Active control of functionally graded laminated cylindrical shells

Composite Structures 90 (2009) 448–457

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Active control of functionally graded laminated cylindrical shells G.G. Sheng a,b, X. Wang a,* a b

School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha, Hunan 410076, People’s Republic of China

a r t i c l e

i n f o

Article history: Available online 14 April 2009 Keywords: Functionally graded laminated shells Active vibration control Thermal and mechanical loads

a b s t r a c t An analytical method on active vibration control of smart FG laminated cylindrical shells with thin piezoelectric layers is presented based on Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell act as distributed sensor and actuator, which are used to control vibration of the smart FG laminated cylindrical shell under thermal and mechanical loads. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of the smart FG laminated cylindrical shell with thin piezoelectric layers. Constant-gain negative velocity feedback approach is used for active vibration control with the structures subjected to impact, step and harmonic excitations. The influences of different piezoelectric materials (PZT-4, BaTiO3 and PZT-5A) and various loading forms on the active vibration control are described in the numerical results. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In the recent years, a great deal of research on the distributed piezoelectric sensors and actuators has already been carried out for active vibration control of light-weight smart structures such as aerospace, hydrospace, nuclear and automotive structural applications. The piezoelectric materials have the property to generate electrical charge under mechanical load or deformation, and the reverse, applying an electrical field to the piezoelectric material results in mechanical strains or stresses. Due to the coupled electromechanical properties of piezoelectric materials and their availability in the form of thin sheets, the piezoelectric layers embedded on structures are well suited for use as distributed sensors and actuators. In order to achieve the most effective actuation and control, extension piezoelectric actuators are usually placed on the surface of a structure at selected optimal locations [1]. A smart structure that contains the main structure and the distributed piezoelectric sensor/actuators can sense the excitations induced by its environment and can also generate control forces to eliminate the undesirable effects or to enhance the desirable effects. Yang et al. [2] developed a generic electromechanical impedance model for the two-dimensional PZT–structure interaction systems. To closely simulate the real situation, the PZT transducers were assumed to interact with the host structure at four edges. The results for a plate structure were in good agreement with the experimental measurements. Yang and Hu [3] presented an electromechanical impedance model for health monitoring of * Corresponding author. E-mail address: [email protected] (X. Wang). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.04.017

cylindrical shell structures. By investigating the interaction between the PZT transducers and a typical cylindrical shell structure, the electromechanical impedance of the PZT transducers is obtained. Reddy [4] presented the Navier solution and finite element models based on the classical and shear deformation plate theories for the analysis of laminated composite plates with integrated sensors and actuators, in which a simple negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects was used to active control the time response of an integrated structure. Account for the coupling of mechanical, electrical, and thermal effect, Lee and Saravanos [5] presented analytical formulations of piezoelectric composite shell structures. This laminate theory is formulated using curvilinear coordinates and the principles of linear thermopiezoelectricity. Utilizing finite element formulations, a plate/shell structure with thin PZT piezoceramic layers embedded on top and bottom surfaces to act as distributed sensor and actuator was considered in Ref. [6]. Based on the kinematic assumption of the Love–Kirchhoff thin plate theory and a quadratic variation of the electric potential along the thickness direction of the piezoelectric parts, Fernandes and Pouget [7] investigated dynamic response of composite plates with piezoelectric actuators, where the spectra of vibration for the plate with a time-dependent electric potential are computed. Vel and Baillargeon [8] presented an analytical solution for the static deformation and steady-state vibration of simply supported hybrid cylindrical shells consisting of fiber-reinforced layers with embedded piezoelectric shear sensors and actuators. Suitable displacement and electric potential functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the governing equations of static deformation and steady-state vibrations. Using piezoelectric fiber reinforced composite

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Nomenclature L length of the shell; R middle surface radius of the FGM layer thickness of the sensor layer hs h thickness of the FGM layer thickness of the actuator layer ha V(x, h, t) applied electric potential plane stress-reduced stiffness of the piezoelectric layers Q iije effective piezoelectric constant for plane stress problem eiije effective permittivity constant for plane stress problem niije pixe ; pihe ; pize effective pyroelectric constant for plane stress problem ws in-plane electric field induced by the deformation of the sensor layer

materials, the active constrained layer damping of laminated thin composite shells was investigated by Ray and Reddy [9]. Oh and Lee [10] investigated the nonlinear deformation and its attendant vibration characteristics of un-symmetrically deposited clamped– clamped micro-beams under piezoelectric and thermal actuations, in which the multi-layered micro-resonators with initial imperfections were modeled by using refined layerwise theory. Utilizing the Hamilton’s principle and finite element methods, the linear response of piezothermoelastic plate is obtained in Ref. [11]. Due to the advantages of being able to withstand severe hightemperature gradient while maintaining structural integrity, functionally graded material (FGM) has been receiving much more attention in engineering communities, especially in applications for high-temperature environment such as nuclear reactors, space planes and chemical plants [12–18]. Therefore, to investigate active vibration control of FGM cylindrical shells with thin piezoelectric layers is very significant. In the present work, FG laminated cylindrical shells are considered as simply supported and temperature distribution across the shell thickness is considered as nonlinear. The dynamic characteristics of FG laminated cylindrical shells with thin piezoelectric layers embedded on inner and outer surfaces to act as distributed sensor and actuator (smart FG laminated cylindrical shells) are investigated under thermal and mechanical loads. Based on the first-order shear deformation theory and the Hamilton’s principle, the coupling equations to govern the electric potential and the flexural deflection of the smart FG laminated cylindrical shell in thermal environments are derived, in which the stiffness and inertial contribution of the piezoelectric layers are considered. The modal analysis technique and Newmark’s direct time integration method are used to obtain the response history of the smart FG laminated cylindrical shell, based on an expansion of the loads, displacements in the double Fourier series that satisfy the boundary conditions. A constant-gain negative velocity feedback approach is utilized to control vibrational characteristics of FGM structures subjected to impact, step and harmonic excitations. The results obtained show that the active control from piezoelectric layers can significantly improves the damping effect of FGM cylindrical shells; The influence of active control to the dynamic responses of smart FG laminated cylindrical shells is dependent on the material property of piezoelectric layer; the active control to the response amplitudes of smart FG laminated cylindrical shells greatly depends on the composition of the metal–ceramic constituents of FGM layer. Finally, the present approach is validated by comparing the natural frequencies of a simple supported piezoelectric cylindrical shell with the results from other researchers.

wa

in-plane electric field induced by the deformation of the actuator layer qs mass density of the sensor layer qa mass density of the actuator layer q(z) mass density of the FGM layer Ii(i = 1, 2, 3) mass moments of inertia for the FGM layer Iai ði ¼ 1; 2; 3Þ mass moments of inertia for the actuator layer Isi ði ¼ 1; 2; 3Þ mass moments of inertia for the sensor layer G control gain m,n wave numbers

2. Theoretical formulations Fig. 1 shows smart FG laminated cylindrical shells, where (x, h, z) denote the orthogonal curvilinear coordinates such that x and h curves are the lines of curvature on the middle surface (z = 0) of FGM layer. The smart FG laminated cylindrical shell is made of a FGM layer and two thin piezoelectric layers embedded on inner and outer surfaces to act as distributed sensor and actuator. Form the first-order shear deformation theory, the displacements (u1, v1, w1) of a point (x, h, z) in the smart FG laminated cylindrical shell are expressed as sum of the mid-surface displacements (u, v, w) along the x, h and z direction, and rotations (/1, /2) of the normals to the mid-surface along x and h axes, as follows

u1 ðx; h; z; tÞ ¼ uðx; h; tÞ þ z/1 ðx; h; tÞ;

v 1 ðx; h; z; tÞ ¼ v ðx; h; tÞ þ z/2 ðx; h; tÞ;

ð1Þ

w1 ðx; h; z; tÞ ¼ wðx; h; tÞ: Here, the material properties of FGM layer are considered as graded distribution along the thickness direction according to a power law in terms of the volume fractions (power law exponent U) of the constituents [19]. The stress–strain relation including the temperature effects is given by

re ¼ Ce ðzÞ½e  ae ðzÞDT;

ð2Þ

where DT(x, h, z, t) is the temperature change referenced to the stress free state ðDTðx; h; z; tÞ ¼ Cðx; h; tÞTðzÞÞ. The elasticity matrix Ce(z) and thermal expansion coefficients matrix ae(z) of the FGM layer are given by Kadoli and Ganesan [15]. For the actuator and sensor layers, the relations between the field variables are given by the piezoelectric constitutive equations:

z,w z,w

Actuator Layer

ha

ha

θ,v

x,u Sensor Layer

R

R

h

hs hs

FGM Layer

L

h

Fig. 1. Coordinate system of a smart FG laminated cylindrical shell.

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G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

ri ¼ Ci ½e  ai DT  ei Ei ;

ð3Þ

Di ¼ eTi e þ ni Ei þ DTpi ;

ð4Þ

1 H¼ 2

K¼

Q i11e

6 i 6 Q 12e 6 6 Ci ¼ 6 0 6 6 40 2

0 ni11e

6 ni ¼ 4 0

0

Q i12e

0

0

0

¼ a; sÞ:

Q i22e

0

0

0

0

Q i66e

0

0

0

0

Q i44e

0

0

0

0

0

0

3

ni22e

0

7 5;

0

ni33e

3 7 7 7 7 7; 7 7 5

2

0

6 60 6 ei ¼ 6 60 6 40

Q i55e 2 i 3 pxe 6 7 pi ¼ 4 pihe 5:

ei15e

0 0 0 ei24e 0

ei31e

3

7 ei32e 7 7 0 7 7; 7 0 5 0

pize

In the above formulas, the effective constants of actuator and sensor layers are described in Appendix A. The poling direction of the piezoelectric layer is coincident with the thickness direction. The electric field vectors of actuator and sensor layers are the negative gradients of the electric potentials:

Ea



 ¼

Es



rua : rus

ð5Þ

"

 2 # z h ua ¼ 2 a Vðx; h; tÞ þ z2a  a wa ðx; h; tÞ; ha 2

ð6Þ

V N0 ¼

"

us ¼ z2s 

 2 # hs ws ðx; h; tÞ; 2

ð7Þ

where zs is the local thickness coordinate with respect to the sensor layer mid-plane, zs = z + (h + hs)/2. Considering the contribution of the actuator and sensor on the mass and stiffness of the smart FG laminated cylindrical shell, the dynamic governing equations of the smart FG laminated cylindrical shell are derived utilizing the total potential energy H and the kinetic energy K of the actuator layer, sensor layer and the FGM layer as follows [9]

hþh a 2 h

A

# DTa Ea Rdz

dhdx

2 Z "Z

rTs Rdz

hþh a 2



Z "Z

qa u_ 21 þ v_ 21 þ w_ 21 Rdz dhdx

2h

2hhs

A

Z "Z

h 2

2h

A



#

q



_2 s u1

þv þ _ 21

_ 21 w

#  Rdz dhdx

#  2 2 2 _ _ _ qðzÞ u1 þ v 1 þ w1 Rdz dhdx; 

ð9Þ

1 2

Z A

N0

 2 @w Rdhdx: @x

ð10Þ

Based on the first-order shear deformation theory, the strain vector e of an arbitrary point in the smart FG laminated cylindrical shell is related to the mid-surface strains (ex, eh, cxh, cxz and chz) and curvatures (jx, jh and jxh) (see Eq. (1)), which can be defined as

  1 @v @ v 1 @u ; þ w ; cxh ¼ þ R @h @x R @h @w 1 @w @/1 cxz ¼ /1 þ ; chz ¼ /2 þ ; jx ¼ ; @x R @h @x 1 @/2 @/ 1 @/1 jh ¼ ; jxh ¼ 2 þ : R @h R @h @x @u ; @x

eh ¼

ð11Þ

To derive the equations of motion and the charge equations, the formulation is based on Hamilton’s principle and the first-order shear deformation theory extended to piezoelectricity. The variational principle can be stated as

0

t

ðdK  dH  dV N0 Þdt ¼ 0:

ð12Þ

The equations of motion for the smart FG laminated cylindrical shell under the axial loading N0 are given by

@ðNx  NTx þ NPx Þ 1 @ðNxh  NTxh þ NPxh Þ þ @x R @h     €1; € þ I2 þ Ia2 þ Is2 / þ fx ðx; h; tÞ ¼ I1 þ Ia1 þ Is1 u   T P T P @ Nxh  Nxh þ Nxh 1 @ Nh  Nh þ Nh 1 dv : Q h  Q Th þ Q Ph þ þ R R @h @x     € 2; þ fh ðx; h; tÞ ¼ I1 þ Ia1 þ Is1 v€ þ I2 þ Ia2 þ Is2 /   T P @ Q x  Q Tx þ Q Px 1 @ Qh  Qh þ Qh 1 dw : Nh  NTh þ NPh þ  R R @h @x  @2w  a s € þ fz ðx; h; tÞ þ N0 2 ¼ I1 þ I1 þ I1 w; @x   T P @ Mx  M Tx þ M Px 1 @ Mxh  M xh þ M xh þ d/1 : @x @h  R T P  Q x  Q x þ Q x þ mx ðx; h; tÞ     € 1; € þ I3 þ Ia3 þ Is3 / ¼ I2 þ Ia2 þ Is2 u du :

where za is the local thickness coordinate with respect to the actuator layer mid-plane, za = z  (h + ha)/2. When the external voltage applied V is zero, the piezoelectric layer can be taken as a sensor. From Eq. (6), the electric potential us induced by elastic deformation in the sensor layer is yielded by

Z "Z

where u is the mechanical displacement vector and W is the electric potential vector. q and p are the surface forces intensities (fx, fh, fz, mx and mh) and the surface electric charge intensities (pa and ps), respectively. The potential energy V N0 of the axial loading (N0) is taken as [20]

Z

For the actuator layer, taking into account both the direct piezoelectric effect and the converse piezoelectric effect, a layerwise quadratic distribution of the electric potential ua is given by [7]

e

2h

h 2

A

1 þ 2

ex ¼ ð4bÞ



1 2

Z "Z

1 þ 2 ð4aÞ

2

2 Z "Z

1 dhdx  2 #

ð8Þ

i ¼ a; sÞ;

According to the state of generalized plane stress of thin shell, the normal stress is zero in the thickness direction, so that the elasticity, piezoelectricity, permittivity and pyroelectric matrices of actuator and sensor layers are defined as

rTa Rdz

# 2h 1 T e dhdx  D Es Rdz dhdx 2 A h2hs s A h2hs " # Z Z h Z Z 2 1 rTe eRdz dhdx  qT uRdhdx þ pT WRdhdx þ V N0 ; þ 2 A h2 A A

ri ¼ ½rix

Ei ¼ ½Eix Eih Eiz T ðelectricfields; i ¼ a; sÞ; Di ¼ ½Dix Dih Diz T ðelectricdisplacements; pi ¼ ½pix pih piz T ðpyroelectricconstants; i

#

hþh a 2 h

A

1 þ 2

where subscript i = a denotes actuator layer, subscript i = s denotes sensor layer and subscript i = e denotes FGM layer (see Eq. (2)). The column matrices in Eqs. (2)–(4) are given by

rih sixh sihz sixz T ðstresses; i ¼ e; a; sÞ; xh c hz c xz T ðstrainsÞ; e ¼ ½ex eh c ai ¼ ½aixxe aihhe 0 0 0T ðthermalexpansioncoefficients; i ¼ e; a; sÞ;

Z "Z

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G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

d/2 :

1 @ðM h  M Th þ MPh Þ @ðM xh  MTxh þ M Pxh Þ  þ  Q h  Q Th þ Q Ph R @h @x     € 2: þ mh ðx; h; tÞ ¼ I2 þ Ia2 þ Is2 v€ þ I3 þ Ia3 þ Is3 / ð13Þ

where superscripts T denote resultants due to thermal effects, superscripts P denote resultants due to piezoelectric effects, and Nx, Nh, Nxh, Mx, Mh, Mxh, Qx and Qh are the usual stress resultants. The thermal stress resultants, piezoelectric stress resultants and usual stress resultants are computed using the constitutive relationships Eqs. (2) and (3), which are described in Appendix B. Utilizing Appendix B and Eq. (11), the equations of motion can be expressed in terms of generalized displacements (u, v, w, /1, /2) as follows:

a1 C0x þ fx ¼

M X N X

qmn1 ðtÞ cos km x cos nh;

m¼0 n¼0

a2 C0h þ fh ¼

M X N X

qmn2 ðtÞ sin km x sin nh;

m¼1 n¼1

a3 C þ fz ¼

M X N X

qmn3 ðtÞ sin km x cos nh;

m¼1 n¼0

a4 C0x þ mx ¼

M X N X

qmn4 ðtÞ cos km x cos nh;

ð18Þ

m¼0 n¼0

L11 u þ L12 v þ L13 w þ L14 /1 þ L15 /2 þ L16 wa þ þL17 ws     € € þ I2 þ Ia2 þ Is2 /; þ L18 V þ a1 C0x þ fx ¼ I1 þ Ia1 þ Is1 u

a5 C0h þ mh ¼

L21 u þ L22 v þ L23 w þ L24 /1 þ L25 /2 þ L26 wa þ L27 ws     € 2; þ L28 V þ a2 C0h þ fh ¼ I1 þ Ia1 þ Is1 v€ þ I2 þ Ia2 þ Is2 / L31 u þ L32 v þ L33 w þ L34 /1 þ L35 /2 þ L36 wa þ L37 ws   € þ L38 V þ a3 C þ fz ¼ I1 þ Ia1 þ Is1 w;

tric charge intensities p, and can also be expanded in double Fourier series

M X N X

qmn5 ðtÞ sin km x sin nh;

m¼1 n¼1

a6 C þ pa ¼

M X N X

qmn6 ðtÞ sin km x cos nh;

m¼1 n¼0

ð14Þ

M X N X

L41 u þ L42 v þ L43 w þ L44 /1 þ L45 /2 þ L46 wa þ L47 ws     €1; € þ I3 þ Ia3 þ Is3 / þ L48 V þ a4 C0x þ mx ¼ I2 þ Ia2 þ Is2 u

a7 C þ ps ¼

L51 u þ L52 v þ L53 w þ L54 /1 þ L55 /2 þ L56 wa þ þL57 ws     € 2: þ L58 V þ a5 C0h þ mh ¼ I2 þ Ia2 þ Is2 v€ þ I3 þ Ia3 þ Is3 /

Substituting Eqs. (17) and (18) into Eqs. (14) and (15), and making use of the orthogonality conditions, yields a set of equations as follows:

From Eq. (12), the charge equilibrium equations are given by



dwa : L61 u þ L62 v þ L63 w þ L64 /1 þ L65 /2 þ L66 wa þ L67 V dws : L71 u þ L72 v þ L73 w þ L74 /1 þ L75 /2 þ L76 ws þ a7 C þ ps ¼ 0: ð15Þ where operators Lij, a6 and a7 in Eqs. (14) and (15) are defined in Appendix C. ai (i = 1, 2, . . . , 5) can be obtained from the thermal stress resultants. C0x and C0h represent the partial derivatives of thermal field C(x, h, t) with respect to coordinates x and h, respectively. In order to supply the control voltage V(x, h, t) for distributed piezoelectric actuator, a constant-gain negative velocity feedback control law has been employed [6]. According to this law, the control voltage for actuator can be expressed in terms of the distributed piezoelectric sensor voltage as follows:

ð16Þ

mn



M X N X

umn ðtÞ cos km x cos nh;

v¼

m¼0 n¼0

w¼

M X N X

M X N X

wmn ðtÞ sin km x cos nh; /1 ¼  mn ðtÞ sin km x sin nh; w ¼ / a

m¼1 n¼1

ws ¼

N X

mn

mn2 ðtÞ;

ð20Þ 

 € mn ðtÞ þ T 30 w_ smn ðtÞ þ T 31 umn ðtÞ þ T 32 v mn ðtÞ I1 þ Ia1 þ Is1 w 2  mn ðtÞ þ ðT 33 þ k No Þwmn ðtÞ þ T 34 / ðtÞ þ T 35 / mn

m

þ T 36 wamn ðtÞ þ T 37 wsmn ðtÞ ¼ qmn3 ðtÞ; 

ð21Þ

   € mn ðtÞ þ T 40 w_ s ðtÞ € mn ðtÞ þ I3 þ Ia3 þ Is3 / I2 þ Ia2 þ Is2 u mn þ T 41 umn ðtÞ þ T 42 v mn ðtÞ þ T 43 wmn ðtÞ þ T 44 /mn ðtÞ  mn ðtÞ þ T 46 wa ðtÞ þ T 47 ws ðtÞ ¼ q ðtÞ; þ T 45 / mn4 mn mn



ð22Þ

  €  mn ðtÞ þ T 50 w_ s ðtÞ I2 þ Ia2 þ Is2 v€ mn ðtÞ þ I3 þ Ia3 þ Is3 / mn

/mn ðtÞ cos km x cos nh;

 mn ðtÞ þ T 56 wa ðtÞ þ T 57 ws ðtÞ ¼ q ðtÞ; þ T 55 / mn5 mn mn

ð23Þ

m¼0 n¼0

M X N X

M X

M X N X

  €  mn ðtÞ þ T 20 w_ s ðtÞ I1 þ Ia1 þ Is1 v€ mn ðtÞ þ I2 þ Ia2 þ Is2 / mn   2 þ T 21 umn ðtÞ þ T 22 þ km No v mn ðtÞ þ T 23 wmn ðtÞ  mn ðtÞ þ T 26 wa ðtÞ þ T 27 ws ðtÞ ¼ q þ T 24 / ðtÞ þ T 25 / mn

ð19Þ

mn1

þ T 51 umn ðtÞ þ T 52 v mn ðtÞ þ T 53 wmn ðtÞ þ T 54 /mn ðtÞ

m¼1 n¼1

m¼1 n¼0

/2 ¼

v mn ðtÞ sin km x sin nh;

mn

mn

Here, the two ends of the smart FG laminated cylindrical shell are considered as simply supported, so that a solution for Eqs. (14) and (15) can be described by

u¼

   € mn ðtÞ þ T 10 w_ s ðtÞ € mn ðtÞ þ I2 þ Ia2 þ Is2 / I1 þ Ia1 þ Is1 u mn þ T 11 umn ðtÞ þ T 12 v mn ðtÞ þ T 13 wmn ðtÞ þ T 14 /mn ðtÞ  mn ðtÞ þ T 16 wa ðtÞ þ T 17 ws ðtÞ ¼ q ðtÞ; þ T 15 /

þ a6 C þ pa ¼ 0;

Vðx; h; tÞ ¼ Gw_ s ðx; h; tÞ:

qmn7 ðtÞ sin km x cos nh:

m¼1 n¼0

M X N X

wamn ðtÞ sin km x cos nh;

 mn ðtÞ T 61 umn ðtÞ þ T 62 v mn ðtÞ þ T 63 wmn ðtÞ þ T 64 /mn ðtÞ þ T 65 / a s þ T 66 w ðtÞ ¼ T 67 Gw_ ðtÞ  q ðtÞ; mn

m¼1 n¼0

mn

mn6

ð24Þ

 mn ðtÞ T 71 umn ðtÞ þ T 72 v mn ðtÞ þ T 73 wmn ðtÞ þ T 74 /mn ðtÞ þ T 75 /

wsmn ðtÞ sin km x cos nh;

m¼1 n¼0

ð17Þ  mn ; /a ; /s Þ are coefficients to where km ¼ mLp ; ðumn ; v mn ; wmn ; /mn ; / mn mn be determined. The generalized loads in Eqs. (14) and (15) are related to the thermal loads DT(x, h, z, t), the surface forces intensities q and elec-

þ T 76 wsmn ðtÞ ¼ qmn7 ðtÞ;

ð25Þ

where the unknown constants Tij can be determined by operators Lij in Eqs. (14) and (15). From Eqs. (24) and (25), the Fourier coefficients of the induced electric potentials are given by

452

wsmn ðtÞ ¼

G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

1 ½qmn7 ðtÞ  T 71 umn ðtÞ  T 72 v mn ðtÞ  T 73 wmn ðtÞ T 76  mn ðtÞ;  T 74 / ðtÞÞ  T 75 / mn

wamn ðtÞ ¼

1 ½qmn6 ðtÞ  T 61 umn ðtÞ  T 62 v mn ðtÞ  T 63 wmn ðtÞ T 66  mn ðtÞ þ T 67 G w_ s ðtÞ:  T 64 /mn ðtÞÞ  T 65 / T 66 mn

( 4a qmn3 ðtÞ ¼ ð26Þ qmn6 ðtÞ ¼ ( ð27Þ

Substituting Eqs. (26) and (27) into Eqs. (19)–(23), a system of govern equations of the smart FG laminated cylindrical shell can be expressed in the matrix form:

 €g þ G½Cfqg _ þ ½K uu  þ ½K uu  fqg ¼ fFgq þ fFgT þ fFgp ½Mfq

ð28Þ

T

 mn g , the active damping effect where fqg ¼ fumn v mn wmn /mn / G[C] is from the negative velocity feedback of the actuator voltage (see Eq. (16)). {F}q, {F}T and {F}p are related to the applied mechanical forces, thermal loads and applied electrical charges, respectively, and the generalized forces are obtained from Eq. (18). [M] is the mass matrix obtained from the mass moments of inertia ðIi ; Iai ; Isi ; i ¼ 1; 2; 3Þ, and [Kuu] and [Kuu] are the elastic stiffness matrix and the elastic–electric coupling stiffness matrix related to the coefficients Tij in Eqs. (19)–(25). The solutions of Eq. (28) can be obtained using Newmark’s direct time integration method. 3. Results and discussions In this study, in-plane uniform distribution of temperature is considered as

Cðx; h; tÞ ¼ F 1 ðtÞðDTðx; h; z; tÞ ¼ TðzÞF 1 ðtÞÞ:

ð29Þ

The temperature distribution along the thickness of the smart FG laminated cylindrical shell can be obtained by solving a steady-state heat transfer equation [16]. For the sensor and actuator layers, temperature distribution across the thickness is considered as linear:

TðzÞ ¼ T m þ

TðzÞ ¼ T c þ

  T0  Tm h z  T0 2 ha

  Tc  T1 h zþ  T0 2 hs



 h h 6 z 6 þ ha ; 2 2

ð30Þ

  h h   hs 6 z 6  2 2

ð31Þ

and for the FGM layer:

TðzÞ ¼ T c  R h

Tc  Tm

2

2h

dz=keff ðzÞ

Z

z

h2

dz  T0 keff ðzÞ

  h h ;  6z6 2 2

(

ð32Þ

where Tc,Tm are the temperature of the inner surface and the outer surface of the FGM layer, respectively. keff is the effective thermal conductivity of the FGM layer. T0 is the room temperature (zero thermal stress state, T0 = 300 K), and T1 is the inner surface temperature of the sensor layer. Tc and Tm can be obtained by utilizing the temperature continuous conditions between layers and the heat transfer equation. The temperature difference between the inner surface and the outer surface of the smart FG laminated cylindrical shell is TO1 = T1  T0. Piezoelectric material possesses Curie temperature that limits their use much below the Curie temperature. In this paper, the Curie temperature of piezoelectric materials is around 350 °C [23]. The electric charge intensities (pa and ps) applied are zero [7]. The applied mechanical load considered in this paper is the uniform load F2(t) over a small rectangular area 21  2n. The area of applied load is variable and the center point (x0, h0) of the area can be everywhere on the surface of the smart FG laminated cylindrical shell. Fourier coefficients related to the applied mechanical force and the thermal load are expressed as

qmn7 ðtÞ ¼

m 21 mpx0 mpn 3 pm ½1  ð1Þ F 1 ðtÞ þ p2 mR sin L sin L F 2 ðtÞ mpx0 n1 mpn 4 p2 mn sin L sin L sin R cos nh0 F 2 ðtÞ n – 0 4a6 pm

½1  ð1Þm F 1 ðtÞ n ¼ 0

0n – 0

n¼0

;

;

½1  ð1Þm F 1 ðtÞ n ¼ 0 ; 0 n–0 4a7 pm

qmn1 ðtÞ ¼ qmn2 ðtÞqmn4 ðtÞ ¼ qmn5 ðtÞ ¼ 0:

ð33Þ

where F1(t) = sin25t expresses the thermal loading function, and F2(t) expresses the applied mechanical forcing function. It is considered that the center point (x0, h0) of the distributed load is (0.5L, 0.1p), and the rectangular area 21  2n is 0.2L  0.2L. The dynamic response w(x0, h0, t) of the center point is obtained with various applied mechanical pulse loads, thermal loads and different piezoelectric materials. The FGM layer is Zirconia rich at the inner surface and Aluminum rich at the outer surface, and the material constants are given by Reddy [19]. The sensor and actuator layers are assumed to be made of the same piezoelectric material (the piezoelectric material is PZT-4 in Figs. 2–5 and 7). The piezoelectric material constants (PZT-4, BaTiO3 and PZT-5A) are given by Oh and Lee [10], Dong and Wang[21], Hussein and Heyliger [22], Ganesan and Kadoli [23] and Ramirez et al. [24]. In this calculation, the axial load (see Eq. (13)) is non-dimensionalized as

 0 ¼ N0 =N0cr ; N

ð34Þ

where N0cr expresses the axial buckling load of cylindrical shell [25] and E, t are the material constants of Aluminum. The other calculation data are given by

R ¼ 1m;

h=R ¼ 0:01;

L=R ¼ 6;

U ¼ 0:5;

ha =R ¼ 0:001;

N0 ¼ 0:5N0cr :

hs =R ¼ 0:001; ð35Þ

Fig. 2 shows convergence of dynamic response w(x0, h0, t) for variable M and N. The dynamic response is calculated for M,N = 2, M,N = 10, M,N = 16 and M,N = 24 in model expansion, respectively. As shown in Fig. 2, for the transient analysis the use of M,N = 16 series number is adequate for the converged results. The effectiveness of the active control in controlling the dynamic response w(x0, h0, t) of the smart FG laminated cylindrical shell is described in Fig. 3 wherein harmonic load, F2(t) = 50cos25t (KN). It is seen from Fig. 3 that the dynamic response w(x0, h0, t) of the smart FG laminated cylindrical shell under mechanical loading exhibits a constant amplitude with a time period, and as the gain increases, the amplitude of vibration decreases. It can also be observed that the phase angles of the vibration are different for the different gain in Fig. 3. Fig. 4 shows the active control of smart FG laminated cylindrical shells under an external Step pulse load of 50KN for 0.015s duration. It is seen from Fig. 4 that since the pulse load applied on the smart FG laminated cylindrical shell is withdraw after some time, the smart FG laminated cylindrical shell exhibits a free vibration after some time, thus the response amplitude w(x0, h0, t) of the smart FG laminated cylindrical shell gradually declines to zero as the gain increases. Fig. 5 describes the effect of the material property of FGM layer on dynamic characteristics of the shell. It is seen from Fig. 5 that the response amplitude of the smart FG laminated cylindrical shell subjected to Sine load (F2(t) = 50sin25t KN) decreases with the increase of volume fraction exponent U of the FGM layer when the volume fraction exponent U is less than 5. But when the volume fraction exponent U is larger than 5, the volume fraction exponent U of the FGM layer does not effect the dynamic characteristics of

453

G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

0.009

M=2, N=2 M=16, N=16

M=10, N=10 M=24, N=24

0.006

Φ= 0.5 Φ= 10.

0.008

0.003

0.004

w (m)

w (m)

Φ= 0. Φ= 5.

0.012

0.000

0.000 -0.004

-0.003

-0.008

-0.006

-0.012

-0.009 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0

0.1

0.2

0.3

t (s) Fig. 2. The convergence of dynamic responses w(x0,h0,t) for various M and N(F2(t) = 50sin25t KN, G=400, TO1 = 2000C).

0.0020

G=1000 G=5000 G=10000

0.0015 0.0010

w (m)

0.0005 0.0000 -0.0005 -0.0010 -0.0015 -0.0020 0.2

0.4

0.6

0.4

0.5

t (s)

0.8

1.0

1.2

t (s) Fig. 3. Dynamic response w(x0,h0,t) of the smart FG laminated cylindrical shell subjected to the applied mechanical Cosine load and active control with different gains.

0.08

G=0. G=50. G=100.

0.06

Fig. 5. Influence of the volume fraction exponent U on the dynamic response w(x0,h0,t) of the smart FG laminated cylindrical shell for the applied mechanical Sine load (G=400, TO1 = 2000C).

their metal counterpart (Aluminum). The material properties exhibit small variations when the volume fraction exponent U is larger, and when U = 1, the FGM layer is fully Zirconia. Fig. 6 shows the influences of the different piezoelectric materials (PZT-4, BaTiO3 and PZT-5A, respectively) on the dynamic response w(x0, h0, t) of the smart FG laminated cylindrical shell subjected to impact load and the active control. It can be noted that for the same gain values and different piezoelectric layers, the control effect of the PZT-5A layer is more obvious than those of PZT-4 and BaTiO3 layers. This is due to the fact that the piezoelectric coefficient e31 (absolute value) of PZT-5A is larger than those of PZT-4 and BaTiO3, and the piezoelectric coefficient with highest impact on the first three vibration modes is e31 [24]. Fig. 7 shows the influence of various thermal loads on the dynamic response w(x0, h0, t) of the smart FG laminated cylindrical shell subjected to mechanical Sine load. The temperature differences To1 between the inner surface and the outer surface of the smart FG laminated cylindrical shell are taken as 50 °C, 100 ° C and 200 °C, respectively. As the thermal loading increases, the amplitudes of vibration also increase. To verify the present analysis, vibration frequencies of a simple supported piezoelectric (BaTiO3) cylindrical shell are computed using the present method (see Eq. (28), G = 0., and the applied

w (m)

0.04 0.04 0.02 0.00

BaTiO3

0.02

PZT-4 PZT-5A

0.01

w (m)

-0.02 -0.04 0.0

0.03

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.00

-0.01

t (s) -0.02 Fig. 4. Dynamic response w(x0,h0,t) of the smart FG laminated cylindrical shell subjected to the applied mechanical Step pulse load and active control with different gains.

-0.03 -0.04 0.0

the shell, such as for U = 5 and U = 10, the results are almost equal in Fig. 5. This is due to the fact that the ceramic (Zirconia) content in FGM layer increases as the value of U increases, and the elastic modulus of the ceramic, Zirconia, is much larger as compared to

0.1

0.2

0.3

0.4

t (s) Fig. 6. Influence of different piezoelectric materials (PZT-4, BaTiO3 and PZT-5A) on the dynamic response w(x0,h0,t) of the smart FG laminated cylindrical shell subjected to 50KN impact load and active control (G=400, TO1 = 2000C).

454

G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

0

0.009

0

To1=50 C

metal–ceramic constituents. Under the identical control gain from piezoelectric layers, the responses amplitudes of shells decrease with increase of the volume fraction exponent. (3) The influence of active control from piezoelectric layers embedded on FGM shell is dependent on the material property of piezoelectric layer. The control effect of the PZT-5A is more obvious than those of PZT-4 and BaTiO3, because the piezoelectric coefficient e31 (absolute value) of PZT-5A is larger than those of PZT-4 and BaTiO3, in which the piezoelectric coefficient e31 has the highest impact on the first three vibration modes of structure. (4) The responses amplitudes of shells under identical gain control are significantly influenced by temperature change in smart FG laminated cylindrical shells.

0

To1=100 C

To1=200 C

0.006

w (m)

0.003 0.000

-0.003 -0.006 -0.009 0.0

0.2

0.4

0.6

0.8

1.0

t (s) Fig. 7. Influence of the different applied thermal loads on the dynamic response w(x0,h0,t) of the smart FG laminated cylindrical shell subjected to the applied mechanical Sine loading (F2(t) = 50sin 25t KN, G=400).

Acknowledgements

1800 Bhangale and Ganesan, R/h=200 Bhangale and Ganesan, R/h=100 Bhangale and Ganesan, R/h=50 present, R/h=200 present, R/h=100 present, R/h=50

1600 1400 1200

ω (Hz)

Because the engineering applications of FGM structures could be extremely sensitive to their vibration modes and amplitudes, the investigation on the influence of active control to dynamic characteristics of smart FG laminated cylindrical shells under various loads is very significant.

1000

The authors thank the reviewers for their valuable comments, and the supports of the National Science Foundation of China (10872127) and the 863 Project (2006AA09A103-2).

Appendix A

800 600

Q i13 Q i13

Q i11e ¼ Q i11 

400 200

Q i33 Q i23 Q i23

Q i22e ¼ Q i22 

Q i33

0 0

2

4

6

8

10

12

14

16

18

;

Q i12e ¼ Q i12 

;

Q i44e ¼ Q i44 ;

20

m Fig. 8. The fundamental frequency versus axial half-waves for the BaTiO3 piezoelectric cylindrical shell, where x denotes the fundamental frequency of the shell.

mechanical forces, thermal loads and applied electrical charge are zero). As seen in Fig. 8, there is good agreement between the present results and those from Bhangale and Ganesan [26]. 4. Conclusions A theoretical analysis on active vibration control of smart FG laminated cylindrical shells based on Hamilton’s principle and the first-order shear deformation theory was presented. Some main conclusions obtained from the numerical analysis are summarized as follows: (1) The active control from piezoelectric layers can significantly improves the damping effect of FG cylindrical shells, so that it is easily to control the dynamic characteristics of smart FG laminated cylindrical shells by adjusting the gain from piezoelectric layers according to the requirement of engineering applications. (2) When the FGM layer of the shell is ceramic rich at the inner surface and metal rich at the outer surface, the active control to the response amplitudes of smart FG laminated cylindrical shells greatly depends on the composition of the

Q i66e ¼ Q i66 ;

aixxe ¼ aixx 

Q i23 Q i33

ei33 ;

pize ¼ piz þ

Q i33

;

Q i55e ¼ Q i55 ;

ðQ i31 aixx þ Q i32 aihh Þ;

ei15e ¼ ei15 ;

ni22e ¼ ni22 ;

n11e ¼ n11 ; pihe ¼ pih ;

Q i33

Q i23  i i i i Q a þ Q a ; xx hh 31 32 Q i33

aihhe ¼ aihh  ei32e ¼ ei32 

Q i13

Q i13 Q i23

ei31e ¼ ei31 

Q i13 Q i33

ei33 ;

e24e ¼ e24 ;

ni33e ¼ ni33 þ

ei33 ei33 Q i33

;

pixe ¼ pix ;

ei33  i i Q 31 a11 þ Q i32 ai22 þ Q i33 ai33 ði ¼ a; sÞ; i Q 33

where Q imn ; eimn ; nimn and aimn denote the elastic constants, piezoelectric constants, permittivity constants and thermal expansion coefficients of actuator and sensor layers, respectively; pix ; pih and piz denote the pyroelectric constants.

Appendix B

ðI1 ; I2 ;I3 Þ ¼

Z

h=2

h=2

 s s s I 1 ; I2 ; I3 ¼

Z









qðzÞ 1;z; z2 dz; Ia1 ; Ia2 ;Ia3 ¼

h=2

h=2hs





qs 1; z; z2 dz;

Z

h=2þha h=2





qa 1; z;z2 dz;

455

G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

8 T N > > < x NTh > > : T Nxh

9 MTx > > = MTh > > ; MTxh

ðAij ; Bij ; D1ij Þ ¼

9 8 Q ðzÞaexxe ðzÞ þ Q 12 ðzÞaehhe ðzÞ > Z h> = < 11 2 ¼ Cðx; h; tÞ Q 12 ðzÞaexxe ðzÞ þ Q 22 ðzÞaehhe ðzÞ TðzÞð1 zÞdz > > 2h : ; 0 9 8 a Q aa þ Q a12e aahhe > > > Z hþha > = < 11e xxe 2 þ Cðx; h; tÞ Q a12e aaxxe þ Q a22e aahhe TðzÞð1 zÞdz > > h > > 2 ; : 0 9 8 s s Q a þ Q s12e ashhe > Z h > = < 11e xxe 2 þ Cðx; h; tÞ Q s12e asxxe þ Q s22e ashhe TðzÞð1 zÞdz; > 2hhs > ; : 0

Q Tx ¼ Q Th ¼ 0; 9 8 a 9 MPx > Z hþha > = = < e31e > 2 2 P ¼ ea32e ð1 zÞdzV Mh > > > ; ha 2h ; : 0 MPxh 9 9 8 a 8 s e31e za > e31e zs > Z hþha > Z h > = = < < 2 2 þ2 ea32e za ð1 zÞdzwa þ 2 es32e zs ð1 zÞdzws ; > > > h 2hhs > ; ; : : 2 0 0

8 P > < Nx NPh > : P Nxh

Q Px

Z

2ea ¼ 15e ha

h 2

Z

þ es15e

Q Ph ¼ ea24e

hþh a 2

2h

2hhs

Z

hþh a 2

h 2

þ es24e

Z

@V za dz þ ea15e @x Pðzs Þdz

2h

3 P i 6 A11 6 i¼1 6 3 6P Ai21 9 6 8 6 N x > i¼1 6 > > > > 6 > > > 6 > Nh > > > > 6 0 > > = 6 xh 6 ¼6 3 > 6P M x > > > > > 6 Bi > > 11 > > 6 > Mh > > > 6 i¼1 > > ; 6 : 3 6P Mxh 6 Bi21 6 6 i¼1 6 4 0

i¼1 3 P i¼1

0

2 Qx Qh

3 P

C i44

6 6 i¼1 ¼6 4 0

Z

hþh a 2

h 2

Pðza Þ @wa dz Rþz @h

i¼1 3 P i¼1

Ai22

0 3 P

3 P

Bi22

0

i¼1

C i55

3 P

Ai66

0

0

3 P

i¼1

Bi12

0

Eeff 2ð1 þ meff Þ

Q 44 ¼ j

A;

Eeff ; 2ð1 þ meff Þ

h 2

2h

Q 55 dz; C 155 ¼

Z

h 2

2h

Q 44 dz;

,

meff Eeff Eeff A; Q 21 ¼ ; Q 22 ¼ 1  m2eff 1  m2eff

C 44 ¼

Z

h 2

Q 44 dz;

2h

Q 55 ¼ Q 44 =A;

C 55 ¼

Z

h 2

h2

, A;

Q 55 dz;

A ¼ 1 þ z=R;

where Eeff and meff are the effective elastic modulus and effective Poisson’s ratio of FGM layer, respectively. k is the shear correction factor introduced by Kadoli and Ganesan [15], Reddy [19] and is equal to 5/6.

 Z A2ij ; B2ij ; D2ij ¼ C 255 ¼

Z

hþh a 2

h 2

2h

h2hs

Q aije ð1; z; z2 Þdz;

C 244 ¼

Z

hþh a 2 h 2

Q a55e dz;

Q a44e dz;

ðA3ij ; B3ij ; D3ij Þ ¼ Z

hþh a 2

h 2

Z

2h

Q sije ð1; z; z2 Þdz;

h2hs

C 344 ¼

Z

2h

2hhs

Q s55e dz;

Q s44e dz:

3 P 3 P i¼1

Bi66

3 7 c  7 xz ; 7 5 chz

Bi21

Di11 Di21 0

Akij ;

k¼1 3 X

3 X

Bkij ;

k¼1

!

3 X

Dkij

; i; j ¼ 1; 2; 6;

k¼1

C kii ; i ¼ 4; 5;

k¼1 2 2  66 @ ;  11 @ þ 1 A L11 ¼ A @x2 R2 @h2 3 P i¼1 3 P i¼1

0

i¼1

3 P

Bi11

3 X

  ij ; D  ij ¼ Aij ; B

 2 hs ; 2

i¼1

i¼1 3 P

Q 66 ¼

 ii ¼ C

Ai12

0

2h

Z

Q ij ð1; z; z2 Þdz; C 144 ¼

Appendix C

Pðzs Þ ¼ z2s 

3 P

h 2

Eeff meff Eeff ; Q 12 ¼ Q 11 ¼ 1  m2eff 1  m2eff

C 355 ¼

@ws ; @x

i¼1



h 2

@w Pðza Þdz a @x

Pðzs Þ @ws dz ; Rþz @h

 2 ha ; 2

2



hþh a 2

2za @V dz þ ea24e ðR þ zÞha @h

2hhs

Pðza Þ ¼ z2a 

Z

Z

Bi12 Bi22

0 3 P i¼1 3 P i¼1

Di12 Di22 0

3 0

7 7 7 7 0 7 78 ex 9 7> > > > 7> > 3 > 7> > eh > P > i 7> > B66 7> > = i¼1 7 xh ; 7 > 7> > > > jx > 0 7 > > > 7> > > jh > 7> > ; : 7> 7 jxh 7 0 7 7 7 3 P i 5 D66

i¼1

 11 L14 ¼ B

L12 ¼

 12 þ A  66 @ 2 A ; R @x@h

L13 ¼

 12 @ A ; R @x

Z hþha  66 @ 2  66 @ 2  12 þ B 2 @2 B @ B þ 2 2 ; L15 ¼ za dz ; ; L16 ¼ 2ea31e 2 h @x R @x@h @x R @h 2

L17 ¼ 2es31e

Z

2h

2hhs

zs dz

@ 2 ; L18 ¼ @x ha

Z

hþh a 2

h 2

ea31e dz

 12 @ 2  66 þ A @ A ; L21 ¼ ; @x R @x@h

2 2 2        66 @ þ A22 @  C 55 ; L23 ¼ A22 þ C 55 @ ; L24 ¼ B66 þ B12 @ L22 ¼ A 2 @h @x2 R2 @h2 R2 R @x@h R

2 2    66 @ þ B22 @ þ C 55 ; L25 ¼ B 2 2 2 @x R R @h " # Z h Z h 2ea32e 2þha ea24e 2þha Pðza Þ @ za dz þ dz ; L26 ¼ h h R þ z @h R R 2 2

"

L27

2es32e ¼ R

Z

2h

es zs dz þ 24e h R 2hs

Z

h2 2hhs

# Pðzs Þ @ dz ; Rþz @h

456

G.G. Sheng, X. Wang / Composite Structures 90 (2009) 448–457

L28 ¼

L31

Z

2 Rha

hþh a 2

ea32e dz

h 2

 21 @ A ¼ ; R @x

L32

ea þ 24e R

Z

! 2za @ dz ; @h ðR þ zÞha

hþh a 2

h 2

L62 ¼ 2e32e

 22 @  55 @ A C ¼ 2  2 ; @h R R @h

L63 ¼ 2e32e

2 2 2    44 @ þ C 55 @  A22 þ N0 @ ; L33 ¼ C 2 2 2 2 @x @x2 R @h R

L34

  44 @  B21 @ ; ¼C @x R @x Z

ea24e

L36 ¼

R

ea24e R

Pðza Þ @ þ ea15e dz R þ z @h2

Z

2h

2

Pðzs Þ @ þ es15e dz R þ z @h2

h2hs

Z

hþh a 2

h 2

 66 @ 2 @2 B þ 2 ; 2 @x R @h2   B12  @ ¼ ;  C 44 @x R

 11 L44 ¼ D

L46 ¼

Z

2ea31e

Z

2es31e

2 L48 ¼ ha

Z

hþh a 2

h2

Z

hþh a 2

2

Pðza Þdz

Z

2ea32e

@  @x2 R

2h

h 2

Z

2

@ 2es Pðzs Þdz 2  32e h @x R 2hs Z

hþh a 2

h 2

za dz

@2 2  @x2 Rha

Z

es15e

2ea ea31e zdz  15e ha

Z

h 2

# Pðza Þdz

h 2

hþh a 2

L67 ¼ R þ

@ ; @x

@ ; Pðzs Þdz h @x 2hs

Z

hþh a 2

h 2

!

h 2

L72 ¼ 2e32e

L73 ¼ 2e32e

Z

L52 L54

2hhs

Z

 e24e

 55 R @  22  C B ¼ ; 2 @h R

L75 ¼ 2e32e

 22 @ 2 D

L57 ¼

L58 ¼

" # Z h Z h 2 2es32e 2 Pðzs Þ @ zs zdz  es24e dz ; @h R 2hhs h2hs R þ z 2 Rha

Z

L61 ¼ 2e31e R

hþh a 2 h 2

Z h 2

ea32e zdz

hþh a 2

za dz



@ ; @x

ea24e

Z h 2

hþh a 2

! 2za @ dz ; @h ðR þ zÞha

Z

Z

2h

2h

2hhs 2h

2hhs

2h

2hhs

Z

Pðza Þdz

@ ; @x

# Pðza Þ @ dz ; Rþz @h

hþh a 2

h 2

Z

#

hþh a 2

Pðza Þ2 2

ðR þ zÞ

h 2

@2

dz

@h2

Z

hþh a 2

h 2

@2 @h2

za Pðza Þdz

@2 @x2

;

L71 ¼ 2e31e R

Z

2h

h2hs

zs dz

@ ; @x

@ ; @h Z

2h

Pðzs Þdz

h2hs

@2 @x2

Pðzs Þ @2 ; dz R þ z @h2

h2hs

Z

 4n33 R

a7 ¼ 2R

Z

dz

zs dz  e15e R

2h

"

L76 ¼ n11 R

zs dz

2hhs

L74 ¼ 2e31e R

za Pðza Þ

paze za TðzÞdz;

2h

@2 @x2

z2a dz;

ðR þ zÞ

2hhs

Z

Z

@2 þ n22 R @x2

2

2h

 66 þ B  12 @ 2 B @ za dz ; L51 ¼ ; @x R @x@h

2  55 þ D  66 @ ;  C @x2 R2 @h2 " # Z hþha Z hþh 2 2ea32e 2 a Pðza Þ @ ¼ za zdz  ea24e dz ; h h R þ z @h R 2 2

L55 ¼ L56

L53

hþh a 2

hþh a 2

h 2

4na33e 2Rn11 za dz þ ha ha

h 2

hþh a 2

Z

zza dz  e15e R

Pðza Þ2 dz hþh a 2

Z

2Rn22 ha

Pðza Þdz

h 2

zza dz  Re24e

h 2

hþh a 2

"  55 C  66 @ þ ¼B þ ; 2 2 2 @x R R @h  12 þ D  66 @ 2 D ¼ ; R @x@h  22 @ 2 B

2

Z

h 2

Z

hþh a 2

hþh a 2

h 2

Z

hþh a 2

h 2

Z

hþh a 2

Pðza Þ @ ; dz R þ z @h2

h 2

Z

Z

2

h 2

 4n33 R

ea32e dz;

#

2h

hþh a 2

Z

@ ; @h

za dz  e15e R

h 2

zs dz;

 12 þ D  66 @ 2 D ; R @x@h

hþh a 2

hþh a 2

Z

L65 ¼ 2e32e

L66 ¼ n11 R

2h

2hhs

Z

za dz;

 66 @ 2  12 þ B B ; R @x@h

L45 ¼

Z

" hþh a 2

za dz

h 2

L64 ¼ 2e31e R

a6 ¼ 2R

zs zdz 

2hhs

h 2

h 2

@ za zdz  ea15e @x

h 2

" L47 ¼

L42 ¼

 66 @ 2 D @2  44 ; þ 2 C 2 @x R @h2

"

hþh a 2

2za @ 2 2ea dz 2 þ 15e ðR þ zÞha @h ha

 11 L41 ¼ B L43

Z

hþh a 2

"

 22 @ B C 55 @ ¼  2 ; R @h R @h

2

h 2

es L37 ¼ 24e R

L38 ¼

hþh a 2

L35

 e24e

Z

zzs dz  Re24e

Pðzs Þ2 dz 2h

h2hs

Z

zzs dz  e15e R

#

h2 2hhs

Z

2h

h2hs

@2 þ n22 R @x2

Z

Pðzs Þdz

@ ; @x

# Pðzs Þ @ dz ; Rþz @h h2

2hhs

Pðzs Þ2 2

ðR þ zÞ

dz

@2 @h2

z2s dz;

psze zs TðzÞdz:

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