Research Article Free Vibration Analysis of Fiber Metal ...downloads.hindawi.com/journals/amse/2014/602708.pdf · Research Article Free Vibration Analysis of Fiber Metal Laminate

Research ArticleFree Vibration Analysis of Fiber Metal Laminate AnnularPlate by State-Space Based Differential Quadrature Method

G. H. Rahimi,1 M. S. Gazor,1 M. Hemmatnezhad,2 and H. Toorani1

1 Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran2 Faculty of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran

Correspondence should be addressed to G. H. Rahimi; rahimi [email protected]

Received 6 May 2013; Revised 30 September 2013; Accepted 2 October 2013; Published 2 January 2014

Academic Editor: Jianqiao Ye

Copyright © 2014 G. H. Rahimi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A three-dimensional elasticity theory bymeans of a state-space based differential quadrature method is presented for free vibrationanalysis of fiber metal laminate annular plate. The kinds of composite material and metal layers are considered to be S2-glass andaluminum, respectively. A semianalytical approach which uses state-space in the thickness and differential quadrature in the radialdirection is implemented for evaluating the nondimensional natural frequencies of the annular plates. The influences of changes inboundary condition, plate thickness, and lay-up direction on the natural frequencies are studied. A comparison is also made withthe numerical results reported by ABAQUS software which shows an excellent agreement.

1. Introduction

Recently, fiber metal laminates (FML), due to their excellentmechanical properties as well as low density, have gainedmuch attention for aircraft structures. Till now, severalresearch papers have been conducted on the vibrationalbehavior of these structures.Using the free vibration dampingtests, Botelho et al. [1] obtained the elastic and viscousresponses for aluminum 2024-T3 alloy, carbon fiber/epoxycomposites, carbon fiber/aluminum 2024-T3/epoxy hybridcomposites, and glass fiber/aluminum2024-T3/epoxy hybridcomposites. They also compared the elastic and viscousresponses of these new materials with those of conventionalpolymer composites. Reyes and Cantwell [2] investigated thequasistatic and impact properties of a novel fiber/metallaminate system based on a tough glass-fiber-reinforced pol-ypropylene. Their testing showed that, by incorporating aninterlayer based on a maleic-anhydride modified polypropy-lene copolymer at the interface between the composite andaluminum layers, one can reach to excellent adhesion prop-erties. Based on the first-order shear deformation theory,Shooshtari and Razavi [3] solved the linear and nonlinearvibrations of FML plate using the multiple time scalesmethod. Khalili et al. [4] studied the dynamic response of

FML cylindrical shells subjected to initial combined axialload and internal pressure. They implemented the Galerkinmethod for solving the governing equations. They examinedthe influences of FML parameters and arrived at the pointthat the FML layup has a significant effect on the naturalfrequencies of vibration. In recent years, several research-ers have implemented the differential quadrature method(DQM) for investigating the free vibration and static analysesof engineering structures. Using the three-dimensional the-ory of elasticity, Alibeigloo and Shakeri [5] combined thestate-space and differential quadrature method (DQM) forinvestigating the free vibration analysis of crossply laminatedcylindrical panels. Based on the theory of elasticity, Li and Shi[6] extended a state-space based DQM for investigating thefree vibrational behavior of functionally graded piezoelectricmaterial (FGPM) beam under various boundary conditions.Alibeigloo and Madoliat [7] gave a three-dimensional solu-tion for the static analysis of crossply rectangular plates withintegrated surface piezoelectric layers using DQM and theFourier series approach. Also, the static and free vibrationcharacteristics of anisotropic laminated cylindrical shellshave been studied by applying the state-space in conjunctionwith DQM [8]. Yas and Aragh [9] investigated the freevibration characteristics of rectangular continuous grading

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014, Article ID 602708, 11 pageshttp://dx.doi.org/10.1155/2014/602708

2 Advances in Materials Science and Engineering

fiber reinforced (CGFR) plates resting on elastic foundationsbased on the three-dimensional, linear, and small strainelasticity theory and using DQM.

Nallim and Grossi [10] performed the free transversevibration analysis of symmetrically laminated solid andannular elliptic and circular plates using Rayleigh-Ritzmethod. Ovesy and Fazilati [11] applied the finite stripmethod based upon a Reddy type, third-order shear defor-mation theory for investing the buckling and free vibrationalbehavior of thick plates containing internal cutouts. Thebuckling behavior of laminated composite circular plates hav-ing circular holes and subjected to uniform radial load wasinvestigated using the finite element method by Baltaci et al.[12]. They also studied the influences of changes in the holesize, location of the hole, thickness, and boundary conditionson the buckling load. Based on the three-dimensional theoryof elasticity and a combination of state-space method andDQM,Nie and Zhong [13] used a semianalytical approach forobtaining the vibration frequencies and dynamic response offunctionally graded circular plates. Seifi et al. [14] studied thebuckling behavior of composite annular plates under uniforminternal and external radial edge loadswhich have been inves-tigated using energy method. Jodaei et al. [15] used a state-space based DQM to analyze the free vibrational behaviorof functionally graded annular plates. They also modeledthe plate by artificial neural network for different boundaryconditions. Further, the influences of thickness of the annularplate, material property graded index, and circumferentialwave number on the nondimensional natural frequencies ofthe annular plates with different boundary conditions wereinvestigated.

In this paper the free vibrational behavior of FML platewith central hole is investigated based on the theory of elas-ticity. The plate is considered asymmetric in the tangentialdirection which means that the displacements, stresses, andstrains are functions of the tangential component. A semi-analytical method which is a combination of DQM, state-space, and the Fourier series methods is applied for solvingthe governing equations of motion. By applying DQM in theradial direction, the derivatives in radius direction convert toalgebraic expressions. By using the Fourier series in tangentialdirection, the displacement and stress parameters lose thedependency of the tangential component and the equationswill contain only derivatives in the thickness direction.Therefore, state-spacemethod is used for solving the problemandobtaining the natural frequencies.Thekinds of compositematerial and metal layers are considered to be S2-glass andaluminum, respectively.The boundary conditions consideredhere are clamped-clamped and simply supported-simply sup-ported. The influences of variations in the plate thickness,radius of the plate, layup of composite layers, and radius of thehole on the natural frequencies are investigated. The resultsobtained show that this method has high precision as well asconvergence. The results are compared with those obtainedvia ABAQUS software. Comparison of the results demon-strates the high accuracy of the solutions and confirms theaccuracy of the present results.

a

b

r

𝜃

hc

hc

ha

ha

Aluminium

Aluminium

S2-glassS2-glass

o

z

Figure 1: Schematic viewof a circular fibermetal laminate platewitha central hole.

2. Basic Equations

Figure 1 depicts the schematic view of a circular fiber metallaminate platewith a central hole. 𝑎 stands for the outer radiusof the plate and 𝑏 is the hole radius. The plate is composedof four layers with two T2024 aluminum plates on the topand the bottom and two S2-glass inner layers. Based on theelasticity theory, the governing equations of motion in polarcoordinates can be written as

𝜕𝜎𝑟

𝜕𝑟

+

1

𝑟

𝜕𝜏𝑟𝜃

𝜕𝜃

+

𝜕𝜏𝑟𝑧

𝜕𝑧

+

𝜎𝑟− 𝜎𝜃

𝑟

= 𝜌

𝜕2

𝑢𝑟

𝜕𝑡2,

𝜕𝜏𝑟𝜃

𝜕𝑟

+

1

𝑟

𝜕𝜎𝜃

𝜕𝜃

+

𝜕𝜏𝜃𝑧

𝜕𝑧

+

2𝜏𝑟𝜃

𝑟

= 𝜌

𝜕2

𝑢𝜃

𝜕𝑡2

,

𝜕𝜏𝑟𝑧

𝜕𝑟

+

1

𝑟

𝜕𝜏𝜃𝑧

𝜕𝜃

+

𝜕𝜎𝑧

𝜕𝑧

+

𝜏𝑟𝑧

𝑟

= 𝜌

𝜕2

𝑢𝑧

𝜕𝑡2

,

(1)

where 𝜎𝑟, 𝜎𝜃, and 𝜎

𝑧are the normal stresses in the radial,

tangential, and thickness directions, respectively, 𝜏𝑟𝜃, 𝜏𝑟𝑧, and

𝜏𝜃𝑧

are the shear stresses, and 𝑢𝑟, 𝑢𝜃, and 𝑢

𝑧describe the

displacement components along radial, tangential, and thick-ness directions, respectively. The strain-displacement rela-tions are given as

𝜀𝑟=

𝜕𝑢𝑟

𝜕𝑟

, 𝛾𝑟𝜃

=

1

𝑟

𝜕𝑢𝑟

𝜕𝜃

+

𝜕𝑢𝜃

𝜕𝑟

−

𝑢𝜃

𝑟

,

𝜀𝜃=

𝑢𝑟

𝑟

+

1

𝑟

𝜕𝑢𝜃

𝜕𝜃

, 𝛾𝑟𝑧

=

𝜕𝑢𝑟

𝜕𝑧

+

𝜕𝑢𝑧

𝜕𝑟

,

𝜀𝑟=

𝜕𝑢𝑧

𝜕𝑧

, 𝛾𝜃𝑧

=

𝜕𝑢𝜃

𝜕𝑧

+

1

𝑟

𝜕𝑢𝑧

𝜕𝜃

.

(2)

For a linear elastic material, the structural relationshipbetween stress and strain is given as

𝜎𝑖𝑗

= 𝐶𝑖𝑗𝑘𝑙

𝜀𝑘𝑙, (3)

Advances in Materials Science and Engineering 3

which can be written in the matrix form for the polar systemas the following [16]:

[

[

[

[

[

[

[

[

𝜎𝑟

𝜎𝜃

𝜎𝑧

𝜏𝑟𝜃

𝜏𝑟𝑧

𝜏𝜃𝑧

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

𝑄11

𝑄12

𝑄13

𝑄14

0 0

𝑄12

𝑄22

𝑄23

𝑄24

0 0

𝑄13

𝑄23

𝑄33

𝑄34

0 0

𝑄14

𝑄24

𝑄34

𝑄44

0 0

0 0 0 0 𝑄55

𝑄56

0 0 0 0 𝑄56

𝑄66

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

𝜀𝑟

𝜀𝜃

𝜀𝑧

𝛾𝑟𝜃

𝛾𝑟𝑧

𝛾𝜃𝑧

]

]

]

]

]

]

]

]

, (4)

where 𝑄𝑖𝑗are the plane stress-reduced stiffnesses for the

composite material and their values as a function of materialproperties and fiber angle are given in Appendix A.

3. Boundary Conditions

In the present work two kinds of boundary conditions areconsidered for the annular plate. The clamped-clampedboundary condition considers the plate around the hole andthe outer radius to be fixed.These end conditions are demon-strated by the following equation:

𝑢𝑟= 𝑢𝜃= 𝑢𝑧= 0 at 𝑟 = 𝑎, 𝑏. (5)

Another boundary condition considered here is a kind ofsimple-simple boundary condition. In this kind of boundarycondition, the plate is assumed tomove freely along the radiusdirection, whereas in two other directions it is considered tobe fixed. This can be stated through the following:

𝜎𝑟= 𝑢𝜃= 𝑢𝑧= 0 at 𝑟 = 𝑎, 𝑏. (6)

Also, it should be noted that these end conditions are evenalong the thickness at the ends.

4. Solution Method

The displacement components can be assumed in the fol-lowing forms which simultaneously satisfy the equilibriumequations and the boundary conditions [15]:

𝑢𝑟(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

�̂�𝑟(𝑟, 𝑧) cos (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝑢𝜃(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

�̂�𝜃(𝑟, 𝑧) sin (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝑢𝑧(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

�̂�𝑧(𝑟, 𝑧) sin (𝑚𝜃) 𝑒𝑖𝜔𝑡.

(7)

Also, the stress components can be assumed as

𝜎𝑟(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

�̂�𝑟(𝑟, 𝑧) cos (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝜎𝜃(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

�̂�𝜃(𝑟, 𝑧) cos (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝜎𝑧(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

�̂�𝑧(𝑟, 𝑧) cos (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝜏𝜃𝑧

(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

𝜏𝜃𝑧

(𝑟, 𝑧) sin (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝜏𝑟𝑧

(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

𝜏𝑟𝑧

(𝑟, 𝑧) cos (𝑚𝜃) 𝑒𝑖𝜔𝑡,

𝜏𝑟𝜃

(𝑟, 𝜃, 𝑧, 𝑡) =

∞

∑

𝑚=0

𝜏𝑟𝜃

(𝑟, 𝑧) sin (𝑚𝜃) 𝑒𝑖𝜔𝑡,

(8)in which𝑚 = 0, 1, 2, . . . ,∞. In this analysis,𝑚 = 0 associatedwith the axisymmetric vibration gives the first mode ofvibration and 𝑚 = 1 and 𝑚 = 2 present the first and secondmodes of vibration, respectively. Also, 𝜔 denotes the naturalfrequency of the plate. Introducing the following dimension-less parameters:

𝑧 =

𝑧

ℎ

, 𝑟 =

𝑟

𝑎

, Ω = 𝜔ℎ√

𝜌𝑎

𝑄11𝑎

,

(𝑢𝑟, 𝑢𝜃, 𝑢𝑧) =

(�̂�𝑟, �̂�𝜃, �̂�𝑧)

ℎ

, 𝑄𝑖𝑗

=

𝑄𝑖𝑗

𝑄11𝑎

,

(𝜎𝑟, 𝜎𝜃, 𝜎𝑧, 𝜏𝑟𝜃, 𝜏𝑟𝑧, 𝜏𝜃𝑧) =

(�̂�𝑟, �̂�𝜃, �̂�𝑧, 𝜏𝑟𝜃, 𝜏𝑟𝑧, 𝜏𝜃𝑧)

𝑄11𝑎

.

(9)

Equation (1) can be rewritten in the following form:

1

𝑎

𝜕𝜎𝑟

𝜕𝑟

+

𝑚

𝑎

𝜏𝑟𝜃

𝑟

+

1

ℎ

𝜕𝜏𝑟𝑧

𝜕𝑧

+

1

𝑎

(

𝜎𝑟− 𝜎𝜃

𝑟

) = −

𝜌𝑖

𝜌𝑎

Ω2

ℎ

𝑢𝑟,

1

𝑎

𝜕𝜏𝑟𝜃

𝜕𝑟

−

𝑚

𝑎

𝜎𝜃

𝑟

+

1

ℎ

𝜕𝜏𝜃𝑧

𝜕𝑧

+

2

𝑎

𝜏𝑟𝜃

𝑟

= −

𝜌𝑖

𝜌𝑎

Ω2

ℎ

𝑢𝜃,

1

𝑎

𝜕𝜏𝑟𝑧

𝜕𝑟

+

𝑚

𝑎

𝜏𝜃𝑧

𝑟

+

1

ℎ

𝜕𝜎𝑧

𝜕𝑧

+

1

𝑎

𝜏𝑟𝑧

𝑟

= −

𝜌𝑖

𝜌𝑎

Ω2

ℎ

𝑢𝑧,

(10)

where 𝜌𝑎is the density of aluminum,𝑄

11𝑎is the first element

of the stiffness matrix for aluminum, and ℎ is the totalthickness of the plate. In terms of the above dimensionlessparameters, the strain components can be reformed as

𝜀𝑟=

ℎ

𝑎

𝜕𝑢𝑟

𝜕𝑟

, 𝛾𝑟𝜃

= −

𝑚ℎ

𝑎

𝑢𝑟

𝑟

+

ℎ

𝑎

𝜕𝑢𝜃

𝜕𝑟

−

ℎ

𝑎

𝑢𝜃

𝑟

,

𝜀𝜃=

ℎ

𝑎

𝑢𝑟

𝑟

+

𝑚ℎ

𝑎

𝑢𝜃

𝑟

, 𝛾𝑟𝑧

=

𝜕𝑢𝑟

𝜕𝑧

+

ℎ

𝑎

𝜕𝑢𝑧

𝜕𝑟

,

𝜀𝑧=

𝜕𝑢𝑧

𝜕𝑧

, 𝛾𝜃𝑧

=

𝜕𝑢𝜃

𝜕𝑧

−

𝑚ℎ

𝑎

𝑢𝑧

𝑟

.

(11)

Applying the method of the Fourier series and separationof components of displacement and stress at the parameters,these parameters become a function of the thickness andradius. The differential governing equations for vibrationanalysis of plate has three equations with two variables. Tosolve these equations, there are different ways but one of thebest as well as effective ways is the combination of differentialquadrature and state-space methods. This semianalyticalapproach has a high rate of convergence.


5. Differential Quadrature Method (DQM)

In order to solve the governing differential equations ofmotion, DQM is applied along the radius direction.Thus, theexpressions containing the first and second order derivativesof the displacements are replaced by differential quadraturefunctions with certain amount of points. For a circular plateof radius 𝑎, containing a central hole of radius 𝑏, the selectedpoints in the differential quadrature method are chosen as

𝑟𝑖=

𝑎 − 𝑏

2

(1 − cos( 𝑖 − 1𝑁 − 1

𝜋)) + 𝑏, (𝑖 = 1, . . . , 𝑁) .(12)

Based on DQM, the 𝑛th-order partial derivative of acontinuous function as 𝑓(𝑟, 𝑧) with respect to 𝑟 at a givenpoint 𝑟

𝑖is approximated by a linear summation of weighting

function values at all points in the domain of 𝑟 as

𝜕𝑛

𝑓 (𝑟𝑖, 𝑧)

𝜕𝑟𝑛

=

𝑁

∑

𝑗=1

𝑔(𝑛)

𝑖𝑗𝑓 (𝑟𝑖, 𝑧) ,

(𝑖 = 1, . . . , 𝑁, 𝑛 = 1, . . . , 𝑁 − 1) ,

(13)

where 𝑁 is the number of sample points and 𝑔(𝑛)𝑖𝑗

are theweighting coefficients related to 𝑟

𝑖defined as

𝑔(1)

𝑖𝑗=

𝑀(𝑟𝑖)

(𝑟𝑖− 𝑟𝑗)

, 𝑖, 𝑗 = 1, . . . , 𝑁 (𝑖 ̸= 𝑗) , (14)

in which

𝑀(𝑟𝑖) =

𝑁

∏

𝑗=𝑖,𝑖 ̸= 𝑗

(𝑟𝑖− 𝑟𝑗) , 𝑔

(1)

𝑖𝑖= −

𝑁

∑

𝑗=1,𝑖 ̸= 𝑗

𝑔(1)

𝑖𝑗. (15)

For higher order derivatives, the values of weighting func-tions can be obtained from the following formula:

𝑔(𝑛)

𝑖𝑗= −∑𝑛(𝑔

(𝑛−1)

𝑖𝑗𝑔(1)

𝑖𝑗−

𝑔(𝑛−1)

𝑖𝑗

𝑟𝑖− 𝑟𝑗

) ,

𝑖, 𝑗 = 1, . . . , 𝑁, 𝑖 ̸= 𝑗.

(16)

6. State-Space Equations Derivation

Using (4)–(10) and (11) and applyingDQMas proposed in thisequations, the state-space equations for the 𝑖th sample pointcan be derived as [5]

𝑑𝜎𝑧𝑖

𝑑𝑧

= −

𝜌𝑖

𝜌𝑎

Ω2

𝑢𝑧𝑖

−

ℎ

𝑎

𝑁

∑

𝑗=1

𝑔(1)

𝑖𝑗𝜏𝑟𝑧𝑗

−

𝑚ℎ

𝑎

𝜏𝜃𝑧𝑖

𝑟𝑖

−

ℎ

𝑎

𝜏𝑟𝑧𝑖

𝑟𝑖

,

𝑑𝑢𝑟𝑖

𝑑𝑧

= −

ℎ

𝑎

𝑁

∑

𝑗=1

𝑔(1)

𝑖𝑗𝑢𝑧𝑗

+

1

𝑄55

𝜏𝑟𝑧𝑖

,

𝑑𝑢𝜃𝑖

𝑑𝑧

=

𝑚ℎ

𝑎

𝑢𝑧𝑖

𝑟𝑖

+

1

𝑄55

𝜏𝜃𝑧𝑖

,

𝑑𝑢𝑧𝑖

𝑑𝑧

=

𝜎𝑧𝑖

𝑄33

−

𝑄13

𝑄33

ℎ

𝑎

𝑁

∑

𝑗=1

𝑔(1)

𝑖𝑗𝑢𝑟𝑗

−

𝑄13

𝑄33

ℎ

𝑎

𝑢𝑟𝑖

𝑟𝑖

−

𝑚ℎ

𝑎

𝑢𝜃𝑖

𝑟𝑖

,

𝑑𝜏𝑟𝑧𝑖

𝑑𝑧

= −

𝑄13

𝑄33

ℎ

𝑎

𝑁

∑

𝑗=1

𝑔(1)

𝑖𝑗𝜎𝑧𝑗

+

ℎ

𝑎

(𝑄23

− 𝑄13)

𝜎𝑧𝑖

𝑄33

−

ℎ2

𝑎2(𝑄11

−

𝑄

2

13

𝑄33

)

𝑁

∑

𝑗=1

𝑔(2)


+

ℎ2

𝑎2(−𝑄11

+

𝑄

2

13

𝑄33

)

1

𝑟𝑖

𝑁

∑

𝑗=1

𝑔(1)


+

ℎ2

𝑎2(𝑄22

−

𝑄

2

23

𝑄33

+ 𝑚2

𝑄44)

𝑢𝑟𝑖

𝑟𝑖

2

−

𝑚ℎ2

𝑎2

(𝑄12

−

𝑄13𝑄23

𝑄33

+ 𝑄44)

1

𝑟𝑖

𝑁

∑

𝑗=1

𝑔(1)

𝑖𝑗𝑢𝜃𝑗

+

𝑚ℎ2

𝑎2

(𝑄22

−

𝑄

2

23

𝑄33

+ 𝑄44)

𝑢𝜃𝑖

𝑟𝑖

2−

𝜌𝑖

𝜌𝑎

Ω2

𝑢𝑟𝑖,

𝑑𝜏𝜃𝑧𝑖

𝑑𝑧

=

𝑄23

𝑄33

𝑚ℎ

𝑎

𝜎𝑧𝑖

𝑟𝑖

+

𝑚ℎ2

𝑎2

(𝑄12

−

𝑄13𝑄23

𝑄33

+ 𝑄44)

×

1

𝑟𝑖

𝑁

∑

𝑗=1

𝑔(1)


+

𝑚ℎ2

𝑎2

(𝑄22

−

𝑄13𝑄23

𝑄33

+ 𝑄44)

𝑢𝑟𝑖

𝑟𝑖

2

−

ℎ2

𝑎2𝑄44

𝑁

∑

𝑗=1

𝑔(2)


−

ℎ2

𝑎2𝑄44

1

𝑟𝑖

𝑁

∑

𝑗=1

𝑔(1)


+

ℎ2

𝑎2(𝑚2

(𝑄22

−

𝑄

2

23

𝑄33

) + 𝑄44)

𝑢𝜃𝑖

𝑟𝑖

2−

𝜌𝑖

𝜌𝑎

Ω2

𝑢𝑟𝑖.

(17)

The above six equations are the state-space equationswhich must be solved to obtain the natural frequencies of theplate. After applying the method of Fourier series and DQM,the derivatives along the tangential and radial directions areremoved from the final equation and only the first-orderderivatives with respect to thickness will remain.

The state-space equations can be found as below

𝑑

𝑑𝑧

𝛿 (𝑧) = 𝑀𝑘

𝛿 (𝑧) , (18)

where

𝛿 (𝑧) = [[𝜎𝑧𝑖] [𝑢𝑟𝑖] [𝑢𝜃𝑖] [𝑢𝑧𝑖] [𝜏𝑟𝑧𝑖

] [𝜏𝜃𝑧𝑖

]]𝑇

. (19)

In (22),

[𝜎𝑧𝑖] = [𝜎

𝑧1⋅ ⋅ ⋅ 𝜎𝑧𝑁

]𝑇

, [𝑢𝑟𝑖] = [𝑢

𝑟1⋅ ⋅ ⋅ 𝑢𝑟𝑁

]𝑇

, . . . .

(20)

The above equations which form a system of ordinarydifferential equations can be solved to give

𝛿 (𝑧) = exp (𝑀𝑧) 𝛿 (𝑧 = 0) , (21)


(a) (b)

Figure 2: Mode shapes associated with (a) first and (b) second modes of vibration of an annular plate.

in which 𝛿(𝑧 = 0) is the state-space vector at the bottom ofthe plate [5]. Using this equation, the stress and displacementcomponents in the state-space vectors are extracted in eachpoint in the thickness direction and the other components ofstress can also be calculated. Matrix 𝑀 is different for eachboundary condition and it is given in Appendix B.

The state-space equations for each layer are writtenseparately. Therefore, for a laminated plate, these equationsare combined based on the following relation:

𝛿 (𝑧 = ℎ𝑇) =

1

∏

𝑘=𝑛

exp (𝑀ℎ𝑘) 𝛿 (0) ,

𝑇 =

1

∏

𝑘=𝑛

exp (𝑀ℎ𝑘) .

(22)

Expanding the previous equation in the matrix form, wearrive at the following state-space equations for fiber metallaminate as

[

[

[

[

[

[

[

[

[𝜎𝑧𝑖]

[𝑢𝑟𝑖]

[𝑢𝜃𝑖]

[𝑢𝑧𝑖]

[𝜏𝑟𝑧𝑖

]

[𝜏𝜃𝑧𝑖

]

]

]

]

]

]

]

]

]𝑧=ℎ𝑇

=

[

[

[

[

[

[

[

[

𝑇11

𝑇12

𝑇13

𝑇14

𝑇15

𝑇16

𝑇21

𝑇22

𝑇23

𝑇24

𝑇25

𝑇26

𝑇31

𝑇32

𝑇33

𝑇34

𝑇35

𝑇36

𝑇41

𝑇42

𝑇43

𝑇44

𝑇45

𝑇46

𝑇51

𝑇52

𝑇53

𝑇54

𝑇55

𝑇56

𝑇61

𝑇62

𝑇63

𝑇64

𝑇65

𝑇66

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[𝜎𝑧𝑖]

[𝑢𝑟𝑖]

[𝑢𝜃𝑖]

[𝑢𝑧𝑖]

[𝜏𝑟𝑧𝑖

]

[𝜏𝜃𝑧𝑖

]

]

]

]

]

]

]

]

]𝑧=0

.

(23)

Since, the top and bottom surfaces of the plate are freeof static and surface forces, as a result, the normal andshear stresses are zero at these surfaces. Thus, the state-space

equations for the free vibration analysis of plate can be recastto the following:

[

[

[

[

[

[

[

[

0

[𝑢𝑟𝑖]

[𝑢𝜃𝑖]

[𝑢𝑧𝑖]

0

0

]

]

]

]

]

]

]

]𝑧=ℎ𝑇

=

[

[

[

[

[

[

[

[

𝑇11

𝑇12

𝑇13

𝑇14

𝑇15

𝑇16

𝑇21

𝑇22

𝑇23

𝑇24

𝑇25

𝑇26

𝑇31

𝑇32

𝑇33

𝑇34

𝑇35

𝑇36

𝑇41

𝑇42

𝑇43

𝑇44

𝑇45

𝑇46

𝑇51

𝑇52

𝑇53

𝑇54

𝑇55

𝑇56

𝑇61

𝑇62

𝑇63

𝑇64

𝑇65

𝑇66

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

0

[𝑢𝑟𝑖]

[𝑢𝜃𝑖]

[𝑢𝑧𝑖]

0

0

]

]

]

]

]

]

]

]𝑧=0

.

(24)

Expanding the first, fifth, and sixth rows and rewriting ina matrix form, we reach to

[

[

0

0

0

]

]𝑧=ℎ𝑇

=[

[

𝑇12

𝑇13

𝑇14

𝑇52

𝑇53

𝑇54

𝑇62

𝑇63

𝑇64

]

]

[

[

[𝑢𝑟𝑖]

[𝑢𝜃𝑖]

[𝑢𝑧𝑖]

]

]𝑧=0

. (25)

To obtain the nontrivial solution of (25), the determinantof the coefficient matrix must set to be zero, which yields acharacteristic equation whose roots are the natural frequen-cies of the annular plate.

7. Results and Discussions

The type of plate considered is GLARE2 and composed offour layers [17]. The layers in the top and bottom surfacesof the plate are composed of aluminum T2024 with materialproperties given in Table 1. Also, the inner layers are com-posed of S2-glass with material properties given in Table 2.

In order to examine the convergence rate of the presentanalytic procedure, Table 3 lists the nondimensional naturalfrequencies of a clamped-clamped GLARE2 annular plate.Table 4 presents the nondimensional natural frequencies


0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

1

2

3

×10−4

Ω

Ω1Ω2

a (m)

Figure 3: Variation of the first and secondmode natural frequencieswith the radius of the plate (𝑏 = 0.05m and ℎ

𝑐= 0.0005m).

2

3

4

5

6

7

8

9

10

11

12

×10−4

Ω1

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

b = 0.025

b = 0.05

b = 0.075

a (m)

Figure 4: Variation of the first mode natural frequency with theradius of the plate for different hole radiuses and clamped-clampedboundary conditions (ℎ

𝑐= 1mm).

associated with the first mode of vibration for a clamped-clamped annular plate with different values of ℎ

𝑐. The top

and bottom aluminum layers are assumed to be 0.5mmthick. While estimating the natural frequencies, the value of𝑁 is set to be equal to 11. Results are also compared withthose reported via ABAQUS software. The plate is meshedby shell-type elements. As would be observed, an excellent

2

3

4

5

6

7

8

9

10

11

12×10

−4

Ω2

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

b = 0.025

b = 0.05

b = 0.075

a (m)

Figure 5: Variation of the second mode natural frequency with theradius of the plate for different hole radiuses and clamped-clampedboundary conditions (ℎ

𝑐= 1mm).

Table 1: Mechanical properties of aluminum T2024.

𝜌 (kg/m3) ] 𝐸 (GPa)2780 0.33 72.2

agreement has been achieved. Table 5 gives the similar resultsfor the second mode of vibration. The first and second modeshapes of vibration for the clamped-clamped annular plate aredepicted in Figure 2.

Figure 3 illustrates the variation of the natural frequenciescorresponding to the first and second vibration modes withthe radius of the plate having clamped-clamped end condi-tions. The thickness of aluminum layers on the top and bot-tom surfaces is taken as 0.5mm.The layup of composite layersis unidirectional. As can be seen from this figure the naturalfrequency decreases as the radius increases. This is becauseof the decrease in the plate stiffness. Figure 4 clarifies thevariation of the natural frequency associated with the firstvibration mode with the radius of the plate for different holeradiuses. Figure 5 is the similar one to the second mode ofvibration. As seen from these figures, the natural frequencyincreases by an increment in the hole radius. This fact ismainly due to the fact that by an increase in the hole radius,the effective radius of the plate decreases and this leads toan increase in the plate stiffness and the frequency value asa consequence. Also, by increasing the radius of the hole,the mass decreases which results into an increase in thenatural frequency of the plate. Figures 6 and 7 illustrate thenatural frequency variations of the first and second modeswith respect to the plate radius for three different thicknessesof the composite layer. As can be seen, the natural frequency


0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

h = 0.002

h = 0.003

h = 0.005

2.5

2

1

1.5

0.5

0

Ω1

×10−3

a (m)

Figure 6: Variation of the first mode natural frequency with theradius of the plate for different composite layer thicknesses andclamped-clamped end conditions (𝑏 = 0.05m).

increases as the composite layer thickness increases. How-ever, by increasing the plate thickness, the mass and stiffnessvalues of the plate increase, but this increase is more signifi-cant for the bending stiffness.

The natural frequencies of the plate for GLARE3 are alsocalculated. The layup of the composite layers is considered ascross ply. Figures 8 and 9 exhibit the variation of the naturalfrequencies associated with the first and second vibrationmodes with plate radius for unidirectional and cross plylayups. The natural frequencies of plate with unidirectionallayup are greater than those of the cross ply counterpart.Thisis because of the greatness of the bending stiffness for theunidirectional layup over the cross ply case.

Figure 10 presents the variation of natural frequenciesof the first and second vibration modes for the plate withsimple-simple boundary condition. Figures 11 and 12 exhibitthe variations of the natural frequencies associated with thefirst and second vibration modes for annular plate withunidirectional layup and different boundary conditions.

8. Conclusions

Based on the theory of elasticity, free vibration analysis ofcircular fiber metal composite plate with a central hole hasbeen performed. The governing equations derived using theelasticity theory will then be solved using a combination ofdifferential quadrature method, state-space, and the Fourierseries in order to obtain the natural frequencies of the plate.The composite metal plate is made up of GLARE and twokinds of GLARE2 and GLARE3 are chosen for the vibrationanalysis. Plate is composed of four layers with aluminum

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

h = 0.002

h = 0.003

h = 0.005

2.5

2

1

1.5

0.5

0

Ω2

×10−3

a (m)

Figure 7: Variation of the second mode natural frequency withthe radius of the plate for different composite layer thicknesses andclamped-clamped end conditions (𝑏 = 0.05m).

0/00/90

2

3

4

5

6

7

8

9

10×10

−4

Ω1

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

a (m)

Figure 8: Variation of the first mode natural frequency withthe radius of the plate for different layups and clamped-clampedboundary condition (𝑏 = 0.05m, ℎ

𝑐= 0.001m).

layers on the top and bottom surfaces and inner compositelayers. Vibration frequencies of circular fibermetal compositeplates with central holes for both the clamped-clamped andsimply supported boundary conditions were presented. Also,effects of layup, hole radius, and plate thickness on the nat-ural frequencies are studied. Results obtained from present


Table 2: Mechanical properties of S2-glass.

𝜌 (kg/m3) ]23

]13

]12

𝐺23(GPa) 𝐺

12(GPa) 𝐺

13(GPa) 𝐸

33(GPa) 𝐸

22(GPa) 𝐸

11(GPa)

1980 0.32 0.25 0.25 7 7 7 17 17 52

Table 3: A convergence study for the nondimensional natural frequencies associated with the first and second vibrationmodes of a clamped-clamped annular plate (𝑎 = 40 cm and 𝑏 = 5 cm).

m h (mm) 𝑁 = 7 𝑁 = 8 𝑁 = 9 𝑁 = 10 𝑁 = 11

02 0.0001919 0.0001907 0.0001910 0.0001909 0.00019103 0.00042822 0.00042540 0.0004262 0.0004260 0.00042625 0.00115613 0.00114870 0.00115090 0.0011502 0.0011508

12 0.0002075 0.0002046 0.0002052 0.0002050 0.00020513 0.00046116 0.00045416 0.0004553 0.0004549 0.00045525 0.00123097 0.00121644 0.0012193 0.00121824 0.0012190

0/00/90

2

3

4

5

6

7

8

9

10×10

−4

Ω2

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

a (m)

Figure 9: Variation of the second mode natural frequency withthe radius of the plate for different layups and clamped-clampedboundary condition (𝑏 = 0.05m and ℎ

𝑐= 0.001m).

Table 4: Comparison of the frequency value associatedwith the firstvibration mode for a clamped-clamped annular plate (𝑎 = 50 cm,𝑏 = 5 cm, and 𝑚 = 0).

ℎ𝑐(mm) State-space DQM ABAQUS Error (%)

0.5 0.000115 0.0001156 0.51 0.0002576 0.00025011 2.52 0.0006955 0.0006415 8

semianalytical approach have been compared with thosereported by ABAQUS software. This comparison shows thatthe present solution is of high accuracy.The results show thatthe natural frequency of the plate increases with an increment

0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

1

1.2

0.8

0.6

1.4

1.6

1.8

2

2.2

×10−4

Ω

Ω1Ω2

a (m)

Figure 10: Variation of the first and second mode natural frequen-cies with the radius of the plate for the simple-simple boundarycondition (𝑏 = 0.05m and ℎ

𝑐= 0.0005m).

Table 5: Comparison of the frequency value associated with thesecond vibration mode for a clamped-clamped annular plate (𝑎 =50 cm, 𝑏 = 5 cm, and 𝑚 = 1).

ℎ𝑐(mm) State-space DQM ABAQUS Error (%)

0.5 0.0001246 0.0001213 2.61 0.0002767 0.00025997 62 0.0007406 0.00065769 11

in the radius of the hole. This is due to the fact that theeffective radius of the plate is reduced and this increases theplate stiffness and the natural frequency as a consequence.It was observed that the natural frequency of the GLARE2


0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

3.5

2.5

2

3

1.5

1

0.5

0.34

Ω1

×10−4

S-SC-C

a (m)

Figure 11: Variation of the first mode natural frequency with theradius of the plate with different boundary conditions (𝑏 = 0.05mand ℎ

𝑐= 0.0005m).

0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

2.8

2.6

2.4

2.2

2

1.8

1.6

1.4

1.2

1

0.8

Ω2

S-SC-C

×10−4

a (m)

Figure 12: Variation of the second mode natural frequency with theradius of the plate with different boundary conditions (𝑏 = 0.05mand ℎ

𝑐= 0.0005m).

for unidirectional composite layers arrangement is morecompared with the GLARE3 with cross ply composite layers.Furthermore, the effect of boundary conditions on the natu-ral frequencies was studied which illustrated that the naturalfrequencies of the plate with clamped-clamped boundarycondition are higher than those of the simply supported case.

This is due to the fact that in the clamped-clamped case thedegree of freedom is less and this causes an increase in theplate stiffness.

Appendices

A. Plane Stress-Reduced Stiffnesses forthe Composite Material

𝑄11

= 𝑚4

𝐶11

+ 2𝑚2

𝑛2

(𝐶12

+ 2𝐶44) + 𝑛4

𝑄22,

𝑄12

= 𝑚2

𝑛2

(𝑄11

+ 𝑄22

− 4𝑄44) + (𝑚

4

+ 𝑛4

) 𝐶12,

𝑄13

= 𝑚2

𝐶13

+ 𝑛2

𝐶23,

𝑄14

= 𝑚𝑛 [(𝐶11

− 𝐶12

− 2𝐶44)𝑚2

+(𝐶11

− 𝐶12

+ 2𝐶44) 𝑛2

] ,

𝑄22

= 𝑛4

𝐶11

+ 2𝑚2

𝑛2

(𝐶12

+ 2𝐶44) + 𝑚4

𝐶22,

𝑄23

= 𝑚2

𝐶23

+ 𝑛2

𝐶13,

𝑄24

= 𝑚𝑛 [(𝐶11

− 𝐶12

− 2𝐶44) 𝑛2

+ (𝐶11

− 𝐶12

+ 2𝐶44)𝑚2

] ,

𝑄33

= 𝐶33,

𝑄56

= 𝑚𝑛 (𝐶31

− 𝐶32) ,

𝑄44

= 𝑚2

𝑛2

(𝐶11

− 2𝐶12

+ 𝐶22

− 2𝐶44) +(𝑚

4

+ 𝑛4

) 𝐶44,

𝑄55

= 𝑚2

𝐶55

+ 𝑛2

𝐶66,

𝑄56

= 𝑚𝑛 (𝐶55

− 𝐶66) ,

𝑄66

= 𝑛2

𝐶55

+ 𝑚2

𝐶66,

(A.1)

in which 𝑚 = cos(𝛼) and 𝑛 = sin(𝛼), where 𝛼 is theangle between the direction of the principal axis and the fiberdirection.The values of𝐶

𝑖𝑗constants only depend on the kind

of material and they are given as follows:

𝐶11

=

𝐸11

(1 − ]23]32)

Δ

; 𝐶12

=

𝐸11

(]21

+ ]31]23)

Δ

𝐶13

=

𝐸11

(]31

+ ]21]32)

Δ

; 𝐶22

=

𝐸22

(1 − ]31]13)

Δ

𝐶23

=

𝐸22

(]32

+ ]12]13)

Δ

; 𝐶33

=

𝐸33

(1 + ]12]21)

Δ


Δ = (1 − ]12]21

− ]23]32

− ]13]31

− 2]12]23]31)

𝐶44

= 𝐺12

𝐶55

= 𝐺13

𝐶66

= 𝐺23.

(A.2)

B. 𝑀 Matrices for DifferentBoundary Conditions

𝑀 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 −

𝜌

𝜌𝑎

Ω2

𝐼𝑁

−

ℎ𝑇

𝑎

𝑔1

𝑖𝑗−

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁

−𝑚

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁

0 0 0 −

ℎ𝑇

𝑎

𝑔1

𝑖𝑗

𝐼𝑁

𝑄55

0

0 0 0 𝑚

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁

0

𝐼𝑁

𝑄66

𝐼𝑁

𝑄33

−

𝑄13

𝑄33

ℎ𝑇

𝑎

𝑔1

𝑖𝑗−

𝑄23

𝑄33

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁

−𝑚

ℎ𝑇

𝑎

𝑄23

𝑄33

1

𝑟𝑖

𝐼𝑁

0 0 0

𝑠1

𝑠2−

𝜌

𝜌𝑎

Ω2

𝐼𝑁

𝑠3

0 0 0

𝑠4

𝑠5

𝑠6−

𝜌

𝜌𝑎

Ω2

𝐼𝑁

0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

𝑖, 𝑗 = 1, . . . , 𝑁,

𝑠1= −

𝑄13

𝑄33

ℎ𝑇

𝑎

𝑔1

𝑖𝑗+

ℎ𝑇

𝑎

(

𝑄23

− 𝑄13

𝑄33

) 𝐼𝑁,

𝑠2= −(

ℎ𝑇

𝑎

)

2

(𝑄11

−

𝑄

2

13

𝑄33

)𝑔2

𝑖𝑗+ −(

ℎ𝑇

𝑎

)

2

(𝑄11

−

𝑄

2

13

𝑄33

)

1

𝑟𝑖

𝐼𝑁𝑔1

𝑖𝑗

+ 𝑚(

ℎ𝑇

𝑎

)

2

(𝑚2

𝑄44

+ 𝑄22

−

𝑄

2

23

𝑄33

)

1

𝑟2

𝑖

𝐼𝑁,

𝑠3= −𝑚(

ℎ𝑇

𝑎

)

2

(𝑄44

+ 𝑄12

−

𝑄13𝑄23

𝑄33

)

1

𝑟𝑖

𝐼𝑁𝑔1

𝑖𝑗

+ 𝑚(

ℎ𝑇

𝑎

)

2

(𝑄44

+ 𝑄22

−

𝑄

2

23

𝑄33

)

1

𝑟2

𝑖

𝐼𝑁,

𝑠4= 𝑚

ℎ𝑇

𝑎

𝑄23

𝑄33

1

𝑟𝑖

𝐼𝑁,

𝑠5= 𝑚(

ℎ𝑇

𝑎

)

2

(𝑄44

+ 𝑄12

−

𝑄13𝑄23

𝑄33

)

1

𝑟𝑖

𝐼𝑁𝑔1

𝑖𝑗+ 𝑚(

ℎ𝑇

𝑎

)

2

(𝑄44

+ 𝑄22

−

𝑄

2

23

𝑄33

)

1

𝑟2

𝑖

𝐼𝑁,

𝑠6= −(

ℎ𝑇

𝑎

)

2

𝑄44𝑔2

𝑖𝑗− (

ℎ𝑇

𝑎

)

2

𝑄44

1

𝑟𝑖

𝐼𝑁𝑔1

𝑖𝑗

+ (

ℎ𝑇

𝑎

)

2

(𝑄44

+ 𝑚2

(𝑄22

−

𝑄

2

23

𝑄33

))

1

𝑟2

𝑖

𝐼𝑁,


𝑀𝐶

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 −

𝜌

𝜌𝑎

Ω2

𝐼𝑁−2

−𝑄55(

ℎ𝑇

𝑎

)

2

𝑓𝑐𝑐

−

ℎ𝑇

𝑎

𝑔1

𝑖𝑗−

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁−2

−𝑚

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁−2

0 0 0 −

ℎ𝑇

𝑎

𝑔1

𝑖𝑗

𝐼𝑁−2

𝑄55

0

0 0 0 𝑚

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁−2

0

𝐼𝑁−2

𝑄66

𝐼𝑁

𝑄33

−

𝑄13

𝑄33

ℎ𝑇

𝑎

𝑔1

𝑖𝑗−

𝑄23

𝑄33

ℎ𝑇

𝑎

1

𝑟𝑖

𝐼𝑁−2

−𝑚

ℎ𝑇

𝑎

𝑄23

𝑄33

1

𝑟𝑖

𝐼𝑁−2

0 0 0

𝑠1

𝑠2−

𝜌

𝜌𝑎

Ω2

𝐼𝑁−2

𝑠3

0 0 0

𝑠4

𝑠5

𝑠6−

𝜌

𝜌𝑎

Ω2

𝐼𝑁−2

0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

𝑓𝑐𝑐

= 𝑔1

𝑖1𝑔1

1𝑗+ 𝑔1

𝑖𝑁𝑔1

𝑖𝑗, 𝑖, 𝑗 = 2, . . . , 𝑁 − 1.

(B.1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] E. C. Botelho, A. N. Campos, E. de Barros, L. C. Pardini, andM.C.Rezende, “Damping behavior of continuous fiber/metal com-posite materials by the free vibration method,” Composites PartB, vol. 37, no. 2-3, pp. 255–263, 2005.

[2] V. G. Reyes and W. J. Cantwell, “The mechanical properties offibre-metal laminates based on glass fibre reinforced polypropy-lene,”Composites Science andTechnology, vol. 60, no. 7, pp. 1085–1094, 2000.

[3] A. Shooshtari and S. Razavi, “A closed form solution for linearand nonlinear free vibrations of composite and fiber metallaminated rectangular plates,” Composite Structures, vol. 92, no.11, pp. 2663–2675, 2010.

[4] S. M. R. Khalili, K. Malekzadeh, A. Davar, and P. Mahajan,“Dynamic response of pre-stressed fibre metal laminate (FML)circular cylindrical shells subjected to lateral pressure pulseloads,” Composite Structures, vol. 92, no. 6, pp. 1308–1317, 2010.

[5] A. Alibeigloo and M. Shakeri, “Elasticity solution for the freevibration analysis of laminated cylindrical panels using thedifferential quadrature method,” Composite Structures, vol. 81,no. 1, pp. 105–113, 2007.

[6] Y. Li and Z. Shi, “Free vibration of a functionally graded piezo-electric beam via state-space based differential quadrature,”Composite Structures, vol. 87, no. 3, pp. 257–264, 2009.

[7] A. Alibeigloo and R. Madoliat, “Static analysis of cross-ply lam-inated plates with integrated surface piezoelectric layers usingdifferential quadrature,” Composite Structures, vol. 88, no. 3, pp.342–353, 2009.

[8] A. Alibeigloo, “Static and vibration analysis of axi-symmetricangle-ply laminated cylindrical shell using state space differen-tial quadraturemethod,” International Journal of PressureVesselsand Piping, vol. 86, no. 11, pp. 738–747, 2009.

[9] M.H.Yas andB. S. Aragh, “Free vibration analysis of continuousgrading fiber reinforced plates on elastic foundation,” Interna-tional Journal of Engineering Science, vol. 48, no. 12, pp. 1881–1895, 2010.

[10] L. G. Nallim and R. O. Grossi, “Natural frequencies of sym-metrically laminated elliptical and circular plates,” InternationalJournal of Mechanical Sciences, vol. 50, no. 7, pp. 1153–1167, 2008.

[11] H. R. Ovesy and J. Fazilati, “Buckling and free vibration finitestrip analysis of composite plates with cutout based on twodifferent modeling approaches,” Composite Structures, vol. 94,no. 3, pp. 1250–1258, 2012.

[12] A. Baltaci, M. Sarikanat, and H. Yildiz, “Static stability of lami-nated composite circular plates with holes using shear deforma-tion theory,” Finite Elements in Analysis and Design, vol. 43, no.11-12, pp. 839–846, 2007.

[13] G. J. Nie and Z. Zhong, “Semi-analytical solution for three-dimensional vibration of functionally graded circular plates,”Computer Methods in Applied Mechanics and Engineering, vol.196, no. 49–52, pp. 4901–4910, 2007.

[14] R. Seifi, N. Khoda-Yari, andH.Hosseini, “Study of critical buck-ling loads and modes of cross-ply laminated annular plates,”Composites Part B, vol. 43, no. 2, pp. 422–430, 2012.

[15] A. Jodaei, M. Jalal, and M. H. Yas, “Free vibration analysis offunctionally graded annular plates by state-space based differ-ential quadrature method and comparative modeling by ANN,”Composites Part B, vol. 43, no. 2, pp. 340–353, 2012.

[16] J. N. Reddy,Mechanics of LaminatedComposite Plates and Shells:Theory and Analysis, CRC Press, London, UK, 2nd edition,2004.

[17] E. C. Botelho, R. A. Silva, L. C. Pardini, and M. C. Rezende,“A review on the development and properties of continuousfiber/epoxy/aluminum hybrid composites for aircraft struc-tures,”Materials Research, vol. 9, no. 3, pp. 247–256, 2006.

Submit your manuscripts athttp://www.hindawi.com

ScientificaHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation http://www.hindawi.com Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials

Research Article Free Vibration Analysis of Fiber Metal ...downloads.hindawi.com/journals/amse/2014/602708.pdf · Research Article Free Vibration Analysis of Fiber Metal Laminate

Documents