Research ArticleForced Response of Polar Orthotropic Tapered Circular PlatesResting on Elastic Foundation

A H Ansari

Department of Mathematics AlBaha University Al Baha 1988 Saudi Arabia

Correspondence should be addressed to A H Ansari ansaridranwarrediffmailcom

Received 10 April 2016 Accepted 9 June 2016

Academic Editor Mohammad Tawfik

Copyright copy 2016 A H Ansari This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Forced axisymmetric response of polar orthotropic circular plates of linearly varying thickness resting on Winkler type of elasticfoundation has been studied on the basis of classical plate theory An approximate solution of problem has been obtained byRayleigh Ritz method which employs functions based upon the static deflection of polar orthotropic circular plates The effectof transverse loadings has been studied for orthotropic circular plate resting on elastic foundation The transverse deflections andbendingmoments are presented for various values of taper parameter rigidity ratio foundation parameter and flexibility parameterunder different types of loadings A comparison of results with those available in literature shows an excellent agreement

1 Introduction

The study of vibrational characteristics of plates has beenof great interest due to their increasing use in various engi-neering applications The development of fibre-reinforcedcomposite materials and their extensive use in fabricationof plate type structural components in aerospace oceanengineering and electronic and mechanical components hasled to the study of dynamic response of anisotropic platesTheconsideration of thickness variation together with anisotropynot only reduces the size and weight of components but alsomeets the desirability of high strength corrosion resistancehigh temperature performance and economy One can fabri-cate the orthotropic nature of plate in different ways that isrectangular and polar orthotropy The consideration of polarorthotropy in circular and annular plates provides the bestapproximation to the results because polar coordinate axesare also the axes for the materials symmetry An excellentreview of vibration of plates has been given by Leissa in hismonograph [1] and a series of review articles [2ndash7] Reddy [8]and Lal and Gupta [9] have presented an up-to-date surveyanalysing transverse vibrations of nonuniform rectangularorthotropic plates In a recent study Sayyad and Ghugal[10] presented a review of recent literatures on free vibrationanalysis of Laminated Composite and Sandwich Plates

A considerable amount of work dealing with vibrationof polar orthotropic circular and annular plates of uni-formnonuniform thickness in presenceabsence of elasticfoundation has been done by a number of workers and isreported in [11ndash31] to mention a few The orthotropic platesresting on an elastic foundation find their application in foun-dation engineering such as reinforced concrete pavements ofhigh runways foundation of deep wells and storage tanksand slabs of buildings (Lekhnitskii [11] p 136) In this con-nection various models such as Winkler [12ndash17] Pasternak[18] and Vlasov [19] have been proposed in the literatureLaura et al [13] studied the effect of Winkler Foundation onvibrations of solid circular plate of linearly varying thicknessGupta et al [14] analysed the effect of elastic foundation onaxisymmetric vibrations of polar orthotropic circular platesof variable thickness Further the work has been extendedto asymmetric vibration of polar orthotropic plates restingon elastic foundation by Ansari and Gupta [15] Gupta etal [16] studied buckling and vibration of polar orthotropiccircular plate attached to Winkler Foundation Orthotropiccircularannular plates such as deck diaphragm and bulkheads are used as structural components in launch vehicles

In some technological situations a plate is exposed totransverse loads on the surfaces with a downward andorupward thrust that is ship container and aeronautical

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2016 Article ID 1492051 10 pageshttpdxdoiorg10115520161492051

2 Advances in Acoustics and Vibration

structural components The stability of these componentsincreases with the support of elastic foundations Thus thestudy of the combined effect of transverse loads and elasticfoundation on vibrational characteristics of plates is of prac-tical importance Gupta et al [16] analysed the complicatingeffect of elastic foundation on elastic properties of the platesA number of research papers are available showing thestudy of transverse deflection and bending moments of polarorthotropicisotropic plates of variable thickness resting onelastic foundation [20ndash31] with complicating effects

The present work is concerned with the study of trans-verse loads on vibration of polar orthotropic circular platesof linearly varying thickness resting on Winkler type ofelastic foundationThe support of elastic foundation providesgreater stability to these structural components exposed totransverse loads An approximate solution is obtained by Ritzmethod employing functions based on static deflection givenbyLekhnitskii [11]Thepresent choice of functions has a fasterrate of convergence as compared to polynomial coordinatefunctions employed by Laura et al [13 21 26 27]

2 Analysis

Consider a thin circular plate of radius 119886 and variable thick-ness ℎ = ℎ(119903) resting on elastic foundation of modulus 119896119891elastically restrained against rotation by springs of stiffness119896120601 and subjected to 119875(119903) cos120596119905 type of excitation extendingfrom 119903 = 1199030 to 119903 = 1199031 Let (119903 120579) be the polar coordinates ofany point on the neutral surface of the circular plate referredto as the centre of the plate as origin (shown in Figure 1(a))

The maximum kinetic energy of the plate is given by

119879max =1

2

1205881205962int

119886

0int

2120587

0ℎ1199082119903119889120579 119889119903 (1)

where 119908 is the transverse deflection 120588 the mass density and120596 the frequency in radians per second

The maximum strain energy of the plate is given by

119880max =1

2

int

119886

0int

2120587

0[119863119903 (

1205972119908

1205971199032)

2

+ 2120592120579

1205972119908

1205971199032(

1

119903

120597119908

120597119903

)

+ 119863120579 (1

119903

120597119908

120597119903

)

2

+ 1198961198911199082] 119903 119889119903 119889120579 +

1

2

sdot 119886119896120601 int

2120587

0(

120597119908 (119886 120579)

120597119903

)

2

119889120579

(2)

where 1119896120601 is the rotational flexibility of the spring andflexural rigidities of the plate are

119863119903 =119864119903ℎ3

12 (1 minus 120592119903120592120579)

119863120579 =119864120579ℎ3

12 (1 minus 120592119903120592120579)

(3)

The work done by the external force 119875(119903) acting on the platein the direction parallel to 119911-axis is given by

119881max = int2120587

0119889120579int

1199031

1199030

119908119875 (119903) 119903 119889119903 (4)

3 Method of Solution Ritz Method

Ritz method requires that the functional

119869 (119908) = 119880max minus 119881max minus 119879max =1

2

sdot int

119886

0int

2120587

0[119863119903 (

1205972119908

1205971199032)

2

+ 2120592120579

1205972119908

1205971199032(

1

119903

120597119908

120597119903

)

+ 119863120579 (1

119903

120597119908

120597119903

)

2

+ 1198961198911199082] 119903 119889119903 119889120579 +

1

2

sdot 119886119896120601 int

2120587

0(

120597119908 (119886 120579)

120597119903

)

2

119889120579 minus int

2120587

0119889120579int

1199031

1199030

119908119875 (119903)

sdot 119903 119889119903 minus

1

2

1205881205962int

119886

0int

2120587

0ℎ1199082119903 119889120579 119889119903

(5)

be minimisedUniformly distributed load 1198750 extending from 1199030 to 1199031 is

given by

119875 (119903) = 1199020 =1198750

120587 (11990321 minus 11990320)[119880 (119903 minus 1199030) minus 119880 (119903 minus 1199031)] (6)

Introducing the nondimensional variables 119877 = 119903119886 1198771 =

1199031119886 1198770 = 1199030119886 and assuming the deflection function as119882(119877) = 119908(119886

41199020119863119903

0

)

119882 (119877) =

119898

sum

119894=0

119860 119894119865119894 (119877) =

119898

sum

119894=0

119860 119894 (1 + 1205721198941198774+ 1205731198941198771+119901) 1198772119894 (7)

where 119860 119894 are undetermined coefficients 1199012 = 119864120579119864119903 and 120572119894120573119894 are unknown constants to be determined from boundaryconditions (Leissa [1] p 14)

119870120601

119889119865119894 (1)

119889119877

= minus (1 minus 120572)3[

1198892119865119894

1198891198772+ 120592120579 (

1

119877

119889119865119894

119889119877

)]

119877=1

119865119894 (1) = 0

(8)

Linearly thickness variation of the plate is assumed as ℎ =ℎ0(1minus120572119877) where 120572 and ℎ0 are taper parameter and thicknessof the plate at the centre respectively

The choice of the function approximating the deflectionof the plate 119882(119877) given in (7) is based upon the staticdeflection polar orthotropic plates (Lekhnitskii [11]) whichhas faster rate of convergence (Gupta et al [16]) as comparedto the polynomial coordinate functions used by earlierresearchers [13 21 26 27]

Advances in Acoustics and Vibration 3

r0r1

k120601

h0

kfkf

r0r1

r1

k120601k120601

h0

a

k120601

(a)

Disk loaded plate Annular loaded plate Uniformly loaded plate

(b)

Figure 1 (a) Plate geometry with side and surface views (b) Loading structure of the plate

Introducing the nondimensional variables 119882 and 119877 in(5) the functional 119869(119882) becomes

119869 (119882) = 1205871198631199030

[int

1

0(1 minus 120572119877)

3(

1205972119882

1205971198772)

2

+ 2120592120579

1205972119882

1205971198772(

1

119877

120597119882

120597119877

) + 1199012(

1

119877

120597119882

120597119877

)

2

+ 1198701198911198822

sdot 119877 119889119877 + 119870120601 int

2120587

0(

120597119882 (1)

120597119877

)

2

119889120579 minus int

1198771

1198770

2119882 (119877) 119877 119889119877

minus Ω2int

1

0int

2120587

0(1 minus 120572119877)119882

2119877119889119877]

(9)

where

1198631199030

=

119864119903ℎ03

12 (1 minus 120592119903120592120579)

Ω2=

11988641205962120588ℎ0

1198631199030

119870119891 =

1198864119896119891

1198631199030

119870Φ =119886119896Φ

1198631199030

(10)

The minimisation of the functional 119869(119882) given by (9)requires

120597119869 (119882)

120597119860 119894

= 0 119894 = 0 1 2 3 119898 (11)

This leads to a system of nonhomogeneous equations in 119860119895

(119886119894119895 minus 119887119894119895)119860119895 = 119862119894 119894 119895 = 0 1 119898 (12)

where 119860 = [119886119894119895] and 119861 = [119887119894119895] are square matrices of order119898 + 1 given by

119886119894119895 = int

1

0(1 minus 120572119877)

3[11986510158401015840119894 11986510158401015840119895 + 2120592120579119865

10158401015840119894 (

1198651015840119895

119877

)

+ 1199012(

1198651015840119894

119877

)(

1198651015840119895

119877

) + 119870119891119865119894119865119895]119877119889119877 + 1198701206011198651015840119894 (1)

sdot 1198651015840119895 (1)

119887119894119895 = int

1

0(1 minus 120572119877) 119865119894119865119895119877119889119877

119862119894119895 = int

1

02119865119894119877119889119877

(13)

The solution of system of (12) gives the value of 119860 119894 and thusthe transverse deflection 119882 and the radial and transversebending moments

119872119903

11988621199020

= minus (1 minus 120572119877)3[

1198892119882

1198891198772+ 120592120579

1

119877

119889119882

119889119877

]

119872120579

11988621199020

= minus (1 minus 120572119877)3[120592120579

1198892119882

1198891198772+ 1199012 1

119877

119889119882

119889119877

]

(14)

are computed

4 Numerical Results

Numerical results have been calculated for different valuesof taper parameter 120572 (=plusmn03) 119864120579119864119903 (=50) and foundation

4 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005W

a4q0D

r 0

(a) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(b) Disk loading (CL-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

0005

Wa

4q0D

r 0

(c) Annular loading (SS-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(d) Annular loading (CL-plate)

Figure 2 Radius vector versus deflection parameter119882119886411990201198631199030 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

modulus 119870119891 (=001 002 003) for forced vibration of polarorthotropic circular plate with simply supported (SS-plate)and clamped (CL-plate) edges The plate is subjected underdifferent types of loadings such as uniform loading onentire plate annular loading and disk loading The naturalfrequencies for free vibration are obtained by putting 119875(119903) =0 In case of forced vibration the nondimensional frequencyparameter is taken as Ω = 120578Ω00 for Ω lt Ω00 and Ω =

Ω00 + 120578(Ω01 minus Ω00) for Ω00 lt Ω lt Ω01 where 120578 = 02 Thenormalised deflection and bendingmoments are obtained forfixed value of Poissonrsquos ratio as 03

5 Discussion

Forced axisymmetric response of polar orthotropic circularplates of linearly varying thickness has been analysed forvarious values of plate parameters Transverse deflection andbendingmoments are obtained for circumferentially stiffenedplate that is 119864120579 gt 119864119903 Bending moments of the platecannot be obtained for radially stiffened plate 119864120579 gt 119864119903because infinite stress is developed at the centre in this case(Lekhnitskii [11] p 372) Transverse deflection and bendingmoments are presented here under different types of loadings

Advances in Acoustics and Vibration 5

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0002

0004

0006

0008

Wa

4q0D

r 0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005

0006

0007

Wa

4q0D

r 0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0002

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

Mra

2q0

(c) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus003

minus002

000

000

001

002

Mra

2q0

(d) Disk loading (CL-plate)

Figure 3 Radius vector versus radial bending moment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

(a) When load is distributed uniformly on the diskextending from 1198770 = 00 to 1198771 = 05 (disk loadedplate)

(b) When load is distributed uniformly on the annularregion extending from 1198770 = 03 to 1198771 = 07 (annularloaded plate)

(c) When load is distributed uniformly on the entireregion extending from 1198770 = 00 to 1198771 = 10

(uniformly loaded plate)The total load on the plate is the same in all three cases (shownin Figure 1(b))

The whole analysis of numerical results is presented inFigures 2 3 4 5 6(a) and 6(b) for Ω lt Ω00 and 119864120579119864119903 = 50for different values of plate parameters of simply supported(SS) and clamped (CL) edges The transverse deflection forall three cases of loading for SS- and CL-plate is presented inFigures 2 3(a) and 3(b) Figure 2(a) presents the transversedeflection of polar orthotropic simply supported plates alongthe radius vector 119877 for taper parameter 120572 = plusmn03 andfoundation parameter 119870119891 = 001 002 and 003 under diskloading The transverse deflection is maximum at the centreof the plate which decreases as the foundation parameter119870119891 increases keeping other plate parameters fixed Thus the

6 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0005

0000

0005

0010

0015

0020

0025

0030M

ra

2q0

(a) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus004

minus003

minus002

000

000

001

002

Mra

2q0

(b) Annular loading (CL-plate)

Mra

2q0

01 02 03 04 05 06 07 08 0900 10

Radius R

000

001

002

003

004

(c) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

minus009

minus008

minus007

minus006

minus005

minus004

minus003

minus002

minus001

000

000

002

Mra

2q0

(d) Uniform loading (CL-plate)

Figure 4 Radius vector versus radial bendingmoment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

elastic stability of the plate under loading is more if someelastic foundation to the structural component is providedFigure also shows that the transverse deflection for taperparameter 120572 = 03 (centrally thicker plate) is greater than thatfor120572 = minus03 (centrally thinner plate) the reason being greatermass attribution at the centre in case of centrally thickerplate The centrally thicker plate having higher amplitudeof deflection can be protected from being damaged usingfoundationThis type of structural components is highly usedin civil mechanical aeronautical and modern technologyThe idea of structural stability in these fields is very essentialfor design engineer to know before Figure 2(b) presents thattransverse deflection for clamped plate under disk loading

for the same plate parameters as in Figure 2(a) It has beenobserved that the transverse deflection gradually decreases asthe flexibility parameter119870120601 increases for elastically restrainedplate Thus the transverse deflection for clamped plate isalways less than that for simply supported plate keeping otherplate parameters fixed The effect of foundation parameter119870119891 on transverse deflection for clamped plate shows thatdeflection at the centre decreases more rapidly for centrallythicker plate than that for centrally thinner plate the reasonbeing more mass attribution for centrally thicker plateThough values are nearly close the foundation parameter119870119891 increases nature of variation of transverse deflection ofCL-plate which is found to be different Figures 2(c) and

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

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2 Advances in Acoustics and Vibration

structural components The stability of these componentsincreases with the support of elastic foundations Thus thestudy of the combined effect of transverse loads and elasticfoundation on vibrational characteristics of plates is of prac-tical importance Gupta et al [16] analysed the complicatingeffect of elastic foundation on elastic properties of the platesA number of research papers are available showing thestudy of transverse deflection and bending moments of polarorthotropicisotropic plates of variable thickness resting onelastic foundation [20ndash31] with complicating effects

The present work is concerned with the study of trans-verse loads on vibration of polar orthotropic circular platesof linearly varying thickness resting on Winkler type ofelastic foundationThe support of elastic foundation providesgreater stability to these structural components exposed totransverse loads An approximate solution is obtained by Ritzmethod employing functions based on static deflection givenbyLekhnitskii [11]Thepresent choice of functions has a fasterrate of convergence as compared to polynomial coordinatefunctions employed by Laura et al [13 21 26 27]

2 Analysis

Consider a thin circular plate of radius 119886 and variable thick-ness ℎ = ℎ(119903) resting on elastic foundation of modulus 119896119891elastically restrained against rotation by springs of stiffness119896120601 and subjected to 119875(119903) cos120596119905 type of excitation extendingfrom 119903 = 1199030 to 119903 = 1199031 Let (119903 120579) be the polar coordinates ofany point on the neutral surface of the circular plate referredto as the centre of the plate as origin (shown in Figure 1(a))

The maximum kinetic energy of the plate is given by

119879max =1

2

1205881205962int

119886

0int

2120587

0ℎ1199082119903119889120579 119889119903 (1)

where 119908 is the transverse deflection 120588 the mass density and120596 the frequency in radians per second

The maximum strain energy of the plate is given by

119880max =1

2

int

119886

0int

2120587

0[119863119903 (

1205972119908

1205971199032)

2

+ 2120592120579

1205972119908

1205971199032(

1

119903

120597119908

120597119903

)

+ 119863120579 (1

119903

120597119908

120597119903

)

2

+ 1198961198911199082] 119903 119889119903 119889120579 +

1

2

sdot 119886119896120601 int

2120587

0(

120597119908 (119886 120579)

120597119903

)

2

119889120579

(2)

where 1119896120601 is the rotational flexibility of the spring andflexural rigidities of the plate are

119863119903 =119864119903ℎ3

12 (1 minus 120592119903120592120579)

119863120579 =119864120579ℎ3

12 (1 minus 120592119903120592120579)

(3)

The work done by the external force 119875(119903) acting on the platein the direction parallel to 119911-axis is given by

119881max = int2120587

0119889120579int

1199031

1199030

119908119875 (119903) 119903 119889119903 (4)

3 Method of Solution Ritz Method

Ritz method requires that the functional

119869 (119908) = 119880max minus 119881max minus 119879max =1

2

sdot int

119886

0int

2120587

0[119863119903 (

1205972119908

1205971199032)

2

+ 2120592120579

1205972119908

1205971199032(

1

119903

120597119908

120597119903

)

+ 119863120579 (1

119903

120597119908

120597119903

)

2

+ 1198961198911199082] 119903 119889119903 119889120579 +

1

2

sdot 119886119896120601 int

2120587

0(

120597119908 (119886 120579)

120597119903

)

2

119889120579 minus int

2120587

0119889120579int

1199031

1199030

119908119875 (119903)

sdot 119903 119889119903 minus

1

2

1205881205962int

119886

0int

2120587

0ℎ1199082119903 119889120579 119889119903

(5)

be minimisedUniformly distributed load 1198750 extending from 1199030 to 1199031 is

given by

119875 (119903) = 1199020 =1198750

120587 (11990321 minus 11990320)[119880 (119903 minus 1199030) minus 119880 (119903 minus 1199031)] (6)

Introducing the nondimensional variables 119877 = 119903119886 1198771 =

1199031119886 1198770 = 1199030119886 and assuming the deflection function as119882(119877) = 119908(119886

41199020119863119903

0

)

119882 (119877) =

119898

sum

119894=0

119860 119894119865119894 (119877) =

119898

sum

119894=0

119860 119894 (1 + 1205721198941198774+ 1205731198941198771+119901) 1198772119894 (7)

where 119860 119894 are undetermined coefficients 1199012 = 119864120579119864119903 and 120572119894120573119894 are unknown constants to be determined from boundaryconditions (Leissa [1] p 14)

119870120601

119889119865119894 (1)

119889119877

= minus (1 minus 120572)3[

1198892119865119894

1198891198772+ 120592120579 (

1

119877

119889119865119894

119889119877

)]

119877=1

119865119894 (1) = 0

(8)

Linearly thickness variation of the plate is assumed as ℎ =ℎ0(1minus120572119877) where 120572 and ℎ0 are taper parameter and thicknessof the plate at the centre respectively

The choice of the function approximating the deflectionof the plate 119882(119877) given in (7) is based upon the staticdeflection polar orthotropic plates (Lekhnitskii [11]) whichhas faster rate of convergence (Gupta et al [16]) as comparedto the polynomial coordinate functions used by earlierresearchers [13 21 26 27]

Advances in Acoustics and Vibration 3

r0r1

k120601

h0

kfkf

r0r1

r1

k120601k120601

h0

a

k120601

(a)

Disk loaded plate Annular loaded plate Uniformly loaded plate

(b)

Figure 1 (a) Plate geometry with side and surface views (b) Loading structure of the plate

Introducing the nondimensional variables 119882 and 119877 in(5) the functional 119869(119882) becomes

119869 (119882) = 1205871198631199030

[int

1

0(1 minus 120572119877)

3(

1205972119882

1205971198772)

2

+ 2120592120579

1205972119882

1205971198772(

1

119877

120597119882

120597119877

) + 1199012(

1

119877

120597119882

120597119877

)

2

+ 1198701198911198822

sdot 119877 119889119877 + 119870120601 int

2120587

0(

120597119882 (1)

120597119877

)

2

119889120579 minus int

1198771

1198770

2119882 (119877) 119877 119889119877

minus Ω2int

1

0int

2120587

0(1 minus 120572119877)119882

2119877119889119877]

(9)

where

1198631199030

=

119864119903ℎ03

12 (1 minus 120592119903120592120579)

Ω2=

11988641205962120588ℎ0

1198631199030

119870119891 =

1198864119896119891

1198631199030

119870Φ =119886119896Φ

1198631199030

(10)

The minimisation of the functional 119869(119882) given by (9)requires

120597119869 (119882)

120597119860 119894

= 0 119894 = 0 1 2 3 119898 (11)

This leads to a system of nonhomogeneous equations in 119860119895

(119886119894119895 minus 119887119894119895)119860119895 = 119862119894 119894 119895 = 0 1 119898 (12)

where 119860 = [119886119894119895] and 119861 = [119887119894119895] are square matrices of order119898 + 1 given by

119886119894119895 = int

1

0(1 minus 120572119877)

3[11986510158401015840119894 11986510158401015840119895 + 2120592120579119865

10158401015840119894 (

1198651015840119895

119877

)

+ 1199012(

1198651015840119894

119877

)(

1198651015840119895

119877

) + 119870119891119865119894119865119895]119877119889119877 + 1198701206011198651015840119894 (1)

sdot 1198651015840119895 (1)

119887119894119895 = int

1

0(1 minus 120572119877) 119865119894119865119895119877119889119877

119862119894119895 = int

1

02119865119894119877119889119877

(13)

The solution of system of (12) gives the value of 119860 119894 and thusthe transverse deflection 119882 and the radial and transversebending moments

119872119903

11988621199020

= minus (1 minus 120572119877)3[

1198892119882

1198891198772+ 120592120579

1

119877

119889119882

119889119877

]

119872120579

11988621199020

= minus (1 minus 120572119877)3[120592120579

1198892119882

1198891198772+ 1199012 1

119877

119889119882

119889119877

]

(14)

are computed

4 Numerical Results

Numerical results have been calculated for different valuesof taper parameter 120572 (=plusmn03) 119864120579119864119903 (=50) and foundation

4 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005W

a4q0D

r 0

(a) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(b) Disk loading (CL-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

0005

Wa

4q0D

r 0

(c) Annular loading (SS-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(d) Annular loading (CL-plate)

Figure 2 Radius vector versus deflection parameter119882119886411990201198631199030 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

modulus 119870119891 (=001 002 003) for forced vibration of polarorthotropic circular plate with simply supported (SS-plate)and clamped (CL-plate) edges The plate is subjected underdifferent types of loadings such as uniform loading onentire plate annular loading and disk loading The naturalfrequencies for free vibration are obtained by putting 119875(119903) =0 In case of forced vibration the nondimensional frequencyparameter is taken as Ω = 120578Ω00 for Ω lt Ω00 and Ω =

Ω00 + 120578(Ω01 minus Ω00) for Ω00 lt Ω lt Ω01 where 120578 = 02 Thenormalised deflection and bendingmoments are obtained forfixed value of Poissonrsquos ratio as 03

5 Discussion

Forced axisymmetric response of polar orthotropic circularplates of linearly varying thickness has been analysed forvarious values of plate parameters Transverse deflection andbendingmoments are obtained for circumferentially stiffenedplate that is 119864120579 gt 119864119903 Bending moments of the platecannot be obtained for radially stiffened plate 119864120579 gt 119864119903because infinite stress is developed at the centre in this case(Lekhnitskii [11] p 372) Transverse deflection and bendingmoments are presented here under different types of loadings

Advances in Acoustics and Vibration 5

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0002

0004

0006

0008

Wa

4q0D

r 0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005

0006

0007

Wa

4q0D

r 0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0002

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

Mra

2q0

(c) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus003

minus002

000

000

001

002

Mra

2q0

(d) Disk loading (CL-plate)

Figure 3 Radius vector versus radial bending moment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

(a) When load is distributed uniformly on the diskextending from 1198770 = 00 to 1198771 = 05 (disk loadedplate)

(b) When load is distributed uniformly on the annularregion extending from 1198770 = 03 to 1198771 = 07 (annularloaded plate)

(c) When load is distributed uniformly on the entireregion extending from 1198770 = 00 to 1198771 = 10

(uniformly loaded plate)The total load on the plate is the same in all three cases (shownin Figure 1(b))

The whole analysis of numerical results is presented inFigures 2 3 4 5 6(a) and 6(b) for Ω lt Ω00 and 119864120579119864119903 = 50for different values of plate parameters of simply supported(SS) and clamped (CL) edges The transverse deflection forall three cases of loading for SS- and CL-plate is presented inFigures 2 3(a) and 3(b) Figure 2(a) presents the transversedeflection of polar orthotropic simply supported plates alongthe radius vector 119877 for taper parameter 120572 = plusmn03 andfoundation parameter 119870119891 = 001 002 and 003 under diskloading The transverse deflection is maximum at the centreof the plate which decreases as the foundation parameter119870119891 increases keeping other plate parameters fixed Thus the

6 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0005

0000

0005

0010

0015

0020

0025

0030M

ra

2q0

(a) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus004

minus003

minus002

000

000

001

002

Mra

2q0

(b) Annular loading (CL-plate)

Mra

2q0

01 02 03 04 05 06 07 08 0900 10

Radius R

000

001

002

003

004

(c) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

minus009

minus008

minus007

minus006

minus005

minus004

minus003

minus002

minus001

000

000

002

Mra

2q0

(d) Uniform loading (CL-plate)

Figure 4 Radius vector versus radial bendingmoment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

elastic stability of the plate under loading is more if someelastic foundation to the structural component is providedFigure also shows that the transverse deflection for taperparameter 120572 = 03 (centrally thicker plate) is greater than thatfor120572 = minus03 (centrally thinner plate) the reason being greatermass attribution at the centre in case of centrally thickerplate The centrally thicker plate having higher amplitudeof deflection can be protected from being damaged usingfoundationThis type of structural components is highly usedin civil mechanical aeronautical and modern technologyThe idea of structural stability in these fields is very essentialfor design engineer to know before Figure 2(b) presents thattransverse deflection for clamped plate under disk loading

for the same plate parameters as in Figure 2(a) It has beenobserved that the transverse deflection gradually decreases asthe flexibility parameter119870120601 increases for elastically restrainedplate Thus the transverse deflection for clamped plate isalways less than that for simply supported plate keeping otherplate parameters fixed The effect of foundation parameter119870119891 on transverse deflection for clamped plate shows thatdeflection at the centre decreases more rapidly for centrallythicker plate than that for centrally thinner plate the reasonbeing more mass attribution for centrally thicker plateThough values are nearly close the foundation parameter119870119891 increases nature of variation of transverse deflection ofCL-plate which is found to be different Figures 2(c) and

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 3

r0r1

k120601

h0

kfkf

r0r1

r1

k120601k120601

h0

a

k120601

(a)

Disk loaded plate Annular loaded plate Uniformly loaded plate

(b)

Figure 1 (a) Plate geometry with side and surface views (b) Loading structure of the plate

Introducing the nondimensional variables 119882 and 119877 in(5) the functional 119869(119882) becomes

119869 (119882) = 1205871198631199030

[int

1

0(1 minus 120572119877)

3(

1205972119882

1205971198772)

2

+ 2120592120579

1205972119882

1205971198772(

1

119877

120597119882

120597119877

) + 1199012(

1

119877

120597119882

120597119877

)

2

+ 1198701198911198822

sdot 119877 119889119877 + 119870120601 int

2120587

0(

120597119882 (1)

120597119877

)

2

119889120579 minus int

1198771

1198770

2119882 (119877) 119877 119889119877

minus Ω2int

1

0int

2120587

0(1 minus 120572119877)119882

2119877119889119877]

(9)

where

1198631199030

=

119864119903ℎ03

12 (1 minus 120592119903120592120579)

Ω2=

11988641205962120588ℎ0

1198631199030

119870119891 =

1198864119896119891

1198631199030

119870Φ =119886119896Φ

1198631199030

(10)

The minimisation of the functional 119869(119882) given by (9)requires

120597119869 (119882)

120597119860 119894

= 0 119894 = 0 1 2 3 119898 (11)

This leads to a system of nonhomogeneous equations in 119860119895

(119886119894119895 minus 119887119894119895)119860119895 = 119862119894 119894 119895 = 0 1 119898 (12)

where 119860 = [119886119894119895] and 119861 = [119887119894119895] are square matrices of order119898 + 1 given by

119886119894119895 = int

1

0(1 minus 120572119877)

3[11986510158401015840119894 11986510158401015840119895 + 2120592120579119865

10158401015840119894 (

1198651015840119895

119877

)

+ 1199012(

1198651015840119894

119877

)(

1198651015840119895

119877

) + 119870119891119865119894119865119895]119877119889119877 + 1198701206011198651015840119894 (1)

sdot 1198651015840119895 (1)

119887119894119895 = int

1

0(1 minus 120572119877) 119865119894119865119895119877119889119877

119862119894119895 = int

1

02119865119894119877119889119877

(13)

The solution of system of (12) gives the value of 119860 119894 and thusthe transverse deflection 119882 and the radial and transversebending moments

119872119903

11988621199020

= minus (1 minus 120572119877)3[

1198892119882

1198891198772+ 120592120579

1

119877

119889119882

119889119877

]

119872120579

11988621199020

= minus (1 minus 120572119877)3[120592120579

1198892119882

1198891198772+ 1199012 1

119877

119889119882

119889119877

]

(14)

are computed

4 Numerical Results

Numerical results have been calculated for different valuesof taper parameter 120572 (=plusmn03) 119864120579119864119903 (=50) and foundation

4 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005W

a4q0D

r 0

(a) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(b) Disk loading (CL-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

0005

Wa

4q0D

r 0

(c) Annular loading (SS-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(d) Annular loading (CL-plate)

Figure 2 Radius vector versus deflection parameter119882119886411990201198631199030 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

modulus 119870119891 (=001 002 003) for forced vibration of polarorthotropic circular plate with simply supported (SS-plate)and clamped (CL-plate) edges The plate is subjected underdifferent types of loadings such as uniform loading onentire plate annular loading and disk loading The naturalfrequencies for free vibration are obtained by putting 119875(119903) =0 In case of forced vibration the nondimensional frequencyparameter is taken as Ω = 120578Ω00 for Ω lt Ω00 and Ω =

Ω00 + 120578(Ω01 minus Ω00) for Ω00 lt Ω lt Ω01 where 120578 = 02 Thenormalised deflection and bendingmoments are obtained forfixed value of Poissonrsquos ratio as 03

5 Discussion

Forced axisymmetric response of polar orthotropic circularplates of linearly varying thickness has been analysed forvarious values of plate parameters Transverse deflection andbendingmoments are obtained for circumferentially stiffenedplate that is 119864120579 gt 119864119903 Bending moments of the platecannot be obtained for radially stiffened plate 119864120579 gt 119864119903because infinite stress is developed at the centre in this case(Lekhnitskii [11] p 372) Transverse deflection and bendingmoments are presented here under different types of loadings

Advances in Acoustics and Vibration 5

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0002

0004

0006

0008

Wa

4q0D

r 0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005

0006

0007

Wa

4q0D

r 0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0002

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

Mra

2q0

(c) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus003

minus002

000

000

001

002

Mra

2q0

(d) Disk loading (CL-plate)

Figure 3 Radius vector versus radial bending moment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

(a) When load is distributed uniformly on the diskextending from 1198770 = 00 to 1198771 = 05 (disk loadedplate)

(b) When load is distributed uniformly on the annularregion extending from 1198770 = 03 to 1198771 = 07 (annularloaded plate)

(c) When load is distributed uniformly on the entireregion extending from 1198770 = 00 to 1198771 = 10

(uniformly loaded plate)The total load on the plate is the same in all three cases (shownin Figure 1(b))

The whole analysis of numerical results is presented inFigures 2 3 4 5 6(a) and 6(b) for Ω lt Ω00 and 119864120579119864119903 = 50for different values of plate parameters of simply supported(SS) and clamped (CL) edges The transverse deflection forall three cases of loading for SS- and CL-plate is presented inFigures 2 3(a) and 3(b) Figure 2(a) presents the transversedeflection of polar orthotropic simply supported plates alongthe radius vector 119877 for taper parameter 120572 = plusmn03 andfoundation parameter 119870119891 = 001 002 and 003 under diskloading The transverse deflection is maximum at the centreof the plate which decreases as the foundation parameter119870119891 increases keeping other plate parameters fixed Thus the

6 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0005

0000

0005

0010

0015

0020

0025

0030M

ra

2q0

(a) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus004

minus003

minus002

000

000

001

002

Mra

2q0

(b) Annular loading (CL-plate)

Mra

2q0

01 02 03 04 05 06 07 08 0900 10

Radius R

000

001

002

003

004

(c) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

minus009

minus008

minus007

minus006

minus005

minus004

minus003

minus002

minus001

000

000

002

Mra

2q0

(d) Uniform loading (CL-plate)

Figure 4 Radius vector versus radial bendingmoment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

elastic stability of the plate under loading is more if someelastic foundation to the structural component is providedFigure also shows that the transverse deflection for taperparameter 120572 = 03 (centrally thicker plate) is greater than thatfor120572 = minus03 (centrally thinner plate) the reason being greatermass attribution at the centre in case of centrally thickerplate The centrally thicker plate having higher amplitudeof deflection can be protected from being damaged usingfoundationThis type of structural components is highly usedin civil mechanical aeronautical and modern technologyThe idea of structural stability in these fields is very essentialfor design engineer to know before Figure 2(b) presents thattransverse deflection for clamped plate under disk loading

for the same plate parameters as in Figure 2(a) It has beenobserved that the transverse deflection gradually decreases asthe flexibility parameter119870120601 increases for elastically restrainedplate Thus the transverse deflection for clamped plate isalways less than that for simply supported plate keeping otherplate parameters fixed The effect of foundation parameter119870119891 on transverse deflection for clamped plate shows thatdeflection at the centre decreases more rapidly for centrallythicker plate than that for centrally thinner plate the reasonbeing more mass attribution for centrally thicker plateThough values are nearly close the foundation parameter119870119891 increases nature of variation of transverse deflection ofCL-plate which is found to be different Figures 2(c) and

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005W

a4q0D

r 0

(a) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(b) Disk loading (CL-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

0005

Wa

4q0D

r 0

(c) Annular loading (SS-plate)

1002 03 04 05 06 07 08 090100

Radius R

0000

0000

0002

0003

0004

Wa

4q0D

r 0

(d) Annular loading (CL-plate)

Figure 2 Radius vector versus deflection parameter119882119886411990201198631199030 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

modulus 119870119891 (=001 002 003) for forced vibration of polarorthotropic circular plate with simply supported (SS-plate)and clamped (CL-plate) edges The plate is subjected underdifferent types of loadings such as uniform loading onentire plate annular loading and disk loading The naturalfrequencies for free vibration are obtained by putting 119875(119903) =0 In case of forced vibration the nondimensional frequencyparameter is taken as Ω = 120578Ω00 for Ω lt Ω00 and Ω =

Ω00 + 120578(Ω01 minus Ω00) for Ω00 lt Ω lt Ω01 where 120578 = 02 Thenormalised deflection and bendingmoments are obtained forfixed value of Poissonrsquos ratio as 03

5 Discussion

Forced axisymmetric response of polar orthotropic circularplates of linearly varying thickness has been analysed forvarious values of plate parameters Transverse deflection andbendingmoments are obtained for circumferentially stiffenedplate that is 119864120579 gt 119864119903 Bending moments of the platecannot be obtained for radially stiffened plate 119864120579 gt 119864119903because infinite stress is developed at the centre in this case(Lekhnitskii [11] p 372) Transverse deflection and bendingmoments are presented here under different types of loadings

Advances in Acoustics and Vibration 5

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0002

0004

0006

0008

Wa

4q0D

r 0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005

0006

0007

Wa

4q0D

r 0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0002

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

Mra

2q0

(c) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus003

minus002

000

000

001

002

Mra

2q0

(d) Disk loading (CL-plate)

Figure 3 Radius vector versus radial bending moment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

(a) When load is distributed uniformly on the diskextending from 1198770 = 00 to 1198771 = 05 (disk loadedplate)

(b) When load is distributed uniformly on the annularregion extending from 1198770 = 03 to 1198771 = 07 (annularloaded plate)

(c) When load is distributed uniformly on the entireregion extending from 1198770 = 00 to 1198771 = 10

(uniformly loaded plate)The total load on the plate is the same in all three cases (shownin Figure 1(b))

The whole analysis of numerical results is presented inFigures 2 3 4 5 6(a) and 6(b) for Ω lt Ω00 and 119864120579119864119903 = 50for different values of plate parameters of simply supported(SS) and clamped (CL) edges The transverse deflection forall three cases of loading for SS- and CL-plate is presented inFigures 2 3(a) and 3(b) Figure 2(a) presents the transversedeflection of polar orthotropic simply supported plates alongthe radius vector 119877 for taper parameter 120572 = plusmn03 andfoundation parameter 119870119891 = 001 002 and 003 under diskloading The transverse deflection is maximum at the centreof the plate which decreases as the foundation parameter119870119891 increases keeping other plate parameters fixed Thus the

6 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0005

0000

0005

0010

0015

0020

0025

0030M

ra

2q0

(a) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus004

minus003

minus002

000

000

001

002

Mra

2q0

(b) Annular loading (CL-plate)

Mra

2q0

01 02 03 04 05 06 07 08 0900 10

Radius R

000

001

002

003

004

(c) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

minus009

minus008

minus007

minus006

minus005

minus004

minus003

minus002

minus001

000

000

002

Mra

2q0

(d) Uniform loading (CL-plate)

Figure 4 Radius vector versus radial bendingmoment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

elastic stability of the plate under loading is more if someelastic foundation to the structural component is providedFigure also shows that the transverse deflection for taperparameter 120572 = 03 (centrally thicker plate) is greater than thatfor120572 = minus03 (centrally thinner plate) the reason being greatermass attribution at the centre in case of centrally thickerplate The centrally thicker plate having higher amplitudeof deflection can be protected from being damaged usingfoundationThis type of structural components is highly usedin civil mechanical aeronautical and modern technologyThe idea of structural stability in these fields is very essentialfor design engineer to know before Figure 2(b) presents thattransverse deflection for clamped plate under disk loading

for the same plate parameters as in Figure 2(a) It has beenobserved that the transverse deflection gradually decreases asthe flexibility parameter119870120601 increases for elastically restrainedplate Thus the transverse deflection for clamped plate isalways less than that for simply supported plate keeping otherplate parameters fixed The effect of foundation parameter119870119891 on transverse deflection for clamped plate shows thatdeflection at the centre decreases more rapidly for centrallythicker plate than that for centrally thinner plate the reasonbeing more mass attribution for centrally thicker plateThough values are nearly close the foundation parameter119870119891 increases nature of variation of transverse deflection ofCL-plate which is found to be different Figures 2(c) and

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 5

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0002

0004

0006

0008

Wa

4q0D

r 0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

0000

0000

0002

0003

0004

0005

0006

0007

Wa

4q0D

r 0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0002

0000

0002

0004

0006

0008

0010

0012

0014

0016

0018

Mra

2q0

(c) Disk loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus003

minus002

000

000

001

002

Mra

2q0

(d) Disk loading (CL-plate)

Figure 3 Radius vector versus radial bending moment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

(a) When load is distributed uniformly on the diskextending from 1198770 = 00 to 1198771 = 05 (disk loadedplate)

(b) When load is distributed uniformly on the annularregion extending from 1198770 = 03 to 1198771 = 07 (annularloaded plate)

(c) When load is distributed uniformly on the entireregion extending from 1198770 = 00 to 1198771 = 10

(uniformly loaded plate)The total load on the plate is the same in all three cases (shownin Figure 1(b))

The whole analysis of numerical results is presented inFigures 2 3 4 5 6(a) and 6(b) for Ω lt Ω00 and 119864120579119864119903 = 50for different values of plate parameters of simply supported(SS) and clamped (CL) edges The transverse deflection forall three cases of loading for SS- and CL-plate is presented inFigures 2 3(a) and 3(b) Figure 2(a) presents the transversedeflection of polar orthotropic simply supported plates alongthe radius vector 119877 for taper parameter 120572 = plusmn03 andfoundation parameter 119870119891 = 001 002 and 003 under diskloading The transverse deflection is maximum at the centreof the plate which decreases as the foundation parameter119870119891 increases keeping other plate parameters fixed Thus the

6 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0005

0000

0005

0010

0015

0020

0025

0030M

ra

2q0

(a) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus004

minus003

minus002

000

000

001

002

Mra

2q0

(b) Annular loading (CL-plate)

Mra

2q0

01 02 03 04 05 06 07 08 0900 10

Radius R

000

001

002

003

004

(c) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

minus009

minus008

minus007

minus006

minus005

minus004

minus003

minus002

minus001

000

000

002

Mra

2q0

(d) Uniform loading (CL-plate)

Figure 4 Radius vector versus radial bendingmoment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

elastic stability of the plate under loading is more if someelastic foundation to the structural component is providedFigure also shows that the transverse deflection for taperparameter 120572 = 03 (centrally thicker plate) is greater than thatfor120572 = minus03 (centrally thinner plate) the reason being greatermass attribution at the centre in case of centrally thickerplate The centrally thicker plate having higher amplitudeof deflection can be protected from being damaged usingfoundationThis type of structural components is highly usedin civil mechanical aeronautical and modern technologyThe idea of structural stability in these fields is very essentialfor design engineer to know before Figure 2(b) presents thattransverse deflection for clamped plate under disk loading

for the same plate parameters as in Figure 2(a) It has beenobserved that the transverse deflection gradually decreases asthe flexibility parameter119870120601 increases for elastically restrainedplate Thus the transverse deflection for clamped plate isalways less than that for simply supported plate keeping otherplate parameters fixed The effect of foundation parameter119870119891 on transverse deflection for clamped plate shows thatdeflection at the centre decreases more rapidly for centrallythicker plate than that for centrally thinner plate the reasonbeing more mass attribution for centrally thicker plateThough values are nearly close the foundation parameter119870119891 increases nature of variation of transverse deflection ofCL-plate which is found to be different Figures 2(c) and

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Advances in Acoustics and Vibration

01 02 03 04 05 06 07 08 09 1000

Radius R

minus0005

0000

0005

0010

0015

0020

0025

0030M

ra

2q0

(a) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus004

minus003

minus002

000

000

001

002

Mra

2q0

(b) Annular loading (CL-plate)

Mra

2q0

01 02 03 04 05 06 07 08 0900 10

Radius R

000

001

002

003

004

(c) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

minus009

minus008

minus007

minus006

minus005

minus004

minus003

minus002

minus001

000

000

002

Mra

2q0

(d) Uniform loading (CL-plate)

Figure 4 Radius vector versus radial bendingmoment11987211990311988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation parameter

119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

elastic stability of the plate under loading is more if someelastic foundation to the structural component is providedFigure also shows that the transverse deflection for taperparameter 120572 = 03 (centrally thicker plate) is greater than thatfor120572 = minus03 (centrally thinner plate) the reason being greatermass attribution at the centre in case of centrally thickerplate The centrally thicker plate having higher amplitudeof deflection can be protected from being damaged usingfoundationThis type of structural components is highly usedin civil mechanical aeronautical and modern technologyThe idea of structural stability in these fields is very essentialfor design engineer to know before Figure 2(b) presents thattransverse deflection for clamped plate under disk loading

for the same plate parameters as in Figure 2(a) It has beenobserved that the transverse deflection gradually decreases asthe flexibility parameter119870120601 increases for elastically restrainedplate Thus the transverse deflection for clamped plate isalways less than that for simply supported plate keeping otherplate parameters fixed The effect of foundation parameter119870119891 on transverse deflection for clamped plate shows thatdeflection at the centre decreases more rapidly for centrallythicker plate than that for centrally thinner plate the reasonbeing more mass attribution for centrally thicker plateThough values are nearly close the foundation parameter119870119891 increases nature of variation of transverse deflection ofCL-plate which is found to be different Figures 2(c) and

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 7

M120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(a) Disk loading (SS-plate)M

120579a

2q0

01 02 03 04 05 06 07 08 09 1000

Radius R

000

002

004

006

(b) Disk loading (CL-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

000

002

004

006

008

M120579a

2q0

(c) Annular loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus002

minus001

000

001

002

003

004

005

006M

120579a

2q0

(d) Annular loading (CL-plate)

Figure 5 Radius vector versus tangential bending moment11987212057911988621199020 for taper parameter 120572 = minus03 mdash and 120572 = 03 Keys foundation

parameter 119870119891 = 001 ∙ ∙ ∙ 119870119891 = 002 zzz and 119870119891 = 003

2(d) present the transverse deflection when load is annularextending from 1198770 = 03 to 1198771 = 07 for SS- and CL-platesrespectively Transverse deflection for annular loaded plate isfound to be greater than that for disk loaded plate of courseit depends on the position of annular region of loading Theeffect on deflection for simply supported plate is more thanthat of clamped plate Transverse deflection for uniformlyloaded plate is observed to be maximum as compared to diskand annular loaded plates as shown in Figures 3(a) and 3(b)

The radial bending moments for disk annular and uni-formly loaded SS- andCL-plates are presented in Figures 3(c)3(d) 4(a) 4(b) 4(c) and 4(d) Radial bending moment for

disk loaded plate is presented in Figures 3(c) and 3(d) Figuresshow that radial bending moment for 120572 = minus03 is greaterthan that for 120572 = 03 other plate parameters being fixedIt is observed that peak of the bending moment decreasesas the foundation parameter 119870119891 increases Maximum radialbending moment occurs at 119877 = 035 for disk loaded platefor these plate parameter values whereas in case of annularloaded plate peak of the bending moment occurs at 119877 = 05Peak of the bending moment shifts towards the edge 119877 = 10for uniformly loaded plate (shown in Figures 4(a) and 4(b))Radial bendingmoment at the edge increases as the flexibilityparameter 119870120601 increases and is maximum for C1 plate Radial

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Acoustics and Vibration

1002 03 04 05 06 07 08 090100

Radius R

000

002

004

006

008

010

012

014M

120579a

2q0

(a) Uniform loading (SS-plate)

01 02 03 04 05 06 07 08 09 1000

Radius R

minus001

000

001

002

003

004

005

006

007

008

009

M120579a

2q0

(b) Uniform loading (CL-plate)

01 02 03 04 05 06 07 08 0900 10

Radius R

000

002

004

006

008

010

012

014

Wa

4q0D

r 0

(c) Radius vector versus transverse deflection

01 02 03 04 05 06 07 08 0900 10

Radius R

minus015

000

015

030

045

Mra

2q0

(d) Radius vector versus bending moment

Figure 6 (∙) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate mdash and CL-plate respectively

bendingmoment for 120572 = minus03 is greater than that for 120572 = 03which decreases as the foundation parameter 119870119891 increasesMaximum bending moment at the edge occurs when theplate is uniformly loaded (Figures 4(c) and 4(d)) compared tothose of disk and annular loaded plates Tangential bendingmoments are also presented in Figures 5(a) 5(b) 5(c) 5(d)6(a) and 6(b) under three types of loadings Radial bendingmoment at the edge for SS orthotropic plate is zero but thetangential bending moment is nonzero Tangential bendingmoment at the edge for SS-plate is greater than that of C1plate for corresponding value of plate parametersThepresentchoice of function considered in the study has faster rate

of convergence as compared to the polynomial coordinatefunction used by earlier researchers [13 21 26 27] (seeFigure 7)

Table 1 compares the deflection and radial bendingmoment at the centre of the plate for Ω lt Ω00 and Ω00 ltΩ lt Ω01 for uniform isotropic plate obtained by Laura etal [24] using Ritz method Table 2 compares the transversedeflection of simply supported plate of variable thicknessFigure 6(c) compares the transverse deflection with the resultof Laura et al [26] for uniformly loaded isotropic circu-lar plate of constant thickness with simply supported andclamped edge respectively obtained by Glarkinrsquos method

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 9

4 6 8 10 12 142

Number of terms

0999

1001

1003

1005

1007

1009

1011

1013

1015

1017

1019

ΩΩ

lowast

Figure 7 Normalised frequency parameter ΩΩlowast for first three natural frequencies in fundamental mode ◻◻◻ second Mode andthird mode for CL-plate with 119864120579119864119903 = 50 119899 = 1 120572 = 03 Keys present method mdash and polynomial coordinate method whereΩlowast is the frequency obtained using 15 terms

Table 1 Comparison of displacement and radial bending moment at the centre as a function of 120578 (= Ω lt Ω00) and (= Ω00 lt Ω lt Ω01) foruniform isotropic circular plate

120578

Ω lt Ω00 Ω00 lt Ω lt Ω01

119882119886411990201198631199030

11987211990311988621199020 119882119886

411990201198631199030

11987211990311988621199020

Reference [24] Present Reference [24] Present Reference [24] Present Reference [24] Present

CL-plateΩ00 = 102158Ω01 = 39771

02 00160 001630 0085 008524 00110 001181 0084 008468

04 001872 001873 0099 009952 00055 000553 0051 005121

06 002479 002471 0135 013533 00040 000404 0039 003840

08 00446 004461 0253 025281 00044 000446 0080 008001

SS-plateΩ00 = 49351Ω01 = 29721

02 00664 006640 0215 021554 00226 002263 0094 009407

04 00760 007603 0248 024874 00094 000946 0052 005224

06 01001 010010 0331 033179 00062 0006238 0049 004948

08 01787 017877 0603 060332 00063 0006329 0076 007653

Table 2 Comparison of displacements for simply supported circu-lar plate of linearly varying thickness for ]120579 = 025

Taper parameter 120572 Reference [26] Presentminus05708 01817 01815minus00 006568 00656minus033333 01143 01127

The bending moments for same plate parameter values astaken by Laura et al [26] have been also compared andpresented in Figure 6(d) Comparison of these obtainedresults shows an excellent agreement with the result availablein the literature

6 Conclusion

The forced vibrational characteristics of polar orthotropiccircular plates resting on Winkler type of elastic foundationhave been studied using Ritz method The function basedon static deflection of polar orthotropic plates has been usedto approximate the transverse deflection of the plate Thepresent choice of function has faster rate of convergence ascompared to the polynomial coordinate function used byearlier researchers Study shows that the deflection attainsmaximum value at the centre of the plate under uniformloading The effect is influenced by thickness variation Cen-trally thicker plate has more deflection than that of centrallythinner plate the region being the more mass attribution atthe centre in case of centrally thicker plate While designing

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Advances in Acoustics and Vibration

a structural component stability is one of the importantfactors to be considered The result analysis shows that theconsideration of elastic foundation can protect the systemfrom fatigue failure under heavy loads

Competing Interests

The author declares that they have no competing interests

References

[1] A W Leissa ldquoVibration of platesrdquo Tech Rep sp-160 NASA1969

[2] A W Leissa ldquoRecent research in plate vibrations classicaltheoryrdquo The Shock and Vibration Digest vol 9 no 10 pp 13ndash24 1977

[3] A W Leissa ldquoRecent research in plate vibrations 1973ndash1976complicating effectsrdquoTheShock andVibrationDigest vol 10 no12 pp 21ndash35 1978

[4] A W Leissa ldquoLiterature review plate vibration research 1976ndash1980 classical theoryrdquo The Shock and Vibration Digest vol 13no 9 pp 11ndash22 1981

[5] AW Leissa ldquoPlate vibrations research 1976ndash1980 complicatingeffectsrdquoTheShock andVibrationDigest vol 13 no 10 pp 19ndash361981

[6] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partI classical theoryrdquoThe Shock and Vibration Digest vol 19 no 2pp 11ndash18 1987

[7] A W Leissa ldquoRecent studies in plate vibrations 1981ndash1985 partII complicating effectsrdquoThe Shock and Vibration Digest vol 19no 3 pp 10ndash24 1987

[8] J N Reddy Mechanics of Laminated Composite Plates Theoryand Analysis CRP Press Boca Raton Fla USA 1997

[9] R Lal and U S Gupta ldquoChebyshev polynomials in the studyof transverse vibrations of non-uniform rectangular orthotropicplatesrdquoTheShock andVibrationDigest vol 33 no 2 pp 103ndash1122001

[10] A S Sayyad and Y M Ghugal ldquoOn the free vibration analysisof laminated composite and sandwich plates a review of recentliterature with some numerical resultsrdquo Composite Structuresvol 129 pp 177ndash201 2015

[11] S G Lekhnitskii Anisotropic Plates Breach Science New YorkNY USA 1985

[12] S Chonan ldquoRandomvibration of an initially stressed thick plateon an elastic foundationrdquo Journal of Sound and Vibration vol71 no 1 pp 117ndash127 1980

[13] P A A Laura R H Gutierrez R Carnicer and H C SanzildquoFree vibrations of a solid circular plate of linearly varying thick-ness and attached to awinkler foundationrdquo Journal of Sound andVibration vol 144 no 1 pp 149ndash161 1991

[14] U S Gupta R Lal and S K Jain ldquoEffect of elastic foundationon axisymmetric vibrations of polar orthotropic circular platesof variable thicknessrdquo Journal of Sound and Vibration vol 139no 3 pp 503ndash513 1990

[15] A H Ansari and U S Gupta ldquoEffect of elastic foundation onasymmetric vibration of polar orthotropic parabolically taperedcircular platerdquo Indian Journal of Pure and Applied Mathematicsvol 30 no 10 pp 975ndash990 1999

[16] U S Gupta A H Ansari and S Sharma ldquoBuckling andvibration of polar orthotropic circular plate resting on Winkler

foundationrdquo Journal of Sound and Vibration vol 297 no 3ndash5pp 457ndash476 2006

[17] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadraturemethod forMindlin plates onWinkler foundationsrdquoInternational Journal of Mechanical Sciences vol 38 no 4 pp405ndash421 1996

[18] T M Wang and J E Stephens ldquoNatural frequencies of Timo-shenko beams on pasternak foundationsrdquo Journal of Sound andVibration vol 51 no 2 pp 149ndash155 1977

[19] B Bhattacharya ldquoFree vibration of plates on Vlasovrsquos founda-tionrdquo Journal of Sound andVibration vol 54 no 3 pp 464ndash4671977

[20] A K Upadhyay R Pandey and K K Shukla ldquoNonlineardynamic response of laminated composite plates subjectedto pulse loadingrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4530ndash4544 2011

[21] P A A Laura G C Pardoen L E Luisoni and D AvalosldquoTransverse vibrations of axisymmetric polar orthotropic circu-lar plates elastically restrained against rotation along the edgesrdquoFibre Science and Technology vol 15 no 1 pp 65ndash77 1981

[22] D J Gunaratnam and A P Bhattacharya ldquoTransverse vibra-tions and stability of polar orthotropic circular plates high-levelrelationshipsrdquo Journal of Sound andVibration vol 132 no 3 pp383ndash392 1989

[23] G N Weisensel and A L Schlack Jr ldquoAnnular plate responseto circumferentially moving loads with sudden radial positionchangesrdquo The International Journal of Analytical and Experi-mental Modal Analysis vol 5 no 4 pp 239ndash250 1990

[24] P A A Laura D R Avalos and H A Larrondo ldquoForcevibrations of circular stepped platesrdquo Journal of Sound andVibration vol 136 no 1 pp 146ndash150 1990

[25] G C Pardoen ldquoAsymmetric vibration and stability of circularplatesrdquo Computers and Structures vol 9 no 1 pp 89ndash95 1978

[26] P A A Laura C Filipich and R D Santos ldquoStatic and dynamicbehavior of circular plates of variable thickness elasticallyrestrained along the edgesrdquo Journal of Sound and Vibration vol52 no 2 pp 243ndash251 1977

[27] R H Gutierrez E Romanelli and P A A Laura ldquoVibrationsand elastic stability of thin circular plates with variable profilerdquoJournal of Sound andVibration vol 195 no 3 pp 391ndash399 1996

[28] B Bhushan G Singh and G Venkateswara Rao ldquoAsymmetricbuckling of layered orthotropic circular and annular plates ofvarying thickness using a computationally economic semiana-lytical finite element approachrdquo Computers and Structures vol59 no 1 pp 21ndash33 1996

[29] U S Gupta and A H Ansari ldquoAsymmetric vibrations andelastic stability of polar orthotropic circular plates of linearlyvarying profilerdquo Journal of Sound and Vibration vol 215 no 2pp 231ndash250 1998

[30] C S Kim and S M Dickinson ldquoThe flexural vibration ofthin isotropic and polar orthotropic annular and circular plateswith elastically restrained peripheriesrdquo Journal of Sound andVibration vol 143 no 1 pp 171ndash179 1990

[31] S Sharma R Lal and N Singh ldquoEffect of non-homogeneity onasymmetric vibrations of non-uniform circular platesrdquo Journalof Vibration and Control 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of