Free Vibration Analysis of FunctionallyHeterogeneous Hollow Cylinder Based onThree Dimensional Elasticity TheoryMasoud AsgariFaculty of Mechanical Engineering, K. N. Toosi University of Technology, Pardis Street, Molla-Sadra Avenue,Vanak Square, Tehran, Iran

(Received 16 November 2014; accepted 13 January 2016)

A two-dimensional functionally heterogeneous thick hollow cylinder with a finite length is considered and itsnatural modes and frequencies are determined. Since mode shapes of a thick cylinder are three-dimensional, evenwith axisymmetric conditions, three-dimensional theory of elasticity implemented for problem formulation. Theaxisymmetric conditions are assumed for the cylinder. The material properties of the two-dimensional functionallygraded material (2D-FGM) cylinder are varied in the radial and axial directions with power law functions. Effectsof volume fraction distribution on the different types of anti-symmetric mode shape configuration and vibrationalbehaviour of a simply supported cylinder are analyzed. Three-dimensional equations of motion are used and theeigen value problem is developed based on direct variational method. The study shows that the 2D-FGM cylinderexhibit interesting vibrational characteristics and mode shapes when the constituent volume fractions are varied.

1. INTRODUCTION

Recently, the composition of several different materials isoften used in structural components in order to optimize theresponses of structures subjected to severe loadings. For re-ducing the local stress concentrations induced by abrupt tran-sitions in material properties, the transition between differentmaterials is made gradually. This idea leads to the concept offunctionally graded materials (FGMs).1 The mechanical prop-erties of FGMs vary continuously between several differentmaterials. Most research in this area is concerned with thermo-elastic and residual stress analysis. In many applications ofthese materials the vibrational characteristics are of great im-portance in addition to stress considerations. The vibration ofFGM cylindrical structures has been studied by a number ofresearchers.2–7 Different studies on the vibration of cylindricalshells made of a FGM based on Love’s and some other clas-sical shell theories have been done and usually Rayleigh-Ritzand finite element methods used for solving governing equa-tions.2–7

While the vibrational behaviour of a thick walled cylinder isof considerable engineering importance, all of the previouslydiscussed papers are mainly focused on cylindrical shells us-ing the classical equations of thin shell theories, except Chenet al.7 who investigated a thick hollow cylinder using equa-tions of piezoelasticity based on laminate model as opposedto classical shell theories. Classical or thin-shell theories arebased on the simplifying assumptions of Kirchhof-Love’s hy-pothesis. This omission makes the thin-shell theories highlyinadequate for the analysis of even slightly thick shells.8 Thehigher order shell theories are better than the thin-shell theo-ries for the analysis of slightly thick shells but are still inade-quate for the analysis of moderately thick shells. To analyzemoderately thick shells, the transverse normal stress and straincomponents, which are ignored in the higher ordered shell the-ories, have to be accounted for and only an analysis based onthe three-dimensional theory of elasticity would account for allthe transverse stress and strain components.8

Few studies have been conducted for thick hollow circularcylinders. They require a three-dimensional analysis, whichis based on the theory of elasticity. As a result, in the lit-eratures, the study of free vibrations of thick circular cylin-ders using three-dimensional theory of elasticity is relativelyscarce in comparison to the study of thin shells using othershell theories. Studies on shells based on three-dimensionaltheory of elasticity have been presented by some researchersfor infinitely long cylindrical shells.9–12 For finite-length thickcylindrical shells, different methods such as the finite elementmethod, series solution, and the Ritz energy method have beenused by some researchers for both solid and hollow homoge-neous cases.13–17 A three-dimensional energy formulation wasused by Liew et. al.18 to compute frequencies and developgraphical representation for three-dimensional mode shapesof a homogeneous hollow cylinder. Loy and Lam8 also pre-sented an approximate analysis by using a layerwise approachto study the vibration of thick circular cylindrical shells on thebasis of three-dimensional theory of elasticity. Buchanan andYii19 investigated the effect of symmetrical boundary condi-tions on the vibration of thick hollow cylinders using finite ele-ment method. Other researchers20–23 have also presented stud-ies using three-dimensional theory of elasticity based on theRayleigh-Ritz method for homogeneous and laminated cylin-ders.

Most of the referred studies considered ways of determiningthe frequencies of cylinders. However, very few of the au-thors give a description of the mode shapes of the thick cylin-ders. While mode shapes are also very important sources ofinformation for understanding and controlling the vibration ofa structure.26 Singal and Williams24 combined experimentalresults with a Ritz energy method of analysis to compare fre-quencies for free–free cylinders. They gave a description forthe mode shapes of thick- walled hollow cylinders and rings.Wang and Williams25 have studied extensively frequencies andmode shapes of finite length hollow cylinder using a commer-cial finite element code and three-dimensional block elementsfor their analysis. Singhal et al.26 presented theoretical and

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 151–160) https://doi.org/10.20855/ijav.2017.22.2460 151

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

experimental modal analysis by using a thick-walled circu-lar cylinder model to obtain its natural frequencies and modeshapes.

On the other hand, in the previous discussed literature, vi-bration analysis of moderately thick- walled hollow cylindersare limited to isotropic and laminated cylinders and function-ally graded thick hollow cylinders with finite length were notseen in the literatures. Also the functionally graded cylin-drical shells considered using thin shell theories. So, inves-tigation of a functionally graded thick hollow cylinder canbe of great importance. Additionally, in all of the discussedcases, the variation of volume fraction and properties of theFGMs are one-dimensional and the properties vary continu-ously from the inner surface to the outer one with a prescribedfunction. But a conventional functionally graded material mayalso not be so effective in some design problems since allouter or inner surfaces will have the same composition dis-tribution while in advanced machine elements, load distribu-tion may change in two or three directions.27 Therefore, ifthe FGM has two-dimensional dependent material propertiesmore effective material resistance can be obtained. Based onthis fact, a two-dimensional FGM whose material propertiesare bi-directionally dependent is introduced. Recently a fewauthors have investigated 2D-FGM especially its stress analy-sis.27–36

The author also has investigated the natural frequencies ofa thick hollow cylinder with finite length made of 2D-FGM.37

The influence of constituent volume fractions on natural fre-quencies was studied by varying the volume fractions of theconstituent metals and ceramics. Furthermore, the effects oflength and thickness of the cylinder on fundamental natu-ral frequency were considered in different types of 2D-FGMcylinder.

So far, investigation of mode shapes configuration of FGMthick finite cylinder has not been considered in previous stud-ies. Hence, in order to extend the previous studies and toinvestigate the mode shape configuration of an FGM thickhollow cylinder, the free vibration analysis and all types ofmode shapes of a thick hollow cylinder with finite lengthmade of 2D-FGM on the basis of three- dimensional theoryof elasticity has been considered in this study. The mate-rial properties of the cylinder are varied in the radial and ax-ial directions with power law functions. Effects of volumefraction distribution and FGM configuration on the naturalfrequencies and three-dimensional mode shapes of a simplysupported functionally graded thick hollow cylinder are ana-lyzed. The influence of constituent volume fractions is stud-ied by varying the volume fractions of the constituent met-als and ceramics. A functionally graded cylinder with two-dimensional gradation of distribution profile has been investi-gated as well as the one-dimensional gradation of material dis-tribution. Three-dimensional equations of motion are used andthe eigen value problem is developed based on Rayleigh-Ritzvariational method and all non- axisymmetric mode shapes areconsidered. The finite element method with graded materialcharacteristics within each element of the structure is used forthe solution. Using conventional finite element formulationssuch that the property field is constant within an individual ele-ment for dynamic problems leads to significant discontinuitiesand inaccuracies.38 These inaccuracies will be more significantin 2D-FGM cases. On the other hand, by using graded finiteelement in which the material property field is graded continu-

Figure 1. Axisymmetric cylinder with two dimensional material distribu-tions.37

ously through the elements, accuracy can be improved withoutrefining the mesh size.39, 40 Based on these facts the graded fi-nite element developed by the author41 is used for modeling ofthe present problem.

2. PROBLEM FORMULATION

In this section, the volume fraction distributions in the tworadial and axial directions are introduced. The 3D govern-ing equations of motion in cylindrical coordinates are obtainedand the graded finite element is used for modeling the non-homogeneity of the material.

2.1. Volume Fraction and MaterialDistribution in 2D-FGM Cylinder

In the conventional one dimensional functionally gradedcylinder, the cylinder’s material is graded through the radialdirection. The cylinder is made of a combined metal-ceramicmaterial for which the mixing ratio is varied continuously inthe r-direction from pure ceramic in the inner surface to puremetal in the outer surface, or vice versa. In such cases, thevolume fraction variation of the metal is proposed as a powerlaw relation.37 Using the rule of mixtures, the distribution ofmaterial properties is obtained.

Significant advances in fabrication and processing tech-niques have made it possible to produce FGMs using processesthat allow FGMs with complex properties and shapes, in-cluding two-and three- dimensional gradients using computer-aided manufacturing techniques. 2D-FGMs are usually madeby continuous gradation of three or four distinct materialphases that one or two of them are ceramics and the others aremetal alloy phases. The volume fractions of the constituentsvary in a predetermined composition profile. Now considerthe volume fractions of 2D-FGM at any arbitrary point in theaxisymmetric cylinder of internal radius ri, external radius ro,and finite length L shown in Fig. 1 In the present cylinder theinner surface is made of two distinct ceramics and the outersurface from two metals. The variables c1, c2, m1, and m2denote first ceramic, second ceramic, first metal and secondmetal, respectively.

The volume fraction of the first ceramic material is changedfrom 100% at the lower surface to zero at the upper surfaceby a power law function. Additionally, this volume fraction ischanged continuously from inner surface to the outer surface.The volume fractions of the other materials change similar tothe mentioned one in two directions. The function of volumefraction distribution of each material can be explained as37

Vc1(r, z) =

[1−

(r − rir0 − ri

)nr] [

1−( zL

)nz]

; (1a)

152 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 1. Basic constituents of the 2D-FGM cylinder.

Constituents Material E (Gpa) ρ (kg/m3)m1 Ti6Al4V 115 2715m2 Al 1100 69 4515c1 SiC 440 3210c2 Al2O3 150 3470

Vc2(r, z) =

[1−

(r − rir0 − ri

)nr] [( z

L

)nz]

; (1b)

Vm1(r, z) =

(r − rir0 − ri

)nr [1−

( zL

)nz]

; (1c)

Vm2(r, z) =

(r − rir0 − ri

)nr ( zL

)nz

; (1d)

where nr and nz are non-zero parameters that represent thebasic constituent distributions in r and z directions. Materialproperties at each point can be obtained by using the linear ruleof mixtures, in which a material property P at any arbitrarypoint (r, z) in the 2D-FGM cylinder is determined by linearcombination of volume fractions and material properties of thebasic materials as37

P = Pc1Vc1 + Pc2Vc2 + Pm1Vm1 + Pm2Vm2. (2)

The basic constituents of the 2D-FGM cylinder are pre-sented in Table 1. It should be noted that Poisson’s ratio isassumed to be constant through the body. This assumption isreasonable because of the small differences between the Pois-son’s ratios of basic materials.

2.2. Governing EquationsConsider a 2D-FGM thick hollow cylinder of internal ra-

dius, ri external radius ro, and finite length L. Because of theaxisymmetry of geometry and material distribution profile, co-ordinates r, z, and θ are used in the analysis. Neglecting bodyforces, the equations of motion in cylindrical coordinates areobtained as

∂σrr∂r

+∂τrz∂z

+∂τrθr∂θ

+σrr − σθθ

r= ρ(r, z)

∂2u

∂t2; (3a)

∂τrθ∂r

+∂σθθr∂θ

+∂τzθ∂z

+2τrθr

= ρ(r, z)∂2v

∂t2; (3b)

∂τrz∂r

+∂τzθr∂θ

+∂σzz∂z

+τrzr

= ρ(r, z)∂2w

∂t2; (3c)

where u, v, and w are radial, circumferential, and axial com-ponents of displacement respectively those are functions of(r, z, θ, t) and ρ(r, z) is the mass density that depends on r andz coordinates. The constitutive equations for FGM are writtenas

σij = [D(r, z)]εij; (4)

where the stress and strain components and the coefficients ofelasticity are

σij =[σrr σθθ σzz τzθ τrz τrθ

]; (5)

εij =[εrr εθθ εzz γzθ γrz γrθ

]; (6)

[D(r, z)] =E(r, z)

(1 + ν)(1− 2ν)1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 00 0 0 1−2ν

2 0 00 0 0 0 1−2ν

2 00 0 0 0 0 1−2ν

2

; (7)

where ν denotes the Poison’s ratio, which is uniform throughthe cylinder and E(r, z) is Young’s modulus that depends on rand z coordinates. The strain-displacement equations are:42

εr =∂u

∂r; (8a)

εθ =∂v

r∂θ+u

r; (8b)

εz =∂w

∂z; (8c)

γrz =∂u

∂z+∂w

∂r; (8d)

γzθ =∂v

∂z+∂w

r∂θ; (8e)

γrθ =∂u

r∂θ+∂v

∂r− v

r. (8f)

The cylinder is simply supported on its two end edges. Somechanical boundary conditions on upper and lower edges areassumed as

v(r, 0, θ, t) = v(r, L, θ, t) = w(r, 0, θ, t) = w(r, L, θ, t) = 0.(9)

A solution that satisfies the circumferential displacement anddefines a circular frequency is19

u(r, θ, z, t) = ψ1(r, z) cos(mθ)eiωt; (10a)

v(r, θ, z, t) = ψ2(r, z) sin(mθ)eiωt; (10b)

w(r, θ, z, t) = ψ3(r, z) cos(mθ)eiωt; (10c)

wherem is the circumferential wave number and ω is the circu-lar frequency. Also considering the circumferential symmetryof the cylinder about the coordinate θ, the displacement ampli-tude functions can be written asψ1(r, z), ψ2(r, z) andψ3(r, z).It is obvious that m = 0, which donates the axisymmetric vi-bration. Certain specified uniform boundary conditions alongthe two ends can be satisfied by choosing the displacement am-plitude functions properly.

In order to solve the governing equations the finite elementmethod with graded element properties is used. For this pur-pose, the variational formulation is considered. In conven-tional finite element formulations a predetermined set of mate-rial properties are used for each element such that the propertyfield is constant within an individual element. For modelinga continuously non-homogeneous material, the material prop-erty function must be discretized according to the size of ele-ments mesh. This approximation can provide significant dis-continuities. In addition, variation of material properties in

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 153

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

two directions such as the present problem makes this effectmore considerable. Based on these facts the graded finite ele-ment is strongly preferable for modeling of the present prob-lem. Hamilton’s principle for the present problem is

t2∫t1

δ(Π− T )dt = 0; (11)

where Π and T are potential energy and kinetic energy respec-tively. These functions and their variations are

T =1

2

∫∫∫vol

ρ(r, z)

((∂u

∂t

)2

+

(∂v

∂t

)2

+

(∂w

∂t

)2)dV ;

(12)

δT =

∫∫∫vol

ρ(r, z)

((∂2u

∂t2

)δu+

(∂2v

∂t2

)δv+

+

(∂2w

∂t2

)∂w

)dV ; (13)

Π =1

2

∫∫∫vol

σijT εijdV ; (14)

δΠ =1

2

∫∫∫vol

σijT δεijdV ; (15)

where V denotes the area and volume of the domain under con-sideration. Substituting Eqs. (12) to (15) in Hamilton’s princi-ple and applying side conditions, δu(t1) = δu(t2) = δv(t1) =δv(t2) = δw(t1) = δw(t2) = 0, by part integration have∫∫∫

vol

σijT δεijdV +

∫∫∫vol

ρ(r, z)

((∂2u

∂t2

)δu+

+

(∂2v

∂t2

)δv +

(∂2w

∂t2

)∂w

)dV = 0; (16)

The strain-displacement relations can be written as43

ε = [L]u; (17)

where [L] relates displacements into strain components. And

u =

uvw

. (18)

Four node tetrahedral element is used to discrete the domain.By taking the nodal values of u, v, and w as the degrees offreedom a linear displacement model can be assumed asuv

w

e

= [N ]Qe0eiωt; (19)

where [N ] is the matrix of assumed shape functions and satisfycertain specified boundary conditions and Qe0 is the nodaldisplacement vector of element. The matrix of interpolationfunctions corresponding to elements which derived in termsof global coordinates of nodes of elements its components are

three-dimensional and axisymmetric. Vector of nodal displace-ments (degrees of freedom) is

Qe =ui vi wi uj vj wj uk vk wk ul vl wl

T;

(20)

where subscripts i, j, k, l are related to four nodes of eachelement.

The cylinder will be divided into some brick-like subdivi-sions in radial, axial, and circumferential directions as wellas making a tetrahedral mesh through the use of brick sub-divisions. In this case 10*10*12 brick subdivisions producedthat leads to 2400 elements including 4356 degrees of free-dom. Applying Hamilton’s principle for each element, it canbe achieved as

δQeT∫∫∫

V e

[B(r, z, θ)]T [D(r, z)][B(r, z, θ)]dV

Qe++δQeT

∫∫∫V e

ρ(r, z)[N ]T [N ]dV

Qe = 0;

(21)

where V is the volume of element and [B] is the operationmatrix of strain-nodal displacement. In graded finite element,the interpolation function for the displacements within the el-ements and strain-displacement relations are the same as stan-dard conventional finite. In this way the constitutive relation is

σij = [D(r, z)]εij; (22)

where the components of [D(r, z)] could be explicit func-tions describing the actual material property gradient in whichE(r, z) is determined at each point through the element usingdistribution function of this property based on rule of mixturesas

E(r, z) = Ec1Vc1(r, z) + Ec2Vc2(r, z)+

+Em1Vm1(r, z) + Em2Vm2(r, z). (23)

Also, the mass density ρ(r, z) is, in general, a function ofposition as well as the mechanical properties. Therefore, in thegraded finite element the mass density distribution should beassigned into the element formulation as

ρ(r, z) = ρc1Vc1(r, z) + ρc2Vc2(r, z)+

+ρm1Vm1(r, z) + ρm2Vm2(r, z). (24)

Since δQe is the variational displacement of the nodal pointsand is arbitrary, it can be omitted from Eq. (23), and then thisequation can be written as

[M ]eQe+ [K]eQe = 0; (25)

where the characteristic matrices are given as

[K]e =

∫∫∫V e

[B(r, z, θ)]T [D(r, z)][B(r, z, θ)]dV ; (26)

[M ]e =

∫∫∫V e

[N(r, z, θ)]T [N(r, z, θ)]ρ(r, z)dV. (27)

154 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 2. Dimensionless frequencies φ = (ωH/π)√ρ/G of axisymmetric vibration for freely supported isotropic cylinders (m = 0, H/R = 0.4, ν = 0.3).

H/L Method φ1 φ2 φ3 φ4 φ5 φ6(Armenakas et al., 1969) 0.20495 0.34765 1.07205 1.82336 2.09588 3.00073

0.2 (Cheung and Wu, 1972) 0.20495 0.34765 1.07312 1.82688 2.10257 3.02850Present method 0.20492 0.34763 1.07309 1.82686 2.10254 3.02848

(Armenakas et al., 1969) 0.27540 0.67185 1.23591 1.76178 2.25224 2.999800.4 (Cheung and Wu, 1972) 0.27544 0.67188 1.23712 1.76587 2.25874 3.02742

Present method 0.27539 0.67182 1.23704 1.76580 2.25868 3.02735(Armenakas et al., 1969) 0.42022 0.98133 1.44860 1.75005 2.43159 3.00691

0.6 (Cheung and Wu, 1972) 0.42038 0.98145 1.45008 1.75433 2.43851 3.03437Present method 0.42032 0.98138 1.45001 1.75421 2.43842 3.03428

For finding the components of characteristic matrices, theintegral must be taken over the elements’ volume consideringEqs. (23) and (24). As [D(r, z)] and ρ(r, z) are not constant,these matrices are evaluated by numerical integration for eachelement.

Now by assembling the element matrices, the global matrixequation for the structure can be obtained as

[M ]Q+ [K]Q = 0. (28)

Once the finite element equations are established, Q =Q0e

iωt was substituted into Eq. (28) which resulted in an eigenvalue equation that can be solved using standard eigen valueextraction procedures. To get a better illustration of the modeshapes, the numerical results of the displacements relate toeach eigen value on each node were transferred into the propercoordinate system and the radial, tangential, and longitudinaldirections of the nodes determined.

3. NUMERICAL RESULTS AND DISCUSSION

To verify the present solution, as similar works to the presentwork are few, a finite length of a homogeneous thick cylinderthat can be found with the existing literature is used. A fi-nite element for axisymmetric elasticity is formulated directlyin the cylindrical coordinates to study the vibration of hol-low, isotropic, and homogeneous finite length cylinders andfrequencies are computed for free-free end boundary condi-tions in the reference19 and compared with the reference.16 Forsolving the aforementioned problem using the graded finite el-ement method developed here, we considered a thick hollowcylinder with freely supported end conditions in which the ma-terial distribution is uniform. Therefore, the volume fractionexponent and property coefficients in the 2D-FGM are takenas: nz = 0, nr = 0, Pc1 = Pc2 = Pm1 = Pm2 = P , whereP is a uniform material properties of the cylinder. Comparisonof the results for this case in Table 2 shows good agreementbetween the two methods again.

A thick hollow cylinder of inner radius ri = 0.5 m, outerradius ro = 1 m and length L = 2 m is considered. Effectsof volume fraction distribution on the natural frequencies andmode shapes configuration of a simply supported functionallygraded thick hollow cylinder are analyzed. The influence ofconstituent volume fractions is studied by varying the volumefractions of the constituent metals and ceramics. A function-ally graded cylinder with two-dimensional gradation of distri-bution profile has been investigated as well as the case wherethe axial power law exponent is assumed to be zero, i.e., nz =0, and the results of one-dimensional gradation of material dis-tribution can be obtained in the hollow cylinder. The basicmaterials are as explained in the previous section. Constituent

materials are two distinct ceramics and two distinct metals de-scribed in Table 1 and Poison’s ratio ν = 0.3. Volume fractionsof materials are distributed according to Eqs. (1a) to (1d). Vi-bration characteristics of cylinder for some different powersof material composition profile n r and n z are presented andcompared. Dimensionless frequency parameter is assumed as

Ω = (roω)

√ρ

G. (29)

The longitudinal and radial modes are uncoupled from puretorsional modes when the circumferential wave number istaken as m = 0.19 For symmetric boundary conditions, themode shapes are either symmetric or antisymmetric.

In order to investigate the effect of material distribution pro-file in the case of non-symmetrical modes, some selected modeshapes were computed for comparison, are shown in followingTables.

Three-dimensional antisymmetric vibrational modes relatedto non-zero values for circumferential wave numbers (m 6= 0)will be described in the following illustrations. Mode shapesare according to the classification of the modes of thick cylin-ders used by Wang and Williams.25 Axial bending modes,where the circumferential cross section segments bend oppo-sitely in the axial direction, and the radial motion with shear-ing modes; for this kind the cylinder no longer retains a con-stant cross sectional along its length. Circumferential in thesemodes means adjacent segmental elements expand or contractone by one in the circumferential direction. The median cir-cumferential length of an expanding segment becomes longerand the length of the contracting segment becomes shorter.Global modes for these modes is the thick cylinder can be con-sidered to behave as a simple beam vibrating in a transversedirection, a bar vibrating in torsion or as a rod vibrating in alongitudinal direction.

The variations of the mode shapes the circumferential wavenumbers m for different values of longitudinal mode numbersfor nr = 0.2 and nz = 0 is shown in Table 3. Although thedisplacements of each nodes of the cylinder are calculated, justsome cross sections of the cylinder on inner and outer radii areshown due to the nature of the illustration.

The variations of the natural dimensionless frequencies pa-rameter with the circumferential wave numbers, m for differ-ent values of radial power exponent while the axial power ex-ponent nz = 0 is shown in Fig. 2. It is clear that the effectof variation of radial power exponent is more considerable forhigher natural frequencies. The same results for nr = 5 andnz = 0 is shown in Table 4.

The effect of variation of axial power exponent is consideredin Tables 5 and 6. Mode shapes for two different axial powerexponents while the radial power exponent is zero (nr = 0)

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 155

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 3. Antisymmetric modes in 1D-FGM cylinder, nz = 0, nr = 0.2.

Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3

Table 4. Antisymmetric modes in 1D-FGM cylinder, nz = 0, nr = 5.

Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3

are shown in these tables. The same investigation for naturalfrequency is also indicated in Fig. 3. It is clear that effect ofaxial power exponent on the natural frequencies is insignificantwhen the radial power exponent is zero.

In order to further study the mode configuration due to thepower exponent, the 2D-FGM cylinder with both non-zero ex-ponents is considered in Tables 7 and 8.

Variation of natural frequencies with a circumferential wavenumber for different values of axial power exponent, while theradial power exponent is not zero (nr = 2) is shown in Fig. 4;

the influence of the value of nz can be seen here. It is clear thatas nz increased, the natural frequencies increased. The naturalfrequencies and the mode shapes of different kinds of modesvary with the changing material’s distribution profile.

It is clear from the results that the natural frequencies andmode shapes, are strongly influenced by the material compo-sition profile. The constituent volume fractions of the con-stituent materials affect antisymmetric mode shapes particu-larly in higher mode numbers. It should be noted that althoughthe manufacturing of multidimensional FGM may seem to

156 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 5. Antisymmetric modes in 1D-FGM cylinder, nr = 0, nz = 0.2.

Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3

Table 6. Antisymmetric modes in 1D-FGM cylinder, nr = 0, nz = 5.

Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3

be costly or difficult, these technologies are relatively new,processes such as three-dimensional printing (3DPTM) andLaser Engineering Net Shaping (LENS R©) can currently pro-duce FGMs with relatively arbitrary tree-dimensional grad-ing.33 With further refinement, FGM manufacturing meth-ods may provide designers with more control of the composi-tion profile of functionally graded components with reasonablecost.

4. CONCLUSIONS

Based on the three-dimensional mode shapes, a study onthe free vibration of simply supported thick hollow cylinderwith finite length made of 2D-FGM is presented. Materialproperties are graded in the thickness and longitudinal direc-tions of the cylinder according to a volume fraction powerlaw distribution. The equations of motion are based on three-dimensional elasticity theory and the graded finite elementmethod, which has some advantages to the conventional finite

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 157

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 7. Antisymmetric modes in 2D-FGM cylinder, nr = 0.5, nz = 5.

n=1 n=2 n=3 n=4

m=1

m=2

m=3

Table 8. Antisymmetric modes in 2D-FGM cylinder, nr = 5, nz = 0.5.

n=1 n=2 n=3 n=4

m=1

m=2

m=3

element method and is employed for the solution. The effectsof two-dimensional material distribution on the mode shapescharacteristics are considered and compared with conventionalone-dimensional FGM. Variation of natural frequencies andanti-symmetric three-dimensional mode shapes with circum-ferential wave number associated with the numerous values ofradial and axial power exponents are calculated for 1D- FGMcylinder as well as 2D-FGM cylinder. The study shows that the2D-FGM cylinder exhibits an interesting frequency and modeshape characteristics when the constituent volume fractions are

varied. Based on the achieved results, 2D-FGMs have a power-ful potential for designing and optimization of structures undermulti-functional requirements.

REFERENCES1 Koizumi, M. The concept of FGM Ceramic Transaction,

Functionally Graded Materials, 34, 3–10, (1993).

2 Loy, C. T., Lam, K. Y., and Reddy, J. N. Vibra-

158 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Figure 2. Natural frequencies for various radial power law exponents in 1D-FGM cylinder, nz = 0.37

Figure 3. Natural frequencies for various axial power law exponents in 1D-FGM cylinder, nr = 0.

tion of functionally graded cylindrical shells, Int Jour-nal of Mechanical Sciences, 41, 309–324, (1999).https://dx.doi.org/10.1016/s0020-7403(98)00054-x

3 Pradhan, S. C., Loy, C. T., Lam, K. Y., andReddy, J. N. Vibration characteristics of function-ally graded cylindrical shells under various boundaryconditions, Applied Acoustics, 61, 111–129, (2000).https://dx.doi.org/10.1016/s0003-682x(99)00063-8

4 Zhao, X., Ng, T. Y., and Liew K. M. Free vi-bration of two-side simply-supported laminated cylin-drical panels via the mesh-free kp-Ritz method, IntJournal of Mechanical Sciences, 46, 123–142, (2004).https://dx.doi.org/10.1016/j.ijmecsci.2004.02.010

5 Patel, B. P., Gupta, S. S., Loknath, M. S., andKadu, C. P. Free vibration analysis of functionallygraded elliptical cylindrical shells using higher-ordertheory, Composite Structures 69, 259–270, (2005).https://dx.doi.org/10.1016/j.compstruct.2004.07.002

6 Kadoli, R. and Ganesan, N. Buckling and free vibra-tion analysis of functionally graded cylindrical shells sub-jected to a temperature specified boundary condition,Journal of Sound and Vibration, 289, 450–480, (2006).https://dx.doi.org/10.1016/j.jsv.2005.02.034

Figure 4. Natural frequencies for various axial power law exponents in 2D-FGM cylinder, nr = 2.

7 Chen, W. Q., Bian, Z. G., Lv, C. F., and Ding, H. J. 3Dfree vibration analysis of a functionally graded piezoelec-tric hollow cylinder filled with compressible fluid, Inter-national Journal of Solids and Structures, 41, 947–964,(2004). https://dx.doi.org/10.1016/j.ijsolstr.2003.09.036

8 Loy, C. T. and Lam, K. Y. Vibration of thick cylindricalshells on the basis of three-dimensional theory of elasticity,Journal of Sound and vibration, 226 (4), 719–737, (1999).https://dx.doi.org/10.1006/jsvi.1999.2310

9 Greenspon, J. E. Flexural vibrations of a thick-walledcylinder according to the exact theory of elasticity,Journal of Aerospace Sciences, 27, 1365–1373, (1957).https://dx.doi.org/10.2514/8.8370

10 Gazis, D. C. Three-dimensional investigation of the prop-agation of waves in hollow circular cylinders, Journal ofthe Acoustical Society of America, 31, 568–578, (1959).https://dx.doi.org/10.1121/1.1907754

11 Nelson, R. B., Dong, S. B., and Kalra R. D. Vibra-tions and waves in laminated orthotropic circular cylin-ders, Journal of Sound and Vibration, 18, 429–444, (1971).https://dx.doi.org/10.1016/0022-460x(71)90714-0

12 Armenakas, A. E., Gazis, D. S., and Herrmann G.Free vibrations of Circular Cylindrical Shells, PergamonPress, Oxford, (1969). https://dx.doi.org/10.1016/b978-0-08-011732-4.50012-x

13 Gladwell, G. and Vijay, D. K. Natural frequen-cies of free finite-length circular cylinders, Journalof Sound and Vibration, 42 (3), 387–397, (1975).https://dx.doi.org/10.1016/0022-460x(75)90252-7

14 Gladwell, G. and Vijay, D. K. Vibration analysis of ax-isymmetric resonators, Journal of Sound and Vibration,42 (2), 137–145, (1975). https://dx.doi.org/10.1016/0022-460x(75)90211-4

15 Hutchinson, J. R. and El-Azhari, S. A. Vibrations of freehollow circular cylinders, Journal of Applied Mechanics,53, 641–646, (1986). https://dx.doi.org/10.1115/1.3171824

16 So, J. and Leissa, A. W. Free vibrations of thick hollowcircular cylinders from three-dimensional analysis, TransASME, Journal of Vibrations and Acoustics, 119, 89–95,(1997). https://dx.doi.org/10.1115/1.2889692

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 159

M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

17 So, J. Three dimensional vibration analysis of elastic bodiesof revolution, PhD dissertation, The Ohio State University,Columbus, (1993).

18 Liew, K. M., Hung, K. C., and Lim, M. K. Vibration ofstress-free hollow cylinders of arbitrary cross section, TransASME, Journal of Applied Mechanics, 62, 718–724, (1995).https://dx.doi.org/10.1115/1.2897005

19 Buchanan, G. R. and Yii, C. B. Y., Effect of symmet-rical boundary conditions on the vibration of thick hol-low cylinders, Applied Acoustics, 63, 547–566, (2002).https://dx.doi.org/10.1016/s0003-682x(01)00048-2

20 Cheung, Y. K. and Wu, C. I., Free vibrations of thick, lay-ered cylinders having finite length with various boundaryconditions, Journal of Sound and Vibration, 24, 189–200,(1972). https://dx.doi.org/10.1016/0022-460x(72)90948-0

21 Soldatos, K. P. and Hadjigeorgiou, V. P. Three-dimensionalsolution of the free vibration problem of homoge-neous isotropic cylindrical shells and panels, Jour-nal of Sound and Vibration, 137, 369–384, (1990).https://dx.doi.org/10.1016/0022-460x(90)90805-a

22 Jiang, X. Y. 3-D vibration analysis of fiber rein-forced composite laminated cylindrical shells, Jour-nal of Vibration and Acoustics, 119, 46–51, (1997).https://dx.doi.org/10.1115/1.2889686

23 Ye, J. Q. and Soldatos, K. P. Three-dimensional vibrationsof cross-ply laminated hollow cylinders with clamped edgeboundaries, Journal of Vibration and Acoustics, 119, 317–323, (1997). https://dx.doi.org/10.1115/1.2889726

24 Singal, R. K. and Williams, K. A theoretical and experimen-tal study of vibrations of thick circular cylindrical shellsand rings, Trans ASME, Journal of Vibrations, Acoustics,Stress, and Reliability in Design, 110, 533-537, (1988).https://dx.doi.org/10.1115/1.3269562

25 Wang, H. and Williams, K. Vibrational modesof thick cylinders of finite length, Journal ofSound and Vibration, 191 (5), 955–971, (1996).https://dx.doi.org/10.1006/jsvi.1996.0165

26 Singhal, R. K., Guan, W., and Williams, K. Modal anal-ysis of a thick-walled circular cylinder, Mechanical Sys-tems and Signal Processing, 16 (1), 141–153, (2002).https://dx.doi.org/10.1006/mssp.2001.1420

27 Nemat-Alla, M. Reduction of thermal stresses by de-veloping two dimensional functionally graded materi-als, Int. J. Solids Structures, 40, 7339–7356, (2003).https://dx.doi.org/10.1016/j.ijsolstr.2003.08.017

28 Dhaliwal, R. S. and Singh, B. M. On the theory of elasticityof a non-homogeneous medium, J Elasticity, 8, 211–219,(1978). https://dx.doi.org/10.1007/bf00052484

29 Clements, D. L. and Budhi, W. S. A boundary elementmethod for the solution of a class of steady-state prob-lems for anisotropic media, Heat Transfer, 121, 462–465,(1999). https://dx.doi.org/10.1115/1.2826000

30 Abudi, J. and Pindera, M. J. Thermoelastic theory forthe response of materials functionally graded in two di-rections, Int. J. Solids Structures, 33, 931–966, (1996).https://dx.doi.org/10.1016/0020-7683(95)00084-4

31 Cho, J. R. and Ha, D.Y. Optimal tailoring of 2D volume-fraction distributionsfor heat-resisting functionally gradedmaterials using FDM, Comput Method Appl Mech Eng,191, 3195–3211, (2002). https://dx.doi.org/10.1016/s0045-7825(02)00256-6

32 Hedia, H. S., Midany, T. T., Shabara, M. N., andFouda N. Development of cementless metal-backed ac-etabular cup prosthesis using functionally graded ma-terial, Int J Mech Mater Des, 2, 259–267, (2005).https://dx.doi.org/10.1007/s10999-006-9006-y

33 Goupee, A. J. and Vel, S. Optimization of natu-ral frequencies of bi-directional functionally gradedbeams, Struct Multidisc Optim, 32 (6), (2006).https://dx.doi.org/10.1007/s00158-006-0022-1

34 Asgari, M., Akhlaghi M., and Hosseini S. M. Dy-namic Analysis of Two-Dimensional Functionally GradedThick Hollow Cylinder with Finite Length under Im-pact Loading, Acta Mechanica, 208, 163–180, (2009).https://dx.doi.org/10.1007/s00707-008-0133-4

35 Asgari, M. and Akhlaghi M. Thermo-Mechanical Analysisof 2D-FGM Thick Hollow Cylinder using Graded Finite El-ements, Advances in Structural Engineering, 14 (6), (2011).https://dx.doi.org/10.1260/1369-4332.14.6.1059

36 Asgari, M. and Akhlaghi, M. Transient thermal stresses intwo-dimensional functionally graded thick hollow cylinderwith finite length, Arch Appl Mech, 80, 353–376, (2010).

37 Asgari M. and Akhlaghi M. Free Vibration Anal-ysis of 2D-FGM Thick Hollow Cylinder Based onThree Dimensional Elasticity Equations, EuropeanJournal of Mechanics A/Solids, 30, 72–81, (2011).https://dx.doi.org/10.1016/j.euromechsol.2010.10.002

38 Santare, M. H., Thamburaj, P., and Gazonas, A. Theuse of graded finite elements in the study of elasticwave propagation in continuously non-homogeneous ma-terials, Int J Solids Structures, 40, 5621–5634, (2003).https://dx.doi.org/10.1016/s0020-7683(03)00315-9

39 Santare, M. H. and Lambros, J. Use of a graded finiteelement to model the behavior of non-homogeneousmaterials, J Appl Mech, 67, 819–822, (2000).https://dx.doi.org/10.1115/1.1328089

40 Kim, J. H. and Paulino, G. H. Isoparametric gradedfinite elements for nonhomogeneous isotropic and or-thotropic materials, J Appl Mech, 69, 502–514, (2002).https://dx.doi.org/10.1115/1.1467094

41 Asgari, M. and Akhlaghi M. Thermo-mechanicalStresses in Two-Dimensional Functionally GradedThick Hollow Cylinder with Finite Length, Ad-vances in Structural Engineering, 14 (6), (2012).https://dx.doi.org/10.1177/0021998315622051

42 Boresi, B., Peter, A., and Ken, P. Elasticity in EngineeringMechanics, 2nd ed. Wiley, New York, (1999).

43 Zienkiewicz, O. C. and Taylor, R. L. The Finite elementmethod, VII: Soloid mechanics, 5th edition, Butterworth-Heinemann, Oxford, (2000).

160 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017