Dynamic analysis and active control of smart doubly curved FGM panels

Composite Structures 102 (2013) 205–216

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Dynamic analysis and active control of smart doubly curved FGM panels Y. Kiani ⇑, M. Sadighi, M.R. Eslami Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Available online 14 March 2013 Keywords: Doubly-curved panel Hybrid functionally graded materials Hybrid Fourier–Laplace transformation Modified sander’s shell theory Velocity feedback controller

a b s t r a c t In this paper, active control and dynamic analysis of shallow doubly curved functionally graded material (FGM) panels integrated with sensor/actuator piezoelectric layers are presented, analytically. Properties of the FGM panel are dictated using a simple power law model across the thickness. Based on the modified Sander’s shell theory combined with first order shear deformation theory, total potential energy of the system is derived. Five mechanical equilibrium equations and two electrical equations are established as the governing equations. The classical negative velocity feedback controller rule is implemented to suppress the vibration response of the panel. For both the active and passive cases, established equations are reduced into new five equations in terms of displacements and rotations. Employing the combined analytical Fourier–Laplace transformation, consistent with the panels with movable simply-supported edges, an accurate closed-form solution is resulted for response of the panel in both active and passive cases in real time domain. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Panels made of a rectangular planform are one of the most practical tools of solid structures that are used widely in mechanical and civil engineering. Therefore, theoretical analysis of panels to reach a reliable design has been the subject of many researches for a long period on time. Based on the three-dimensional elasticity equations, Chern and Chao [1] developed a power series solution in depth coordinate combined with the Navier method to study the free vibration of laminated doubly curved panels. Based on a higher order displacement field theory, consistent with sandwich structures, a Navier solution is performed by Khare et al. to study the thermoelastic bending [2] and free vibration [3] of laminated doubly-curved panels. Employing the modified Sander’s shell theory, Reddy [4] developed the first order theory-based analysis of doubly curved laminated panels. As reported, unlike plates, antisymmetric doubly curved panels are not consistent with the conventional Navier solution. A Ritz-based solution is reported by Qatu and Asadi to obtain the natural frequencies of thin doubly-curved shallow panels [5]. Three classes of panels combined with all 21 possible combinations of boundary conditions are reported in this work. Using an iso-parametric doubly curved quadrilateral shear flexible element, Chandrashekhara [6] solved the free vibration problem of shear deformable panels in an anisotropic laminated configuration. Reddy and Chandrashekhara [7] studied the von-Karman-based geometrically non-linear transient response of panels based on an iterative finite element method. ⇑ Corresponding author. Tel.: +98 2164543448; fax: +98 2166419736. E-mail address: [email protected] (Y. Kiani). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.02.031

Most recently, Hosseini Hashemi and Fadaee [8] reported the free vibration analysis of Levy-type spherical panels based on an exact closed form solution. In this analysis, Five FSDT-based partially differential equations are uncoupled and solved analytically to present the characteristic equations of free vibration problem, exactly. A complete review of dynamic analysis of doubly curved composite panels in recent decade is reported by Qatu et al. [9]. Functionally graded materials (FGMs) are a class of novel materials in which, unlike composites, properties of material vary smoothly in one or more specific direction. In comparison with the analysis of doubly-curved panels in composite configuration, researches on panels made of FGMs are so rare. Besides, most of the investigations are accomplished numerically and analytical solutions on the behaviour of FGM shell-panels are limited in number. Tornabene and Viola [10] implemented the generilized differential quadrature method to study the free vibration of doubly curved panel with arbitrary class of boundary conditions. An asymptotic perturbation method is developed by Zhang et al. [11] to study the behaviour of an FGM plate subjected to through-the-thickness thermal loading combined with external mechanical loadings. Zhao et al. [12] presented an element-free kp-Ritz method to investigate the linear free vibration and static analysis of cylindrical panels. Displacement field of the shell is given in the form of a finite set of mesh-free kernel particle functions and the methodology is capable of handling various types of boundary conditions. Besides, Zhao and Liew [13] developed the previous formulation to study the non-linear analysis of cylindrical panels in the framework of von-Karman geometrical non-linearity. Alijani et al. [14,15] presented the non-linear vibration of panels including thermal effects [15] when all edges are simply-supported

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with moving capability in normal to side planes directions. Established analysis of [14] is based on the classical shell theory of Donnell type. Time-dependent non-linear differential equations are obtained by means of Galerkin procedure and the solution in time domain is obtained. In contrast, in [15] a higher order shear deformation shell theory, i.e. Reddy-Amabili theory is used to calculate the total energy function of the shell and admissible motion conditions are established with the aid of Lagrange equations. A linear finite element formulation obtained based on a third-order inplane displacement field is developed in [16] to study the free vibration problem. Since the method is numerical, both clamped and simply-supported edges are taken into account. Pradyumna et al. [17] developed the previous formulation to study the dynamic response of a panel under the applied out of plane sudden load. To investigate the free vibration and parametric resonance of a cylindrical shell, a combined Galerkin-Generalised differential Quadrature (GDQ) method is carried out by Yang and Shen [18]. Based on the first five/six eigenfunctions of a beam suitable for various boundary conditions in axial direction, governing PDEs are reduced into new equations as functions of time and width direction. Solution of the space domain is obtained by means of GDQ method. As one of the most widely used smart materials, piezoelectric materials have gain incredible attention in recent decades. This type of smart structures are extensively adopted to control the deformation, vibration and buckling of various types of solid structures. A review of static and dynamic shape control of structures coupled with piezoelectric actuation is reported by Irschik [19]. In last decade, hybrid FGM structures, i.e. FGM structures combined with piezoelectric layers are under develop since they have the advantages of both piezoelectric and FGM structures, linked together. Some authors have obtained the general formulation to control the dynamic buckling of FGM shells [20] and plates [21] via the non-linear finite element method. In a series of works, Liew and his co-authors, presented the linear finite elements formulation to control the static and dynamic response of FGM cylindrical shell [22], plates under dynamic loading [23], Thermoelastic bending of cylindrical panels [24] and dynamic behaviour of spherical panels in frequency domain [25]. To control the natural frequencies of an FGM plate, Mirzaeifar et al. [26] developed an optimisation strategy for collocated piezo-FGM plates. Javanbakht et al. [27] developed a combined Fourier-layerwise formulation to investigate the response of shallow and non-shallow panels under the action of dynamic loads. Since the panel is integrated with piezoelectric layers, electric excitation is also considered as an external force. Sheng and Wang [28,29] performed the active control of FGM cylindrical shells integrated with perfectly bonded sensor and actuator layers. A constant-gain negative velocity feedback control law has been employed and various loading types such as step loading [28], moving load [29], and thermal shock [29] are analysed. To control the large amplitude vibration of the FGM rectangular plate, a non-linear finite elements formulation is presented by Fakhari and Ohadi [30]. In this research, two types of controller schemes, i.e. classical displacement-velocity feedback control and robust H2 control are analysed. A linear finite elements formulation is developed by Zheng et al. [31] to control the static and dynamic responses of an FGM cylindrical shell under thermo-mechanical loading. However, various studies are available on active control of FGM plates and shells, but within the literature survey of the present authors, there is no work reported for doubly-curved panels, especially in an analytical method. In what follows, FSDT-based formulation is presented to obtain the equations of motion for a hybrid doubly curved FGM panel via the modified Sander’s shell theory assumptions. As all edges of the shell are simply-supported with moving capability in normal to side planes, conventional Navier solution omits the dependency of the governing equations to the

space variables. Panel is assumed to be mechanically induced. Various types of mechanical loading are considered. Performing the Laplace transformation on the resulted equations, solution of the displacements and rotations are described in the Laplace domain as a function of Laplace parameter. An analytic procedure is developed to re-transfer the Laplace domain equations into the real time domain. In dynamic analysis, to reach a reliable accuracy, sufficient terms in Fourier series expansion have to be considered. Reported results cover plates, spherical panels, cylindrical panels, and hyperbolic paraboloidals as especial cases. The obtained results are validated with the available data in the open literature. Influences of various involved parameters are examined via different parametric studies. 2. Basic equations Consider a doubly-curved panel made of FGMs collocated with a pair of sensor/actuator piezoelectric layers. Length, width,host thickness, sensor thickness and actuator thickness are indicated as a, b, hH, hs, and ha, respectively. Total shell thickness is indicated as h = ha + hs + hH. Hybrid panel is referred to an orthogonal curvilinear coordinate system (x, y, z) as shown in Fig. 1. Radius of principal curvatures of the middle surface are assumed to be Rx and Ry which are overlapped on coordinate axes. In this work, same as those used in definition of each lamina thickness, superscripts a, s and H refer to the actuator, sensor and the FGM layer, respectively. In general, distribution of material properties in an FGM media is stated as a mathematical position-dependent function. In this work, following Reddy [32,33], distribution of ceramic volume fraction, Vc for the host layer is dictated using a simple power law form, such that

Vc ¼

 k 1 z þ H 2 h

ð1Þ

where k is a power law index and depicts the intensity of property dispersion. Material non-homogeneous properties of a panel may be obtained by means of the Voigt rule of mixture [34–37]. Thus, using Eq. (1), each property of the host layer, P, as a function of thickness coordinate becomes

PðzÞ ¼ Pm þ Pcm



1 z þ 2 hH

k ð2Þ

where Pcm = Pc  Pm, and Pm and Pc are the corresponding properties of the metal and ceramic, respectively. In the present work, we assume that the elasticity modulus, E and density of the substrate

Fig. 1. Coordinate system and geometry of a hybrid doubly curved FGM panel.

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layer, q are described by Eq. (2), while Poisson’s ratio, m, is considered to be constant across the thickness of the middle layer [38–40]. Reddy and Chandreshekhara developed a FSDT for analysis of panel-type structures as an extension of Sander’s shell theory [7]. Based on this theory that is consistent with the moderately thick class of shells, displacements on an arbitrary point of the panel ; v  ; wÞ  are described as a function of mid-surface displacements ðu (u, v, w) and mid-surface rotations (uy, ux) [4,7] such that

8 9  8 8 9 9 > 1 þ Rzx uðx; y; tÞ > > >  u ðx; y; z; tÞ > > > > > < ux ðx; y; tÞ > < = = = <   ðx; y; z; tÞ ¼ v þ z uy ðx; y; tÞ z v ðx; y; tÞ 1 þ > > > > > Ry : > : ; > ; > > > 0 wðx; y; z; tÞ : ; wðx; y; tÞ

2

6 6 6 6 cyz ¼ 6 6 > > > 6 > > cxz > > > 6 > > > > : ; 4

cxy

@ @x

0

1 Rx

@ z @x

ð3Þ

0

@ @y

1 Ry

0

0

 R1y

@ @y

0

 R1x

0

@ @x

1

0

@ z @y

@ @ ð1 þ zc0 Þ @y ð1  zc0 Þ @x

ð4Þ

theory characteristic which is adopted to account the condition of zero strain for rigid body motion [4,17]. Constitutive equations of a panel including the linear piezoelectroelastic effects may be written as [44]

38 9 2 0 0 > exx > Q 11 Q 12 0 0 > > > 7> 6 6 > > > > e 0 0 7> 6 Q 21 Q 22 0 6 0 yy > < > = 6 7> 6 7 6 syz ¼ 6 0 7 cyz  6 0 0 Q 44 0 6 0 > > > 7> 6 6 > > > > > >c > 6 0 > > > 6 sxz > 0 0 Q 55 0 7 > > > 5> 4 4 e15 xz > > > > > > > > > : : ; ; c sxy 0 0 0 0 Q 66 0 xy 2

8 9 2 0 > < Dx > = 6 Dy ¼ 4 0 > : > ; Dz e31

0

0

e15

0 e32

e24 0

0 0

8 9 exx > > > > > > 3> 2 > > g1 0 > > < eyy > = 7 c 6 0 05 þ 4 yz > >c > > > 0 0 > > xz > > > > > : ;

0

0 e31

3

78 9 0 e32 7> Ex > 7< = 7 e24 0 7 Ey 7> : > ; 0 0 7 5 Ez 0

0

0

g2 0

38 9 > < Ex > = 7 0 5 Ey > > g3 : Ez ; 0

cxy

ð5Þ where rij, sij, Qij, eij, Ei, gi, Di stand for the normal stress components, shear stress components, elastic constants, piezoelectric constants, elements of the electric field vector, dielectric permittivity coefficients and electric displacement coefficients, respectively. For each of the piezoelectric layers, elements of the electric field are obtained as the negative gradient of the electric potential [20,45]. When piezoelectric layers are thin enough, electric potential may be considered as a linear function of the thickness coordinate [20,22,45]. This assumption is validated for thin piezoelectric layers in [46] based on a three-dimensional layerwise finite elements analysis. Furthermore components of in-plane electrical field through the piezoelectric layers are neglected since only the transverse one is dominant [20–22,24,44,45]. Therefore electric field for both layers may be written as follows:

3

7 07 7 7( ) a 07 7 U 7 7 0 7 Us 7 7 07 5

6 60 6 6 6 1a 6h z ¼6 6 s > > > > 60 Ex > > > > 6 > > > > > > 6 > 60 > > Esy > > > 4 > > > > > > : s; 0 E

ð6Þ

1 hs

Here, Us and Ua stands as the electric potential difference function between two surfaces of sensor and actuator layers, respectively. When the bottom surface of each piezoelectric layer is grounded, Us and Ua describe the electric potential of the sensor top surface and actuator top surface, respectively. Based on the FSDT of shallow shells (which accepts the conditions h/Rx, h/Ry < 0.05 [47]) when z/Rx and z/Ry are negligible in comparison with unity, the stress resultants are related to the stresses by the equations [4,47]

Z

ðNxx ; Nyy ; Nxy Þ ¼

ðQ xz ; Q yz Þ ¼ K s

1 2

hH þha

12 hH hs

ðM xx ; Myy ; M xy Þ ¼

where eij and cij are normal and shear strain components respec  tively. Besides, c0 ¼ 12 R1x  R1y . This constant is the Sander’s shell

8 rxx 9 > > > > > > > > > > r > yy > > > < =

0

3

8 9 7> > u > > > @ 7> z @y 7> > > v> > > < = 7 7 1 7 w > > > 7> > > ux > > 0 7 > > > 5> : ; u @ y z @x 0

2

z

In general, Donnell–Mushtari–Vlasov (DMV), Reissner and Sanders theories are especial cases of the modified sanders shell theory that is used in the present work. For a brief overview of DMV, Reissner, Sanders and modified Sanders shell theories one may refer to the works of Chaudhuri and Kabir [41,42]. The linear strain–displacement relations using the modified Sander’s theory assumptions [4,6,43] and displacement field (3) are obtained as follows

8 9 exx > > > > > >e > > > > > < yy > =

8 a9 Ex > > > > > > > > > a> > > > > E > y> > > > > > > > > a > =

Z

Z

1 2

ðrxx ; ryy ; sxy Þdz

hH þha

12 hH hs 1 2

zðrxx ; ryy ; sxy Þdz

hH þha

12 hH hs

ðsxz ; syz Þdz

ð7Þ

where Ks is the shear correction factor. It is well-accepted that the value of 5/6 can be used as an approximate value of Ks for the FGM or composite plates and panels [43,48], therefore, in this research, the shear correction factor is taken as Ks = 5/6 for the hybrid FGM shell-panel. Substituting Eqs. (3)–(6) into Eq. (7), yields the stress resultants in terms of the mid-plane displacement as

8 9 Nxx > > > > > > > > > > > > N > > yy > > > > > > > > > > Nxy > > > > > > > > > > > > > > > < Mxx > =

2

A11 A12

0

B11 B12

0

0

0

3

7 6 6 A21 A22 0 B21 B22 0 0 0 7 7 6 7 6 7 6 6 0 0 A66 0 0 B66 0 0 7 7 6 7 6 6B B 0 0 7 7 6 11 12 0 D11 D12 0 7 6 ¼6 7 > > > > 7 6 B M B 0 D D 0 0 0 yy 21 22 21 22 > > > > 7 6 > > > > 7 6 > > >M > > 6 0 0 B > 0 0 D66 0 0 7 > xy > 66 7 6 > > > > 7 > > 6 > > 7 6 > > > > 7 6 0 0 0 0 0 0 K Q A 0 > > s 44 yz > > 4 5 > > > > : ; 0 0 0 0 0 0 0 K s A55 Q xz 8 9 2 3 u;x þ Rwx ea31 Ha1 es31 Hs1 > > > > > > > > 7 6 > > > > 6 ea Ha es Hs 7 > > v ;y þ Rwy > > 7 6 > > 1 1 32 32 > > 7 6 > > > > 7 6 > > > > u þ v 7 6 0 > > ;y ;x 0 > > 7 6 > > > > 7 ( ) 6 > > > > a s > > 6 a s ux;x < = 6 e31 H2 e31 H2 7 Ua 7 7  þ6 6 a a s s7 > > uy;y > > 6 e32 H2 e32 H2 7 Us > > > > 7 6 > > > > 7 6 > > > 6 0 > ux;y þ uy;x  c0 ðv ;x  u;y Þ > > 0 7 > > 7 6 > > > > 7 6 > > > > 6 7 v > > > > uy þ w;y  Ry 6 0 0 7 > > > > 5 4 > > > > > > : ; u ux þ w;x  Rx 0 0 where Aij, Bij and Dij stand for the stretching, bending and coupling stretching-bending stiffnesses, respectively. Since the considered FGM is an isotropic layer and both of piezoelectric layers are isotropic, aforementioned functions are calculated as follows

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A12 ¼ A21 A44 ¼ A55

B66 D11 D12 D66

E1

2

s

a

s

In function dV, p and qa are the external mechanical applied load and surface charge density applied to the actuator layer, respectively. In general sensor layer is not actuated externally, hence this term is not included in dV. Recalling Eq. (6), substituting Eq. (12) into Eq. (11), and performing the Green-Gauss theorem to relieve the virtual displacements, following system of equations of motion are resulted

þ

Ea h Ha2 Es h Hs1 þ 1  ma2 1  ms2 1m mH E2 ma Ea ha Ha2 ms Es hs Hs2 ¼ B21 ¼ þ þ 2 H 1  ma2 1  m s2 1m a a a s s s E2 E h H2 E h H2 ¼ þ þ 2ð1 þ mH Þ 2ð1 þ ma Þ 2ð1 þ ms Þ

B11 ¼ B22 ¼ B12

a

Ea h Ha1 Es h Hs1 þ 1  m a2 1  m s2 1  mH m H E1 ma Ea ha Ha1 ms Es hs Hs1 ¼ þ þ 2 H 1  m a2 1  ms2 1m s a a a E1 E h H1 Es h Hs1 ¼ A66 ¼ þ þ H a 2ð1 þ m Þ 2ð1 þ m Þ 2ð1 þ ms Þ

A11 ¼ A22 ¼

E2

H2

    Q xz 2 1 € þ S2 þ S3 u€x ¼ S1 þ S2 u Rx Rx Rx     Q yz 2 1 dv : Nxy;x þ Nyy;y  c0 M xy;x þ ¼ S1 þ S2 v€ þ S2 þ S3 u€y Ry Ry Ry

þ

a

du : Nxx;x þ Nxy;y þ c0 Mxy;y þ

ð9Þ

and the followings are pre-defined

E1 ¼ E2 ¼ E3 ¼

Z

1 2

12

Z

1 2

12

Z

1 2

12



 s

  Ecm H EðzÞdz ¼ h Em þ kþ1 hH   hH 1 1 H2 2zEðzÞdz ¼ h Ecm  k þ 2 2k þ 2 hH    hH 1 1 1 1 H3 Em þ Ecm  þ z2 EðzÞdz ¼ h 12 k þ 3 k þ 2 4k þ 4 hH

Ha1 ; H1 ¼ ð1; 1Þ  1 H a H s Ha2 ; Hs2 ¼ ðh þ h ; h  h Þ 2  a s 1 H2 H a a2 H2 H s s2 H3 ; H 3 ¼ ð3h þ 6h h þ 4h ; 3h þ 6h h þ 4h Þ 12

ð10Þ

where

1 2 1 2

Z

hH þha

hH 12 hH

12 hH hs

Daz dz ¼ qa Dsz dz ¼ 0 a

ð13Þ

s

Si ¼ Ii þ h Hai þ h Hsi

ð11Þ

0

where virtual strain energy, virtual electrical energy, virtual work done by external applied forces and virtual kinetic energy of the structure are denoted by dUs, dUe, dV and dK, respectively. The aforesaid functions for a panel-type structure become [6,20,47]

Z Z

ðrxx dexx þ ryy deyy þ rxy dcxy þ K s sxz dcxz þ K s syz dcyz ÞdAdz Z Z  a a  dU e ¼  Dz dEz þ Dsz dEsz dAdz z A Z dV ¼  ðpdw þ qa dUa ÞdA A

and

Ii ¼

R 12

hH

12 hH

zi1 qðzÞdz; i ¼ 1; 2; 3.

Closed form expressions for these integrals may be obtained, same with those exist in Eq. (10) for Ei’s. In this work, we only focus on the movable simply-supported (SS1) edge conditions. Mathematical expressions for these class of edge supports may be written in the following form [50]

uðx; 0; tÞ ¼ uðx; b; tÞ ¼ 0

wðx; 0; tÞ ¼ wðx; b; tÞ ¼ 0

t1

z

1 s h

Z

ux ðx; 0; tÞ ¼ ux ðx; b; tÞ ¼ 0

Equations of motion for a hybrid doubly curved FGM panel are obtained by means of the Hamilton principle. According to this principle, an equilibrium position in the structures holds when the following equality occurs [49,50]

ðdU e þ dU s þ dV  dKÞdt ¼ 0

dUs :

1 a h

v ð0; y; tÞ ¼ v ða; y; tÞ ¼ 0

Myy ðx; 0; tÞ ¼ M yy ðx; b; tÞ ¼ 0

3. Equations of motion

dU s ¼

dUa :

hH



Z

Nxx N yy €  þ p ¼ S1 w Rx Ry   1 € þ S3 u€x dux : Mxx;x þ M xy;y  Q xz ¼ S2 þ S3 u Rx   1 duy : Mxy;x þ M yy;y  Q yz ¼ S2 þ S3 v€ þ S3 u€y Ry dw : Q xz;x þ Q yz;y 

s

Ea h Ha3 Es h Hs3 ¼ D22 ¼ þ þ 2 H 1  ma2 1  ms2 1m mH E3 ma Ea ha Ha3 ms Es hs Hs3 ¼ D21 ¼ þ þ 2 H 1  ma2 1  ms2 1m a a a s s s E3 E h H3 E h H3 ¼ þ þ 2ð1 þ mH Þ 2ð1 þ ma Þ 2ð1 þ ms Þ E3

Nyy ðx; 0; tÞ ¼ Nyy ðx; b; tÞ ¼ 0

M xx ð0; y; tÞ ¼ M xx ða; y; tÞ ¼ 0

uy ð0; y; tÞ ¼ uy ða; y; tÞ ¼ 0 wð0; y; tÞ ¼ wða; y; tÞ ¼ 0 Nxx ð0; y; tÞ ¼ Nxx ða; y; tÞ ¼ 0

ð14Þ

4. Controller scheme In this study, a constant-gain negative velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed-loop system to provide feedback control of the integrated FGM doubly curved structure. This controller is widely used in vibration and buckling control of various types of FGM structures [20–25,28–30,51]. Mathematical statement of the aforesaid control law is

Ua ¼ Gv U_ s

ð15Þ

Here, the velocity feedback control gain is indicated as Gv. 5. Space solution

A

    z z _x _x u_ þ zu 1þ du_ þ zdu q 1þ dK ¼ Rx Rx z A     z z _ w _ dAdz v_ þ zu_ y 1 þ dv_ þ zdu_ y þ wd þ 1þ Ry Ry Z Z

ð12Þ

Equations of motion (13) are stated in terms of stress resultants and electric displacements. When system of Eqs. (8) is imported into the Eq. (13), seven coupled partially differential equations are established in terms of mid-plane displacements, mid-plane rotations and sensor/actuator electrical fields. Considering the assumed boundary conditions (14), following set of trigonometric expansions, is consistent with the resulted system of equations

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uðx; y; tÞ ¼

v ðx; y; tÞ ¼

1 X 1 X U mn ðtÞ cosðkm xÞ sinðln yÞ n¼1 m¼1 1 X 1 X

if both of the piezoelectric layers act as sensors, F U mn is equal to zero, whereas for the case when they act as actuators F U mn – 0. For the case, when piezoelectric layers act as sensor, the sensory voltage is obtained using Eq. (19) as

V mn ðtÞ sinðkm xÞ cosðln yÞ

n¼1 m¼1 1 X 1 X

h i1 h i fUmn g ¼ K UU K UmnD mn

W mn ðtÞ sinðkm xÞ sinðln yÞ

wðx; y; tÞ ¼

22

n¼1 m¼1 1 X 1 X

X mn ðtÞ cosðkm xÞ sinðln yÞ

ux ðx; y; tÞ ¼

n¼1 m¼1 1 X 1 X uy ðx; y; tÞ ¼ Y mn ðtÞ sinðkm xÞ cosðln yÞ

ð16Þ

n¼1 m¼1

where km ¼ map and ln ¼ nbp are pre-assumed. Due to the presence of external load, p, and the actuator electrical force qa, resulted equations are non-homogeneous. Besides, because of the coupling of the functions Ua and Us with displacement field in equilibrium equations, these functions, also, have to be expanded in a suitable form of the trigonometric functions. The next set of sinusoidal Fourier expansions is proper for the aforesaid functions 1 X 1 X pðx; y; tÞ ¼ pmn ðtÞ sinðkm xÞ sinðln yÞ n¼1 m¼1 1 X 1 X

a

q ðx; y; tÞ ¼

Ua ðx; y; tÞ ¼

qamn ðtÞ sinðkm xÞ sinð n¼1 m¼1 1 X 1 X

h i1s h is fSmn g ¼ K UU K UmnD mn

U ðx; y; tÞ ¼

15

11

ð21Þ

fDmn g51

ð22Þ

The actuated voltage, considering the controller law definition (16) is obtained as

ln yÞ

h i1s h is fAmn g ¼ Gv K UU K UmnD mn

15

11

Amn ðtÞ sinðkm xÞ sinðln yÞ

1 X 1 X

fDmn g51

When the controller rule (15) is settled between piezoelectric layers (i.e. the active case [26]), one layer acts as the sensor (the bottom one) and the other one as the actuator (the top one). Since the panel is excited mechanically, electric charges are induced and collected in the sensor layer. Through the closed loop control described in Eq. (15), the charges give rise to electric potentials, which are subsequently amplified and converted into the open circuit voltage. The signal is then fed back into the distributed actuator. Due to the effect of piezoelectric layers observed in the sixth of Eqs. (13), stresses and strains are generated. The resultant force can actively control the dynamic response of the structure. Since no applied external charge is considered for the sensor layer as input, the generated potential on the sensor is given from Eq. (13)

fD_ mn g51

ð23Þ

Combining the Eqs. (18), (22), and (23) gives us

n¼1 m¼1 s

25

Smn ðtÞ sinðkm xÞ sinðln yÞ

ð17Þ

b mn  fD_ mn g þ ½ K b mn  fD b mn  fDmn g ¼ f b € mn g þ ½ C ½M F mn g51 51 51 51 55 55 55

n¼1 m¼1

ð24Þ

Considering the Eqs. (16) and (17), the first five equations of motion (13) may be written in the following form h i DD

  DU € mn g þ K DD Mmn 55 fD fUmn g21 ¼ F Dmn 51 51 mn 55 fDmn g51 þ K mn 52

ð18Þ T

Here, {Dmn} = < Umn, Vmn, Wmn, Xmn, Ymn > is the displacement vector and {Umn} = < Amn, Smn > T is the electrical potential vector. Furh i

DD DU thermore, the matrices, MDD mn ; K mn and K mn stand for the mass,

where the following definitions are needed



b mn  ¼ MDD ½M mn 55 h ia h i1s h is

DD DU b mn  ¼ a M DD ½C K UU K UmnD mn 55 þ b K mn 55  Gv K mn mn 51 15 11 h is h i1s h is DD DU UU UD b ½ K mn  ¼ K mn 55 þ K mn K mn K mn 51 15 11  D  fb F mn g ¼ F mn 51 ð25Þ

stiffness and piezoelectric matrices, respectively. Elements of these matrices are given in appendix A. Besides, if Rayleigh damping is of h i 



 interest, the term C DD ¼ a MDD þ b K DD fD_ mn g should be

Eq. (24) is implemented for vibration control of hybrid FGM doubly-curved panels.

added to the left hand side of the above equation. Here, constants a and b are Rayleigh constants [52]. Similar to the mechanical equations, electrical equations presented as the sixth and seventh equations of system (13), may be written as

6. Laplace transform

mn

h

K UmnD

i 25

mn

h i fDmn g51  K UU mn

22

51

mn

n o fUmn g21 ¼ F Umn

21

ð19Þ

h i h iT h i D UD is the permittivity matrix that its Here, K U and K UU mn ¼ K mn mn elements are presented in Appendix A. When {Umn} is eliminated between Eqs. (18) and (19), displacement vector is revealed through the next system of equations



h i

DD € MDD fD_ mn g51 mn 55 fDmn g51 þ C mn 55  h i h i1 h i

DU þ K DD K UU K UmnD mn 55 þ K mn mn 22

52

¼





F Dmn 51

h i þ K DmnU

h 52

UU

K mn

i1 n 22

U

F mn

25

o 21

 fDmn g51 ð20Þ

The above equation is obtained when there is no controller scheme between the piezoelectric layers (i.e. passive case [26]). In this case,

When shell-panel is subjected to an out-of-plane dynamic loading, system of Eqs. (20) or (24) have to be solved. To treat the dynamic analysis of the panel, initial conditions through the panel should be prescribed. When panel is initially at rest, following initial condition describe the panel conditions prior to loading.

U mn ð0Þ ¼ V mn ð0Þ ¼ W mn ð0Þ ¼ X mn ð0Þ ¼ Y mn ð0Þ ¼ 0 _ mn ð0Þ ¼ X_ mn ð0Þ ¼ Y_ mn ð0Þ ¼ 0 U_ mn ð0Þ ¼ V_ mn ð0Þ ¼ W

ð26Þ

Dynamic response of equations of motion is followed, when Eq. (20) or (24) is transformed into the Laplace domain. Let’s assume s is the Laplace transformation parameter. In such a case, considering the initial conditions (26), performing the Laplace transform on Eq. (20) or (24) yields a new system of equations in which time dependency is eliminated. For instance, for the case of active control, equations in Laplace domain are written as

h

b mn þ s2 M b mn þ s C b mn K

i

55

F mn g51 fDmn g51 ¼ f b

ð27Þ

Here, a line over each quantity indicates its Laplace transform function.

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When system of Eqs. (27) is solved, each of the elements of the displacement vector is obtained in a closed form expression in Laplace domain. Laplace inverse definition is needed to re-transfer the displacement vector from Laplace domain into the time domain. 7. Analytical laplace inversion At the end of previous section, each of the five components of the displacement vector was obtained in the Laplace domain. In this section, an analytical Laplace inversion method is employed to express the displacement vector in the time domain. In this analysis, various types of dynamic loading are considered for the panel. Some typical types of dynamic loading and their Laplace transform functions are given in Table 1. Assume a function f(t), whose Laplace transformed function is given by f ðsÞ ¼ AðsÞ. Each element of the vector Dmn may be exBðsÞ

pressed in this form. The function B(s) is a polynomial function, while for some cases of applied loads, the function A(s) may contain non-polynomial (exponential) functions. Let’s divide the function A(s) as a multiplication of two new functions i.e. A(s) = APol(s)ANonPol(s). Here, APol(s) is a part of function A(s) that is only polynomial and the remaining is settled in ANonPol(s) (As an especial case when A(s) is completely polynomial, the function ANonPol(s) is equal to unity). The aim of this section is to present an analytical solution to the Laplace inverse function of f ðsÞ ¼ APol ðsÞ  ANonPol ðsÞ ¼ f1 ðsÞf2 ðsÞ. BðsÞ

For various types of loadings given in Table 1, the function f2 ðsÞ ¼ ANonPol ðsÞ and its Laplace inversion are given in Table 2. Pol The first part of the function f ðsÞ, i.e. f1 ðsÞ is equal to AðsÞ . Both BðsÞ

functions B(s) and A(s)Pol are polynomials. Let’s assume that the roots of the function B(s) are known. Some of the roots are complex, call them ci’s, and the others are real and indicated by ri’s. Number of ci’s and ris are represented by nc and nr, respectively. When all roots are simple, which happens in all the cases of this study, inverse of the function f 1 ðsÞ, i.e. f1(t) is obtained as [53,54]:

f1 ðtÞ ¼ Re

nc X APol ðci Þ i¼1

B0 ðci Þ

! ci t

e

þ

nr X APol ðrj Þ j¼1

B0 ðr j Þ

e

rj t

f ðtÞ ¼

Mathematical expression

Laplace transform

I

p(t) = PH(t)

II

p(t) = P(H(t)  H(t  t0))

III

pðtÞ ¼ Pð1  et=t0 ÞHðtÞ

IV

p(t) = Pt(H(t)  H(t  t0))

ðsÞ ¼ 1s P p  t s  ðsÞ ¼ 1es 0 P p   0 ðsÞ ¼ 1s  t tsþ1 p P 0  t s  t s 0 se 0 ðsÞ ¼ 1e 0 st p P 2

Load type

f2 ðsÞ

f2(t)

I II III IV

1 1  et0 s 1

d(t) d(t)  d(t  t0) d(t) H(t)  H(t  t0)  t0d(t  t0)

1et0 s t 0 set0 s s

ð29Þ

8. Results and discussion The procedure outlined in the previous sections is used here to study the free vibration, dynamic response and vibration control of hybrid doubly curved FGM panels. Constituents of the FGM layer are chosen from those exist in Table 3. Both sensor and actuator layers are the same in thickness and chosen as G  1195N. Properties of G  1195N are given in Table 4. 8.1. Comparison studies To show the accuracy and effectiveness of the procedure discussed above, some comparison studies are performed between results of our study and the available data in the literature. In Table 5, fundamental frequency of a thick moderately deep cylindrical/spherical panel is presented and compared with those reported by Matsunaga [55] based on a higher order displacement field, results of Chorfi and Houmat [56] based on the FSDT-based analysis of shells, and those obtained by Alijani et al. [15] based on the Reddy-Amabili shell theory. Good agreement is observed between the obtained results of this study and those reported in [15,55,56]. Especially, results of our study exhibit an excellent agreement with those of Alijani et al. [15], where a higher order shell theory with retaining the 1 + z/R terms in displacement field is considered. Table 3 Properties of the constituents of the FGM panel [12,46]. Property

Al

ZrO2

ZrO2

Al2O3

Ti6Al4V

E (GPa)

70 0.3 2707

151 0.3 5700

151 0.3 3000

380 0.3 3800

105.7 0.298 4429

m q (kg/m3)

Table 4 Properties of G  1195N as sensor and actuator layer [46]. E (GPa)

q (kg/m3)

m

e31 (c/m2)

e32 (c/m2)

g3 (F/m)

63

7600

0.3

22.86

22.86

1.5  108

Table 5 qffiffiffiffi A comparison on fundamental frequency parameter defined by X ¼ xh qEcc for Al/ Al2O3 FGM panels. Dimensions of the structure are a/b = 1, a/h = 10 and R/a = 2. k

Table 2 Laplace transform and associated Laplace inverse functions for non-polynomial parts of the general types of dynamical loads.

f1 ðsÞf2 ðt  sÞds

Following the aforementioned procedure, each of the functions Umn(t), Vmn(t), Wmn(t), Xmn(t), Ymn(t), Smn(t), and Amn(t) are obtained analytically. It is worth noting that, proceeding the above inversion method leads to obtaining the natural frequencies of the system, exactly. As cj = Re(cj) + iIm(cj), j = 1, 2, . . ., nc, are obtained, x[rad/ sec] = jIm(cj)j is one of the natural frequencies.

Table 1 Some practical cases of applied dynamic load to the panel. Load type

t

0

ð28Þ

Here Re(x) stands for the real part of the complex number x. When functions f1(t) and f2(t) are obtained, function f(t) may be written in the following closed-form expression based on the convolution integral definition

Z

Matsunaga [55]

Chorfi and Houmat [56]

Present

Cylindrical panel 0 0.0746 0.5 0.0647 1 0.0589 10 0.0455

Alijani et al. [15]

0.0751 0.0657 0.0601 0.0464

0.0762 0.0664 0.0607 0.0471

0.0746 0.0646 0.0588 0.0455

Spherical panel 0 0.0615 0.5 0.0527 1 0.0476 10 0.0383

0.0622 0.0532 0.0482 0.0387

0.0622 0.0535 0.0485 0.0390

0.0616 0.0527 0.0477 0.0384

Y. Kiani et al. / Composite Structures 102 (2013) 205–216

211

Table 6 Frequency parameter in Hz (defined by X ¼ 21p x) for square hybrid Ti6Al4V plate. Dimensions of the hybrid structure are a = 0.4 m, b = 0.4 m, hH = 0.005 m, ha = hs = 0.0001 m. Mode number

Shakeri and Mirzaeifar [46]

He et al. [22]

Present

1 2 3 4 5 6 7 8 9 10

142.90 357.82 357.83 578.29 721.62 721.64 967.47 968.02 1252.98 1253.00

144.25 359.00 359.00 564.10 717.80 717.80 908.25 908.25 1223.14 1223.14

144.94 359.62 359.62 578.80 714.46 714.46 936.67 936.67 1209.09 1209.09

Table 7 Fundamental frequency parameter in Hz (defined by X ¼ 21p x) for square hybrid Al  ZrO2 spherical panels. Dimensions of the hybrid structure are a = b = Rx = Ry = 0.1 m, hH = 0.005 m, ha = hs = 0.0001 m. k

Liew et al. [25]

Present

0 0.2 0.5 1 5 15 100 1000

10305.3 9898.8 9470.1 9019.7 8049.3 7707.0 7513.7 7478.8

10408.6 9997.7 9564.6 9109.9 8130.7 7785.2 7589.7 7554.3

As an another comparison study, first ten natural frequencies of a hybrid square plate with the host layer made of AlO2 are presented in Table 6. Results of this study are compared with those presented by He et al. [22] based on the finite elements analysis of thin plates and those reported by Shakeri and Mirzaeifar [46] by means of the layerwise finite elements method applied to the three dimensional theory of elasticity equations. A brief review of this table accepts the accuracy of the present method. Finally in Table 7, fundamental natural frequency of various FGM spherical panel bonded with a pair of piezoelectric layers are calculated and compared with those reported in [25]. Since in the analysis of [25], natural frequencies are obtained via a displacement velocity feedback controller algorithm, here the case when the top layer is grounded all around is considered (This assumption is consistent with the case studied in [25] where displacement feedback controller gain is set equal to zero). An overview of the results accepts the accuracy of the present method. The maximum error between the results of this study and FEM-based analysis of [25] is about 1 percent. 8.2. Parametric studies In what follows, various types of Al/ZrO2 FGM panels integrated with G1195  N sensor and actuator piezoelectric layers are analysed. 8.2.1. Passive case When the controller scheme is not included, Eq. (20) yields the displacement field of the structure. Piezoelectric layers may be of the sensor type or the actuator type. In this section we assume that both of smart layers acts as sensor and therefore no electrical n excio tation is considered. Therefore the electrical force vector, F U is mn vanished from Eq. (20). As a first example, an Al/ZrO2 spherical panel, perfectly bonded with two G  1195N piezoelectric layers under the action of dynamic load of type IV is considered. The loading function is expressed as

Fig. 2. Lateral deflection history of a hybrid FGM spherical panel versus various power law indices.

Fig. 3. The time history of induced electric potential at the surface of the top piezoelectric layer.

pðtÞ ¼

Pt

t < 0:005

0

t > 0:005

ð30Þ

where P = 1 MPa/s. Dimensions of the panel are a = 0.1 m, b = 0.1 m, Rx = 1 m, Ry = 1 m, hH = 2 mm, hs = 0.2 mm. Figs. 2 and 3 demonstrate the lateral deflection at the centre point of the panel and sensory voltage of the top piezoelectric layer, respectively. As seen, magnitude of the both sensory voltage and transverse deflection of hybrid FGM panels are bounded by those associated to the case of isotropic homogeneous host layer. Note that, the cyclic behaviour of electric potential is resulted from the cyclic response of deflections. As an another example, to show the effect of structural damping, lateral deflection of centre point of hybrid panels is depicted in Fig. 4. Geometry of the structure is the same with the one studied in Fig. 2. A uniformly distributed load is applied to the upper surface of the panel (type I). Magnitude of the applied uniform pressure is P = 10 kPa. Constants of structural damping are a = 107 and b = 0.965  105. These coefficients are the same with those utilised in [57]. Note that, when damping is accounted in formulation, forced vibration response of the panel is of conver-

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that, a positive curvature is against the deflection direction of the panel. Therefore panel gets stiffer when its principal curvature radii decreases. Among panels, with both curvatures positive, cylindrical panels has got the most deflection. Besides, among panels of this class, cylindrical panels have the most resemblance response to the plate since they have got an infinite principal curvature radii same as the plate does. To demonstrate the effect of the applied load, lateral deflection responses under four types of loading are compared in Fig. 6. Characteristics of the linearly graded panel are the same with the one used in Fig. 4. Each type of loading is defined as

Type I : pðtÞ ¼ PHðtÞ

P t < 0:002 Type II : pðtÞ ¼ 0 t > 0:002 Type III : pðtÞ ¼ Pð1  e3000t Þ

500Pt t < 0:002 Type IV : pðtÞ ¼ 0 t > 0:002 Fig. 4. Lateral deflection history of a hybrid FGM spherical panel versus various power law indexes including structural damping effects. Type I of loading is considered.

gent-type. For each case of power law index depicted in Fig. 4 temporal evolution curve converges to the static deflection of the panel under the static load P = 10 kPa. For the studied cases in Fig. 4, the larger the power law index, the larger the magnitude of central deflection. To assess the influence of curvature on the response of the hybrid FGM shallow shells, dynamic response of various types of panels is depicted in Fig. 5. A uniform load-type is applied to the panel. Magnitude of the uniform applied pressure is P = 10 kPa. Geometry of the structure are a = 0.1 m, b = 0.1 m, hH = 2 mm, hs = 0.2 mm and Ry = 1 m. Therefore, values of Rx =  1, 1, 1, 0.5 m belong to a saddle, spherical, cylindrical and an arbitrary type of doublycurved panel, respectively. The associated plate-type structure is also considered when Ry = Rx = 1 is chosen. Host layer is graded linearly across the thickness. Structural damping is also considered, where constants a and b are chosen the same with those used in Fig. 4. It is concluded that, for panels with both curvatures positive, as one curvature radii increases and other parameters are kept constant, deflection of the panel increases. This is due to the fact

Fig. 5. Lateral deflection history of various hybrid FGM panels including structural damping effects. Type I of loading is considered.

ð31Þ

where P = 10 kPa. As seen, prior to removal of the load, lateral deflection history of the panel under type I and type II of the loads are the same. Temporal evolutions of the panel deflection under the action of type II and type IV, fluctuates around the deflection-less position of the plate. For two other cases, however, panel exhibits a fluctuating response around a point that is the static deflection of the panel under the load P = 10 kPa 8.2.2. Active case In the active case, displacement vector is obtained via the solution of Eq. (24). After that, recalling the third and fourth of series expansion (17), combined with Eqs. (22) and (23), the sensor and actuator voltages are evaluated. In whole of this section, structural damping with constants a = 5  107 and b = 1.5  105 is considered. To assess the effect of gain value on the response of hybrid FGM panels, a square spherical panel made of a linearly graded host layer is considered. The geometric characteristics of the panel are the same with those used in Fig. 2. The panel is initially subjected to a uniformly distributed lateral load p = 10 kPa, which on removal sets the plate into vibration. In such a case, initial conditions should be obtained according to the load effect. Figs. 7–9 depict the temporal evolution of lateral deflection, sensory voltage and

Fig. 6. A comparison on various types of loading.

Y. Kiani et al. / Composite Structures 102 (2013) 205–216

Fig. 7. Plots of lateral deflection response for a linearly graded spherical panel integrated with sensor/actuator smart layers with various gain values.

213

Fig. 9. Plots of piezoelectric actuator response for a linearly graded spherical panel integrated with sensor/actuator smart layers with various gain values.

0.2 mm, the maximum allowable voltage of each layer is about 400 V. A comparison on Figs. 7–9 accepts that, vibrational period of the sensory voltage and actuator voltage are the same with the period of hybrid structure. Furthermore, there is about a p/2 phase difference between the sensed voltage of Fig. 8 and actuator voltage in Fig. 9. This observation is consistent with the definition of velocity feedback controller algorithm. The influence of material composition of the host layer on the damped temporal evolution of lateral deflection, sensory voltage and actuator voltage is presented in Figs. 10–12, respectively. Dimensions of FG media and the smart layers are both the same with the one used in Figs. 7–9. The velocity feedback controller gain is chosen as Gv = 0.0005. Apparently, responses of the FGM panels are bonded with those of ceramic and metal. The more the power law index, the less the suppression time. As the power law index increases, the maximum actuator voltage increases too. Influence of radii of curvature on deflection, sensory voltage and actuator voltage histories of various cases of doubly-curved panels is studied in Figs. 13–15. Properties of the panel are the same with those used in Fig. 5. The properties of the host layer are graded Fig. 8. Plots of piezoelectric sensor response for a linearly graded spherical panel integrated with sensor/actuator smart layers with various gain values.

actuator voltage for various values of gain Gv, respectively. For the studied cases of the closed-loop algorithm, with the introduction of the velocity feedback control gain, the amplitude of vibration suppresses considerably faster with the increase of Gv. As one may see, in Eq. (25), the factor Gv contributes to the damping matrix and therefore the above-mentioned conclusion is expected. For instance, adoption of a PD controller with Gv = 0.0005, vibration response surpasses in approximately 0.005 s which indicates the capability of the model on vibration attenuation. Sensed voltage is depicted in Fig. 8 for three mentioned gain values. As Gv increases, suppression time decreases. In Fig. 9 the actuator response is depicted for various gain values. Similar to the sensor voltage, the actuator voltage has got its higher values in initial stages which is consistent with the expected damping response. Note that, however, suppression time diminishes as the gain value increases, simultaneously, actuated voltage of the smart layer increases. Since piezoelectric layers have got a shakedown (breakdown) voltage, value of Gv cannot be increased up indefinitely. For G  1195 N piezoelectric layers, the maximum allowable electrical field is reported as 2 V/lm [58]. Since the thickness of each layer is

Fig. 10. Plots of lateral deflection response for spherical panels with various power law indices integrated with sensor/actuator smart layers.

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Fig. 11. Plots of piezoelectric sensor response for spherical panels with various power law indices integrated with sensor/actuator smart layers.

Fig. 14. Influence of radii of curvature on sensory voltage response of linearly graded panels integrated with sensor/actuator smart layers.

Fig. 12. Plots of piezoelectric actuator response for spherical panels with various power law indices integrated with sensor/actuator smart layers.

Fig. 15. Influence of radii of curvature on actuator voltage response of linearly graded panels integrated with sensor/actuator smart layers.

linearly across the thickness. A spherical, cylindrical, and saddle panel are compared in these figures. Besides, the associated plate-type is also indicated when Rx = Ry = 1 are considered. Gain is chosen as Gv = 0.0005. A comparison on all of these figures accepts that, among spherical, cylindrical and saddle panels, the latter case has got the most resemblance behaviour to the associated flat panel. Under a unified velocity feedback controller gain, lateral vibrations of spherical panel surpasses faster. The same is true for both sensor and actuator voltages. Since the configuration of a spherical panel is stiffer than a cylindrical one, spherical panel experiences less lateral deflection in comparison with a cylindrical panel. Consequently, the magnitude of both sensor and actuator voltages of spherical panel are less than a cylindrical one. 9. Conclusion

Fig. 13. Influence of radii of curvature on lateral deflection response of linearly graded panels integrated with sensor/actuator smart layers.

In this study, an analytical procedure in both time and space domain is presented to investigate the dynamic analysis of hybrid shear deformable FGM panels. Navier solution in space domain omits the dependency of the governing equations to the space variables. Transferring the time-dependent ODEs into the Laplace do-

Y. Kiani et al. / Composite Structures 102 (2013) 205–216

main omits the dependency of the equation to the time variable. An analytical procedure is also presented to re-transfer the equation of Laplace domain into the real time domain for some practical cases of dynamical loading. Parametric studies are performed to investigate the effects of power law index, velocity feedback gain and radii of curvature. It is concluded that, each of the geometrical or physical parameters have got an influential effect on the response of the hybrid panel in both active or passive cases. Vibration response of the panel can be actively controlled through the constant velocity feedback controller algorithm considering the proper gain coefficient. Gain coefficient has got an important effect on suppression time of the structure. Besides, the actuator voltage is directly depends on the gain value and therefore this constant cannot be exceeded from a certain value since piezoelectric layers have got a breakdown voltage.

K 34 K 35

K 41 ¼ K 14 K 42 ¼ K 24 K 43 ¼ K 34 K 44 ¼ k2m D11  l2n D66  A55 K 45 ¼ km ln ðD12 þ D66 Þ K 51 ¼ K 15 K 52 ¼ K 25 K 53 ¼ K 35 K 54 ¼ K 45

Appendix A

K 55 ¼ k2m D66  l2n D22  A44

A.1. Elements of the mass matrix

A.3. Elements of the piezoelectric matrix

M11 M14 M22 M25

  S2 ¼  S1 þ 2 Rx   S3 ¼  S2 þ Rx   S2 ¼  S1 þ 2 Ry   S3 ¼  S2 þ Ry

(For simplicity the superscript DU is dropped out)

K 11 ¼ ea31 Ha1 km K 12 ¼ ea32 Ha1 ln  a  e ea K 13 ¼  31 þ 32 Rx Ry K 14 ¼ ea31 Ha2 km K 15 ¼ ea32 Ha2 ln

M33 ¼ S1

K 21 ¼ es31 Hs1 km

M41 ¼ M 14

K 22 ¼ es32 Hs1 ln  s  e es K 23 ¼  31 þ 32 Rx Ry

M25 ¼ M 25 M44 ¼ S3

K 24 ¼ es31 Hs2 km

M55 ¼ S3

K 25 ¼ es32 Hs2 ln

Other Mij’s are equal to zero.

A.4. Elements of the permittivity matrix

A.2. Elements of the stiffness matrix (For simplicity the superscript DD is dropped out)

K 11 ¼ k2m A11  l2n A66  c20 l2n D66  K 12 K 13

  ¼ km ln A12 þ A66  c20 D66   A11 A12 A55 ¼ km þ þ Rx Ry Rx

K 14 ¼ k2m B11  l2n B66  c0 l2n D66 þ

A55 R2x

 2c0 l2n B66

K 23 ¼ ln

A12 A22 A44 þ þ Rx Ry Ry



K 22 ¼  A55 Rx

A44 R2y

þ 2c0 k2m B66

A44 Ry

K 31 ¼ K 13 K 32 ¼ K 33 K 33 ¼ k2m A55  l2n A44 

A11 R2x



ga h

a

gs h

s

References

K 24 ¼ km ln ðB12 þ B66  c0 D66 Þ K 25 ¼ k2m B66  l2n B22 þ c0 k2m D66 þ

K 11 ¼  K 21 ¼ 0

K 21 ¼ K 12 K 22 ¼ k2m A66  l2n A22  c20 k2m D66 

(For simplicity the superscript UU is dropped out)

K 12 ¼ 0

K 15 ¼ km ln ðB12 þ B66 þ c0 D66 Þ



215

  B11 B12 ¼ km þ  A55 Rx Ry   B12 B22 ¼ ln þ  A44 Rx Ry

2A12 A22  Rx Ry R2y

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