Characteristic Orthogonal Polynomials in the Study

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Meccanica (2012) 47:175193DOI 10.1007/s11012-011-9430-4

Characteristic orthogonal polynomials in the study

of transverse vibrations of nonhomogeneous rectangularorthotropic plates of bilinearly varying thickness

R. Lal Kumar Yajuvindra

Received: 28 May 2009 / Accepted: 4 April 2011 / Published online: 3 May 2011 Springer Science+Business Media B.V. 2011

Abstract Effect of nonhomogeneity on the vibra-tional characteristics of thin orthotropic rectangularplates of bilinearly varying thickness has been studiedusing boundary characteristic orthogonal polynomialsin the Rayleigh-Ritz method. The thickness variationis taken as the Cartesian product of linear variationsalong two concurrent edges of the plate. The orthog-onal polynomials in two variables are generated usingthe Gram-Schmidt process. The nonhomogeneity ofthe plate material is assumed to arise due to linear vari-

ations in Youngs moduli, shear modulus and densityof the plate with the in-plane coordinates. Numericalresults have been computed for four different combi-nations of clamped, simply supported and free edges.Effect of thickness variation together with varying val-ues of aspect ratio and nonhomogeneity on the naturalfrequencies is illustrated for the first three modes ofvibration. Three dimensional mode shapes have beenpresented. Comparison has been made with the knownresults.

Keywords Nonhomogeneous Rectangular Orthotropic Bilinearly varying thickness

R. Lal () K. YajuvindraDepartment of Mathematics, Indian Institute ofTechnology Roorkee, Roorkee 247 667, Indiae-mail: [email protected]

1 Introduction

In recent years, the desirability of lighter, stiffer andstronger materials particularly in aerospace industryand missile technology lead to the development offiber-reinforced material. The increasing use of suchtailored composite materials in the design of platetype structural elements has necessitated the studyof dynamic behavior of anisotropic plates. In manyengineering applications, these plates with appropri-

ate thickness variation have significantly greater effi-ciency for vibration as compared to the plates of uni-form thickness and also provides the advantage of ma-terial saving and hence the cost requirement. Thus,their design requires an accurate analysis for their vi-bration characteristics.

Up till now, numerous studies on linear vibrationsof orthotropic/composite plates of various geometrieshave been carried out and the work up to 1985 hasbeen reported in a monograph by Leissa [1] and com-prehensive survey papers [27]. Notable contributions

made thereafter are listed in references [825]. Out ofthese, references [23, 25] present a complete surveyof work up to 2007 on circular/annular and rectangu-lar plates. The analysis has been presented using ana-lytical, numerical or approximate methods mainly forplates with unidirectional thickness variations such aslinear, parabolic, stepped, quadratic and exponentialetc. In recent papers, Biancolini [22] used a particu-lar form of Rayleigh-Ritz method to obtain approxi-mate solution for the free vibrations of thin orthotropic
mailto:[email protected]:[email protected]


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rectangular plates, Yunshan et al. [26] analyzed freevibration of Mindlin plates using DSC-Ritz method,Bhasker and Sivaram [27] obtained exact series solu-tion using the principle of virtual work in the vibra-tion analysis of orthotropic rectangular plates, merCivalek [28] obtained fundamental frequency of or-

thotropic rectangular plates employing discrete singu-lar convolution method, Khov et al. [29] used Fouriercosine series for calculating the static and dynamicdeflections and model characteristics of orthotropicplates with general boundary conditions, Xing and Liu[30] presented exact solutions for free vibrations ofthin orthotropic rectangular plates and Zhou et al. [31]studied free vibration of annular sector plates with var-ious boundary conditions using the Chebyshev-Ritzmethod. In this context, authors have come across alimited number of papers in which the thickness of the

plate varies in both the directions and reported in [3240], to report the prominent ones. Of these, Sakiyamaand Huang [35] considered sinusoidal variation, Che-ung and Zhou [36] assumed the power functions ofboth the coordinates and the rest deal with bilinearvariation in thickness.

Orthotropic plates are often nonhomogeneous, ei-ther by design or because of the physical composi-tion and imperfections in the underlying materials.Various models representing the behavior of nonho-mogeneous materials have been proposed in the lit-

erature and a detailed discussion is given in a recentpaper [41]. In these models, the authors have con-sidered the unidirectional variation for nonhomogene-ity of the plate material i.e. mechanical properties arefunctions of only one variable. However, in many prac-tical situations, particularly in modern missile technol-ogy and microelectronics, plate type structural compo-nents have to work under elevated temperature, whichcauses nonhomogeneity of the material. Hence, it doesnot seem appropriate to take unidirectional variationsin the mechanical/elastic properties of the material

with one variable only for nonhomogeneity. Recently,in two papers Chakraverty et al. [42] and Lal et al. [43]have analyzed the vibrational behavior of orthotropicelliptic and isotropic rectangular plates, respectively,assuming that nonhomogeneity of the plate materialdepends upon in-plane variables.

During the above survey of the literature, authorsfound that (i) no work has been done dealing with thevibrations of orthotropic rectangular plates in whichnonhomogeneity is arising due to variations in two

variables (ii) the boundary characteristic orthogonalpolynomials have been used extensively for analyzingthe dynamic behavior of plates of various geometriesunder different boundary conditions. The simplifica-tion of the eigenvalue problem i.e. reduction of a gen-eralized eigenvalue problem into a standard eigenvalue

problem and rapid convergence are the main character-istics of this approach. In the present study boundarycharacteristic orthogonal polynomials have been usedto analyze the vibration characteristics of nonhomoge-neous orthotropic rectangular plates of varying thick-ness employing classical plate theory. For nonhomo-geneity of the plate, it is assumed that Youngs mod-uli, shear modulus and density of the plate material arelinear functions of in-plane variables. The thickness isvarying bilinearly and taken as the Cartesian productof linear variations along two concurrent edges of the

plate. Rayleigh-Ritz method has been used to obtainfirst three natural frequencies for such plates for fourdifferent combinations of the clamped, simply sup-ported and free edge conditions. Graphite-epoxy [42]has been taken as an example of orthotropic material.

2 Formulation and solution of the problem

Consider an orthotropic nonhomogeneous rectangu-lar plate of varying thickness h(x,y), density (x,y)with domain {0 x a, 0 y b} in xy-plane,where a and b are the length and the breadth of theplate, respectively. The x- and y-axes are taken alongthe edges of the plate and axis of z- is perpendicularto the xy-plane. The middle surface being z = 0 andorigin is at one of the corners of the plate as shown inFig. 1(a).

The strain energy and kinetic energy of the plate aregiven by (Lekhnitskii [44])

V =

1

2a

0

b0 [D

x w2xx + 2x Dy wxx wyy + Dy w

2yy

+ 4Dxy w2xy ]dxdy, (1)

T =1

2

a0

b0

h

w

t

2dxdy, (2)

where Dx =Ex h

3

12(1x y ), Dy =

Ey h3

12(1x y ), Dxy =

Gxy h3

12

are bending rigidities, Ex , Ey are Youngs moduliin x- and y-directions, x , y are the Poissons


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Fig. 1 (a) Geometry of the plate and (b) boundary conditions

ratios in the proper directions, t is the time and

Dx y = Dy x .

For harmonic solution, the deflection functionw(x,y,t) is assumed to be

w(x,y,t) = W(x,y) sin t, (3)

where W(x,y) is the maximum transverse displace-

ment at the point (x,y) and is the circular frequency.

Substituting relation (3) in expressions (1) and (2),

maximum strain energy and kinetic energy of the plate

are given by

Vmax = 12

a

0

b

0[Dx W

2xx + 2x Dy Wxx Wyy

+ Dy W2yy + 4Dxy W

2xy ]dxdy, (4)

Tmax =2

2

a0

b0

hW2dxdy. (5)

According to the Rayleigh-Ritz method equating (4)

and (5), one gets the Rayleigh quotient as


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2 =

a0

b0 [Dx W

2xx + 2x Dy Wxx Wyy + Dy W

2yy + 4Dxy W

2xy ]dxdya

0

b0 hW

2dxdy. (6)

Introducing the non-dimensional variables

X = x/a , Y = y/b , W = W /a and H = h/a togetherwith the assumption that Youngs moduli, shear mod-ulus and density of the plate material vary with thespace coordinates by the functional relations

Ex = E1(1 + X), Ey = E2(1 + Y),

Gxy = G0(1 + 1X + 2Y ), (7)

= 0(1 + 1X + 2Y )

and thickness of plate varies linearly in both X- andY-directions given by

h(X,Y) = h0(1 + 1X)(1 + 2Y ). (8)

Equation (6) reduces to

2 = 1

0 1

0 [Dx W2XX + 2x (

ab

)2Dy WXX WY Y + (ab

)4Dy W2Y Y + 4(

ab

)2Dxy W2XY]dXdY

10

10 a50h0(1 + 1X + 2Y )(1 + 1X)(1 + 2Y )W2dXdY. (9)

Now satisfying the essential boundary conditions, letus assume

W(X,Y) =

Nk=1

dk k (X,Y), (10)

where N is the order of approximation to get the de-sired accuracy, dk s are unknowns and k are orthog-onal polynomials which are generated using Gram-Schmidt process as follows:

Orthogonal polynomials k over the region0 X 1, 0 Y 1 have been generated with thehelp of linearly independent set of functions Lk = llk ,k = 1, 2, 3, . . . , with l = Xp1 (1 X)p2 Yp3 (1 Y )p4 ,lk = {1,X,Y,X2,XY,Y2, X3, X2Y,XY2, Y3, . . .},where p1 = 0, 1 or 2 as the edge X = 0 is free, sim-ply supported or clamped. Same justification can begiven to p2, p3 and p4 for the edges X = 1, Y = 0 andY = 1.

1 = L1, k = Lk

k1j=1

kjj,

(11)kj =

Lk , j

j, j,

j = 1, 2, 3, . . . , ( k 1), k = 2, 3, 4, . . . , N .

The inner product of the functions say, 1 and 2 isdefined as

1, 2

=

10

10

(1 + 1X + 2Y )(1 + 1X)(1 + 2Y )

1(X,Y)2(X,Y)dXdY, (12)

where (1 + 1X + 2Y )(1 + 1X)(1 + 2Y ) is theweight function and the norm of the function 1 isgiven by

1 = 1, 11/2

=

10

10

(1 + 1X + 2Y )(1 + 1X)(1 + 2Y )

21 (X,Y)dXdY

1/2. (13)

The normalization can be done by using

k =k

k. (14)

Using expression (10) into (9) and minimization of theresulting expression for 2 with respect to dks leadsto the generalized eigenvalue problem

Nk=1

(aj k 2bj k )dk = 0, j = 1, 2, 3, . . . , N , (15)


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where

aj k = H0

10

10

r1

E1

H0XXj

XXk

+ r2xE2

H0

2(XXj Y Yk +

XXk

Y Yj )

+ 2r3

1 x

E2

H0

2XYj

XYk (16)

+ r24 E

2

H0Y Yj

Y Yk

dXdY,

bj k =

10

10

(1 + 1X + 2Y )(1 + 1X)(1 + 2Y )

jkdXdY.

It is worth to mention that ks are orthonormal poly-nomials with respect to the weight function(1 + 1X + 2Y )(1 + 1X)(1 + 2Y ) so (15) reducesto the standard eigenvalue problem

Nk=1

(aj k 2j k)dk = 0, j = 1, 2, 3, . . . , N , (17)

aj k = H0

10

10

r1

E1

H0XXj

XXk

+ r2xE2

H02(XXj

Y Yk +

XXk

Y Yj )

+ 2r3

1 x

E2

H0

2XYj

XYk

+ r24 E

2

H0Y Yj

Y Yk

dXdY, (18)

r1 = (1 + X)(1 + 1X)3(1 + 2Y )

3,

r2 = (1 + Y)(1 + 1X)3(1 + 2Y )

3,

r3 = (1 + 1X + 2Y )(1 + 1X)3(1 + 2Y )

3,

E1 =E1h

30

12(1 x y ), E2 =

E2h30

12(1 x y ),

G1 =G0h

30

12, H0 = E

2 x + 2G1,

2 =0h0a

42

H0,

=a

band j k =

1, ifj = k

0, ifj = k.

The integrals involved in (18) have been evaluated us-ing the formula

10

10

Xp1 (1 X)p2 Yp3 (1 Y )p4 dXdY

=

10

Xp1 (1 X)p2 dX

10

Yp3 (1 Y )p4 dY

=p1!p2!

(p1 + p2 + 1)!

p3!p4!

(p3 + p4 + 1)!.

3 Boundary conditions

The four boundary conditions namely CCCC, SCSC,FCFC and FSFS have been considered in which Cstands for clamped edge, S for simply supported edgeand F for free edge. The edge conditions are takenin anti-clockwise direction starting at the edge x = 0(Fig. 1(b)) and obtained by assigning various values top1, p2, p3 and p4 as 0, 1, 2 for free, simply supportedand clamped edge conditions, respectively.

4 Results and discussion

The numerical values of the frequency parameter have been obtained by solving (17) employing Ja-cobi method. The lowest three eigenvalues have beenreported as the first three natural frequencies corre-sponding to different boundary conditions consideredhere. The values of various plate parameters for thesethree modes of vibration are taken as follows.

Nonhomogeneity parameters: ,,1, 2, 1,2 = 0.5(0.2)0.5, thickness parameters: 1, 2 =0.5(0.2)0.5 and aspect ratio: a/b = 0.25(0.25)2.00.

The values of elastic constants used for the plate ma-terial (Graphite-epoxy) are taken as E1 /H0 = 13.9,E2 /H0 = 0.79 and x = 0.28 from [42].

To choose the appropriate value of the order of ap-proximation N, a computer program developed for theevaluation of frequency parameter was run for dif-ferent values of N. The numerical values showed aconsistent improvement with the increasing value ofN for different sets of the values of plate parameters.In all the above computations, N = 47 has been fixed,


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Table 1 Convergence of frequency parameter of orthotropicE1 /H0 = 13.9, E

2 /H0 = 0.79, x = 0.28 square (a/b = 1)

plate for = = 1 = 2 = 1 = 2 = 1 = 2 = 0.5

Mode

N I II III

CCCC

10 122.9170 149.7460 222.2350

20 122.9140 149.6140 206.3600

30 122.9130 149.6060 205.2090

40 122.9120 149.6040 205.1810

45 122.9120 149.6040 205.1780

46 122.9120 149.6040 205.1770

47 122.9120 149.6040 205.1770

SCSC

10 63.0089 104.3400 192.5660

20 62.9796 103.6450 174.7320

30 62.9793 103.6400 173.4170

40 62.9793 103.6390 173.3800

45 62.9793 103.6390 173.3780

46 62.9793 103.6390 173.3770

47 62.9793 103.6390 173.3770

FCFC

10 28.1960 37.5307 76.9305

20 28.1865 37.4680 76.4910

30 28.1847 37.4616 76.4680

40 28.1839 37.4561 76.4640

45 28.1826 37.4559 76.4625

46 28.1826 37.4559 76.4625

47 28.1826 37.4559 76.4625

FSFS

10 12.4894 25.8105 59.0049

20 12.3534 25.5784 49.7370

30 12.3534 25.5708 49.6402

40 12.3534 25.5707 49.6391

45 12.3534 25.5707 49.6390

46 12.3534 25.5707 49.6390

47 12.3534 25.5707 49.6390

since further increase in the value of N does not im-prove the results even at the fourth place of decimal inthe third mode. Table 1 shows the convergence of fre-quency parameter with N for a particular set of plateparameters where maximum value ofN was required.All the computations were performed in double preci-sion arithmetic.

A comparison of frequency parameter for homo-geneous ( = = 1 = 2 = 1 = 2 = 0) orthotropicplates of uniform (1 = 2 = 0) and non-uniformthickness with those obtained by exact method [30],Ritz method [33], Kantorovich method [45], Ritzmethod [46] and weighted residuals approach [47] has

been presented in Table 2. A close agreement of resultsis obtained.

Numerical results for various values of plate param-eters have been computed and for a square plate theseare presented in Figs. 28. It is observed that the fre-quency parameter decreases in the order of bound-ary conditions CCCC > SCSC > FCFC > FSFS forthe same set of values of plate parameters. Figure 2shows the effect of nonhomogeneity parameter onthe frequency parameter for = 0.5, 1 = 0.5,2 = 0.5, 1 = 0.5, 2 = 0.5, 1 = 0.5, 2 = 0.5

and a/b = 1 for the first two modes of vibration. Itis observed that the frequency parameter increaseswith increasing values of for all the boundary con-ditions keeping other plate parameters fixed. Further,it also increases with the increasing values of whiledecreases with the increasing value of1 for fixed val-ues of other plate parameters. The rate of increasein with is in the order of the boundary con-ditions SCSC > CCCC > FSFS > FCFC when changes from 0.5 to 0.5 and order becomes CCCC> SCSC > FCFC > FSFS when

1changes from

0.5 to 0.5, other parameters being fixed. This rateis higher in the second mode as compared to the firstmode.

Figure 3 shows the effect of nonhomogeneity pa-rameter on the frequency parameter for = 0.5,1 = 0.5, 2 = 0.5, 1 = 0.5, 2 = 0.5, 1 = 0.5,2 = 0.5 and a/b = 1 for the first two modes of vi-bration. It is observed that the frequency parameter increases with increasing values of for all theboundary conditions keeping other plate parameters

fixed. The rate of increase in with is in the orderof the boundary conditions SCSC > CCCC > FSFS> FCFC when changes from 0.5 to 0.5 and or-der becomes SCSC > FSFS > CCCC > FCFC when1 changes from 0.5 to 0.5, other parameters beingfixed. This rate increases with the increase in the num-ber of modes.

Figure 4 depicts the behavior of the frequency pa-rameter with 1 for = 0.5, = 0.5, 2 = 0.5,1 = 0.5, 2 = 0.5, 1 = 0.5, 2 = 0.5 and a/b = 1


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Table 2 Comparison of frequency parameter for homogeneous ( = = 1 = 2 = 1 = 2 = 0) orthotropic plate

Boundaryconditions

Reference (1, 2) (E1 /H0, E2 H0, x ) a/b Mode

I II III

CCCC [30] 0,0 2, 1, 0.28 1.0 41.4770 76.1285 95.4721

42.4066 76.8540 95.8865

Present 42.3971 76.8447 95.8812

2, 1, 0.28 2.0 99.7360 139.7639 214.7087

100.9338 141.5995 216.1541

Present 100.9220 141.5770 216.1290

2, 2, 0.28 1.0 46.9151 97.9399 97.9399

47.9659 98.5523 98.5523

Present 47.9590 98.5471 98.5471

2, 2, 0.28 2.0 133.9614 165.7204 232.3468

134.9032 167.5871 234.1697

Present 134.8970 167.5730 234.1510

[46] 0.5, 0.5, 0.3 0.5 18.4000

Present 18.3380 25.2581 35.8455

[46] 0.5, 0.5, 0.3 1.0 28.1000

[47] 28.1000

Present 28.0725 56.6299 56.6299

[46] 0.5, 0.5, 0.3 1.5 46.6000

Present 46.4678 74.7039 110.4200

[33] 0.2, 0.2 1.0, 0.5, 0.3 1.0 32.28

Present 31.8100 58.1259 70.3779

0.2, 0.2 1.0, 0.5, 0.3 1.0 39.37

Present 38.9114 71.1149 86.0973

0.4, 0.2 1.0, 0.5, 0.3 1.0 29.40 Present 27.9811 51.0149 61.8339

for the first two modes of vibration. It is seen thatthe frequency parameter increases with increasingvalue of 1 whatever be the values of other plate pa-rameters. The frequency parameter is found to in-crease with the increasing value of 2 keeping otherplate parameters fixed. The value of further in-creases with the increasing value of . The rate of

increase in with 1 is in the order of the bound-ary conditions FCFC > SCSC > FSFS > FCFCwhen 2 changes from 0.5 to 0.5 and order becomesSCSC > CCCC > FCFC > FSFS when changesfrom 0.5 to 0.5. This rate is more pronounced in thesecond mode as compared to the first mode.

Figure 5 demonstrates the effect of density param-eter 1 on the frequency parameter for = 0.5, = 0.5, 1 = 0.5, 2 = 0.5, 2 = 0.5, 1 = 0.5,2 = 0.5 and a/b = 1 for the first two modes. It is no-

ticed that the frequency parameter decreases withincreasing value of1 for all the boundary conditions,other parameters being fixed. The value of increaseswith the increasing value of2 as well as 1. The rateof increase in with 1 is in the order of the bound-ary conditions CCCC > SCSC > FCFC > FSFSwhen 2 changes from 0.5 to 0.5 and order becomes

CCCC > SCSC > FSFS > FCFC when 1 changesfrom 0.5 to 0.5 keeping other parameters fixed. Thisrate is higher in the second mode as compared to thefirst mode.

Figure 6 demonstrates the effect of thickness pa-rameter 1 on the frequency parameter for = 0.5, = 0.5, 1 = 0.5, 2 = 0.5, 1 = 0.5,2 = 0.5, 2 = 0.5 and a/b = 1 for the first twomodes of vibration. It is noticed that the frequencyparameter increases with increasing value of1 for


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Fig. 2 Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plates: for 1 = 2 = 2 = 1 = 2 = 0.5. , firstmode; - - -, second mode; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5


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Fig. 3 Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plates: for 1 = 2 = 2 = 1 = 2 = 0.5. , firstmode; - - -, second mode; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5


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Fig. 4 Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plates: for = 1 = 2 = 1 = 2 = 0.5. , firstmode; - - -, second mode; , 2 = 0.5, = 0.5; , 2 = 0.5, = 0.5; , 2 = 0.5, = 0.5; , 2 = 0.5, = 0.5


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Fig. 5 Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plates: for = = 1 = 2 = 2 = 0.5. , firstmode; - - -, second mode; , 2 = 0.5, 1 = 0.5; , 2 = 0.5, 1 = 0.5; , 2 = 0.5, 1 = 0.5; , 2 = 0.5, 1 = 0.5


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Fig. 6 Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plates: for = 1 = 2 = 1 = 2 = 0.5. , firstmode; - - -, second mode; , 2 = 0.5, = 0.5; , 2 = 0.5, = 0.5; , 2 = 0.5, = 0.5; , 2 = 0.5, = 0.5


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Fig. 7 Frequency parameter for (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS plates: for = 1 = 2 = 1 = 2 = 2 = 0.5. ,first mode; - - -, second mode; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5; , = 0.5, 1 = 0.5


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Fig. 8 First three mode shapes of (a) CCCC, (b) SCSC, (c) FCFC and (d) FSFS orthotropic square plates for = = 1 = 2 = 1 = 2 = 1 = 2 = 0.5

all the boundary conditions for fixed values of all otherplate parameters. The value of increases with the in-creasing value of as well as 2. The rate of increasein with 1 is in the order of the boundary conditions

CCCC > SCSC > FSFS > FCFC when changesfrom 0.5 to 0.5 and order becomes CCCC > SCSC> FCFC > FSFS when 2 changes from 0.5 to 0.5,other parameters being fixed. This rate is higher in thesecond mode as compared to the first mode.

Figure 7 shows the plots for aspect ratio a/b ver-sus frequency parameter for = 0.5, = 0.5,1 = 0.5, 2 = 0.5, 1 = 0.5, 2 = 0.5, 1 = 0.5 and2 = 0.5 for the first two modes of vibration. It is clearthat frequency parameter increases with the increas-

ing value ofa/b for all the boundary conditions, otherplate parameters being fixed. The values of is foundto increase with the increasing values of and 1.The rate of increase in with a/b is in the order of

the boundary conditions SCSC > CCCC > FSFS >FCFC when changes from 0.5 to 0.5 and becomesFCFC > SCSC > CCCC > FSFS when 1 changesfrom 0.5 to 0.5, keeping all other parameters fixed.This rate of increase is much higher for a/b > 1 ascompared to a/b < 1 and increases with the increasein the number of modes.

In case of third mode of vibration the behavior offrequency parameter with other parameters remainsalmost same as that for first two modes except that


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Fig. 8 (Continued)

the rate of increase/decrease with a specific parameteris higher. Three dimensional mode shapes for specificplate have been shown in Fig. 8.

On the suggestion of one of the learned reviewers,the results for CFFF boundary condition have been in-cluded as Figs. 9 and 10. It was found that the valuesof frequency parameter for this case are in the order

of boundary conditions FCFC > CFFF > FSFS forthe same set of the values of plate parameters exceptfor thickness parameters. In this case the behavior offrequency parameter is not monotonous as clear fromFig. 9(e).

5 Conclusions

The effect of non-homogeneity arising due to the de-pendence of Youngs moduli, shear modulus and den-

sity of the plate material on both the variables x andy, on the natural frequencies of orthotropic rectangu-lar plates of bidirectionally varying thickness has beenstudied using boundary characteristic orthogonal poly-nomials in Rayleigh-Ritz method on the basis of clas-sical plate theory. It is observed that the frequency pa-rameter increases as the plate becomes more and

more stiff towards the edge x = a and y = b due to theincreasing values of the parameters , , 1, 2 and itis the reverse with the increasing values of 1 and 2.This also increases as the plate becomes thicker andthicker towards the edge x = a and y = b. Further, anincrease in the value ofa/b increases the frequencies.The percentage variation in the value of frequency pa-rameter for the first mode of vibration are 12.96 to9.10, 8.25 to 6.64, 0.09 to 0.05 and 0.12 to 0.06for CCCC, SCSC, FCFC and FSFS boundary condi-


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Fig. 9 Frequency parameter for CFFF plate. In, (a) Key as Fig. 2; (b) Key as Fig. 3; (c) Key as Fig. 4; (d) Key as Fig. 5; (e) Key asFig. 6; (f) Key as Fig. 7


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Fig. 9 (Continued)

Fig. 10 Mode shapes for CFFF orthotropic square plate. Key as Fig. 8

tions, respectively when the nonhomogeneity arisesdue to the change in only from 0.5 to 0.5. Thepercentage variation in the value of frequency param-eter for the first mode of vibration are 0.5 to 0.5,2.6 to 2.3, 13.9 to 10.3 and 12.7 to 10.3 forCCCC, SCSC, FCFC and FSFS boundary conditions,

respectively when the nonhomogeneity arises due tothe change in only from 0.5 to 0.5. The corre-sponding variations are 14.5 to 10.2, 15 to 10.5,8.6 to 8.3 and 13.9 to 10.3 when 1 changes from0.5 to 0.5. The present analysis will be of great useto the design engineers in obtaining the desired fre-


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192 Meccanica (2012) 47:175193

quency by varying one or more plate parameters con-sidered here.

Acknowledgement The authors wish to express their sin-cere thanks to the learned reviewers for their constructive com-ments/suggestions for improving the paper. One of the authorsYajuvindra Kumar is grateful to Council of Scientific & Indus-

trial Research, India for providing the Senior Research Fellow-ship.

References

1. Leissa AW (1969) Vibration of plates (NASA SP 160). USGovernment Office, Washington, DC

2. Leissa AW (1977) Recent research in plate vibrations: clas-sical theory. Shock Vib Dig 9(10):1324

3. Leissa AW (1978) Recent research in plate vibra-tions 19731976: complicating effects. Shock Vib Dig10(12):2135

4. Leissa AW (1981) Plate vibration research, 19761980:classical theory. Shock Vib Dig 13(9):1122

5. Leissa AW (1981) Plate vibration research, 19761980:complicating effects. Shock Vib Dig 13(10):1936

6. Leissa AW (1987) Recent studies in plate vibrations,19811985 Part I: classical theory. Shock Vib Dig19(2):1118

7. Leissa AW (1987) Recent studies in plate vibrations,19811985 Part II: complicating effects. Shock Vib Dig19(3):1024

8. Ramkumar RL, Chen PC, Sanders WJ (1987) Free vi-bration solution for clamped orthotropic plates using La-grangian multiplier technique. AIAA J 25(1):146151

9. Al-Khaiat H (1989) Free vibration analysis of or-thotropic plates by the initial value method. Comput Struct33(6):14311435

10. Lam KW, Liew KM, Chow ST (1989) Two-dimensional or-thogonal polynomials for vibration of rectangular compos-ite plates. Compos Struct 13:239250

11. Chow ST, Liew KM, Lam KY (1992) Transverse vibrationof symmetrically laminated rectangular composite plates.Compos Struct 20:213226

12. Hung KC, Liew KM, Lim MK, Leong SL (1993) Bound-ary beam characteristics orthogonal polynomials in energyapproach for vibration of symmetric laminates I: classicalboundary conditions. Compos Struct 26:167184

13. Hung KC, Lim MK, LiewKM (1993) Boundary beam char-acteristics orthogonal polynomials in energy approach forvibration of symmetric laminates II: elastically restrainedboundaries. Compos Struct 26:185209

14. Yu SD, Cleghorn WL (1993) Generic free vibration of or-thotropic rectangular plates with clamped and simply sup-ported edges. J Sound Vib 163(3):439450

15. Dalaei M, Kerr AD (1996) Natural vibration analysisof clamped rectangular orthotropic plates. J Sound Vib189(3):399406

16. Bert CW, Malik M (1996) Free vibration analysis of ta-pered rectangular plates by differential quadrature method:a semi-analytical approach. J Sound Vib 190(1):4163

17. Chen YZ (1998) Evaluation of fundamental vibrationfrequency of an orthotropic bending plate by using aniterative approach. Comput Methods Appl Mech Eng161:289296

18. Bambill DV, Rossit CA, Laura PAA, Rossi RE (2000)Transverse vibration of an orthotropic rectangular plate oflinearly varying thickness and with a free edge. J Sound Vib235(3):530538

19. Ashour AS (2001) A semi-analytical solution of the flex-ural vibration of orthotropic plates of variable thickness.J Sound Vib 240(3):431445

20. Hurlebaus S, Gaul L, Wang JTS (2001) An exact se-ries solution for calculating the eigenfrequencies of or-thotropic plates with completely free boundary. J SoundVib 244:747759

21. Bhaskar K, Kaushik B (2004) Simple and exact series solu-tions for flexure of orthotropic rectangular plates with anycombination of clamped and simply supported edges. Com-pos Struct 63:6368

22. Biancolini ME, Brutti C, Reccia L (2005) Approximatesolution for free vibrations of thin orthotropic rectangular

plates. J Sound Vib 288:32134423. Sharma S (2006) Free vibration studies on non-

homogeneous circular and annular plates. PhD Thesis, In-dian Institute of Technology Roorkee, Roorkee, India

24. Bhardwaj N, Gupta AP, Choong KK (2007) Effect of elas-tic foundation on the vibration of orthotropic elliptic plateswith varying thickness. Meccanica 42(4):341358

25. Dhanpati (2007) Free transverse vibrations of rectangularand circular orthotropic plates. PhD Thesis, Indian Instituteof Technology Roorkee, Roorkee, India

26. Yunshan H, Wei GW, Xiang Y (2005) DSC-Ritz method forthe free vibration analysis of Mindlin plates. Int J NumerMethods Eng 62:262288

27. Bhaskar K, Sivaram A (2008) Untruncated infinite se-ries superposition method for accurate flexural analysis ofisotropic/orthotropic rectangular plates with arbitrary edgeconditions. Compos Struct 83:8392

28. Civalek (2009) Fundamental frequency of isotropic andorthotropic rectangular plates with linearly varying thick-ness by discrete singular convolution method. Appl MathModel 33(10):38253835

29. Khov H, Li WL, Gibson RF (2009) An accurate solutionmethod for the static and dynamic deflection of orthotropicplates with general boundary conditions. Compos Struct90:474481

30. Xing YF, Liu B (2009) New exact solutions for free vibra-tions of thin orthotropic rectangular plates. Compos Struct

89(4):56757431. Zhou D, Lo SH, Cheung YK (2009) 3-D vibration analysis

of annular sector plates using the Chebyshev-Ritz method.J Sound Vib 320:421437

32. Laura PAA, Grossi RO (1979) Transverse vibrations ofrectangular plates with thickness varying in two direc-tions and with edges elastically restrained against rotation.J Sound Vib 63(4):499505

33. Grossi RO, Laura PAA (19801981) Transverse vibrationsof orthotropic rectangular plates with thickness varying intwo directions and with edges elastically restrained againstrotation. Fibre Sci Technol 14(4):311317


19/19

Meccanica (2012) 47:175193 193

34. Singh B, Saxena V (1996) Transverse vibration of a rectan-gular plate with bidirectional thickness variation. J SoundVib 198(1):5165

35. Sakiyama T, Huang M (1998) Free vibration analysis ofrectangular plates with variable thickness. J Sound Vib216(3):379397

36. Cheung YK, Zhou D (1999) The free vibrations of taperedrectangular plates using a new set of beam functions withthe Rayleigh-Ritz method. J Sound Vib 223(5):703722

37. Malekzadeh P, Shahpari SA (2005) Free vibration analysisof variable thickness thin and moderately thick plates withelastically restrained edges by DQM. Thin-Walled Struct43(7):10371050

38. Huang M, Ma XQ, Sakiyama T, Matsuda H, Morita C(2005) Free vibration analysis of orthotropic rectangularplates with variable thickness and general boundary con-ditions. J Sound Vib 288(45):931955

39. Huang M, Ma XQ, Sakiyama T, Matsuda H, Morita C(2007) Free vibration analysis of rectangular plates withvariable thickness and point support. J Sound Vib 300(35):435452

40. Gupta AK, Kaur H (2008) Study of the effect of thermalgradient on free vibration of clamped visco-elastic rect-angular plates with thickness variation in both directions.Meccanica 43(4):449458

41. Lal R, Dhanpati (2007) Transverse vibrations of non-homogeneous orthotropic rectangular plates of variablethickness: a spline technique. J Sound Vib 306(12):203214

42. Chakraverty S, Jindal R, Agarwal VK (2007) Vibrationof nonhomogeneous orthotropic elliptic and circular plateswith variable thickness. J Vib Acoust 129:256259

43. Lal R, Kumar Y, Gupta US (2010) Transverse vibrations ofnonhomogeneous rectangular plates of uniform thicknessusing boundary characteristic orthogonal polynomials. IntJ Appl Math Mech 6(14):93109

44. Lekhnitskii SG (1968) Anisotropic plates. Gordon &Breach, London

45. Sakata T, Takahashi K, Bhat RB (1996) Natural frequen-cies of orthotropic rectangular plates obtained by iterativereduction of the partial differential equation. J Sound Vib189:89101

46. Laura PAA, Grossi RO (1979) Transverse vibrations ofrectangular anisotropic plates with elastically restrainededges against rotation. J Sound Vib 64(2):257267

47. Laura PAA, Luisoni LE (1978) Vibrations of orthotropic

rectangular plates with edges possessing different rotationalflexibility coefficients and subjected to in-plane forces.Comput Struct 9(6):527532

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