to Eqs. (1) and (2), the material properties can be expressed as

cijðn3Þ ¼ ccijVc þ cmij ð1� VcÞ ¼ ðccij � cmij Þ2n3 þ h

2h

� �n

þ cmij ; ð4Þ

where ccij and cmij are the corresponding material properties of the ceramic and the metal. The governing

equations are given as follows:

(i) Equations of equilibrium and electrostatics

rij;j þ fbi ¼ q€uui; ð5Þ

Di;i ¼ 0: ð6Þ(ii) Constitutive equations

rij ¼ Cijklekl � eijkEk; ð7Þ

Dk ¼ eijkeij þ kklEl: ð8Þ(iii) Strain and electric field equations

eij ¼ 12ðui;j þ uj;iÞ; ð9Þ

Ei ¼ �/;i: ð10Þ

(iv) Displacement and traction boundary conditions

ui ¼ uui ðon CuÞ; ð11Þ

rijnj ¼ fsi ðon CpÞ: ð12Þ(v) Electric boundary conditions

/ ¼ // ðon C/Þ; ð13Þ

Dini ¼ �q ðon CqÞ: ð14ÞHere, Cijkl represents the elastic constants, eijk denotes the piezoelectric stress constants, and kkl denotes thedielectric permittivity coefficients. The components of the stress tensor that are denoted by rij and eijrepresent the components of the strain tensor. Dk denotes the components of the electric displacement

vector, and the electric field is Ei ¼ �/;i, where / is the electric potential. Integrating over the element

region, Xe, results in the weak forms of Eqs. (5) and (6) for an elementZ t1

t0

ZXðrij;j þ fbi � q€uuiÞdui dXdt ¼ 0; ð15Þ


Z t1

t0

ZXDi;id/dXdt ¼ 0: ð16Þ

Integrating Eqs. (15) and (16) once by parts and noting the boundary conditions of Eqs. (11)–(14), we

obtainZ t1

t0

ZXð�q€uuidui � rijdeij þ fbiduiÞdXdt þ

Z t1

t0

ZCp

fsidui dCdt ¼ 0; ð17Þ

Z t1

t0

ZXð�Did/;iÞdXdt �

Z t1

t0

ZCq

qd/dCdt ¼ 0: ð18Þ

3. Finite element formulation

The displacement field of an arbitrary point in the shell can be expressed as (see [10])

uðn1; n2; n3; tÞ ¼ u0ðn1; n2; tÞ þ n3b1ðn1; n2; tÞ;

vðn1; n2; n3; tÞ ¼ v0ðn1; n2; tÞ þ n3b2ðn1; n2; tÞ;

wðn1; n2; n3; tÞ ¼ w0ðn1; n2; tÞ:

ð19a–cÞ

where u, v, and w are displacements along the curvilinear axes n1, n2, and n3, respectively. u0, v0, and w0 arethe displacements of the neutral surface, and b1 and b2 are the bending rotations. The strains associated

with the displacement field in Eqs. (19a–c) are given by

e11 ¼1

a1

ouon1

þ wR1

¼ 1

a1

ou0on1

þ wR1

þ n3a1

ob1

on1;

e22 ¼1

a2

ovon2

þ wR2

¼ 1

a2

ov0on2

þ wR2

þ n3a2

ob2

on2;

e12 ¼1

a2

ouon2

þ 1

a1

ovon1

¼ 1

a2

ou0on2

þ 1

a1

ov0on1

þ n3a2

ob1

on2þ n3a1

ob2

on1;

e13 ¼1

a1

owon1

þ b1;

e23 ¼1

a2

owon2

þ b2;

ð20a–eÞ

where R1 and R2 are the radii of curvatures, and a1 and a2 are the Lame parameters. The relationship

between the electric fields and the electric potential in the curvilinear coordinate system are defined as

E1 ¼ � 1

a1

oUon1

;

E2 ¼ � 1

a2

oUon2

;

E3 ¼ � oUon

:

ð21a–cÞ

3


The displacements and electric potential at the element level can be defined in terms of nodal variables asfollows:

u0ðn1; n2Þ ¼X4

i¼1

u0iwiðn1; n2Þ;

v0ðn1; n2Þ ¼X4

i¼1

v0iwiðn1; n2Þ;

w0ðn1; n2Þ ¼X4

i¼1

w0iwiðn1; n2Þ;

b1ðn1; n2Þ ¼X4

i¼1

b1iwiðn1; n2Þ;

b2ðn1; n2Þ ¼X4

i¼1

b2iwiðn1; n2Þ;

Uðn1; n2; n3Þ ¼X4

i¼1

Uiðn3Þwiðn1; n2Þ;

ð22a–fÞ

where fuei g ¼ fu0i; v0i;w0i; b1i; b2ig for i ¼ 1; 2; 3; 4, and fUeg ¼ fU1;U2;U3;U4g are the generalized nodal

displacements and the nodal electric potentials in a local coordinate system, respectively. wi are the linearinterpolation functions, which are defined as

wi ¼ 14ð1þ 11iÞð1þ ggiÞ; i ¼ 1; 2; 3; 4 ð23Þ

and

1 ¼ n1a; g ¼ n2

b:

The infinitesimal engineering strains that are associated with the displacements are given by

feg ¼ ½Bu�fueg; ð24Þwhere the strain matrix ½Bu� ¼ ½½Bu1�½Bu2�½Bu3�½Bu4��, ½Bui� ¼ ½Cui� þ n3½Dui� for i ¼ 1; 2; 3 and 4, and

½Cui� ¼

1

a1

owi

on10

wi

R1

0 0

01

a2

owi

on2

wi

R2

0 0

1

a2

owi

on2

1

a1

owi

on10 0 0

0 01

a1

owi

on1wi 0

0 01

a2

owi

on20 wi

2666666666666664

3777777777777775

; ð25Þ

½Dui� ¼

0 0 01

a1

owi

on10

0 0 0 01

a2

owi

on2

0 0 01

a2

owi

on2

1

a1

owi

on10 0 0 0 0

0 0 0 0 0

266666666664

377777777775: ð26Þ


The electric field vector fEg can be expressed in terms of nodal variables as

fEg ¼ �½B/�fUeg ð27Þand

½B/� ¼

1

a1

ow1

on1

1

a1

ow2

on1

1

a1

ow3

on1

1

a1

ow4

on11

a2

ow1

on2

1

a2

ow1

on2

1

a2

ow1

on2

1

a2

ow1

on2o

on3

o

on3

o

on3

o

on3

266666664

377777775: ð28Þ

Substituting Eqs. (7)–(10), (22a–f), (24) and (27) into Eqs. (17) and (18), and assembling the element

equations, yields

½Muu�f€uug þ ½Cs�f _uug þ ½Kuu�fug þ ½Ku/�f/g ¼ fFmg; ð29Þ

½Ku/�Tfug � ½K//�f/g ¼ fFqg; ð30Þwhere ½Muu�, ½Cs�, ½Kuu�, ½Ku/� and ½K//� are the mass, damping, stiffness, piezoelectric, and permittivity

matrices respectively. ½Fq� and ½Fm� are the mechanical excitation and electric excitation vectors in a local

coordinate system. Here it should be noted that the material properties are position-dependent. Substi-

tuting Eq. (30) into Eq. (29), we obtain

½Muu�f€uug þ ½Cs�f _uug þ ð½Kuu� þ ½Ku/�½K//��1½Ku/�TÞfug ¼ fFmg þ ½Ku/�½K//��1fFqg: ð31ÞFor the sensor layer, the applied charge fFqg is zero. Using Eq. (30), the sensor output is

f/gs ¼ ½K//��1

s ½Ku/�Ts fug: ð32ÞHere, the subscript �s� denotes the sensors. ½K//�s and ½Ku/�Ts are the piezoelectric and permittivity matrices

of the sensor layer respectively, and are only integrated across the thickness of the layer. The sensor charge

due to deformation is

fFqgs ¼ ½Ku/�Ts fug: ð33ÞThe control law on f/ga is implemented as (see [30])

f/ga ¼ Gdf/gs þ Gvf _//gs ð34Þwhere Gd and Gv are the displacement and velocity feedback control gains respectively, and the subscript �a�denotes the actuators. Substituting Eqs. (32) and (34) into Eq. (30), the feedback force can be obtained as

follows:

fFqga ¼ ½K/u�afug � Gd½K//�a½K//��1

s ½Ku/�Ts fug � Gv½K//�a½K//��1

s ½Ku/�Ts f _uug: ð35ÞSubstituting Eqs. (33) and (35) into Eq. (31) and rearranging yields

½Muu�f€uug þ ð½Cs� þ ½Cu�Þf _uug þ ½K�uu�fug ¼ fFmg; ð36Þ

where ½Cu� is the active damping matrix

½Cu� ¼ Gv½Ku/�a½K//��1

s ½Ku/�Ts ð37Þand

½K�uu� ¼ ½Kuu� þ Gd½Ku/�a½K//��1

s ½Ku/�Ts : ð38Þ


4. Results and discussions

The purpose of this section is to apply the proposed shell element to the dynamic responses of a spherical

FGM shell with integrated piezoelectric sensor and actuator layers. The verification of the proposed shell

element could be referred to the authors� previous work [20].

A spherical FGM shell with integrated piezoelectric sensor and actuator layers is considered as an

example for the dynamic analysis and control in frequency domain. The shell laminate is simply supported

in four edges. The functionally graded material (FGM) is composed of zirconia and aluminum and itsproperties are graded in the thickness direction according to a volume fraction power-law distribution

described in Eq. (4). G-1195N piezoelectric films are bonded to both the top and bottom surfaces of the

FGM shell. The top piezoelectric layer is used as an integrated actuator and the bottom layer as an

integrated sensor. The geometry of the spherical shell laminate is described in Fig. 1(a). The mid-surface

radius of the shell is 100 mm and the lengths of each side of the shell are 100 mm. The thickness of the FGM

shell is 5 mm and each of the G-1195N piezoelectric layers is 0.2 mm thick. The relevant material properties

for G-1195N and the zirconia/aluminum constituents are given in Table 1. It is noteworthy that the uniform

meshes of 400 (20 · 20) elements were used to model the shell.Tables 2–4 show the effect of the volume fraction power law exponent n and displacement control gain

Gd on the first 10 natural frequencies of the shell laminate. It is observed that the natural frequencies

decrease with increases in the power law exponent n in Eq. (3), which controls the volume fraction of the

constituent materials. As shown in Tables 2–4, the maximum natural frequencies occur at n ¼ 0 for all the

control gain Gd. As n is initially increased, the natural frequencies show significant decrease, but when n is

increased to a sufficiently large value of around n ¼ 100, further increase in n will not result in a significant

further frequency decrease. As the displacement feedback control gain Gd contributes to the stiffness ma-

trix, a certain level of frequency control can be achieved by adjusting Gd, as shown in Fig. 5. The corre-sponding first six vibration modes for the shell laminate are displayed in Figs. 2–4 for n ¼ 1000 and Gd ¼ 0,

500 and 1000, respectively. As shown in Figs. 2 and 3, the first and fifth vibration modes for Gd ¼ 0 and 500

are the first and second bending modes respectively, whereas the third and fifth vibration modes for

Gd ¼ 1000, as shown in Fig. 4, are the first and second bending modes.

Table 1

Mechanical properties of the respective materials

Properties Zirconia [23] Aluminum [23] G-1195N [9]

E11 (N/m2) 151· 109 70 · 109 63· 109E22 (N/m2) 151· 109 70 · 109 63· 109E33 (N/m2) 151· 109 70 · 109 63· 109

G23 (N/m2) – – –

G13 (N/m2) – – –

G12 (N/m2) – – –

m12 0.3 0.3 0.3

m13 0.3 0.3 0.3

m23 0.3 0.3 0.3

q (kg/m3) 3000 2707 7600

k W/mK 2.09 204 0.17

d31 (m/V) – – 254· 10�12

d32 (m/V) – – 254· 10�12

k33 (F/m) – – 15· 10�9

pk (C/m2 K) – – 0.25· 10�4

Table 2

Variation of the natural frequencies (Hz) with the power law exponent n with displacement gain Gd ¼ 0

Mode no. n ¼ 0 n ¼ 0:2 n ¼ 0:5 n ¼ 1 n ¼ 5 n ¼ 15 n ¼ 100 n ¼ 1000

1 10305.3 9898.8 9470.1 9019.7 8049.3 7707.0 7513.7 7478.8

2 12389.3 11872.2 11344.6 10822.4 9817.4 9418.7 9132.4 9073.7

3 12389.3 11872.2 11344.6 10822.4 9817.4 9418.7 9132.4 9073.7

4 15187.7 14524.6 13865.6 13245.7 12178.2 11701.9 11296.7 11207.8

5 17526.4 16746.5 15979.7 15274.8 14125.8 13582.4 13087.8 12976.7

6 17526.4 16746.5 15979.7 15274.8 14125.8 13582.4 13087.8 12976.7

7 19493.9 18759.2 17972.4 17124.7 15207.3 14530.6 14171.8 14109.9

8 20885.2 19939.7 19018.9 18190.1 16910.1 16269.2 15651.3 15509.8

9 20885.2 19939.7 19018.9 18190.1 16910.1 16269.2 15651.3 15509.8

10 26201.5 25002.4 23841.3 22810.2 21276.4 20478.1 19680.5 19495.5

Table 3


Mode no. n ¼ 0 n ¼ 0:2 n ¼ 0:5 n ¼ 1 n ¼ 5 n ¼ 15 n ¼ 100 n ¼ 1000

1 11741.3 11425.0 11089.2 10727.9 9853.0 9478.1 9245.7 9202.2

2 12560.3 12095.2 11622.9 11152.4 10155.1 9681.3 9326.7 9253.4

3 12560.3 12095.2 11622.9 11152.4 10155.1 9681.3 9326.7 9253.4

4 14909.0 14268.5 13634.8 13038.0 11932.6 11364.1 10873.1 10765.3

5 17082.6 16310.2 15553.8 14858.2 13657.5 13019.3 12433.0 12301.4

6 17082.6 16310.2 15553.8 14858.2 13657.5 13019.3 12433.0 12301.4

7 20314.7 19362.9 18438.5 17607.1 16268.6 15533.9 14821.1 14657.8

8 20315.2 19362.9 18438.5 17607.1 16268.6 15533.9 14821.1 14657.8

9 25538.4 24320.3 23142.7 22098.7 20504.8 19617.2 18724.7 18517.6

10 25544.4 24324.2 23144.9 22100.4 20506.4 19618.1 18725.6 18518.6

Table 4


Mode no. n ¼ 0 n ¼ 0:2 n ¼ 0:5 n ¼ 1 n ¼ 5 n ¼ 15 n ¼ 100 n ¼ 1000

1 12630.2 12196.7 11754.7 11306.1 10255.2 9694.8 9266.5 9177.9

2 12630.2 12196.7 11754.7 11306.1 10255.2 9694.8 9266.5 9177.9

3 12633.6 12338.8 12017.0 11656.7 10698.0 10251.7 9969.3 9916.3

4 14577.7 13949.5 13328.5 12739.5 11559.8 10885.1 10295.8 10166.3

5 16593.2 15820.2 15063.8 14365.8 13082.9 12334.0 11640.7 11484.8

6 16593.2 15820.2 15063.8 14365.8 13082.9 12334.0 11640.7 11484.8

7 19702.9 18737.8 17801.1 16957.2 15534.8 14690.2 13865.4 13676.1

8 19702.9 18737.8 17801.1 16957.2 15534.8 14690.2 13865.4 13676.1

9 24838.6 23595.2 22393.6 21328.0 19653.7 18662.8 17659.2 17425.7

10 24838.6 23595.2 22393.6 21328.0 19653.7 18662.8 17659.2 17425.7


To simplify the vibration analysis, modal superposition is used and the first six modes are considered inthis modal space analysis. A harmonic excitation F ¼ eixt is imposed to the center of the shell laminate

along n3 direction to achieve the responses corresponding to the bending mode shape. The responses of

displacement, sensor voltage and actuator voltage of the FGM shell are analyzed using the Newmark-bdirect integration technique and the integration parameters c and b are taken to be 0.5 and 0.25, respec-

tively.

Figs. 5–7 show the first two bending mode responses of displacement and sensor voltage at the center of

the shell laminate at various n values of 0, 1, 1000, respectively. Here, Gd ¼ 0, 500 and 1000 are chosen to

Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

020

4060

0

20

40

60–5

–4

–3

–2

–1

0

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1ξ

2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

10

ξ1

ξ2

w (

mm

)

Fig. 2. Free vibration modes for the FGM shell with n ¼ 1000 and Gd ¼ 0 Mode 1.


Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

020

4060

0

20

40

60–5

–4

–3

–2

–1

0

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

10

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

Fig. 3. Free vibration modes for the FGM shell with n ¼ 1000 and Gd ¼ 500.


Mode 3 Mode 4

Mode 1 Mode 2

Mode 5 Mode 6

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

–4

–3

–2

–1

0

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–10

–5

0

5

ξ1

ξ2

w (

mm

)

020

4060

0

20

40

60–5

0

5

ξ1

ξ2

w (

mm

)

Fig. 4. Free vibration modes for the FGM shell with n ¼ 1000 and Gd ¼ 1000.


0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.01

0.02

Gd = 0Gd = 500Gd = 1000

Frequency (Hz)

Def

lect

ion

ampl

itude

(m

m)

0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

10

20

30

40

Gd =0Gd = 500Gd = 1000

Frequency (Hz)

Sens

or v

olta

ge (

Vol

t)

(a) (b)

Fig. 5. (a) The effects of the displacement gain Gd on the deflection response versus frequency for n ¼ 0 and Gv ¼ 0 and (b) the effects of

the displacement gain Gd on the sensor response versus frequency for n ¼ 0 and Gv ¼ 0.

0.8 1 1.2 1.4 1.6

x 104

0

0.01

0.02

Gd =0Gd = 500Gd = 1000

Frequency (Hz)

Def

lect

ion

ampl

itude

(m

m)

0.8 1 1.2 1.4 1.6

x 104

0

10

20

30

40

Gd =0Gd = 500Gd = 1000

Frequency (Hz)

Sens

or v

olta

ge (

Vol

t)

(a) (b)

Fig. 6. (a) The effects of the displacement gain Gd on the deflection response versus frequency for n ¼ 1 and Gv ¼ 0 and (b) the effects of

the displacement gain Gd on the sensor response versus frequency for n ¼ 1 and Gv ¼ 0.

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

x 104

0

0.01

0.02

0.03

0.04

Gd =0Gd = 500Gd = 1000

Frequency (Hz)

Def

lect

ion

ampl

itude

(m

m)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4x 104

0

20

40

60

80

Gd =0Gd = 500Gd =1 000

Frequency (Hz)

Sens

or v

olta

ge (

Vol

t)

(a) (b)

Fig. 7. (a) The effects of the displacement gain Gd on the deflection response versus frequency for n ¼ 1000 and Gv ¼ 0 and (b) the

effects of the displacement gain Gd on the sensor response versus frequency for n ¼ 1000 and Gv ¼ 0.


examine the effect of the displacement control gain on the bending mode. It is observed from Figs. 5–7 that

as the value of Gd increases, the natural frequency for the first bending mode shifts toward right while the


natural frequency for the second bending mode shifts toward left, i.e., with Gd ¼ 0, 500 and 1000 for n ¼ 0,1 and 1000, the natural frequencies for the first bending mode are increased while the natural frequencies

for the second bending mode are decreased, as shown in Figs. 5–7. It is obvious from Figs. 5–7 that the

resonance region could be avoided by adjusting the value of displacement control gain Gd, especially for the

0.8 1 1.2 1.4 1.6 1.8 2x 104

0.8 1 1.2 1.4 1.6 1.8 2x 104

0 0.5 1 1.5 2 1.5 3x 104

0

0.5

1x 10–3

------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Def

lect

ion

(mm

)

Frequency (Hz)

0

0.5

1------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Sens

or v

olta

ge (

Vol

t)

Frequency (Hz)

0

1

2

3

4

5

6

7------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Frequency (Hz)

Act

uato

r vo

ltage

(V

olt)

(a) (b)

(c)

Fig. 8. The effects of the gain Gv on the first and second bending mode responses under the harmonic force versus frequency for n ¼ 0

and Gd ¼ 0.

0 0.5 1 1.5 2 2.5

x 104

0

2

4

6

8

10------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

0.6 0.8 1 1.2 1.4 1.6 1.8

x 10 4

0

1

2------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Frequency (Hz)

Sens

or v

olta

ge (

Vol

t)

Act

uato

r vo

ltage

(V

olt)

Frequency (Hz)(c)

(b)0.6 0.8 1 1.2 1.4 1.6 1.8

x 10 4

0

0.5

1

1.5x 10–3

------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Def

lect

ion

(mm

)

Frequency (Hz)(a)

Fig. 9. The effects of the gain Gv on the first and second bending mode responses under the harmonic force versus frequency for n ¼ 1

and Gd ¼ 0.

0.6 0.8 1 1.2 1.4 1.6

x 104 x 104

x 104

0

0.5

1x 10–3

Frequency (Hz)

0 0.5 1 1.5 20

1

2

3

4

5

6

7------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Act

uato

r vo

ltage

(V

olt)

Frequency (Hz)

------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Def

lect

ion

(mm

)

0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5------- Gv =0.0001- - - - Gv =0.0005

Gv =0.001

Sens

or v

olta

ge (

Vol

t)

Frequency (Hz)(b)(a)

(c)

Fig. 10. The effects of the gain Gv on the first and second bending mode responses under the harmonic force versus frequency for

n ¼ 1000 and Gd ¼ 0.


first natural frequency as this frequency increases significantly with the increase of displacement control

gain Gd.

The effects of the velocity feedback control gain on the first two bending mode responses of the FGM

shell laminate are shown in Figs. 8–10 for n ¼ 0, 1 and 1000, respectively. To examine the effect of the

feedback control algorithm on the vibration control, Gv ¼ 0:0001, 0.0005 and 0.001 are chosen, respec-

tively, for the vibration control of deflection, sensor voltage and actuator voltage. For n ¼ 0, 1 and 1000,

Figs. 8(a,b)–10(a,b) show the first and second bending mode responses of deflection and sensor voltage in

the frequency domain, respectively. As the velocity feedback control gain Gv increases, the peaks of thedeflection and sensor voltage amplitude are very significantly reduced. Compare with Gv ¼ 0:0001, forexample, when Gv ¼ 0:0005 and 0.001, the amplitude values of the deflection response is reduced about 74%

and 86% respectively for n ¼ 0, as shown in Fig. 8(a); about 82% and about 91% respectively for n ¼ 1, as

shown in Fig. 9(a); about 78% and about 89% respectively for n ¼ 1000, as shown in Fig. 10(a). This

indicates a significant damping effect on the resonance region due to the presence of velocity feedback

control. The first and second bending mode responses of actuator voltage in the frequency domain are

presented in Figs. 9 and 10(c) for n ¼ 0, 1 and 1000, respectively. It is interesting to note that the peak

values of actuator voltage remain the same for various velocity feedback control gain values.

5. Conclusions

Based on the first-order shear deformation theory, a generic finite element algorithm is developed for an

FGM shell with piezoelectric sensors and actuators. A constant displacement and velocity feedback control

algorithm coupling the direct and inverse piezoelectric effects is applied in a closed loop system to provide

feedback control of the frequency response of the shell laminate. A modeling procedure is presented for thevibration control of a shell laminate using self-monitoring sensor and self-controlling actuator layers. The

simulation results demonstrate that mode shapes and resonance frequencies could be controlled by


appropriately adjusting the displacement control gain. The numerical results also show that the velocityfeedback control gain could provide a very active damping on the vibration in resonance region.

Acknowledgements

The work described in this paper was supported by the NTU research grant (contract numbers: RGM

48/01) and the grant awarded by Research Grants Council of the Hong Kong Special Administrative

Region, China (Project No. CityU 1024/01E).

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Documents

Finite element method for the feedback control of FGM shells in the frequency domain via piezoelectric sensors and actuators