free vibration analysis of rotating functionally graded annular...

Scientia Iranica B (2018) 25(2), 728{740

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Free vibration analysis of rotating functionally gradedannular disc of variable thickness using generalizeddi�erential quadrature method

M.H. Jalalia, B. Shahriarib, O. Zargarc, M. Baghanic;�, and M. Baniassadic

a. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran.b. Department of Mechanical and Aerospace Engineering, Malek-Ashtar University of Technology, Isfahan, Iran.c. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.

Received 5 June 2016; received in revised form 22 October 2016; accepted 20 February 2017

KEYWORDSAnnular plate;Functionally gradedmaterial;Generalizeddi�erential quadraturemethod;Natural frequency.

Abstract. In this paper, free vibration analysis of rotating annular disc made ofFunctionally Graded Material (FGM) with variable thickness is presented. Elasticitymodulus, density, and thickness of the disc are assumed to vary radially according to apower low function. The natural frequencies and critical speeds of the rotating FG annulardisc of variable thickness with two types of boundary conditions are obtained employing thenumerical Generalized Di�erential Quadrature Method (GDQM). The boundary conditionsconsidered in the analysis are both edges clamped (C-C): the inner edge clamped andouter edge free (C-F). The inuence of the graded index, thickness variation, geometricparameters, and angular velocity on the dimensionless natural frequencies and criticalspeeds is demonstrated. It is shown that we have higher critical speed and natural frequencyusing a plate with a convergent thickness pro�le, and lower critical speed using a divergentthickness pro�le. It is found that the increase in the ratio of inner-outer radii could increasethe critical speed of the FG annular disk. The results of the present work could improvethe design of the rotating FG annular disk in order to avoid resonance condition.© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Functionally Graded Materials (FGM) are a new typeof composite materials, which have gained considerableattention in various industries recently. Due to thespecial advantages of using composite materials, somemechanical structures, such wind turbine blades, aregrowing in size and becoming structurally more e�-cient [1-3]. Mechanical properties in plates made ofFG materials vary continuously in one or more direc-

*. Corresponding author. Tel.: +98 21 6111 9921;Fax: +98 21 6600 0021E-mail address: [email protected] (M. Baghani)

doi: 10.24200/sci.2017.4325

tions [4,5]. Since circular plates are extensively used inmany engineering applications, many researchers haveperformed vibration analysis of these plates.

Lamb and Southwell [6] investigated vibrationsof spinning uniform disk for the �rst time. Theyobtained the natural frequencies of rotating, homoge-nous, constant thickness circular disk using exactsolution. Southwell [7] extended the Lamb's workand analyzed the e�ects of rotation on the vibrationof uniform homogeneous circular disk, more deeply.Deshpande and Mote [8] presented a model for in-plane vibration of rotating thin disc. Their modelaccounted for the sti�ening of the disk due to theradial expansion resulting from its rotation. Bauerand Eidel [9] obtained the lower approximate naturalfrequencies of a spinning circular plate. In their paper,

M.H. Jalali et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 728{740 729

the plate was assumed uniform and their analysis wasperformed for clamped, guided and simply supportedboundary conditions. Lee and Ng [10] used theassumed modes method to formulate the equations ofmotion of rotating homogeneous annular plates. Theyassumed the thickness of the plate to vary linearly andexponentially and obtained the natural frequencies andcritical speeds of vibration modes consisting of radialnodal lines without any nodal circle.

Singh and Saxena [11] used Rayleigh-Ritz methodto obtain the �rst four natural frequencies, modeshapes, and nodal radii of a circular plate. Theyconsidered the thickness variation of the plate to beexponential in radial direction and the material of theplate to be homogeneous. Taher et al. [12] analyzed thevibration of circular and annular plates with variablethickness and combined boundary conditions. In theirpaper, a 3D elasticity theory was used to obtain theequations, and the plates were assumed to have linearand nonlinear thickness variations.

Kermani et al. [13] used Di�erential QuadratureMethod (DQM) to solve the equations of motion ofrotating FG annular plates with constant thickness.They assumed the variation of the elasticity modulusand density of the plate to be exponential in radialdirection. Horgan and Chan [14] investigated thestress in rotating functionally graded isotropic linearlyelastic disks. The purpose of their paper was mainlydiscussing the e�ect of inhomogeneity on the stressin the rotating solid and annular FG disks. Nieand Batra [15] investigated the stress of the isotropic,linear thermo elastic, and incompressible FG rotatingdisks of variable thickness. They solved the ordinarydi�erential equation analytically and also numericallyusing DQ method. Mohammadsalehi et al. [16] inves-tigated the vibrations of variable thickness rectangularviscoelastic nano plate based on nonlocal theory. Theyused DQ method to solve the equations of motionand evaluated the e�ect of small-scale e�ect on thevibration features. Asghari and Ghafoori [17] presenteda general semi-analytical solution for investigatingelastic response of rotating solid and annular rotatingFG disks. They showed that although the plane-stresssolution satis�es all the governing three-dimensionalequations of motion and boundary conditions, it fails togive a compatible three-dimensional strain �eld. Pengand Li [18] investigated the inuence of orthotropyand gradient on the elastic response of the rotatingannular functionally graded polar orthotropic disks.Bahaloo et al. [19] investigated the vibration and stressof a rotating functionally graded circular disk withan open circumferential crack. Khorasani and Hut-ton [20] investigated the vibration of rotating annulardisks which were elastically restrained with rigid-bodytranslational degrees of freedom. They discussed thee�ect of rotating speed on the natural frequencies of

the disk and also investigated the stability of the disksthoroughly. Guven and C�elik [21] investigated thevibration of rotating functionally graded solid disks.They used Rayleigh-Ritz method to solve the equationsof motion and considered the plate to be isotropic withconstant thickness.

In the past few years, DQM has been appliedextensively to solving engineering problems. Comparedto the other numerical methods, the DQ method canlead to almost accurate results using a considerablysmaller number of grid points and, hence, requiringrelatively little computational e�ort [22,23]. The GDQmethod is an improvement of the DQ method, espe-cially for solving higher order di�erential equations,which is more computationally e�cient and accurate.Unlike the DQ method, the GDQ method considers ageneral situation, where the derivatives of a functionare approximated using a linear weighted sum of allthe functional values and also some derivatives of thefunctional values [23,24]. From the above discussionon the previous papers in the literature and to thebest of the authors' knowledge, the vibration analysisof rotating functionally graded thin disk with variablethickness has not been investigated so far. In thispaper, the generalized di�erential quadrature methodis used for solving the vibration equations of motionof rotating FG annular plates with variable thickness.The variation of the Young's modulus, density, andthickness of the disk in radial direction are assumedto be graded by a power-low function, and the �rstdimensionless natural frequency and dimensionless crit-ical speeds are obtained. The convergence of themethod is shown by the convergence diagrams, andthe boundary conditions of the disk are assumed tobe Clamped-Clamped (C-C) and Clamped-Free (C-F).The e�ects of the graded index, thickness variation,angular velocity, and geometric parameters on the �rstdimensionless natural frequencies and critical speedsare evaluated.

2. Governing equations for vibration analysis

An annular FG plate with outer radius, a, inner radius,b, thickness, h, and outer surface thickness, h0, which isrotating with angular velocity, ~!, is shown in Figure 1.Thickness h can be variable in the radial direction.Theequation of motion for transverse vibration of a ro-tating disc of variable thickness with variable materialproperties can be written as follows [25]:

Dr4w + dDdr

�2@@r

(r2w)

+1r

��@2w@r2

+1r@w@r

+1r2@2w@�2

��

730 M.H. Jalali et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 728{740

Figure 1. FG Rotating annular plate with variablethickness.

+d2Ddr2

�@2w@r2

+ ��

1r@w@r

+1r2@2w@�2

�� 1r@@r

�hr�r

@w@r

�� 1r2h��

@2w@�2

+�h@2w@t2

=0;(1)

where w is the transverse deection, D is the exuralrigidity of the disk, D = Eh

3

12(1��2) r2 = @2@r2 + ( 1r ) @@r +1r2

@2@�2 is the Laplacian operator, and �r and �� are

the radial and circumferential stresses, respectively,and are related to radial displacement by the followingformula:

�r =E

1� �2�dudr

+�ur

�;

�� =E

1� �2��dudr

+ur

�; (2)

where E is the Young's modulus and � is poison ratioassumed to be constant. Kirsho� strain-displacementrelations lead to the expressions below:

"r =dudr; "� =

ur; r� = 0; (3)

where u is the radial displacement. The Young'smodulus, density, and thickness of the disc are assumedto vary in radial direction by the following equations:

E = E0� ra

�n; (4)

� = �0� ra

�n; (5)

h = h0� ra

�m1; (6)

where n is the graded index, and m1 is the parameterfor the variation of the thickness in the radial direction.Thus:

D =E0h30rn+3m1

12(1� v2)an+3m1 : (7)The following boundary conditions must be satis�ed.

Clamped-Clamped (C-C):

r = b; w = 0;@w@r

= 0; (8)

r = a; w = 0;@w@r

= 0: (9)

Clamped-Free (C-F):

r = b; w = 0;@w@r

= 0; (10)

r = a;

Mr = �D�@2w@r2

+ ��

1r@w@r

+1r2@2w@�2

��= 0;

Vr =�D�@3w@r3

+1r2@2w@r2� 1r2@w@r

+2� �r2

@3w@r@�2

� 3� �r3

@2w@�2

�� dD

dr

�@2w@r2

+ ��

1r@w@r

+1r2@2w@�2

��= 0: (11)

The transverse deection can be written as follows:

w = W (r) cos(m�)ei!t; (12)

where m is the circumferential wave number, and !is the natural frequency. By substituting Eqs. (5)-(7),(4), and (12) into Eq. (1), the following equation ofmotion in terms of W (r) is obtained:

r2m1d4Wdr4

+P1d3Wdr3

+P2d2Wdr2

+P3dWdr

+P4W =0;(13)

where:

P1 = 2r2m1�1(n+ 3m1 + 1);

P2 =r2m1�2(�1+(n+3m1)(1+�+n+3m1)�2m2)

� 12a2m1h20

�dudr

+�ur

�;

P3 =r2m1�3�(n+3m1�1)(�1+�(n+3m1)�2m2)�

� 12a2m1h20

�(n+ 1 +m1)

�1rdudr

+�ur2

�+d2udr2� �ur2

+�rdudr

�;


P4 =r2m1�4(m4 �m2(4 + (n+ 3m1)(�3+ �(n+ 3m1 � 1))))

+12m2a2m1h20r2

��dudr

+ur

�� 12�0(1�v2)a2m1

E0h20!2:(14)

In addition, the boundary equations become:


r = b; W = 0;dWdr

= 0; (15)

r = a; W = 0;dWdr

= 0: (16)

Clamped-Free (C-F):

r = b; W = 0;dWdr

= 0; (17)

r = a;d2Wdr2

+ ��

1rdWdr� m2

r2W�

= 0;

d3Wdr3

+1 + n+ 3m1

rd2Wdr2

+��1�m2(2� �) + (n+ 3m1)�

r2

�dWdr

+�m2(3� �)� (n+ 3m1)m2�

r2

�W = 0:

(18)

Introducing the following dimensionless parameters,the motion and boundary equations can be recast ina dimensionless form as follows:

R =ra; �W =

Wh0; �u =

uh0;

=!a2

h0

s12�0(1� �2)

E0;

~ =~!a2

h0

s12�0(1� �2)

E0: (19)

The dimensionless equation of motion is:

R2m1d4 �WdR4

+ �P1d3 �WdR3

+ �P2d2 �WdR2

+ �P3d �WdR

+ �P4 �W =0;(20)

in which:

�P1 = 2R2m1�1(n+ 3m1 + 1);

�P2 =R2m1�2(�1+(n+3m1)(1+�+n+3m1)�2m2)

� 12ah0

�d�udR

+��uR

�;

�P3 =R2m1�3((n+3m1�1)(�1+�(n+3m1)�2m2));

�12ah0

�(n+ 1 +m1)

�1Rd�udR

+�R2

�u�

+d2�udR2

� ��uR2

+�Rd�udR

�;

�P4 =R2m1�4(m4 �m2(4 + (n+ 3m1)(�3

+ �(n+ 3m1 � 1)))) + 12m2a

h0R2

��d�udR

+�uR

�� 2; (21)

and the dimensionless boundary equations are:


R = b=a; �W = 0;d �WdR

= 0; (22)

R = 1; �W = 0;d �WdR

= 0: (23)

Clamped-Free (C-F):

R = b=a; �W = 0;d �WdR

= 0; (24)

R = 1;d2 �WdR2

+ ��

1Rd �WdR� m2R2

�W�

= 0;

d3 �WdR3

+1 + n+ 3m1

Rd2 �WdR2

+��1�m2(2� �) + (n+ 3m1)�

R2

�d �WdR

+�m2(3��)�(n+3m1)�m2

R3

��W =0: (25)

In order to solve the equation of motion (Eq. (20)) andobtaining the dimensionless natural frequency (!), theGDQ method is utilized.

3. Generalized Di�erential QuadratureMethod (GDQM)

In the numerical GDQ method, the solution domain isdivided into points Ri (i = 1; 2; � � � ; N) and derivativesof a function with the weighted summation of thatfunction [22,23,26]. The GDQR expression for the


fourth-order boundary value di�erential equation hasbeen used as follows [23]:

�W (S)(Ri) =ds �W (Ri)dRs

=N+2Xj=1

E(S)ij Uj;

i = 1; 2; � � � ; N; (26)where:

fU1; U2; � � �; UN+2g=n

�W1; �W(1)1 ; �W2; � � �; �WN ; �W (1)N

o;

and �Wj is the function value at point j, and �W(1)1 and

�W (1)N are the �rst-order derivatives of the dimensionlessdisplacement function at the �rst and Nth points [23].E(S)ij are the Sth order Weighting coe�cients at pointsRi. The GDQR explicit weighting coe�cients havebeen derived in [22,23] and used directly in this paper.

In order to discretize the solution interval, twodiscretization schemes (I and II) are used in this paper.

I) Equally spaced points;II) Chebyshev-Gauss-Lobatlo discretization.

Ri =1a

�12

�1� cos (i� 1)�

N � 1�

(a� b) + b�;

i = 1; 2; � � � ; N: (27)Appling GDQM to Eq. (20), the GDQ form of theequation of motion can be obtained as follows:

HijUj = 2 �Wi; i = 2; 3; � � � ; N � 1; (28)where is a (N)�(N+2) matrix. The (N�2) equationsare added by four boundary equations, and (N + 2)algebraic equations are constructed. An assembledform is constituted by arranging them [23]:24[Sbb] [Sbd]

[Sdb] [Sdd]

35�UbUd

�=�

0

2Ud

�; (29)

where:

Ub =

2664�W1

�W (1)1�WN�W (1)N

3775 ; Ud =264 �W2...

�WN�1

375 : (30)By matrix substructuring and manipulation, one ob-tains a standard eigenvalue problem [23]:

[S]Ud = 2Ud; (31)

where:

[S] = [Sdd]� [Sdb] [Sbb]�1 [Sbd] ;with an order (N � 2)� (N � 2).

The dimensionless natural frequency () can beobtained by solving the eigenvalue problem (Eq. (31)).

3.1. Application of boundary conditions inGDQM

The following boundary equations are the GDQ formof Eqs. (45) and (46) for (C-C) boundary conditions.

Clamped-Clamped (C-C):�W1 = 0; U1 = 0;

�W (1)1 = 0; U2 = 0;

�WN = 0; UN+1 = 0;

�W (1)N = 0; UN+2 = 0: (32)

By separating the boundary and domain coe�cientsin Eq. (28), we have:

Hi1U1 +Hi2U2 +HiN+1UN+1 +HiN+2UN+2

+NXj=3

HijUj = 2 �Wi i = 2; � � � ; N � 1:(33)

Thus, the following submatrices are obtained:

Sbb = [I]4�4;

Sbd = [0]4�N�2;

Sdb =�Hi1 Hi2 HiN+1 HiN+2

�;

i = 2; � � � ; N � 1;Sdd=Hij i=2; � � � ; N�1; j=3; � � � ; N: (34)

Therefore, S = Sdd and the eigenvalues of matrixSdd are the dimensionless natural frequencies of (C-C) plate.

The following equations are the GDQ form ofEqs. (27) and (35) for (C-F) boundary conditions.Clamped- Free (C-F):

�W1 = 0; U1 = 0;

�W (1)1 = 0; U2 = 0;

N+2Xj=1

E(2)NjUj +�RN

�W (1)N � �m2

R2N�WN = 0;

N+2Xj=1

E(3)NjUj +1 + n+ 3m1

RN

N+2Xj=1

E(2)NjUj

+��1�m2(2� v) + (n+ 3m1)�

R2N

��W (1)N

+�m2(3�N)� (n+ 3m1)�m2

R3N

��WN = 0:

(35)


Separating the boundary and domain coe�cients inEq. (35) and in light of Eq. (33), the followingsubmatrices are found:

Sbb=

26641 0 0 00 1 0 0

E(2)N1 E(2)N2 E

(2)NN+1� �m2R2N E

(2)NN+2+

�RN

Sbb1 Sbb2 Sbb3 Sbb4

3775 ;where:

Sbb1 = E(3)N1 +

1 + n+ 3m1RN

E(2)N1;

Sbb2 = E(3)N2 +

1 + n+ 3m1RN

E(2)N2 ;

Sbb3 =E(3)NN+1 +

1 + n+ 3m1RN

E(2)NN+1

+m2(3� �)� (n+ 3m1)m2�

R3N;

Sbb4 =E(3)NN+2 +

1 + n+ 3m1RN

E(2)NN+2

+�1�m2(2� �) + (n+ 3m1)�

R2N; (36)

and:

Sbd =

24[0]2�4Sbd3Sbd4

35 ; (37)where Sbd3 = E

(2)Nj j = 3; � � � ; N and Sbd4 = E(3)Nj +

1+n+3m1RN E

(2)Nj j = 3; � � � ; N . In addition, Sdb and

Sdd are the same as Sdb and Sdd in (C-C) boundaryconditions. By obtaining the dimensionless naturalfrequency (), the natural frequency (!) can becalculated. The obtained natural frequency is thenatural frequency in the rotating (non-inertial) co-ordinate system, attached to the rotating plate. Foreach rotating plate in the rotating coordinate system,there are two corresponding natural frequencies inthe stationary (inertial) coordinate system. Thesenatural frequencies are obtained as follows [13]:

!f = ! +m~!;

!b = ! �m~!; (38)where !f and !b are the forward and backwardnatural frequencies, respectively. Whenever !b isequal to zero, the critical speed might occur.

4. Results

Solving the eigenvalue problem of Eq. (31), the di-mensionless natural frequency can be obtained fordi�erent boundary conditions. The Young's modulusand density of the plate at the outer surface are

assumed to be 380 GPa and 3800 kg/m3, respectively,and the poison ratio is assumed to be 0.3 and constant.The considered material properties are the same asthose in [13] in order to validate our results comparedto those in other papers in the literature.

For the discretization of the plate in the radialdirection, two discretization methods are used: Firstequally spaced discretization (I); second, using Eq. (27)(II). Figure 2 shows the convergence diagram for the�rst dimensionless natural frequency using the �rstdiscretization method (equally spaced) and for (C-C) boundary conditions. As can be seen from the�gure, the results could not change appreciably ifthe number of discrete points (N) increases between15 and 25. Figure 3 shows the same diagram, butfor Chebyshev-Gauss-Labatlo discretization (Eq. (27)).Figures 4 and 5 show the convergence diagram for(C-F) boundary conditions. It can be seen from the�gures that the convergence of the method in bothdiscretization methods is guaranteed. It should benoted that in order to obtain the out-of-plane vibration,�rst, the radial displacement should be calculated. Inthis paper, the radial displacement was �rst obtainedusing a numerical method, and the results of that wereused in order to obtain the out-of-plane vibration.

In order to validate our results, Table 1 comparesthe �rst dimensionless natural frequencies obtained

Figure 2. Convergence diagram for (C-C) boundarycondition (discretization method I).

Figure 3. Convergence diagram for (C-C) boundarycondition (discretization method II).


Figure 4. Convergence diagram for (C-F) boundarycondition (discretization method I).

Figure 5. Convergence diagram for (C-F) boundarycondition (discretization method II).

from our study with those obtained in [13]. In thistable, discretization scheme is equally spaced, rotatingspeed is assumed to be zero, and the disc is assumedto be homogenous (n = 0) with the (C-C) boundary

conditions. In Table 2, the same results are presented,but obtained from discretization II.

Tables 3 and 4 present the �rst dimensionlessnatural frequency for (C-F) boundary conditions us-ing discretization I and discretization II, respectively.From Tables 1-4, we can conclude the convergence ofthe results, because the results are not a�ected sensiblyby increasing (N) From 15 to 25. Also, it can validateour results, because there is good agreement betweenthe present results and those obtained in [13]. Inaddition, in order to validate our results compared withthose of rotating disks in the literature, the values ofthe �rst dimensionless natural frequency of the rotatinghomogeneous disk with (C-F) boundary conditionsare presented for various inner-to-outer radii rations;thickness-to-outer radius ratio is presented in Table 5.As seen, the increase of thickness-to-outer radius ratiosdecreases the dimensionless natural frequencies, butthe increase of inner-to-outer radii ratio increases thedimensionless natural frequencies. As obvious, theresults are in good agreement with those in [13].

Figure 6 shows the variation of the �rst dimen-sionless natural frequency with the variation of thegraded index for di�erent thickness pro�les for platewith clamped edges. As can be seen, the increase invalue of m1 from {1 to 1 leads to the decrease in thevalue of the natural frequency. Figure 7 shows the sameresult, but for (C-F) boundary conditions.

Figure 8 shows the variation of dimensionlessforward and backward natural frequencies and that ofthe dimensionless rotating speed for a rotating uniformdisk (m1 = 0) with clamped edges. In this �gure

Table 1. The lowest dimensionless natural frequency against di�erent numbers of nodal circles for plate with (C-C) edges(n = 0, m1 = 0, h0=a = 0:001, and b=a = 0:2) (discretization method I).

m 0 1 2 3 4 5

Present N = 15 34.609 36.203 41.822 52.619 68.360 87.907Ref. [13] N = 15 34.609 36.104 41.821 53.389 70.256 90.854Present N = 20 34.609 36.175 41.820 52.826 68.878 88.712Ref. [13] N = 20 34.609 36.103 41.820 53.388 70.256 90.855Present N = 25 34.609 36.160 41.819 52.946 69.175 89.172Ref. [13] N = 25 34.609 36.103 41.820 53.388 70.256 90.854

Table 2. The lowest dimensionless natural frequency against di�erent numbers of nodal circles for plate with (C-C) edges(n = 0, m1 = 0, h0=a = 0:001, and b=a = 0:2) (discretization method II).

m 0 1 2 3 4 5



Table 3. The lowest dimensionless natural frequency against di�erent numbers of nodal circles for plate with (C-F) edges(n = 0, m1 = 0, h0=a = 0:001, and b=a = 0:2) (discretization method I).

m 0 1 2 3 4 5


Table 4. The lowest dimensionless natural frequency against di�erent numbers of nodal circles for plate with (C-F) edges(n = 0, m1 = 0, h0=a = 0:001, and b=a = 0:2) (discretization method II).

m 0 1 2 3 4 5


Table 5. The lowest dimensionless natural frequency versus di�erent inner-to-outer radii ratios and di�erentthickness-to-outer radius ratios for the rotating functionally graded annular plate with clamped inner edge and free outeredge (m = 0, and ~! = 1000 rpm).

b=a

h0=a0.1 0.2 0.3 0.4

Present Ref. [8] Present Ref. [8] Present Ref. [8] Present Ref. [8]0.005 6.115 6.125 7.400 7.412 9.066 9.077 11.395 11.4010.010 4.309 4.300 5.125 5.241 6.602 6.619 8.712 8.7500.015 3.820 3.843 4.652 4.715 6.030 6.046 8.102 8.1600.020 3.652 3.667 4.502 4.515 5.800 5.832 7.860 7.9430.025 3.595 3.582 4.302 4.419 5.739 5.730 7.789 7.8400.030 3.532 3.534 4.350 4.366 5.596 5.673 7.756 7.784

and also Figures 9-13, the mode shapes are denotedby (x;m) in which x indicates the number of nodaldiameters and m indicates the number of nodal circles(excluding the boundary edges). It should be notedthat all the results in the paper are based on zeronodal diameters. Figure 9 shows the same result for theplate with the inner edge clamped and outer edge free.As it was mentioned, whenever the backward naturalfrequency equals zero, the critical angular velocityoccurs. Figure 10 shows the variation of the �rstdimensionless natural frequency for a rotating disk ofvariable thickness with clamped edges, and Figure 11shows the same result for the plates with the inner edgeclamped and outer edge free. Figures 12 and 13 showthe variation of the dimensionless natural frequencies ofa rotating FG uniform disk with that of the dimension-less rotating speed with clamped edges and clamped-

free edges, respectively. The critical speed is recognizedwhen the backward natural frequency goes to zero. Atthis speed, the disk has zero e�ective bending rigidityand cannot bear any transverse load.

From Figures 8-13, the critical speeds of the rotat-ing plate with the corresponding material and thicknesscharacteristics can be achieved. Table 6 presents the�rst dimensionless critical speed of rotating FG uniformplate with di�erent values of graded index. It can be

Table 6. First dimensionless critical speed of rotating FGplate with clamped edges (m1 = 0, and b=a = 0:2).

m 1 2 3 4

n = 0 28.474 21.017 18.231 17.754n = �1 33.120 22.771 18.878 18.018n = 1 25.394 19.939 17.942 17.714


Figure 6. Fundamental natural frequency against gradedindex for (C-C) plate (b=a = 0:2, ho=a = 0:01, m = 0, and~! = 0).

Figure 7. Fundamental natural frequency against gradedindex for (C-F) plate (b=a = 0:2, ho=a = 0:01, m = 0, and~! = 0).

Figure 8. First dimensionless natural frequency as afunction of dimensionless rotating speed for a rotatingannular disk with clamped edges (n = 0 and m1 = 0).

Figure 9. First dimensionless natural frequency as afunction of dimensionless rotating speed for a rotatingannular disk with clamped-free edges (n = 0 and m1 = 0).

seen that the value of critical speed for n = �1 is largerthan its value for n = 0 and n = 1.

In Table 7, the �rst dimensionless critical speedof the rotating FG uniform plate with (C-F) boundaryconditions is presented. It should be mentioned thatthe critical angular velocity for the wave number one(m = 1) for the plate with clamped-free edges is not

Table 7. First dimensionless critical speed of rotating FGplate with clamped-free edges (m1 = 0 and b=a = 0:2).

m 2 3 4

n = 0 4.510 4.756 5.772n = �1 6.505 5.630 6.344n = 1 3.254 4.176 5.353


Figure 10. First dimensionless natural frequency as afunction of dimensionless rotating speed for a rotatingannular disk with clamped edges (n = 0 and m1 = 0).

Figure 11. First dimensionless natural frequency as afunction of dimensionless rotating speed for a rotatingdisk with clamped-free edges (n = 0 and m1 = 0).

in the interval of the angular velocity considered in theanalysis. The same results are also obtained in [13] for(m = 1).

In Table 8, the values of the �rst dimension-less critical speed for di�erent thickness pro�les androtating homogenous plate with clamped edges arepresented. Table 9 illustrates the same results, butfor (C-F) boundary conditions. From Tables 8 and 9,

Table 8. First dimensionless critical speed of rotatingplate with clamped edges (n = 0 and b=a = 0:2).

m 1 2 3 4

m1 = 0 28.474 21.017 18.231 17.754m1 = �1 43.488 33.951 28.384 26.123m1 = 1 19.920 14.688 12.940 12.795

Figure 12. First dimensionless natural frequency as afunction of dimensionless rotating speed for a rotatingdisk with clamped edges (n = �1 and m1 = 0).

Figure 13. First dimensionless natural frequency as afunction of dimensionless rotating speed for a rotatingdisk with clamped-free edges (n = �1 and m1 = 0).it can be concluded that by fabricating a plate withconvergent thickness pro�le (m1 = �1), we can havea higher critical speed; by fabricating a plate with adivergent thickness pro�le (m1 = �1), we can havea lower critical speed. This conclusion can help toimprove the design of rotating FG disk in order to avoidthe resonance condition.

Figure 14 shows the graph of dimensionless crit-

Table 9. First dimensionless critical speed of rotatingplate with (C-F) edges (n = 0 and b=a = 0:2).

m 2 3 4

m1 = 0 4.510 4.756 5.772m1 = �1 18.055 10.243 9.447m1 = 1 1.896 3.157 4.320


Figure 14. Dimensionless critical speed versusinner-to-outer radius ratio for C-C boundary condition.

Figure 15. Dimensionless critical speed versusinner-to-outer radius ratio for C-C boundary condition.

ical speed versus the ratio of inner-to-outer radius fora FG plate with variable thickness and with clampededges. Figure 15 shows the corresponding graph fora plate with the inner edge clamped and outer edgefree. It can be seen that an increase in the inner-to-outer radii ratio could increase the critical speed. Itsuggests that we can have higher critical speed for therotating annular disk if we design the annular disk witha bigger central hole with respect to the outer radius.It should be mentioned that the authors obtained theirresults for di�erent values of m, n, and m1; the sameconclusion was derived.

5. Summary and conclusions

In this paper, free transverse vibration of rotatingFG thin disc with variable thickness was studied.The thickness pro�le, elastic modulus, and densityof the annular disk were considered to vary in theradial direction by a power function. The numericalgeneralized di�erential quadrature method was used tosolve the equations of motion, and the convergence ofthe method was proved for two types of discretizationmethods. The e�ects of material and geometric pa-rameters on the natural frequencies and critical speeds

of the rotating annular FG plate with (C-C) and (C-F) boundary conditions were investigated, and theobtained results were compared with the available datafrom the literature; the results were validated. Theobtained results can be used to improve the design ofthe rotating FG disk considering the avoidance of theresonance condition. It was shown that:

1. FG annular plate with a convergent thickness pro-�le has higher critical speed than FG annular platewith uniform thickness pro�le, and FG annularplate with uniform thickness pro�le has higher crit-ical speed than FG annular plate with a divergentthickness pro�le. This conclusion can be used toimprove the design of the rotating FG disks in orderto increase or decrease the critical speed;

2. Using a convergent thickness pro�le can lead tohigher natural frequency than using uniform anddivergent thickness pro�les;

3. The increase in the ratio of inner-to-outer radiuscan lead to the increase in the critical speed of therotating FG disk;

4. The increase of thickness-to-outer radius ratiosdecreases the dimensionless natural frequencies, butthe increase of inner-to-outer radii ratio increasesthe dimensionless natural frequencies.

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Biographies

Mohammad Hadi Jalali received BSc and MScdegrees in Mechanical Engineering from Isfahan Uni-versity of Technology in 2011 and 2013, respectively.After the graduation, he performed some industrial andacademic projects in the �elds of rotor dynamics andstructural dynamics. His research interests are rotordynamics, experimental modal analysis, and vibrationsin rotating machinery.

Behrooz Shahriari received BSc degree in Mechani-cal Engineering and MSc and PhD degrees in AerospaceEngineering from Malek-Ashtar University of Technol-ogy in 2002, 2012, and 2016, respectively. He has 13years of experience in research and doing industrial andacademic projects in the �elds of aerospace structuraldesign and analysis, gas turbines, and other aerospacepropulsion systems. His research interests are turbomachinery structural design/test, optimization andaerospace engines design.

Omid Zargar received BSc and MSc degrees inMechanical Engineering from Isfahan University ofTechnology and Tehran University in 2012 and 2015,respectively. After the graduation, he performed someacademic projects in the �elds of micro and nanovibration materials (MEMS and NEMS), biomechanicsand harvesting energy. His research interests arevehicle concept modeling, MEMS, experimental modal


analysis, bio mechanics, and harvesting energy frompiezoelectric materials.

Mostafa Baghani received his BS degree in Mechan-ical Engineering from University of Tehran, Iran in2006, his MS and PhD degrees in Mechanical Engineer-ing from the Department of Mechanical Engineering atSharif University of Technology, Tehran, Iran in 2008and 2012, respectively. He is now an Assistant Profes-sor in School of Mechanical Engineering, University ofTehran. His research interests include solid mechanics,

nonlinear �nite-element method, and shape memorymaterials constitutive modeling.

Majid Baniassadi is working as an Assistant Pro-fessor in the Faculty of Mechanical Engineering atUniversity of Tehran. He received his BSc degree fromIsfahan University of Technology in 2004, his MSc fromUniversity of Tehran in 2006 and PhD from Universityof Strasbourg in 2011. His research interests are mainlyin the area of micromechanics, nano-mechanics, andmaterials design.

free vibration analysis of rotating functionally graded annular...

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