21
Shock and Vibration 20 (2013) 531–550 531 DOI 10.3233/SAV-130766 IOS Press Forced vibration analysis for a FGPM cylindrical shell Hong-Liang Dai a,b,c,and Hao-Jie Jiang a,c a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan, China b Key Laboratory of Manufacture and Test Techniques for Automobile Parts, Ministry of Education, Chongqing University of Technology, Chongqing, China c Department of Engineering Mechanics, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan, China Received 14 June 2012 Revised 8 October 2012 Accepted 7 December 2012 Abstract. This article presents an analytical study for forced vibration of a cylindrical shell which is composed of a functionally graded piezoelectric material (FGPM). The cylindrical shell is assumed to have two-constituent material distributions through the thickness of the structure, and material properties of the cylindrical shell are assumed to vary according to a power-law distribution in terms of the volume fractions for constituent materials, the exact solution for the forced vibration problem is presented. Numerical results are presented to show the effect of electric excitation, thermal load, mechanical load and volume exponent on the static and force vibration of the FGPM cylindrical shell. The goal of this investigation is to optimize the FGPM cylindrical shell in engineering, also the present solution can be used in the forced vibration analysis of cylindrical smart elements. Keywords: Forced vibration, FGPM, cylindrical shell, analytical study, heat conduction 1. Introduction FGPM have experienced a remarkable increase in terms of research and development. By using the continu- ous change in the physical and mechanical properties of a material, it is possible to prevent fracture in composite materials, avoiding the phenomenon of stress concentration and yield in such materials. Vibration of shells is an indispensable branch of research in structural dynamics. Cylindrical shells also have vast range of applications in engineering and technology. Many studies for vibration of cylindrical structures composed of functionally graded materials are available in the literatures, Pradhan et al. [1] investigated vibration characteristics of FGM cylindrical shells under various boundary conditions. By means of the Bolotin’s method, Ng et al. [2] gave a stability analysis of FGM cylindrical shells under harmonic axial loading. Han et al. [3] gave a numerical method for analyzing tran- sient waves in FGM cylindrical shells excited by impact point loads. By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which had been solved successfully by Hou et al. [4]. Accord- ing to a simple power law distribution in terms of the volume fractions of the constituents, Shen [5] presented a Corresponding author: Hong-Liang Dai, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, China. Tel.: +86 0731 8882 2330; Fax: +86 0731 8882 2330; E-mail: [email protected]. ISSN 1070-9622/13/$27.50 c 2013 – IOS Press and the authors. All rights reserved

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Page 1: Forced vibration analysis for a FGPM cylindrical shelldownloads.hindawi.com/journals/sv/2013/512847.pdfForced vibration analysis for a FGPM ... presented linear thermal buckling and

Shock and Vibration 20 (2013) 531–550 531DOI 10.3233/SAV-130766IOS Press

Forced vibration analysis for a FGPMcylindrical shell

Hong-Liang Daia,b,c,∗ and Hao-Jie Jianga,caState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha,Hunan, ChinabKey Laboratory of Manufacture and Test Techniques for Automobile Parts, Ministry of Education, ChongqingUniversity of Technology, Chongqing, ChinacDepartment of Engineering Mechanics, College of Mechanical and Vehicle Engineering, Hunan University,Changsha, Hunan, China

Received 14 June 2012

Revised 8 October 2012

Accepted 7 December 2012

Abstract. This article presents an analytical study for forced vibration of a cylindrical shell which is composed of a functionallygraded piezoelectric material (FGPM). The cylindrical shell is assumed to have two-constituent material distributions throughthe thickness of the structure, and material properties of the cylindrical shell are assumed to vary according to a power-lawdistribution in terms of the volume fractions for constituent materials, the exact solution for the forced vibration problem ispresented. Numerical results are presented to show the effect of electric excitation, thermal load, mechanical load and volumeexponent on the static and force vibration of the FGPM cylindrical shell. The goal of this investigation is to optimize the FGPMcylindrical shell in engineering, also the present solution can be used in the forced vibration analysis of cylindrical smart elements.

Keywords: Forced vibration, FGPM, cylindrical shell, analytical study, heat conduction

1. Introduction

FGPM have experienced a remarkable increase in terms of research and development. By using the continu-ous change in the physical and mechanical properties of a material, it is possible to prevent fracture in compositematerials, avoiding the phenomenon of stress concentration and yield in such materials. Vibration of shells is anindispensable branch of research in structural dynamics. Cylindrical shells also have vast range of applications inengineering and technology. Many studies for vibration of cylindrical structures composed of functionally gradedmaterials are available in the literatures, Pradhan et al. [1] investigated vibration characteristics of FGM cylindricalshells under various boundary conditions. By means of the Bolotin’s method, Ng et al. [2] gave a stability analysisof FGM cylindrical shells under harmonic axial loading. Han et al. [3] gave a numerical method for analyzing tran-sient waves in FGM cylindrical shells excited by impact point loads. By virtue of the introduction of a dependentvariable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problemof a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of thesecond kind about a function with respect to time, which had been solved successfully by Hou et al. [4]. Accord-ing to a simple power law distribution in terms of the volume fractions of the constituents, Shen [5] presented a

∗Corresponding author: Hong-Liang Dai, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University,Changsha 410082, Hunan, China. Tel.: +86 0731 8882 2330; Fax: +86 0731 8882 2330; E-mail: [email protected].

ISSN 1070-9622/13/$27.50 c© 2013 – IOS Press and the authors. All rights reserved

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532 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

postbuckling analysis for a FGM cylindrical shell of finite length subjected to external pressure and in thermal envi-ronments. Utilizing the higher-order theory, Patel et al. [6] studied the free vibration characteristics of functionallygraded elliptical cylindrical shells. Paulino and Silva [7] applied topology optimization to design FGM structuresconsidering a minimum compliance criterion. Bhangale and Ganesan [8] carried out free vibration studies on func-tionally graded materials magneto-elastro-elastic cylindrical shells; they [9] presented linear thermal buckling andfree vibration analysis for functionally graded cylindrical shells with clamped-clamped boundary conditions basedon temperature-dependent material properties. By means of the separation of variables technique as well as the su-perposition method, Wang and Ding [10] investigated transient responses of a special non-homogeneous magneto-electro-elastic hollow cylinder for a fully coupled axisymmetric plane strain problem. By means of the Rayleigh-Ritz method, Arshad et al. [11] performed a frequency analysis for FGM circular cylindrical shells. By means of atwo-dimensional higher-order deformation theory, Matsunaga [12] investigated vibration and buckling problems ofFGM cylindrical shells. Based on the three-dimensional piezoelectricity, Wu and Tsai [13] investigated cylindricalbending vibration of functionally graded piezoelectric shells using the method of perturbation. By using the element-free Kp-Ritz method, Zhao et al. [14] gave thermoelastic and vibration analysis of functionally graded cylindricalshells. Arshad et al. [15] presented a vibration frequency analysis of a bi-layered cylindrical shell composed of twoindependent functionally graded layers. Shah et al. [16] investigated vibrations of functionally graded cylindricalshells based on elastic functions. Based on theory of vibrations of cylindrical shells, Sofiyev [17] presented an an-alytical study on the dynamic behavior of the infinitely-long FGM cylindrical shell under moving loads. By usingthird order shear deformation theory, Daneshjou et al. [18] gave an analytical solution for acoustic transmissionthrough relatively thick FGM cylindrical shells. Based on Sanders’ thin shell theory, the vibrational behavior offunctionally graded cylindrical shells with intermediate ring supports was studied by Rahimi et al. [19]. Based onthe three-dimensional theory of elasticity, free vibration analysis of a functionally graded cylindrical shell embeddedin piezoelectric layers was performed by Alibeigloo et al. [20]. Based on the first order shear deformation theory ofshells, the free vibration analysis of rotating functionally graded cylindrical shells subjected to thermal environmentis investigated by Malekzadeh and Heydarpour [21]. However, As far as we know, the exact solution presented inthis paper, on forced vibration analysis for a FGPM cylindrical shell is a novel and an easily operated method in thisarea.

In the paper, an analytical solution for forced vibration of a FGPM cylindrical shell is presented. This methodis easily understood, and has been validated by comparing the results with Ma and Wang [22]. Numerical resultsare presented to show the effect of electric excitation, thermal load, mechanical load and volume exponent on thestatic and dynamic response of the FGPM cylindrical shell, it will be of great value when engineers design optimumcylindrical smart structures in engineering.

2. Theoretical analysis

Consider a FGPM cylindrical shell subjected to electric excitation φ(z), mechanical load q and thermal load T (z).Its inner radius a, length L and thickness h are shown in Fig. 1. The Cartesian coordinate system (x, y, z) is set onthe mid-plane (z = 0) of the FGPM cylindrical shell, where x and y denotes the axial and circumferential directionsof the mid-plane of the FGPM cylindrical shell, respectively.

2.1. FGPM material properties

The FGPM cylindrical shell is composed of metal and ceramic materials and the material constituents of theshell varies from the outer surface to the inner surface, i.e. the outer surface (z = −h/2) of the cylindrical shell ismetal-rich and the inner surface (z = h/2) is ceramic-rich. In such a way, material properties P (e.g., modulus ofelasticity E, elastic constants cij (i, j = 1, 2), thermal expansion coefficient αi (i = 1, 2), thermal conductivityK , piezoelectric constants ei and pyroelectric coefficients p(z)) of the FGPM cylindrical shell are assumed to varythrough the thickness of the shell. Here, FGPM’s material properties P are related not only to material properties ofthe material constituent, but also to their volume fractions V1 and V2, therefore, one has

P (z) = P1V1 + P2V2 = P2 + (P1 − P2)V1 (1)

where subscript 1 and 2 denotes, respectively, metal material and ceramic material.

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 533

Fig. 1. (a) The geometry of a FGPM cylindrical shell; (b) Fixed of both ends; (c) Simply supported of both ends.

Assuming V1 follows a simple power law distribution [23–32]:

V1 =

(h− 2z

2h

)n

(2)

where n (0 < n < ∞) is the exponent of volume, and n represents the inhomogeneity of FGPMs, and it degeneratesinto metal material at n = 0, when n → ∞, it becomes ceramic material.

2.2. Heat conduction equation

Assuming that the rise of temperature occurs only in the thickness direction of the FGPM cylindrical shell, theone-dimensional steady heat conduction equations is

d

dz

[K(z)

dT (z)

dz

]= 0 (3)

The corresponding thermal boundary conditions of the FGPM cylindrical shell are

T

(−h

2

)= T1, T

(h

2

)= T2 (4)

where T1 and T2 are, respectively, the inner and outer surface temperature of the FGPM cylindrical shell.Solving Eq. (3), solution of the temperature can be written as

T (z) = T1

⎡⎢⎢⎢⎣1 +

(T2

T1− 1

) ∫ z

−h/2

1

K(z)dz

∫ h/2

−h/2

1

K(z)dz

⎤⎥⎥⎥⎦ (5)

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534 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

2.3. Governing equations

According to the geometric symmetry of the FGPM cylindrical shell, the strain-displacement relations are

εx = ε0x + zk0x, εy = ε0y + zk0y, εz = 0 (6)

where

ε0x =∂u

∂x, ε0y = −w

a, k0x =

∂2w

∂x2, k0y =

w

a2(7)

where u and w are the displacements along x and z axes, respectively, ε0x and ε0y are strain components along x andy axes on the middle surface of the shell’s structure, respectively, k0x and k0y are the curvature change of x and ydirections on the middle surface of the shell’s structure, respectively.

The electrothermoelastic constitutive relations of the FGPM cylindrical shell are

σx = c1(z)εx + c2(z)εy + λ(z)T (z)− e1(z)∂φ

∂z(8a)

σy = c2(z)εx + c1(z)εy + λ(z)T (z)− e2(z)∂φ

∂z(8b)

Dz = e1(z)εx + e2(z)εy + p(z)T (z)− g(z)∂φ

∂z(8c)

where σi(i = x, y), Dz , ci(i = 1, 2), ei(i = 1, 2), p(z), λ(z) and φ are the components of stresses, radial electricdisplacement, elastic constants, piezoelectric constants, pyroelectric coefficients, thermal expansion coefficients andelectric potential, respectively, and

c1(z) =E(z)

1− v2, c2(z) =

vE(z)

1− v2, λ(z) = −E(z)

1− vα(z),

e1(z) = e12 + (e11 − e12)V1, e2(z) = e22 + (e21 − e22)V1 (9)

where v denotes Poisson’s ratio, and it is assumed to be a constant in this paper.In absence of free charge density, the charge equation of electrostatics (Heyliger [33]) is expressed as

∂Dz

∂z+

Dz

z − a= 0 (10)

Solving the Eq. (10), it can be obtained as

Dz =kd

a− z(11)

where kd is determined by the boundary conditions, substituting Eq. (11) into Eq. (8c), one gets

∂φ

∂z=

e1(z)εx + e2(z)εyg(z)

− kdg(z)(a− z)

+p(z)T (z)

g(z)(12)

Define the electric potential boundary conditions as:

φ

∣∣∣∣z=−h2

= φ−h2

, φ

∣∣∣∣z=+h2

= φ+

h2

(13)

Substituting Eq. (12) into Eqs (8a) and (8b), yields

σx = Θ11εx +Θ12εy +Θ13 +Θ14 +Θ15 (14a)

σy = Θ21εx +Θ22εy +Θ23 +Θ24 +Θ25 (14b)

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 535

where the expressions of Θij(i = 1, 2; j = 1, 2 · · ·5) are shown in Appendix A.According to the axial-symmetry of the FGPM cylindrical shell, when investigating on the force vibration of this

structure, the equilibrium equations of the shell are

a∂Qx

∂x− Ny + aqi = aq + aqt (15a)

a∂Nx

∂x= 0 (15b)

a∂Mx

∂x+ aQx = 0 (15c)

where Qx, Ni(i = x, y) and Mx denote, respectively, the transversely shear force, axial force of membrane andbending moment, and q, qt and qi denote the static load, dynamic load and inertia force, respectively. When inves-tigating the static problem or the free vibration of the cylindrical shell, the first equilibrium equation of Eq. (15)degenerates into the following form, respectively,

a∂Qx

∂x− Ny = aq (15a,1)

a∂Qx

∂x− Ny + aqi = aq (15a,2)

The corresponding expressions are shown as follows:

Nx =

∫ h/2

−h/2

σx

(1− z

a

)dz (16a)

Ny =

∫ h/2

−h/2

σydz (16b)

Mx = −∫ h/2

−h/2

σxz(1− z

a

)dz (16c)

and

qi =

∫ h/2

−h/2

fz

(1− z

a

)dz (17)

where qi is defined as the inertia force resultant along unit length of the circumferential central line, and

fz = −ρ(z)∂2w

∂t2(18)

The negative sign indicates the opposite direction between the direction of acceleration and that of inertial force.Substituting Eq. (18) into Eq. (17), yields

qi =

∫ h/2

−h/2

[−ρ(z)

∂2w

∂t2

(1− z

a

)]dz = −ρ̄

∂2w

∂t2(19)

where

ρ̄ =

∫ h/2

−h/2

ρ(z)(1− z

a

)dz (20)

Substituting Eqs (15c) and (19) into Eqs (15a) and (15b), yields

a∂2Mx

∂x2+Ny + aρ̄

∂2w

∂t2= aq + aqt (21a)

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536 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

a∂Nx

∂x= 0 (21b)

Then, substituting Eq. (14) into Eq. (16), yields

Nx = A11∂u

∂x+A12

∂2w

∂x2+A13w +A14 +A15 +A16 (22a)

Ny = A21∂u

∂x+A22

∂2w

∂x2+A23w +A24 +A25 +A26 (22b)

Mx = A31∂u

∂x+A32

∂2w

∂x2+A33w +A34 +A35 +A36 (22c)

where Aij(i = 1, 2, 3; j = 1, 2 · · ·6) are shown in Appendix B.

3. Solution of the electrothermoelastic deformation problem

Substituting Eq. (22) to Eq. (21), yields

aA32∂4w

∂x4+ (aA33 +A22)

∂2w

∂x2+A23w + aA31

∂3u

∂x3 (23a)

+A21∂u

∂x+A24 +A25 +A26 + aρ̄

∂2w

∂t2+ aq + aqt = 0

A11∂2u

∂x2+A12

∂3w

∂x3+A13

∂w

∂x= 0 (23b)

Integrating both sides of the Eq. (23b) with respect to x, yields

∂u

∂x= −A12

A11

∂2w

∂x2− A13

A11w − k1

A11(24)

where k1 is an integral constant to be determined by the boundary conditions.Differentiating on both sides of Eq. (23b), yields

∂3u

∂x3= −A12

A11

∂4w

∂x4− A13

A11

∂2w

∂x2(25)

Substituting Eqs (24) and (25) into Eq. (23a), yields

G1∂4w

∂x4+G2

∂2w

∂x2+G3w +G4 + aρ̄

∂2w

∂t2+ aqt = 0 (26)

where

G1 = aA32 − aA31A12

A11, G2 = aA33 +A22 − aA31A13

A11− A21A12

A11,

G3 = A23 − A21A13

A11, G4 = A24 +A25 +A26 − A21

A11k1 + aq (27)

where G1, G2, G3 and ρ̄ are constants, and G4 is a term related to the boundary conditions at the two ends and theelectric field boundary conditions.

First, assuming that the cylindrical shell is only subjected to static mechanical and temperature loads, subsequentlythe cylindrical shell reaches the state of equilibrium, it can be written as

G1∂4we

∂x4+G2

∂2we

∂x2+G3we +G4 + aρ̄

∂2we

∂t2= 0 (28)

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 537

Noticing that the static deflection is time-independent, it gets

∂2we

∂t2= 0 (29)

Then, consider the FGPM cylindrical shell subjected to dynamic load qt and assuming that the transient perturba-tion at any point is wt, then

G1∂4wt

∂x4+G2

∂2wt

∂x2+G3wt +G4 + aqt + aρ̄

∂2wt

∂t2= 0 (30)

Equation (30) minus Eq. (28), yields

G1∂4(wt − we)

∂x4+G2

∂2(wt − we)

∂x2+G3(wt − we) + aqt + aρ̄

∂2(wt − we)

∂t2= 0 (31)

where (wt − we) is the displacement measured from the static equilibrium position, for the sake of analysis, wemark it as wa = wt − we. Then the vibration equation from the static equilibrium position can be obtained

G1∂4wa

∂x4+G2

∂2wa

∂x2+G3wa + aρ̄

∂2wa

∂t2+ aqt = 0 (32)

The perturbation is assumed to have the form as follows

wa(x, t) =

∞∑m=1

wam(x, t) =

∞∑m=1

Sin(mπ

Lx)Hm(t)

=

∞∑m=1

Sin(mπ

Lx)[AmSin(ωmt) +BmCos(ωmt) + τm(t)]

(33)

where ωm is the mth order of natural frequency.Substituting Eq. (33) into Eq. (32), yields

G1

(mπ

L

)4

−G2

(mπ

L

)2

+G3 − aρ̄ω2m + aqt = 0 (34)

For free vibration,

G1

(mπ

L

)4

−G2

(mπ

L

)2

+G3 − aρ̄ω2m = 0 (35)

Then, the natural frequency of the FGPM cylindrical shell is obtained as

ωm =1√aρ̄

√G1

(mπ

L

)4

−G2

(mπ

L

)2

+G3 (36)

Assume the dynamic load as

qt = q0(x)Cos (ωF t) (37)

where ωF is the loading frequency, expanding Eq. (37) into the form of mode shape, yields

qt =

∞∑m=1

qt,m =

∞∑m=1

FmSin(mπ

Lx)Cos (ωF t) (38)

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538 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

where

Fm =2

L

∫ L

0

q0(x)Sin(mπ

Lx)dx (39)

The orthogonality of mode shapes leads to the uncoupled form of equation for motion, in other words, it allowsfor the equations of motion to be written in an uncoupled form. Just take the mth mode shape into discussion, then

wa = wam = Sin(mπ

Lx)Hm(t) (40)

qt = qt,m = FmSin(mπ

Lx)Cos (ωF t) (41)

Substituting Eqs (40) and (41) into Eq. (34), yields[G1

(mπ

L

)4

−G2

(mπ

L

)2

+G3

]Hm(t) + aρ̄

d2Hm(t)

dt2+ aFmCos (ωF t) = 0 (42)

From Eq. (35), one knows

G1

(mπ

L

)4

−G2

(mπ

L

)2

+G3 = aρ̄ω2m (43)

Substituting Eq. (43) into Eq. (42), yields

aρ̄ω2mHm(t) + aρ̄

d2Hm(t)

dt2+ aFmCos (ωF t) = 0 (44)

Then

d2Hm(t)

dt2+ ω2

mHm(t) = −Fm

ρ̄Cos (ωF t) (45)

One particular solution of Eq. (45) is

hm(t) = −FmCos (ωF t)

ρ̄ (ω2m − ω2

F )(46)

Express it as τm(t), gives

τm(t) = hm(t) = −FmCos (ωF t)

ρ̄ (ω2m − ω2

F )(47)

Thus, the deflection may be written as

wa (x, t) =

∞∑m=1

Sin(mπ

Lx)[AmSin (ωmt) +BmCos (ωmt) + τm(t)]

=

∞∑m=1

Sin(mπ

Lx) [

AmSin (ωmt) +BmCos (ωmt)− FmCos (ωF t)

ρ̄ (ω2m − ω2

F )

] (48)

and the velocity may be expressed as

v (x, t) =∂w (x, t)

∂t=

∂wa (x, t)

∂t

=

∞∑m=1

Sin(mπ

Lx)[

[AmCos (ωmt)−BmSin (ωmt)]ωm + ωFFmSin (ωF t)

ρ̄ (ω2m − ω2

F )

] (49)

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 539

Assume the initial condition as

w(x, t)|t=0 = w0(x) (50a)

v(x, t)|t=0 =∂wa (x, t)

∂t

∣∣∣∣t=0

= v0(x) (50b)

The Eqs (50a) and (50b) can be changed into the following series form

w0(x) =

∞∑m=1

CmSin(mπ

Lx)

(51a)

v0(x) =∞∑

m=1

DmSin(mπ

Lx)

(51b)

Applying the orthogonality of trigonometric series results in

Cm =2

L

∫ L

0

w0(x)Sin(mπ

Lx)

(52a)

Dm =2

L

∫ L

0

v0(x)Sin(mπ

Lx)

(52b)

Substituting the initial condition Eq. (50a) into the deflection expression, yields

wa (x, t)|t=0 =

∞∑m=1

Sin(mπ

Lx) [

AmSin(ωmt) +BmCos (ωmt)− FmCos (ωF t)

ρ̄ (ω2m − ω2

F )

]∣∣∣∣∣t=0

=

∞∑m=1

Sin(mπ

Lx) [

Bm − Fm

ρ̄ (ω2m − ω2

F )

] (53)

From Eqs (51a) and (53), it can be obtained as

Bm = Cm +Fm

ρ̄ (ω2m − ω2

F )(54)

Substituting the initial condition Eq. (50b) into the velocity expression, yields

∂wa (x, t)

∂t

∣∣∣∣t=0

=

∞∑m=1

Sin(mπ

Lx) [

[AmCos(t)−BmSin(ωmt)]ωm + ωFFmSin(ωF t)

ρ̄ (ω2m − ω2

F )

]∣∣∣∣∣t=0

=

∞∑m=1

Sin(mπ

Lx)Amωm

(55)

From Eqs (51b) and (55), gives

Am =Dm

ωm(56)

Substituting the expressions of Am and Bm into Eq. (48), gives

wa(x, t) =

∞∑m=1

Sin(mπ

Lx)

×[(

Dm

ωm

)Sin(ωmt) +

(Cm +

Fm

ρ̄ (ω2m − ω2

F )

)Cos (ωmt)− FmCos (ωF t)

ρ̄ (ω2m − ω2

F )

] (57)

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540 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

From Eqs (40) and (57), it can be obtained as

Hm(t) =

[(Dm

ωm

)Sin(ωmt) + CmCos (ωmt)

]+

[Fm

ρ̄ (ω2m − ω2

F )[Cos (ωmt)− Cos (ωF t)]

](58)

There are two terms included in Eq. (58). First consider the first term and simplify it, yields

(Dm

ωm

)Sin(ωmt) + CmCos (ωmt) =

√(Cm)

2+

(Dm

ωm

)2

Sin(ωmt+ ϕ0) (59)

where

ϕ0 = Tan−1

(Cm

Dmωm

)(60)

It is easily noticed that the amplitude in Eq. (59) doesn’t vary with time, it is a term of free vibration.Next, discussing the second term of Eq. (58), noticing that ωF , which is the frequency of the dynamic load qt,

approaches to one of the natural frequency ωm, terms like 0/0 appears. In such case, measures could be taken asfollows:

Fm

ρ̄ (ω2m − ω2

F )[Cos (ωmt)− Cos (ωF t)]

=Fm

ρ̄ (ω2m − ω2

F )

[2Sin

(ωF + ωm) t

2Sin

(ωF − ωm) t

2

] (61)

when ωF → ωm, the term with the corresponding infinitesimal need to be replaced as

Sin(ωF − ωm) t

2=

(ωF − ωm) t

2(62)

Substituting Eq. (62) into Eq. (61), and take the limit ωF → ωm, yields

limωF→ωm

[Fm

ρ̄ · (ω2m − ω2

F )

[2Sin

(ωF + ωm) t

2· Sin (ωF − ωm) t

2

]]

= limωF→ωm

[Fm

ρ̄ (ωF − ωm) (ωF + ωm)

[2Sin

((ωF + ωm) t

2

)(ωF − ωm) t

2

]]

=Fmt

2ρ̄ωmSin (ωmt)

(63)

Substituting Eq. (63) into Eq. (58), yields

Hm(t)|ωF→ωm=

⎡⎣√(Cm)

2+

(Dm

ωm

)2

Sin (ωmt+ ϕ0)

⎤⎦+

[Fmt

2ρ̄ωmSin (ωmt)

](64)

Finally substituting Eq. (64) into Eq. (57), yields

w (x, t)|ωF→ωm=

∞∑m=1

⎛⎝√(Cm)2 +

(Dm

ωm

)2

Sin (ωmt+ ϕ0)

⎞⎠Sin

(mπ

Lx)

+

∞∑m=1

[Fmt

2ρ̄ωmSin (ωmt)

]Sin

(mπ

Lx) (65)

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 541

Fig. 2. Validation of the present method. Fig. 3. Variation law of the fraction of ceramic along z-axis with different k.

Consider the initial state of rest, then

w0(x) = 0, v0(x) = 0 (66)

Substituting Eq. (66) into Eq. (52), yields

Cm =2

L

∫ L

0

w0(x)Sin(mπ

Lx)= 0 (67a)

Dm =2

L

∫ L

0

v0(x)Sin(mπ

Lx)= 0 (67b)

Therefore, Eq. (65) can be turned into

wa (x, t)|ωF→ωm=

∞∑m=1

[Fmt

2ρ̄ωmSin (ωmt)

]Sin

(mπ

Lx)

(68)

4. Numerical examples and discussion

4.1. Simple validation of the present method

In order to validate the analytical method presented in the paper, it was applied to solve the bending and post-buckling problem of a FGM circular plate under mechanical and thermal loadings taken the same condition andparameters as a reference (Ma [22]). Variations of the maximal deflection of the mid-plane for the fixed and simplysupported FGM structures are depicted in Fig. 2, which reveals that the results are nearly the same.

4.2. Static deflection of the FGPM cylindrical shell

Consider an FGPM cylindrical shell with length L = 10 m, inner radius a = 2 m, the ratio of radius to thicknessis 20. In all the following numerical calculations, the following material constants for the FGPM cylindrical shellare shown in Table 1. (Dunn and Taya [34]; Dai and Wang [35]). The variation law of the fraction of ceramic alongz-axis with different k is shown in Fig. 3.

In this section, assuming the two ends of the FGPM cylindrical shell is fixed, then the boundary conditions maybe expressed as

w|x=0,L = 0; u|x=0,L = 0;dw

dx

∣∣∣∣x=0,L

= 0 (69)

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542 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

Table 1Material parameters of FGPM cylindrical shell

Material properties Symbol Unit Almin um/Al (i = 1) Zironia/ZrO2 (i = 2)

Young’s modulus Ei GPa 70 151Coefficient of thermal expansion αi /k 23×10−6 10×10−6

Thermal conductivity Ki W/mk 204 2.9

Piezoelectric constantsei1 C/m2 15.1 15.0ei2 C/m2 −5.2 −5.0

Dielectric constant gi C2/Nm2 5.62×10−9 5.0×10−9

Pyroelectric coefficient pi N/m2k −2.5×10−5 −2.2×10−5

Density ρi kg/m3 2.7×103 5.89×103

Fig. 4. Deflection of the mid-plane for the clamped supported FGPMcylindrical shell imposed by a uniform mechanical load, with differentvolume fraction index n (q = 1 MPa, T1 = 0, T2 = 0, a = 1 m,L = 4 m, a/h = 20, φ1 = 0, φ2 = 0).

Fig. 5. Deflection of the mid-plane for the clamped supported FGPMcylindrical shell imposed by a thermal load, with different volumefraction index n (q = 0 MPa, T1 = 200, T2 = 0, a = 1 m,L = 4 m, a/h = 20, φ1 = 0, φ2 = 0).

Figures 4 and 5 show the distribution of the radial displacement w along x-axis with different volume fractionindex n when the clamped-clamped FGPM cylindrical shell is only loaded by mechanic load or thermal load,respectively. As can be seen from Fig. 4 that w increases with the decrease of n. It depends mostly on the variationof the stiffness of the material. Figure 5 reveals that comparing with the mechanic load, the thermal load exertsmore remarkable influence on w. With the increase of n, the hot deformation decreases when n � 2, but increasesslowly to a stable value when n > 2. That depends mostly on the change of thermal effect of the FGPM. Both theFigs 4 and 5 indicate that when the FGPM cylindrical shell is under either the mechanic load or the thermal load,the maximum value of w along the x-axis happens not in the center but in the places where, respectively, is 0.4 maway from each end. Figure 6 shows the effect of the thermal condition on the radial displacement w along x-axisof the FGPM cylindrical shell when q = 1 MPa, n = 1. It is clearly revealed that the larger the thermal load is, thelarger the radial displacement is.

For the FGPM cylindrical shell, the effect of the electric potential on its response is worthy of concern. Figure 7shows the distribution of the radial displacement w along x-axis of the FGPM cylindrical shell with n = 1, when itis located in different electric field. It is clearly shown that the influence of the electric potential boundary conditionsover w is not obvious. Figure 8 reveals that the distribution of the electric potential does not vary linearly along z-axis, and the maximum value of the electric potential occurs in the points deviated from the neutral surface. Figures 9and 10 show the relationship of the mechanical load or the thermal load with the electric potential distribution,respectively. They indicate that with the increases of the mechanical load or the thermal load, the value of theelectric potential in the same place along the radial direction increases, and the maximum value is obtained in thepoints deviated from the neutral surface.

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 543

Fig. 6. Deflection of the mid-plane for the clamped supported FGPMcylindrical shell imposed by different thermal loads (q = 1 MPa,a = 1 m, L = 4 m, a/h = 20, φ1 = 1000, φ2 = 0, n = 1).

Fig. 7. Deflection of the mid-plane for the clamped supportedFGPM cylindrical shell under different electric fields (q = 1 MPa,T1 = 200, T2 = 0, a = 1 m, L = 4 m, a/h = 20, n = 1).

Fig. 8. Distribution of the electric potential φ along z-axis forthe clamped supported FGPM cylindrical shell with different n(q = 1 MPa, T1 = 200, T2 = 0, a = 1 m, L = 4 m, a/h = 20,φ1 = 1000, φ2 = 0).

Fig. 9. Distribution of the electric potential φ along z-axis for theclamped supported FGPM cylindrical shell imposed by differentmechanical loads (T1 = 200, T2 = 0, a = 1 m, L = 4 m,a/h = 20, φ1 = 1000, φ2 = 0).

4.3. Free vibration of the FGPM cylindrical shell

In this section, the effect of volume fraction index n on the free vibration of the simply supported FGPM cylin-drical shell is studied. All geometric parameters of the FGPM cylindrical shell and material parameters are the sameas the Section 4.2.

Figure 11 gives the effect of volume fraction index n on each order natural frequency ω for the simply supportedFGPM cylindrical shell. Seen from Fig. 11, the effect of volume fraction index n on natural frequency is small whenthe lateral half wave number m < 4. After that the high order natural frequency increase rapidly, and the effect ofvolume fraction index n on high order natural frequency gets larger.

Figure 12 shows the effect of volume fraction index n on the low order natural frequency ω for the simplysupported FGPM cylindrical shell. It can be easily seen that the 1th order, 2th order and 3th order natural frequencyfall abruptly with the increase of volume fraction index n. The slippage reach to the maximum value near n = 5,after that, the curves decrease gently.

Figure 13 presents the effect of volume fraction index n on the high order natural frequency ω for the simplysupported FGPM cylindrical shell. From Fig. 13, the effect of volume fraction index n on the high order naturalfrequency is obviously larger than that the low order natural frequency is. The high order natural frequency ω

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544 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

Fig. 10. Distribution of the electric potential φ along z-axis for theclamped supported FGPM cylindrical shell imposed by differentthermal loads (q = 1 MPa, a = 1 m, L = 4 m, a/h = 20,φ1 = 1000, φ2 = 0).

Fig. 11. The effect of volume fraction index n on each order naturalfrequency ω for the simply supported FGPM cylindrical shell.

Fig. 12. The effect of volume fraction index n on thelow order natural frequency ω for the simply sup-ported FGPM cylindrical shell.

Fig. 13. The effect of volume fraction index n on the high order natural frequency ωfor the simply supported FGPM cylindrical shell.

decrease with the increase of volume fraction index when n < 5, and reach to a minimal value at this location. Afterthat, the natural frequency increase with the increase of volume fraction index, which can’t be found in low orderfrequency. In other words, the constituents of material reach to optimum in high order frequency.

4.4. Force vibration of the FGPM cylindrical shell

In this section, the force vibration of the simply supported FGPM cylindrical shell is investigated. As calculatedin the former Section 4.2, the influence of the electric potential boundary conditions over w is not obvious, andthe thermal boundary conditions have also been discussed in detail. Then in this section the temperatures of innerand outer surfaces are, respectively, taken as T1 = 20◦C and T2 = 20◦C, and the corresponding electric potentialboundary conditions are taken as φ1 = 0, φ2 = 0.

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 545

Fig. 14. Deflection of the mid-plane for the simply supported FGPMcylindrical shell imposed by a uniform mechanical load, whenn = 1.

Fig. 15. Deflection of the mid-plane for the simply supported FGPMcylindrical shell imposed by a uniform mechanical load, whent = 1× 10−4.

The simply supported boundary conditions may be expressed as follows

w|x=0,L = 0, u|x=0,L = 0, Mx|x=0,L = 0 (70)

In the following numerical calculations, three kinds of loading are considered respectively. The driving forces aresupposed to be harmonic forces with the cosine form as Eq. (37).

Case 1 Consider a simply supported FGPM cylindrical shell which is imposed by a uniform distributed load

q0(x) = 105Pa (71)

Figure 14 shows that the instantaneous deflections of the simply supported FGPM cylindrical shell with n = 1 atdifferent time points. From Fig. 14, it is seen that the amplitude value of deflection is equal to zero at two ends of theshell, which satisfies the given boundary conditions. It is also seen from the curves that the amplitude of vibrationfor the simply supported FGPM cylindrical shell increases as the time increases, and the amplitude peak appearsnear two ends of the shell, and the place of the amplitude peak appears further away two ends of the shell as the timegets longer.

Figures 15 and 16 show that the instantaneous deflections of the simply supported FGPM cylindrical shell withvarious gradient indexes n at t = 0.0001 s and t = 0.002 s, respectively. From these figures, it is seen that amplitudevalue of deflection decreases except two ends of the shell as the volume exponent increases. Comparing Fig. 15 withFig. 16, the change of the amplitude value of deflection decreases as the volume exponent increases, and the changetrend of deflection is contrary as the volume exponent increases.

Figure 17 illustrates the change trend of the instantaneous deflections at four different points with the time goes.From Fig. 17, it shows that the pace of the vibration of this four points different from each other, not on the samerhythm. It is also found that the slight variations of the deflection happened at the start-up time, and the form ofvibration changes and the amplitude increases significantly as time goes.

Case 2 Consider a simply supported FGPM cylindrical shell imposed by a sinusoidal distributed load

q0(x) = 105 · Sin( x

L· π

)Pa (72)

Figure 18 shows that variation of the instantaneous deflections of the simply supported FGPM cylindrical shellwith n = 1 at different time points. From Fig. 18, it is found that the sharp of deflection for different time points allare sinusoidal form, which is due to the same form of the driving load and the mode sharp of the cylindrical shell.

Figures 19 and 20 show the instantaneous deflections of the FGPM cylindrical shell imposed a sinusoidal dis-tributed load with various volume exponents n at t = 0.0001 s and t = 0.001 s, respectively. From Fig. 19, it is

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546 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

Fig. 16. Deflection of the mid-plane for the simply supportedFGPM cylindrical shell imposed by a uniform mechanical load,when t = 0.002.

Fig. 17. Deflection of various points for the simply supported FGPM cylin-drical shell imposed by a uniform mechanical load as the time increasing,when n = 1.

Fig. 18. Deflection of the mid-plane for the simply supported FGPMcylindrical shell imposed by a sinusoidal distributed load, whenn = 1.

Fig. 19. Deflection of the mid-plane for the simply supported FGPMcylindrical shell imposed by a sinusoidal distributed load, whent = 0.001.

noticed clearly that the change trend of deflection is similar with various volume exponents, and the deflection ofthe simply supported FGPM cylindrical shell decreases as the volume exponent increases at the same radial point.Comparing Fig. 17 with Fig. 20, one knows the amplitude of deflection of the simply supported FGPM cylindricalshell increases significantly as the time increases.

Figure 21 shows the change trend of various points’ deflection with sinusoidal distributed loads as the timeincreases. From Fig. 21, it is seen easily that the vibrations at different points are in the same step, it means that thedeflections reach the maximum or minimum values at the same time, and the amplitude increases linearly with theincrease of time.

Case 3 Considering a simply supported FGPM cylindrical shell imposed a cosine distributed load.

q0(x) = 105 · Cos( x

L· π

)Pa (73)

Figure 22 shows deflection of the mid-plane for the simply supported FGPM cylindrical shell imposed a cosinedistributed load with n = 1 at different time points. From Fig. 22, it is found that there exists big difference of

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 547

Fig. 20. Deflection of the mid-plane for the simply supportedFGPM cylindrical shell imposed by a sinusoidal distributed load,when t = 0.001.

Fig. 21. Deflection of various points for the simply supported FGPMcylindrical shell imposed by a sinusoidal distributed load as the timeincreasing, when n = 1.

Fig. 22. Deflection of the mid-plane for the simply supported FGPMcylindrical shell imposed by a cosine distributed load, when n = 1.

Fig. 23. Deflection of the mid-plane for the simply supported FGPMcylindrical shell imposed by a cosine distributed load, when t = 0.002.

vibration sharp at different time point. At the beginning, the amplitude is small, and changes gently. However, as thetime goes by, the amplitude turns to be quite large.

Figures 23 and 24 show deflections of the simply supported FGPM cylindrical shell imposed a cosine distributedload with various volume exponents n at t = 0.002 s and t = 0.005 s, respectively. From Fig. 23, it is seen easilythat the amplitude of deflection for the shell decreases from two ends to middle place of the shell, and it decreasesas the volume exponent increases at the same point of the x axis. Comparing Figs 23 and 24, one knows, the changetrend of deflection for the shell is similar.

Figure 25 gives effect of the loading frequency on the mid-plane deflection of the simply supported FGPM cylin-drical shell imposed by a cosine distributed load at x = 3. From Fig. 25, The deflection reach to maximum whenloading frequency ωF equal to ωm (achieved resonance), the greater between the natural frequency ωm and loadingfrequency ωF , the smaller the deflection is by and large.

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548 H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell

Fig. 24. Deflection of the mid-plane for the simply supportedFGPM cylindrical shell imposed by a cosine distributed load,when t = 0.005.

Fig. 25. Effect of the loading frequency on the mid-plane deflection of the simplysupported FGPM cylindrical shell imposed by a cosine distributed load at x = 3.

5. Concluding remarks

The paper presents an analytic study for forced vibration of a FGPM cylindrical shell. To investigate the interac-tion among the effects of electric excitation, thermal load, mechanical load and volume exponent, static deflectionof the FGPM cylindrical shell is first discussed in the numeral examples. As depicted in the relevant section, it isknown that the influence of the electric excitation over the static deflection is not obvious, while with the increaseof the mechanical load or thermal load, the static deflection increases monotonously. The effect of volume fractionindex on high order frequency is larger than that is in low order frequency. In the second example, the force vibrationof this FGPM cylindrical shell is investigated in detail, the influences of vibration amplitude and volume exponenthave been examined. The greater between the natural frequency and loading frequency, the smaller the deflection isby and large. It is confirmed that the characteristics of deflection are significantly influenced by the volume expo-nent and various kinds of loads. Therefore, FGPM cylindrical shell structures deserve special attention in order tooptimize their mechanical response, it is possible to optimize and design the FGPM cylindrical shell by selecting aproper volume exponent and suitable loads.

Acknowledgments

The authors wish to thank reviewers for their valuable comments and the funded by the National Natural Sci-ence Foundation of China (Grant No.11072077), Key Laboratory of Manufacture and Test Techniques for Automo-bile Parts, Ministry of Education, Hunan Provincial Natural Science Foundation for Creative Research Groups ofChina (Grant No.12JJ7001), State key Laboratory of Advanced Design and Manufacturing for Vehicle Body (GrantNo.51075003), and the central colleges of basic scientific research and operational costs (funded by the HunanUniversity).

Appendix A

Θ11 = c1(z)− [e1(z)]2

g(z), Θ12 = c2(z)− e1(z)e2(z)

g(z), Θ13 = λ(z)T (z),

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H.-L. Dai and H.-J. Jiang / Forced vibration analysis for a FGPM cylindrical shell 549

Θ14 =e1(z)kd

g(z)(a− z), Θ15 =

−e1(z)p(z)T (z)

g(z), Θ21 = c2(z)− [e2(z)]

2

g(z),

Θ22 = c1(z)− e1(z)e2(z)

g(z), Θ23 = Θ13, Θ24 =

e2(z)kdg(z)(a− z)

,

Θ25 =−e2(z)p(z)T (z)

g(z).

Appendix B

A11 =

∫ h/2

−h/2

Θ11

(1− z

a

)dz, A12 =

∫ h/2

−h/2

Θ11z(1− z

a

)dz,

A13 =

∫ h/2

−h/2

Θ12

(1− z

a

)(−1

a+

1

a2

)dz, A14 =

∫ h/2

−h/2

Θ13dz,

A15 =

∫ h/2

−h/2

Θ14dz, A16 =

∫ h/2

−h/2

Θ15dz, A21 =

∫ h/2

−h/2

Θ21dz,

A22 =

∫ h/2

−h/2

Θ21zdz, A23 =

∫ h/2

−h/2

Θ22

(−1

a+

1

a2

)dz, A24 =

∫ h/2

−h/2

Θ23dz,

A25 =

∫ h/2

−h/2

Θ24dz, A26 =

∫ h/2

−h/2

Θ25dz, A31 = −∫ h/2

−h/2

Θ11

(1− z

a

)zdz,

A32 = −∫ h/2

−h/2

Θ11

(1− z

a

)z2dz, A33 = −

∫ h/2

−h/2

Θ12

(1− z

a

)(−1

a+

1

a2

)zdz,

A34 = −∫ h/2

−h/2

Θ13

(1− z

a

)zdz, A35 = −

∫ h/2

−h/2

Θ14

(1− z

a

)zdz,

A36 = −∫ h/2

−h/2

Θ15

(1− z

a

)zdz.

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