Scientia Iranica B (2016) 23(6), 2606{2615

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Large-amplitude free vibration ofmagneto-electro-elastic curved panels

A. Shooshtari� and S. Razavi

Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, P.O. Box 65175-4161, Iran.

Received 13 December 2014; received in revised form 20 October 2015; accepted 28 November 2015

KEYWORDSLarge-amplitude freevibration;Magneto-electro-elastic material;Curved panel;Donnell shell theory;Multiple timescalesmethod.

Abstract. In this study, the large-amplitude free vibration of magneto-electro-elasticcurved panels was investigated. The panel was considered to be simply supported on alledges and the magneto-electro-elastic body was subjected to the electric and magnetic �eldsalong z direction. To obtain the governing equations of motion, the Donnell shell theoryand the Gauss's laws for electrostatics and magnetostatics were used. The �rst mode ofvibration of these smart panels was studied in this paper. To this end, the nonlinear partialdi�erential equations of motion were reduced to a nonlinear ordinary di�erential equationby introducing trial functions for displacements and rotations and then by applying thesingle-mode Galerkin method on the obtained equation. The resulting equation was solvedby multiple timescales perturbation method. Some numerical examples were presented tovalidate the study and to investigate the e�ects of several parameters such as geometry ofthe panel and the magneto-electric boundary conditions on the vibration behavior of thesesmart panels.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

Magneto-electro-elastic smart materials exhibit a cou-pling between mechanical, electric, and magnetic �eldsand are able to convert energy among these energyforms, which makes them suitable for energy harvest-ing, vibration control, etc.

Static and dynamic responses of piezoelectricplates and shells have been investigated extensivelyin the past years [1-5]. Pan [6] analyzed the motionof magneto-electro-elastic structures for the �rst time.Ramirez et al. [7] studied the free vibration of magneto-electro-elastic plates. Razavi and Shooshtari [8] stud-ied the free vibration of a single-layered magneto-electro-elastic shell resting on a Pasternak foundation.Bhangale and Ganesan [9] and Annigeri et al. [10]studied free vibration of simply supported and clampedmagneto-electro-elastic cylindrical shells, respectively.

*. Corresponding author. Tel.: +98 8138272410E-mail address: shooshta@basu.ac.ir (A. Shooshtari)

Tsai and Wu [11,12] presented free vibration analysisof doubly curved functionally graded magneto-electro-elastic shells with open-circuit and closed-circuit sur-face conditions, respectively. Chen et al. [13] presentedan analytical solution for the static deformation ofa magneto-electro-elastic hollow sphere. Xin andHue [14] presented a semi-analytical model based onthe three-dimensional elasticity theory to study thefree vibration of simply supported magneto-electro-elastic plates. However, there are few studies dealingwith the nonlinear vibration response of magneto-electro-elastic structures. Xue et al. [15] studied thelarge de ection of a rectangular magneto-electro-elasticthin plate for the �rst time based on the classicalplate theory. Sladek et al. [16] used a meshless localPetrov-Galerkin method to study the large de ectionof magneto-electro-elastic thick plates. Milazzo [17]derived a shear deformable model for the large de- ection analysis of MEE laminated plates. Alaimo etal. [18] presented an original shear deformable �nite-element model for the analysis of large de ections

A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615 2607

of magneto-electro-elastic laminated plates. Rao etal. [19] proposed a �nite-element model for large de ec-tion static analysis of layered magneto-electro-elasticstructures. Razavi and Shooshtari [20] studied non-linear vibration of symmetrically laminated magneto-electro-elastic rectangular plates. They [21,22] alsoinvestigated the e�ects of electric and magnetic po-tentials on the nonlinear vibration response of lami-nated magneto-electro-elastic plates and doubly-curvedshells, respectively, with movable simply supportedboundary condition. Nonlinear forced vibration ofmagneto-electro-elastic nanobeams has also been in-vestigated [23,24]. Kattimani and Ray [25,26] studiedthe active control of large-amplitude vibrations ofmagneto-electro-elastic plates and doubly curved shell,respectively.

According to the published literature, there arenot any studies about the large-amplitude free vi-bration of multiphase magneto-electro-elastic curvedpanels. Thus, this study deals with this topic to�ll the gap. In this paper the large-amplitude freevibration of a multiphase smart curved panel withimmovable simply-supported boundary condition isinvestigated. The panel is considered to be made oftransversely isotropic magneto-electro-elastic material.The magnetic and electric �elds are applied alongz direction. The Donnell shell theory without in-plane and rotary inertias along with Gauss's lawsfor electrostatics and magnetostatics is used to modelthe panel. The �rst mode of vibration is studiedhere. To achieve this goal, after transforming theequations of motion to a nonlinear ordinary di�eren-tial equation by single-mode Galerkin method, it issolved analytically and then a closed-form relation fornonlinear frequency is obtained. This model can beused to study the nonlinear and linear free vibrationsof simply-supported single-layered curved panels withmagneto-electro-elastic, piezoelectric, piezomagnetic,orthotropic, or isotropic material properties. Severalnumerical studies are presented to validate the studyand to investigate the e�ects of several parameters onthe behavior of these smart panels.

2. Theoretical formulations

For a curved panel with Rx and Ry being the radii ofcurvature (Figure 1), the strain-displacement relationsare given as [27]:

"x =�u0;x + w0=Rx +

12w2

0;x

�+ z�x;x;

"y =�v0;y + w0=Ry +

12w2

0;y

�+ z�y;y;

xy = (u0;y + v0;x + w0;xw0;y) + z(�x;y + �y;x);

Figure 1. Schematic of the studied curved panel.

yz = w0;y + �y; xz = w0;x + �x; (1)

where u0, v0, and w0 are the displacements of the mid-surface along x, y, and z directions, respectively, and �xand �y are the rotations of a transverse normal aboutthe y and x directions, respectively.

Assuming that the electric and magnetic �elds areapplied along z-direction, the constitutive equations oftransversely isotropic magneto-electro-elastic panel canbe written in the following form [6]:8>>>><>>>>:

�x�y�xz�yz�xy

9>>>>=>>>>; =

266664C11 C12 0 0 0C12 C22 0 0 00 0 C55 0 00 0 0 C44 00 0 0 0 C66

3777758>>>><>>>>:"x"y xz yz xy

9>>>>=>>>>;+

2666640 0 e310 0 e320 e24 0e15 0 00 0 0

3777758<: 0

0�;z

9=;

+

2666640 0 q310 0 q320 q24 0q15 0 00 0 0

3777758<: 0

0 ;z

9=; ; (2)

8<:DxDyDz

9=; =

24 0 0 0 e15 00 0 e24 0 0e31 e32 0 0 0

358>>>><>>>>:"x"y xz yz xy

9>>>>=>>>>;�24�11 0 0

0 �22 00 0 �33

358<: 00�;z

9=;�24d11 0 0

0 d22 00 0 d33

358<: 00 ;z

9=; ; (3)

2608 A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615

8<:BxByBz9=; =

24 0 0 0 q15 00 0 q24 0 0q31 q32 0 0 0

358>>>><>>>>:"x"y xz yz xy

9>>>>=>>>>;�24d11 0 0

0 d22 00 0 d33

358<: 00�;z

9=;�24�11 0 0

0 �22 00 0 �33

358<: 00 ;z

9=; ; (4)

where��x �y �xz �yz �xy

T is stress vector;�Dx Dy Dz

T and�Bx By Bz

T are the elec-tric displacement and magnetic ux density vectors,respectively; [Cij ], [�ij ], and [�ij ] are the elastic, dielec-tric, and magnetic permeability coe�cient matrices,respectively; [eij ], [qij ], and [dij ] are the piezoelectric,piezomagnetic, and magneto-electric coe�cient matri-ces, respectively; and � and are electric and magneticpotentials.

By neglecting in-plane and rotary inertia e�ects,the equations of motion of a curved panel can beexpressed in the following form [27]:

Nx;x +Nxy;y = 0; (5)

Nxy;x +Ny;y = 0; (6)

Qx;x +Qy;y + (Nxw0;x +Nxyw0;y);x

+ (Nxyw0;x +Nyw0;y);y �Nx=Rx �Ny=Ry= I0w0;tt; (7)

Mx;x +Mxy;y �Qx = 0; (8)

Mxy;x +My;y �Qy = 0; (9)

where I0 =R h=2�h=2 �0dz is the mass moment of inertia,

in which �0 is the density of the material of thepanel. The in-plane force resultants (i.e., Nx, Ny, Nxy),the transverse force resultants (i.e., Qx, Qy) and themoment resultants (i.e., Mx, My, Mxy) are obtainedby the following equations:�

Nx Ny NxyT =

Z h=2

�h=2��x �y �xy

T dz;�Mx My Mxy

T =Z h=2

�h=2��x �y �xy

T zdz;�Qx Qy

T = KZ h=2

�h=2��xz �yz

T dz; (10)

where K is shear correction factor.To obtain the resultants of Eq. (10), the gradients

of the electric and magnetic potentials in Eq. (2) mustbe obtained in terms of z. To this end, Gauss's lawsfor electrostatics and magnetostatics, i.e.:

Dx;x +Dy;y +Dz;z = 0; (11)

Bx;x +By;y +Bz;z = 0; (12)

along with Eqs. (3) and (4) are considered from whichone obtains:

� = M1z2 + �0z + �1; = M2z2 + 0z + 1: (13)

In the above equation, M1 and M2 are obtained by:

M1 =12

(a1w0;xx + a2w0;yy + a3�x;x + a4�y;y) ; (14)

M2 =12

(a5w0;xx + a6w0;yy + a7�x;x + a8�y;y) ; (15)

where:

�1 =d33=(d233��33�33);

�2 =�33=(d233��33�33);

�3 = �33=(d233 � �33�33);

a1 = �1q15 � �2e15;

a2 = �1q24 � �2e24;

a3 = �1(q15 + q31)� �2(e15 + e31);

a4 = �1(q24 + q31)� �2(e24 + e31);

a5 = �1e15 � �3q15; a6 = �1e24 � �3q24;

a7 = �1(e15 + e31)� �3(q15 + q31);

a8 = �1(e24 + e31)� �3(q24 + q31): (16)

On the other hand, the integration constants, i.e. �0,�1, 0, and 1, are obtained by applying the magneto-electric boundary condition on top and bottom surfacesof the plate.

Closed-circuit and open-circuit magneto-electricboundary conditions are considered in this study.These boundary conditions are expressed in the fol-lowing form:

� = = 0; (z = �h=2) (closed-circuit); (17)

Bz = Dz = 0; (z = �h=2) (open-circuit): (18)

Eqs. (5)-(10) and (13) give the Partial Di�erentialEquations (PDEs) of motion in terms of displacements

A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615 2609

Table 1. Dimensionless fundamental frequencies of curved isotropic panels (a = b, a = 10h, and � = 0:3).

Rx=a Ry=bAlijani et al.

[29]Chor� and

Houmat [30]Matsunaga

[31]Bich et al.

[32]Bich et al.

[33]Presentstudy

1 1 0.0597 0.0577 0.0588 0.0597 0.0581 0.058122 1 0.0648 0.0629 0.0622 0.0648 0.0632 0.063272 2 0.0779 0.0762 0.0751 | 0.0767 0.07667-2 2 0.0597 0.0580 0.0563 | 0.0592 0.05812

and rotations. The immovable simply supportedboundary condition is expressed in the following form:

u0 = v0 = w0 = �y = w0;xx = 0 at x = 0; a;

u0 = v0 = w0 = �x = w0;yy = 0 at y = 0; a; (19)

for which the displacements and rotations can beobtained by:8>>>><>>>>:

u0v0w0�x�y

9>>>>=>>>>; =

8>>>><>>>>:hU sin(2�x=a) sin(�y=b)hV sin(�x=a) sin(2�y=b)hW sin(�x=a) sin(�y=b)X cos(�x=a) sin(�y=b)Y sin(�x=a) cos(�y=b)

9>>>>=>>>>; ; (20)

and by applying the Galerkin method to the PDEs ofmotion, one obtains:

L1W 2 + L2W + L3U + L4V = 0; (21)

L5W 2 + L6W + L7U + L8V = 0; (22)

L9W 3 + L10W 2 + L11UW + L12VW + L13V

+L14U+L15X+L16Y +L17W+L18 �W =0; (23)

L19W + L20X + L21Y = 0; (24)

L22W + L23X + L24Y = 0; (25)

where Li (i = 1; 2; � � � ; 24) are constant coe�cientsand are given in Appendix A for two magneto-electricboundary conditions.

Obtaining U and V from Eqs. (21) and (22), andX and Y from Eqs. (24) and (25), and then substitutingthe obtained values into Eq. (23) give:

�W + !20W + �W 2 + �W 3 = 0; (26)

where the coe�cients are given in Appendix B.Eq. (26) can be analytically solved by using

multiple timescales method. To do this, a smalldimensionless parameter (") is inserted to Eq. (26) toscale the nonlinear terms [28]:

�W + !20W = �"�W 2 � "2�W 3: (27)

Solving Eq. (27), one can simply obtain the nonlinear

frequency ratio [28]:

!NL=!L =�1 +

9�!20 � 10�12!4

0p2

0

�1=2

; (28)

where p0 = wmax=h is the dimensionless initial dis-placement.

3. Results and discussion

In the numerical examples, the shear correction factor(K) is taken to be 5/6. Dimensionless fundamentalfrequencies, ! = !0h

p�0=E, of curved isotropic

panels for di�erent radii of curvature are obtainedand compared with the previously published results(Table 1). The results of the presented model arecompared with the results of Alijani et al. [29] based onthe Donnell's nonlinear shallow shell theory, Chor� andHoumat [30] based on the �rst-order shear deformationtheory, Matsunaga [31] based on the two-dimensionalhigher-order theory, and Bich et al. [32,33] based on theclassical shell theory and �rst-order shear deformationtheory, respectively. It is seen that although thepanels are relatively thick, the results are in goodagreement with the higher-order results reported byMatsunaga [31].

Table 2 shows the dimensionless frequenciesof piezoelectric BaTiO3 and piezomagnetic CoFe2O4square thick plates, which are compared with theresults obtained by higher-order shear deformationtheory [34] and 3D approach [35]. For BaTiO3, thematerial properties are: C11 = C22 = 166 GPa,C12 = 77 GPa, C44 = C55 = 43 GPa, C66 = 44:5 GPa,e31 = e32 = �4:4 Cm�2, e15 = e24 = 11:6 Cm�2,�33 = 12:6 � 10�9 C(Vm)�1, �33 = 10 � 10�6

Ns2C�2, and �0 = 5800 kgm�3; and the materialproperties of CoFe2O4 are: C11 = C22 = 286 GPa,C12 = 173 GPa, C44 = C55 = 45:3 GPa, C66 =

Table 2. Dimensionless fundamental frequencies ofpiezoelectric and piezomagnetic thick plates(Rx = Ry =1).

Method BaTiO3 CoFe2O4

HSDT [34] 1.2629 1.13583D [35] 1.2660 1.0212Present study 1.3268 1.1735

2610 A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615

56:5 GPa, q31 = q32 = 580:3 N(Am)�1, q15 = q24 =550 N(Am)�1, �33 = 0:093 � 10�9 C(Vm)�1, �33 =157 � 10�6 Ns2C�2, and �0 = 5300 kgm�3. Thegeometric properties of the plates are a = b = 1 mand h = 0:3 m, and the dimensionless frequenciesare obtained by ! = !0a

p�0=Cmax, where Cmax

denotes the maximum of Cij of the material of theplate. There is a discrepancy between the results ofthe present study with highly accurate HSDT and 3Dresults. This is because the plates are thick and therotary inertia is not included in the formulation. Thus,the proposed approach can predict the vibration ofthin and relatively thick panels with an acceptableprecision.

As the last comparison, the nonlinear frequencyratios (!NL=!L) of an isotropic square plate withimmovable simply-supported boundary condition areobtained and compared with the results obtained by ashear deformable �nite-element model [36] (Table 3).Good agreement is observed between the results of thepresent approach and the ones obtained by Singha andDaripa [36].

In Tables 4 and 5, the nonlinear frequency ratiosof magneto-electro-elastic curved panels with the fol-lowing material properties are presented [37]: C11 =226 GPa, C12 = 124 GPa, C22 = 216 GPa, C44 =C55 = 44 GPa, C66 = 51 GPa, e31 = e32 =�2:2 Cm�2, q31 = q32 = 290:2 N(Am)�1, �33 = 6:35�10�9 C(Vm)�1, d33 = 2737:5�10�12 Ns(CV)�1, �33 =83:5 � 10�6 Ns2C�2, and �0 = 5500 kgm�3. Three

curved panels are considered in these tables, whichare spherical, cylindrical, and hyperbolic paraboloidalpanels. The hyperbolic paraboloidal panel has thehighest and spherical panel has the lowest nonlinearfrequency ratios among the three panels. Moreover, itis seen that the nonlinear frequency ratio is slightlyhigher for the open-circuit magneto-electric bound-ary condition, which means that in the open-circuitboundary condition, the nonlinearity of the panel ismore than that of the closed-circuit case. However,the spherical panel has the same nonlinear behaviorin the open-circuit and closed-circuit cases for vibra-tion amplitudes equal to or smaller than the shellthickness. By contrast, magneto-electric boundarycondition has the greatest e�ect on the response ofhyperbolic paraboloidal panel.

The e�ect of panel thickness on the response hasalso been investigated and the results are shown inTables 6 and 7. For each panel a=b, a=h, Rx=a, andRy=b, ratios are constant whereas h can be changed.It is seen that thinner spherical and cylindrical panelshave higher nonlinear frequency ratios. However, nochange is observed in the nonlinear frequency ratio ofthe hyperbolic paraboloidal panel.

In Figure 2, backbone curves of piezoelec-tric BaTiO3, piezomagnetic CoFe2O4, and magneto-electro-elastic panels are presented. The magneto-electric boundary condition is considered to be closed-circuit. It is observed that for the spherical panel,the backbone curves of magneto-electro-elastic and

Table 3. Nonlinear frequency ratios of an isotropic square plate (a=h = 100, � = 0:3, and Rx = Ry =1).

Method wmax=h0.2 0.4 0.6 0.8 1.0

Singha and Daripa [36] 1.01967 1.07669 1.16597 1.28131 1.41684Present study 1.02085 1.08099 1.17440 1.29389 1.43295

Table 4. Nonlinear frequency ratios of curved panels with open-circuit magneto-electric boundary condition(a = b = 100h).

Rx=a Ry=bwmax=h

0.2 0.4 0.6 0.8 1.0

2 2 1.00013 1.00050 1.00113 1.00201 1.003132 1 1.00036 1.00145 1.00326 1.00579 1.00904-2 2 1.00038 1.00151 1.00339 1.00602 1.00939

Table 5. Nonlinear frequency ratios of curved panels with closed-circuit magneto-electric boundary condition(a = b = 100h).

Rx=a Ry=bwmax=h

0.2 0.4 0.6 0.8 1.0

2 2 1.00013 1.00050 1.00113 1.00201 1.003132 1 1.00036 1.00145 1.00325 1.00578 1.00902-2 2 1.00037 1.00150 1.00336 1.00597 1.00931

A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615 2611

Table 6. Nonlinear frequency ratios of open-circuitmagneto-electric curved panels with di�erent thicknesses(a = b = 100h).

Rx=a Ry=b

wmax=h

0.6 1.0

h h

0.1 1.0 0.1 1.0

2 2 1.00114 1.00113 1.00317 1.00313

2 1 1.00331 1.00326 1.00917 1.00904

-2 2 1.00339 1.00339 1.00939 1.00939

Table 7. Nonlinear frequency ratios of closed-circuitmagneto-electric curved panels with di�erent thicknesses(a = b = 100h).

Rx=a Ry=b

wmax=h

0.6 1.0

h h

0.1 1.0 0.1 1.0

2 2 1.00114 1.00113 1.00317 1.00313

2 1 1.00330 1.00325 1.00915 1.00902

-2 2 1.00336 1.00336 1.00931 1.00931

CoFe2O4 panels are almost identical. Moreover,in cylindrical and hyperbolic paraboloidal panels,CoFe2O4 and BaTiO3 have the most and the leastnonlinearity, respectively.

4. Conclusions

Large-amplitude free vibration of magneto-electro-elastic curved panels is investigated in this paper. TheDonnell shell theory, Gauss's laws for electrostatics andmagnetostatics, and Galerkin and multiple timescalesmethods are used to model and solve the problem.The e�ects of geometry of the panel and the magneto-electric boundary conditions on the vibration behaviorof these smart panels are studied by using somenumerical examples and it is found that:

(a) The hyperbolic paraboloidal panel has the mostand spherical panel has the least nonlinear fre-quency ratio among the spherical, cylindrical, andhyperbolic paraboloidal panels;

(b) The nonlinear frequency ratio is higher for theopen-circuit magneto-electric boundary condition,meaning that in the open-circuit case, the nonlin-earity of the panel is more than that of the closed-circuit case;

(c) Thinner spherical and cylindrical panels havehigher nonlinear frequency ratios;

(d) Backbone curves of magneto-electro-elastic andCoFe2O4 spherical panels are almost identical.

Figure 2. Backbone curves of (a) spherical, (b)cylindrical, and (c) hyperbolic paraboloidal panels withclosed-circuit magneto-electric boundary conditions(a = b = 100h).

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Appendix A

�1 = (d33q31 � �33e31)=(d233 � �33�33);

�2 = (d33e31 � �33q31)=(d233 � �33�33): (A.1)

For the open-circuit boundary condition we have thefollowing equations:

L1 =�2h3

6b(C12 � C66 + e31�1 + q31�2)

� �2bh3

3a2 (C11 + e31�1 + q31�2);

L2 =2bh2

3Rx(C11 + e31�1 + q31�2)

+2bh2

3Ry(C12 + e31�1 + q31�2);

L3 = ��2bh2

a(C11 + e31�1 + q31�2)� �2ah2

4bC66;

L4 = �16h2

9(C12 + C66 + e31�1 + q31�2); (A.2)

L5 =�2h3

6a(C12 � C66 + e32�1 + q32�2)

� �2ah3

3b2(C22 + e32�1 + q32�2);

L6 =2ah2

3Rx(C12 + e32�1 + q32�2)

+2ah2

3Ry(C22 + e32�1 + q32�2);

L7 = �16h2

9(C12 + C66 + e32�1 + q32�2);

L8 =��2ah2

b(C22+e32�1+q32�2)� �2bh2

4aC66; (A.3)

L9 =� �4h4

128ab

"2(C12 + 2C66)

+9b2

a2 (C11 + e31�1 + q31�2)

+9a2

b2(C22 + e32�1 + q32�2) + �1(e31 + e32)

+ �2(q31 + q32)

#;

L10 =� 4h3

9abRxRy

"3b2Ry(C11 + e31�1 + q31�2)

+ 3a2Rx(C22 + e32�1 + q32�2)

+ 3C12(a2Ry + b2Rx) + a2Ry(�1(e31 + 2e32)

+ �2(q31 + 2q32)) + b2Rx(�1(2e31 + e32)

+ �2(2q31 + q32))

#;

L11 =�2h3

3b(C12 � C66 + e32�1 + q32�2)

� 2�2bh3

3a2 (C11 + e31�1 + q31�2);

L12 =�2h3

3a(C12 � C66 + e31�1 + q31�2)

� 2�2ah3

3b2(C22 + e32�1 + q32�2);

L13 =2ah2

3Rx(C12 + e31�1 + q31�2)

2614 A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615

+2ah2

3Ry(C22 + e32�1 + q32�2);

L14 =2bh2

3Rx(C11 + e31�1 + q31�2)

+2bh2

3Ry(C12 + e32�1 + q32�2);

L15 = �14K�bhC55; L16 = �1

4K�ahC44;

L17 =� abh2

4R2x

(C11 + e31�1 + q31�2)

� abh2

4R2y

(C22 + e32�1 + q32�2)

� a�2h2

4bKC44 � b�2h2

4aKC55

� abh2

4RxRy[2C12+�1(e31+e32)+�2(q31+q32)];

L18 = �14I0abh; (A.4)

L19 =�3h4

48b[e24e31�2 � (e24q31 + e31q24)�1

+ q24q31�3]� �bh2

4KC55 +

�3bh4

48a2 [e15e31�2

� (e15q31 + e31q15)�1 + q15q31�3];

L20 =�2bh3

48a[(e2

31+e15e31)�2�C11+(q231+q15q31)�3

� (e15q31 + (q15 + 2q31)e31)�1]� �2ah3

48bC66

� abh4KC55;

L21 =148�2h3[(e2

31 + e24e31)�2 � C12 � C66

+(q231+q24q31)�3�(e24q31+(q24+2q31)e31)�1];

(A.5)

L22 =�3h4

48a[e15e32�2 � (e15q32 + e32q15)�1

+ q15q32�3]� �ah2

4KC44 +

�3ah4

48b2[e24e32�2

� (e24q32 + e32q24)�1 + q24q32�3];

L23 =148�2h3[(e15 + e31)e32�2 � C12 � C66

+ (q15 + q31)q32�3 � (q32(e15 + e31)

+ (q15 + q31)e32)�1];

L24 =�2ah3

48b[(e24 + e31)e32�2 � C22

� (e32(q24 + q31) + (e24 + e31)q32)�1

+ (q24 + q31)q32�3]� �2bh3

48aC66 � abh

4KC44:

(A.6)

For the closed-circuit boundary condition, we have thefollowing equations.

L1 =�2h3

6a2b[a2(C12 � C66)� 2b2C11];

L2 =2bh2

3RxC11 +

2bh2

3RyC12;

L3 = ��2bh2

aC11 � �2ah2

4bC66;

L4 = �16h2

9(C12 + C66); (A.7)

L5 =�2h3

6a(C12 � C66)� �2ah3

3b2C22;

L6 =2ah2

3RxC12 +

2ah2

3RyC22;

L7 = �16h2

9(C12 + C66);

L8 = ��2ah2

bC22 � �2bh2

4aC66; (A.8)

L9 = � �4h4

128ab

�2(C12 + 2C66) +

9b2

a2 C11 +9a2

b2C22

�;

L10 =� 4h3

9abRxRy[3b2RyC11 + 3a2RxC22

+ 3C12(a2Ry + b2Rx)];

L11 =�2h3

3b(C12 � C66)� 2�2bh3

3a2 C11;

L12 =�2h3

3a(C12 � C66)� 2�2ah3

3b2C22;

L13 =2ah2

3RxC12 +

2ah2

3RyC22;

A. Shooshtari and S. Razavi/Scientia Iranica, Transactions B: Mechanical Engineering 23 (2016) 2606{2615 2615

L14 =2bh2

3RxC11 +

2bh2

3RyC12;

L15 = �14K�bhC55; L16 = �1

4K�ahC44;

L17 =� abh2

4R2xC11 � abh2

4R2yC22 � a�2h2

4bKC44

� b�2h2

4aKC55 � abh2

2RxRyC12;

L18 = �14I0abh; (A.9)

L19 =�3h4

48b[e24e31�2 � (e24q31 + e31q24)�1

+ q24q31�3]� �bh2

4KC55 +

�3bh4

48a2 [e15e31�2

� (e15q31 + e31q15)�1 + q15q31�3];

L20 =�2bh3

48a[(e2

31 + e15e31)�2 � C11

+ (q231+q15q31)�3�(e15q31+(q15+2q31)e31)�1]

� �2ah3

48bC66 � abh

4KC55;

L21 =148�2h3[(e2

31 + e24e31)�2 � C12 � C66

+(q231+q24q31)�3�(e24q31+(q24+2q31)e31)�1];

(A.10)

L22 =�3h4

48a[e15e32�2 � (e15q32 + e32q15ht�1

+ q15q32�3]� �ah2

4KC44 +

�3ah4

48b2[e24e32�2

� (e24q32 + e32q24)�1 + q24q32�3];

L23 =148�2h3[(e15 + e31)e32�2 � C12 � C66

+ (q15 + q31)q32�3 � (q32(e15 + e31)

+ (q15 + q31)e32)�1];

L24 =�2ah3

48b[(e24 + e31)e32�2 � C22

� (e32(q24 + q31) + (e24 + e31)q32)�1

+(q24+q31)q32�3]� �2bh3

48aC66� abh4 KC44:

(A.11)Appendix B

g1 = (L21L22 � L19L24)=(L20L24 � L21L23);

g2 = (L19L23 � L20L22)=(L20L24 � L21L23);

g3 = (L4L6 � L2L8)=(L3L8 � L4L7);

g4 = (L4L5 � L1L8)=(L3L8 � L4L7);

g5 = (L2L7 � L3L6)=(L3L8 � L4L7);

g6 = (L1L7 � L3L5)=(L3L8 � L4L7);

!0 =p

(L17+L13g5+L14g3+L15g1+L16g2)=L18;

� = (L10 + L11g3 + L12g5 + L13g6 + L14g4)=L18;

� = (L9 + L11g4 + L12g6)=L18:

Biographies

Alireza Shooshtari is Associate Professor of Me-chanical Engineering at Bu-Ali Sina University. Hereceived his BS degree in Mechanical Engineering fromTehran University, Tehran, Iran; and his MS and PhD(2006) degrees in Applied Mechanics, respectively, fromBu-Ali Sina University, Hamedan, Iran, and TarbiatModarres University, Tehran, Iran. His research inter-ests are nonlinear dynamics of engineering structures,modal analysis, and random vibrations.

Soheil Razavi is candidate for the PhD degree inApplied Mechanics. He received his MSc in AppliedMechanics from Bu-Ali Sina University, Hamedan,Iran, in 2010. His MSc thesis was about single-mode analysis of hybrid laminated plates. His PhDdissertation deals with the analytical study of nonlineardynamics of smart plates and shells.