Research ArticleVibration Analysis of Annular Sector Plates underDifferent Boundary Conditions

Dongyan Shi1 Xianjie Shi1 Wen L Li2 Qingshan Wang1 and Jiashan Han3

1 College of Mechanical and Electrical Engineering Harbin Engineering University Harbin 150001 China2Department of Mechanical Engineering Wayne State University Detroit MI 48201 USA3 Luoyang Sunrui Special Equipment Co Ltd Luoyang 471003 China

Correspondence should be addressed to Xianjie Shi shixianjiehrbeueducn

Received 18 October 2013 Accepted 18 May 2014 Published 9 June 2014

Academic Editor Hassan Haddadpour

Copyright copy 2014 Dongyan Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An analytical framework is developed for the vibration analysis of annular sector plates with general elastic restraints along eachedge of plates Regardless of boundary conditions the displacement solution is invariably expressed as a new form of trigonometricexpansion with accelerated convergence The expansion coefficients are treated as the generalized coordinates and determinedusing the Rayleigh-Ritz technique This work allows a capability of modeling annular sector plates under a variety of boundaryconditions and changing the boundary conditions as easily as modifying the material properties or dimensions of the plates Ofequal importance the proposed approach is universally applicable to annular sector plates of any inclusion angles up to 2120587 Thereliability and accuracy of the current method are adequately validated through numerical examples

1 Introduction

Annular sector plates are one of themost important structuralcomponents used in industrial applications and civil engi-neering The vibrational characteristics of sector plates arethus of great interest to engineers and designers Althoughthere is a vast pool of studies about vibrations of circular andrectangular plates [1] relatively few results are reported forannular sector plates

Over the past decades vibrations of annular sectorplates have been investigated using various analytical ornumerical methods such as the energy method [2] splineelement method [3 4] finite element method [5] integralequation method [6] and so on [7] In particular a generaltechnique was developed by Leissa [1] to obtain exact modalfrequencies for plates which are simply supported along theradial edges and have arbitrary boundary conditions at thecircumferential edges This method utilizes the well-knownBessel function solutions for a circular plate by allowing thefunctions to have noninteger orders His following work [8]using Ritz method advocated that the use of the ordinaryBessel functions solution is incorrect for solid sector thinplates having simply supported radial edges and sector angle

larger than 120587 Liew et al [9] reviewed many investigationsabout the vibration of thick plates published before 1993 Itis shown that a majority of them are focused on the classicalboundary conditions (simply supported clamped or freeedges) In comparison other more complicated boundaryconditions such as elastic boundary supports are rarelyattempted A closed-form solution is proposed by Kim andYoo [10] in which the displacements are expressed in termsof trigonometric and exponential functions under the polarcoordinate system Ramakrishnan and Kunukkasseril [11]solved the vibration problem of an annular sector plate withsimply supported radial edges and arbitrary conditions alongthe circumferential edges Aghdam et al [12] presented anapproximate solution for bending deformation of thin sectorplates using extended Kantorovich method in which thefourth-order governing equation is converted into two ordi-nary differential equations Employing the pb-2 Rayleigh-Ritz method Xiang et al [13] tackled the vibration problemof annular sector Mindlin plates Frequency parameters forannular sector plates with different geometry parametersand boundary conditions were presented Liu and Chen [14]proposed an axisymmetric finite element for axisymmetricvibration analysis of annular and circular plates Civalek

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 517946 11 pageshttpdxdoiorg1011552014517946

2 Shock and Vibration

and Ulker [15] utilized harmonic differential quadrature(HDQ) method to study the linear bending characteristics ofcircular plates Civalek [16] compared the methods of differ-ential quadrature (DQ) and harmonic differential quadrature(HDQ) These methods were utilized for buckling bendingand free vibrations of thin isotropic plates Accurate three-dimensional elasticity solutions of annular sector plates arepresented under arbitrary boundary conditions by Liew et al[17] X Wang and Y Wang [18] extended the differentialquadrature (DQ) method to analyze the free vibrationproblem of thin sector plates Irie et al [19] investigatedthe free vibrations of ring-shaped polar-orthotropic sectorplates using a spline function as an admissible function forthe deflection of the plates In this approach the flexuraltransverse deflection of sector plates is expressed as a seriesof the products of the deflection functions of a sectorial beamand a circular beam that satisfy the similar type of boundaryconditions Three-dimensional vibrations of annular sectorplates with various boundary conditions were studied byZhou et al [20] using Chebyshev-Ritz method Also thesolutions of annular sector plates with reentrant angle arepresented in Zhou et al [20] investigation Baferani et al[21] presented an analytical solution for the free vibration offunctionally graded (FG) thin annular sector plates restingon elastic foundationsThe plates are considered to be simplysupported along radial edges and arbitrarily supported at thecircumferential edges Mirtalaie and Hajabasi [22] studiedthe free vibration analysis of functionally graded (FG) thinannular sector plates with DQ method

It appears that the previous investigations on the annularsector plates are mostly limited to classical edge conditionsIt is widely believed that an exact analytical solution isonly possible for an annular sector plate which is simplysupported along at least two radial edges However a varietyof possible boundary conditions such as elastic restraintsare usually encountered in many engineering applications[2 8 23]Moreover the existing solution procedures are oftenonly customized for a specific kind of boundary conditionsand thus typically require constant modifications of the trialfunctions and corresponding solution procedures to adaptto different boundary conditions As a result the use ofthe existing solution procedures will result in very tediouscalculations and will be easily inundated with a variety ofpossible boundary conditions Therefore it is important todevelop an analytical method which is capable of universallydealing with annular sector plates subjected to differentboundary conditions In addition the results of annularsector plates with reentrant angle are scarce

In this paper an improved Fourier series method (IFSM)previously proposed for the vibration analysis of beams andplates [24ndash28] is extended to annular sector plates underdifferent boundary conditions including the general elasticrestraints The displacement solution of the annular sectorplate regardless of boundary conditions is expressed asa new form of trigonometric expansion with acceleratedconvergence The reliability and accuracy of the proposedsolution technique are validated extensively through numer-ical examples

2 Theoretical Formulations

21 Basic Equations for an Annular Sector Plate An annularsector plate (consisted with two radial and two circum-ferential edges) and the coordinate systems used in thisinvestigation are shown in Figure 1 This plate is of constantthickness ℎ inner radius 119886 outer radius 119887 width 119877 of platein radial direction and sector angle 120601 The plate geometryand dimensions are defined in a cylindrical coordinate system(119903 120579 119911) A local coordinate system (119904 120579 119911) is also shown inFigure 1 which will be used in the analysis The boundaryconditions for the bending motion can be generally specifiedin terms of two kinds of restraining springs (translationaland rotational) along each edge resulting in four sets ofdistributed springs of arbitrary stiffness values

The governing differential equation for the free vibrationof an annular sector plate is given by

119863nabla2

119903nabla2

119903119908 (119903 120579) minus 120588ℎ120596

2119908 (119903 120579) = 0 (1)

where nabla2119903= 12059721205971199032+ 120597119903120597119903 + 120597

211990321205971205792 119908(119903 120579) is the flexural

displacement 120596 is angular frequency and 119863 = 119864ℎ3(12(1 minus1205832)) 120588 and ℎ are the flexural bending rigidity the mass

density and the thickness of the plate respectivelyIn terms of the flexural displacement the bending and

twisting moments and transverse shearing forces can beexpressed as

119872119903= minus 119863[

1205972119908

1205971199032+

120583

119903

(

120597119908

120597119903

+

1

119903

1205972119908

1205971205792)]

119872120579= minus 119863[

1

119903

(

120597119908

120597119903

+

1

119903

1205972119908

1205971205792) + 120583

1205972119908

1205971199032]

119872119903120579= minus (1 minus 120583)

119863

119903

[

1205972119908

120597119903120597120579

minus

1

119903

120597119908

120597120579

]

119876119903= minus 119863

120597 (nabla2119908)

120597119903

119876120579= minus 119863

1

119903

120597 (nabla2119908)

120597120579

(2)

where 120583 is Poissonrsquos ratioThe boundary conditions for an elastically restrained

annular sector plate are given as

119896119903119886119908 = 119876

119903

119870119903119886120597119908

120597119903

= minus119872119903

at 119903 = 119886

119896119903119887119908 = minus 119876

119903

119870119903119887120597119908

120597119903

= 119872119903

at 119903 = 119887

1198961205790119908 = 119876

120579

1198701205790120597119908

119903120597119903

= minus119872120579

at 120579 = 0

1198961205791119908 = minus 119876

120579

1198701205791120597119908

119903120597119903

= 119872120579

at 120579 = 120601

(3)

where 119896119903119886

and 119896119903119887

(1198961205790

and 1198961205791) are translational spring

constants and 119870119903119886

and 119870119903119887

(1198961205790

and 1198961205791) are the rotational

Shock and Vibration 3

a

b

Os

120579

r O998400

120601

(a)

r

z

O

R

h

(b)

Figure 1 Geometry and dimensions of an annular sector plate (a) Annular sector plate (b) Cross section of the annular sector plate

spring constants at 119903 = 119886 and 119887 (120579 = 0 and 120601) respectivelyAll the classical homogeneous boundary conditions can besimply considered as special cases when the spring constantsare either extremely large or substantially smallThe units forthe translational and rotational springs are Nm andNmradrespectively

The solution for the vibration problem of an annularsector plate can be generally written in the forms of Besselfunctions [1]

119908 (119903 120579) =

infin

sum

119899=0

[119860119899119869119899(119896119903) + 119861

119899119884119899(119896119903) + 119862

119899119868119899(119896119903)

+119863119899119870119899(119896119903)] cos 119899120579

+

infin

sum

119899=1

[119860lowast

119899119869119899(119896119903) + 119861

lowast

119899119884119899(119896119903) + 119862

lowast

119899119868119899(119896119903)

+ 119863lowast

119899119870119899(119896119903)] sin 119899120579

(4)

where 119896119903 equiv 120582 119869119899and 119884

119899are the Bessel functions of the first

and second kinds respectively and 119868119899and 119870

119899are modified

Bessel functions of the first and second kinds respectivelyThe coefficients 119860

119899 119863

119899 119860lowast119899 119863

lowast

119899 which determine

the shape of a mode are to be solved from the boundaryconditions

If the boundary conditions are symmetric with respect toone or more diameters of the plate the terms involving sin 119899120579are not needed and the solution (4) is simplified to

119908 (119903 120579) =

infin

sum

119899=0

[119860119899119869119899(119896119903) + 119861

119899119884119899(119896119903) + 119862

119899119868119899(119896119903)

+119863119899119870119899(119896119903)] cos 119899120579

(5)

The characteristic equation is derived by substitutingthe solution into the boundary conditions and setting thedeterminant of the resulting coefficient matrix equal to zeroThe eigenvalues are obtained as the roots of characteristicequation using an appropriate nonlinear root-searching algo-rithmThe eigenvalues can also be found approximately sincethe Bessel functions are tabulated in many mathematical

books or handbooks Regardless of what procedures areadopted the results are understandably dependent on thespecific set of boundary conditions involved The modalproperties for annular plates are comprehensively reviewed in[1] for various boundary conditions or complicating factorsHowever annular sector plates are rarely dealt with in theliterature This is evident from the fact that reviewing thevibrations of sector plates occupies only less than two pagesin the classical monograph [1]

22 An Accelerated Trigonometric Series Representation for theDisplacement Function In the previous papers [26 29] eachdisplacement component of a rectangular plate is expressedas a 2D Fourier cosine series supplemented by eight auxiliaryterms which are introduced to accelerate the convergence ofthe series expansion In this study a similar butmuch simplerand more concise form of series expansion is employed toexpand the flexural displacement of an annular sector platein local coordinate system (119904 120579 119911)

119908 (119904 120579) =

infin

sum

119898119899=minus4

119860119898119899120593119898(119904) 120593119899(120579) (119904 = 119903 minus 119886) (6)

where 119860119898119899

denotes the series expansion coefficients and

120593119898(119904) =

cos 120582119898119904 119898 ge 0

sin 120582119898119904 119898 lt 0

(120582119898=

119898120587

119877

) (7)

The basis function120593119899(120579) in the 120579-direction is also given by

(7) except for 120582119899= 119899120587120601 The sine terms in the above equa-

tion are introduced to overcome the potential discontinuitiesalong the edges of the plate of the displacement functionwhen it is periodically extended and sought in the form oftrigonometric series expansion As a result the Gibbs effectcan be eliminated and the convergence of the series expansioncan be substantially improved

To clarify this point consider a function 119891(119909) having119862119899minus1 continuity on the interval [0 120587] and the 119899th derivative

is absolutely integrable (the 119899th derivative may not exist at

4 Shock and Vibration

certain points) Denote the partial sum of the trigonometricseries as

F1198722119875[119891] (119909) =

119872

sum

119898=minus2119875

119886119898120593119898(119909) (8)

It can then be mathematically proven that the seriesexpansion coefficients satisfy

lim119898rarrinfin

1198861198981198982119875= 0 (for 2119875 le 119899) (9)

if the negatively indexed coefficients 119886119898(119898 lt 0) are

calculated from

119886119898=

119875

sum

119896=1

[(minus1)119898119891(2119896minus1)

(120587) + 119891(2119896minus1)

(0)]

times

sum1le1198951ltsdotsdotsdotlt119895

119875minus119896le1198751198951119895119875minus119896= 1198941199092

1198951

sdot sdot sdot 1199092

119895119875minus119896

119909119894prod119875

119895=1119895 = 119894(1199092

119895minus 1199092

119894)

119898 = 119909119894=

2119894 minus 1 if 119898 is odd2119894 if 119898 is even

(119894 = 1 2 119875)

(10)

More explicitly the convergence estimate (9) can beexpressed as

119886119898= O (119898

minus(2119875+1)) for 2119875 le 119899 (11)

which means

max0le119909le120587

1003816100381610038161003816119891 (119909) minus F

1198722119875[119891] (119909)

1003816100381610038161003816= O (119872

minus2119875) (12)

It is seen that convergence can be drastically improvedvirtually at no extra cost It should be pointed out thatthe convergence rate of the series expansion (8) can becontrolled by setting 119875 to any appropriate value In realityhowever the smoothness of the solution required for a givenboundary value problem is mathematically dictated by thehighest order of derivatives that appeared in the governingdifferential equation Take the current plate problem forexample The plate equation demands that the third-orderderivatives are continuous and the fourth-order derivativesexist everywhere over the surface area of the plate Accord-ingly one needs to set 119875 = 2 in seeking for a strong 1198623solution or 119875 = 1 for 1198621 solution in a weak formulationBecause the smoothness (or explicitly the convergence rate)of the current series expansion can be managed at willover the solution domain the unknown series expansioncoefficients can be obtained from either a weak or strongformulation In seeking for a strong form of solution theseries is required to simultaneously satisfy the governingequation and the boundary conditions exactly on a point-wise basis As a consequence the expansion coefficients arenot totally independent the negatively indexed coefficientsare related to the others via the boundary conditions Ina weak formulation such as the Rayleigh-Ritz techniquehowever all the expansion coefficients are considered asthe generalized coordinates independent from each other

The strong and weak solutions are mathematically equivalentif they are constructed with the same degree of smoothnessover the solution domain The Rayleigh-Ritz technique willbe adopted in this study since the solution can be obtainedmuch easily More importantly such a solution process ismore suitable for future modeling of built-up structures

23 Final System for an Annular Sector Plate For a purelybending plate the total potential energy can be expressed as

119881119901

=

119863

2

int

120601

0

int

119877

0

[(

1205972119908

1205971199042+

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

2

minus 2 (1 minus 120583)

1205972119908

1205971199042(

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

+2 (1 minus 120583)

120597

120597119904

(

1

119904 + 119886

120597119908

120597120579

)

2

2

](119904+119886) 119889119904 119889120579

(13)

By neglecting rotary inertia the kinetic energy of theannular sector plate is given by

119879 =

1

2

120588ℎ1205962int

119877

0

int

120601

0

1199082(119904 + 119886) 119889119904 119889120579 (14)

The potential energies stored in the boundary springs arecalculated as

119881119904=

1

2

int

120601

0

119886[1198961199031198861199082+ 119870119903119886(

120597119908

120597119904

)

2

]

119904=0

+119887[1198961199031198871199082+ 119870119903119887(

120597119908

120597119904

)

2

]

119904=119877

119889120579

+

1

2

int

119877

0

[11989612057901199082+ 1198701205790(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=0

+[11989612057911199082+ 1198701205791(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=120601

119889119904

(15)

The Lagrangian for the annular sector plate can begenerally expressed as

119871 = 119881119901+ 119881119904minus 119879 (16)

By substituting (6) into (16) and minimizing Lagrangianagainst all the unknown series expansion coefficients oneis able to obtain a system of linear algebraic equations in amatrix form as

(K minus 1205962M)E = 0 (17)

where E is a vector which contains all the unknown seriesexpansion coefficients and K and M are the stiffness andmass matrices respectively For conciseness the detailed

Shock and Vibration 5

Table 1 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for a completely clamped annular sector plate (119886119887 = 04 120601 = 1205873 and 120583 = 033)

Mode number1 2 3 4 5 6

119872 = 119873 = 5 85267 15014 19429 24366 26621 35817119872 = 119873 = 6 85253 15013 19427 24361 26620 35807119872 = 119873 = 7 85257 15010 19423 24361 26607 35805119872 = 119873 = 8 85251 15010 19423 24359 26607 35803119872 = 119873 = 9 85251 15010 19422 24359 26605 35803119872 = 119873 = 10 85250 15010 19422 24359 26605 35803119872 = 119873 = 11 85250 15010 19422 24359 26604 35802119872 = 119873 = 12 85250 15010 19422 24359 26604 35802FEM 85230 15008 19429 24369 26592 35856Reference [22] 85250 15010 19422 24359 26604 mdash

Table 2 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an annular sector plate with radial edges simply supported (119886119887 = 05 120601 = 1205874 and120583 = 03)

Circumferential edges Source Mode number1 2 3 4 5 6

Free

119872 = 119873 = 5 21069 66726 81606 14642 17612 17691119872 = 119873 = 7 21067 66723 81604 14641 17612 17690119872 = 119873 = 9 21067 66722 81604 14641 17612 17690119872 = 119873 = 11 21067 66722 81604 14641 17612 17690119872 = 119873 = 12 21067 66722 81604 14641 17612 17690Reference [22] 21067 66722 81604 14641 17612 17690Reference [30] 21067 66722 81604 14641 17612 17690

Simply supportedPresent 68379 15098 18960 27839 28359 38762

Reference [22] 68379 15098 18960 27839 28359 38762Reference [30] 68379 15098 18960 27839 28359 38764

ClampedPresent 10756 17882 26948 30584 34644 47629

Reference [22] 10757 17882 26949 30584 34646 47630Reference [30] 10758 17882 26949 30584 34646 47630

Table 3 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for completely free annular sector plates (119886119887 = 04 and 120583 = 03)

120601 Source Mode number1 2 3 4 5 6

1205876Present 61515 67249 11397 14951 17191 24495FEM 61516 67230 11397 14946 17194 24492

1205872Present 15647 23576 38428 53649 63390 70739FEM 15646 23572 38425 53645 63389 70740

21205873Present 10148 16348 24588 35673 44649 59663FEM 10148 16345 24584 35672 44645 59663

120587Present 70437 76871 15378 17404 28449 28559FEM 70435 76858 15374 17404 28445 28558

71205876Present 54291 64773 13094 13136 21512 24262FEM 54281 64769 13093 13131 21511 24258

31205872Present 28863 52324 88762 97849 14001 17608FEM 28858 52320 88759 97822 13999 17606

161205879Present 18048 38657 73593 77069 11229 13075FEM 18044 38653 73581 77066 11227 13074

2120587Present 12956 29318 57558 71511 99277 10581FEM 12963 29334 57582 71519 99283 10584

6 Shock and Vibration

Table 4 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for fully clamped annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205876

02 Present 18821 30011 41761 42957 57784 59988

04Present 18837 30506 41747 46114 60003 67178FEM 18841 30511 41776 46126 60035 67218

Reference [13] 18836 30504 41739 46100 59616 6720106 Present 21612 42267 45432 66294 72863 82091

1205872

02 Present 50283 87826 11399 13690 16536 19550

04Present 69822 95706 13896 17979 19538 20776FEM 69839 95709 13898 17985 19549 20784

Reference [13] 60835 95701 13896 17979 19551 2078206 Present 14427 15926 18724 22834 28534 35181

21205873

02 Present 41835 63382 93964 10476 12964 1335804 Present 65700 78393 10137 13372 17432 1757106 Present 14205 14960 16344 18416 21292 24907

120587

02 Present 37061 45338 59618 78667 98872 10043FEM 37043 45334 59767 78671 98962 10081

04 Present 63331 68008 76617 89647 10722 12875FEM 63329 68006 76606 89639 10714 12875

06 Present 14080 14364 14903 15659 16753 18184FEM 14064 14368 14998 15685 16762 18155

71205876

02 Present 36241 41835 52102 66284 83552 98083FEM 36244 41844 52088 66314 83364 98114

04 Present 62904 66110 72109 80996 93532 10883FEM 62900 66142 72044 81035 93320 10884

06 Present 14034 14253 14635 15185 15958 17005FEM 14036 14253 14628 15178 15925 16888

31205872

02 Present 35495 38369 44053 52411 63089 7571704 Present 62412 64212 67419 72312 79187 8787606 Present 14000 14083 14354 14675 15099 15598

161205879

02 Present 35192 37153 40797 46446 54017 6321204 Present 62298 63555 65696 69042 73579 7927306 Present 13994 14166 14246 14456 14748 15101

2120587

02 Present 35061 36520 39252 43464 49307 56495FEM 35056 36502 39208 43417 49199 56438

04 Present 62188 63167 64820 67263 70707 75100FEM 62192 65153 64821 67294 70663 75033

06 Present 13972 14068 14175 14344 14558 14837FEM 13988 14057 14174 14341 14561 14838

expressions for the stiffness and mass matrices are not shownhere

The eigenvalues (or natural frequencies) and eigenvectorsof annular sector plates can now be easily and directlydetermined from solving a standard matrix eigenvalue prob-lem (17) For a given natural frequency the correspondingeigenvector actually contains the series expansion coefficientswhich can be used to construct the physical mode shapebased on (6) Although this investigation is focused on thefree vibration of an annular sector plate the response ofthe annular sector plate to an applied load can be easilyconsidered by simply including the work done by this load

in the Lagrangian eventually leading to a force term onthe right side of (17) Since the displacement is constructedwith the same smoothness as required of a strong form ofsolution other variables of interest such as shear forces andpower flows can be calculated directly and perhaps moreaccurately by applying appropriate mathematical operationsto the displacement function

3 Result and Discussion

To demonstrate the accuracy and usefulness of the proposedtechnique several numerical examples will be presented in

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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2 Shock and Vibration

and Ulker [15] utilized harmonic differential quadrature(HDQ) method to study the linear bending characteristics ofcircular plates Civalek [16] compared the methods of differ-ential quadrature (DQ) and harmonic differential quadrature(HDQ) These methods were utilized for buckling bendingand free vibrations of thin isotropic plates Accurate three-dimensional elasticity solutions of annular sector plates arepresented under arbitrary boundary conditions by Liew et al[17] X Wang and Y Wang [18] extended the differentialquadrature (DQ) method to analyze the free vibrationproblem of thin sector plates Irie et al [19] investigatedthe free vibrations of ring-shaped polar-orthotropic sectorplates using a spline function as an admissible function forthe deflection of the plates In this approach the flexuraltransverse deflection of sector plates is expressed as a seriesof the products of the deflection functions of a sectorial beamand a circular beam that satisfy the similar type of boundaryconditions Three-dimensional vibrations of annular sectorplates with various boundary conditions were studied byZhou et al [20] using Chebyshev-Ritz method Also thesolutions of annular sector plates with reentrant angle arepresented in Zhou et al [20] investigation Baferani et al[21] presented an analytical solution for the free vibration offunctionally graded (FG) thin annular sector plates restingon elastic foundationsThe plates are considered to be simplysupported along radial edges and arbitrarily supported at thecircumferential edges Mirtalaie and Hajabasi [22] studiedthe free vibration analysis of functionally graded (FG) thinannular sector plates with DQ method

It appears that the previous investigations on the annularsector plates are mostly limited to classical edge conditionsIt is widely believed that an exact analytical solution isonly possible for an annular sector plate which is simplysupported along at least two radial edges However a varietyof possible boundary conditions such as elastic restraintsare usually encountered in many engineering applications[2 8 23]Moreover the existing solution procedures are oftenonly customized for a specific kind of boundary conditionsand thus typically require constant modifications of the trialfunctions and corresponding solution procedures to adaptto different boundary conditions As a result the use ofthe existing solution procedures will result in very tediouscalculations and will be easily inundated with a variety ofpossible boundary conditions Therefore it is important todevelop an analytical method which is capable of universallydealing with annular sector plates subjected to differentboundary conditions In addition the results of annularsector plates with reentrant angle are scarce

In this paper an improved Fourier series method (IFSM)previously proposed for the vibration analysis of beams andplates [24ndash28] is extended to annular sector plates underdifferent boundary conditions including the general elasticrestraints The displacement solution of the annular sectorplate regardless of boundary conditions is expressed asa new form of trigonometric expansion with acceleratedconvergence The reliability and accuracy of the proposedsolution technique are validated extensively through numer-ical examples

2 Theoretical Formulations

21 Basic Equations for an Annular Sector Plate An annularsector plate (consisted with two radial and two circum-ferential edges) and the coordinate systems used in thisinvestigation are shown in Figure 1 This plate is of constantthickness ℎ inner radius 119886 outer radius 119887 width 119877 of platein radial direction and sector angle 120601 The plate geometryand dimensions are defined in a cylindrical coordinate system(119903 120579 119911) A local coordinate system (119904 120579 119911) is also shown inFigure 1 which will be used in the analysis The boundaryconditions for the bending motion can be generally specifiedin terms of two kinds of restraining springs (translationaland rotational) along each edge resulting in four sets ofdistributed springs of arbitrary stiffness values

The governing differential equation for the free vibrationof an annular sector plate is given by

119863nabla2

119903nabla2

119903119908 (119903 120579) minus 120588ℎ120596

2119908 (119903 120579) = 0 (1)

where nabla2119903= 12059721205971199032+ 120597119903120597119903 + 120597

211990321205971205792 119908(119903 120579) is the flexural

displacement 120596 is angular frequency and 119863 = 119864ℎ3(12(1 minus1205832)) 120588 and ℎ are the flexural bending rigidity the mass

density and the thickness of the plate respectivelyIn terms of the flexural displacement the bending and

twisting moments and transverse shearing forces can beexpressed as

119872119903= minus 119863[

1205972119908

1205971199032+

120583

119903

(

120597119908

120597119903

+

1

119903

1205972119908

1205971205792)]

119872120579= minus 119863[

1

119903

(

120597119908

120597119903

+

1

119903

1205972119908

1205971205792) + 120583

1205972119908

1205971199032]

119872119903120579= minus (1 minus 120583)

119863

119903

[

1205972119908

120597119903120597120579

minus

1

119903

120597119908

120597120579

]

119876119903= minus 119863

120597 (nabla2119908)

120597119903

119876120579= minus 119863

1

119903

120597 (nabla2119908)

120597120579

(2)

where 120583 is Poissonrsquos ratioThe boundary conditions for an elastically restrained

annular sector plate are given as

119896119903119886119908 = 119876

119903

119870119903119886120597119908

120597119903

= minus119872119903

at 119903 = 119886

119896119903119887119908 = minus 119876

119903

119870119903119887120597119908

120597119903

= 119872119903

at 119903 = 119887

1198961205790119908 = 119876

120579

1198701205790120597119908

119903120597119903

= minus119872120579

at 120579 = 0

1198961205791119908 = minus 119876

120579

1198701205791120597119908

119903120597119903

= 119872120579

at 120579 = 120601

(3)

where 119896119903119886

and 119896119903119887

(1198961205790

and 1198961205791) are translational spring

constants and 119870119903119886

and 119870119903119887

(1198961205790

and 1198961205791) are the rotational

Shock and Vibration 3

a

b

Os

120579

r O998400

120601

(a)

r

z

O

R

h

(b)

Figure 1 Geometry and dimensions of an annular sector plate (a) Annular sector plate (b) Cross section of the annular sector plate

spring constants at 119903 = 119886 and 119887 (120579 = 0 and 120601) respectivelyAll the classical homogeneous boundary conditions can besimply considered as special cases when the spring constantsare either extremely large or substantially smallThe units forthe translational and rotational springs are Nm andNmradrespectively

The solution for the vibration problem of an annularsector plate can be generally written in the forms of Besselfunctions [1]

119908 (119903 120579) =

infin

sum

119899=0

[119860119899119869119899(119896119903) + 119861

119899119884119899(119896119903) + 119862

119899119868119899(119896119903)

+119863119899119870119899(119896119903)] cos 119899120579

+

infin

sum

119899=1

[119860lowast

119899119869119899(119896119903) + 119861

lowast

119899119884119899(119896119903) + 119862

lowast

119899119868119899(119896119903)

+ 119863lowast

119899119870119899(119896119903)] sin 119899120579

(4)

where 119896119903 equiv 120582 119869119899and 119884

119899are the Bessel functions of the first

and second kinds respectively and 119868119899and 119870

119899are modified

Bessel functions of the first and second kinds respectivelyThe coefficients 119860

119899 119863

119899 119860lowast119899 119863

lowast

119899 which determine

the shape of a mode are to be solved from the boundaryconditions

If the boundary conditions are symmetric with respect toone or more diameters of the plate the terms involving sin 119899120579are not needed and the solution (4) is simplified to

119908 (119903 120579) =

infin

sum

119899=0

[119860119899119869119899(119896119903) + 119861

119899119884119899(119896119903) + 119862

119899119868119899(119896119903)

+119863119899119870119899(119896119903)] cos 119899120579

(5)

The characteristic equation is derived by substitutingthe solution into the boundary conditions and setting thedeterminant of the resulting coefficient matrix equal to zeroThe eigenvalues are obtained as the roots of characteristicequation using an appropriate nonlinear root-searching algo-rithmThe eigenvalues can also be found approximately sincethe Bessel functions are tabulated in many mathematical

books or handbooks Regardless of what procedures areadopted the results are understandably dependent on thespecific set of boundary conditions involved The modalproperties for annular plates are comprehensively reviewed in[1] for various boundary conditions or complicating factorsHowever annular sector plates are rarely dealt with in theliterature This is evident from the fact that reviewing thevibrations of sector plates occupies only less than two pagesin the classical monograph [1]

22 An Accelerated Trigonometric Series Representation for theDisplacement Function In the previous papers [26 29] eachdisplacement component of a rectangular plate is expressedas a 2D Fourier cosine series supplemented by eight auxiliaryterms which are introduced to accelerate the convergence ofthe series expansion In this study a similar butmuch simplerand more concise form of series expansion is employed toexpand the flexural displacement of an annular sector platein local coordinate system (119904 120579 119911)

119908 (119904 120579) =

infin

sum

119898119899=minus4

119860119898119899120593119898(119904) 120593119899(120579) (119904 = 119903 minus 119886) (6)

where 119860119898119899

denotes the series expansion coefficients and

120593119898(119904) =

cos 120582119898119904 119898 ge 0

sin 120582119898119904 119898 lt 0

(120582119898=

119898120587

119877

) (7)

The basis function120593119899(120579) in the 120579-direction is also given by

(7) except for 120582119899= 119899120587120601 The sine terms in the above equa-

tion are introduced to overcome the potential discontinuitiesalong the edges of the plate of the displacement functionwhen it is periodically extended and sought in the form oftrigonometric series expansion As a result the Gibbs effectcan be eliminated and the convergence of the series expansioncan be substantially improved

To clarify this point consider a function 119891(119909) having119862119899minus1 continuity on the interval [0 120587] and the 119899th derivative

is absolutely integrable (the 119899th derivative may not exist at

4 Shock and Vibration

certain points) Denote the partial sum of the trigonometricseries as

F1198722119875[119891] (119909) =

119872

sum

119898=minus2119875

119886119898120593119898(119909) (8)

It can then be mathematically proven that the seriesexpansion coefficients satisfy

lim119898rarrinfin

1198861198981198982119875= 0 (for 2119875 le 119899) (9)

if the negatively indexed coefficients 119886119898(119898 lt 0) are

calculated from

119886119898=

119875

sum

119896=1

[(minus1)119898119891(2119896minus1)

(120587) + 119891(2119896minus1)

(0)]

times

sum1le1198951ltsdotsdotsdotlt119895

119875minus119896le1198751198951119895119875minus119896= 1198941199092

1198951

sdot sdot sdot 1199092

119895119875minus119896

119909119894prod119875

119895=1119895 = 119894(1199092

119895minus 1199092

119894)

119898 = 119909119894=

2119894 minus 1 if 119898 is odd2119894 if 119898 is even

(119894 = 1 2 119875)

(10)

More explicitly the convergence estimate (9) can beexpressed as

119886119898= O (119898

minus(2119875+1)) for 2119875 le 119899 (11)

which means

max0le119909le120587

1003816100381610038161003816119891 (119909) minus F

1198722119875[119891] (119909)

1003816100381610038161003816= O (119872

minus2119875) (12)

It is seen that convergence can be drastically improvedvirtually at no extra cost It should be pointed out thatthe convergence rate of the series expansion (8) can becontrolled by setting 119875 to any appropriate value In realityhowever the smoothness of the solution required for a givenboundary value problem is mathematically dictated by thehighest order of derivatives that appeared in the governingdifferential equation Take the current plate problem forexample The plate equation demands that the third-orderderivatives are continuous and the fourth-order derivativesexist everywhere over the surface area of the plate Accord-ingly one needs to set 119875 = 2 in seeking for a strong 1198623solution or 119875 = 1 for 1198621 solution in a weak formulationBecause the smoothness (or explicitly the convergence rate)of the current series expansion can be managed at willover the solution domain the unknown series expansioncoefficients can be obtained from either a weak or strongformulation In seeking for a strong form of solution theseries is required to simultaneously satisfy the governingequation and the boundary conditions exactly on a point-wise basis As a consequence the expansion coefficients arenot totally independent the negatively indexed coefficientsare related to the others via the boundary conditions Ina weak formulation such as the Rayleigh-Ritz techniquehowever all the expansion coefficients are considered asthe generalized coordinates independent from each other

The strong and weak solutions are mathematically equivalentif they are constructed with the same degree of smoothnessover the solution domain The Rayleigh-Ritz technique willbe adopted in this study since the solution can be obtainedmuch easily More importantly such a solution process ismore suitable for future modeling of built-up structures

23 Final System for an Annular Sector Plate For a purelybending plate the total potential energy can be expressed as

119881119901

=

119863

2

int

120601

0

int

119877

0

[(

1205972119908

1205971199042+

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

2

minus 2 (1 minus 120583)

1205972119908

1205971199042(

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

+2 (1 minus 120583)

120597

120597119904

(

1

119904 + 119886

120597119908

120597120579

)

2

2

](119904+119886) 119889119904 119889120579

(13)

By neglecting rotary inertia the kinetic energy of theannular sector plate is given by

119879 =

1

2

120588ℎ1205962int

119877

0

int

120601

0

1199082(119904 + 119886) 119889119904 119889120579 (14)

The potential energies stored in the boundary springs arecalculated as

119881119904=

1

2

int

120601

0

119886[1198961199031198861199082+ 119870119903119886(

120597119908

120597119904

)

2

]

119904=0

+119887[1198961199031198871199082+ 119870119903119887(

120597119908

120597119904

)

2

]

119904=119877

119889120579

+

1

2

int

119877

0

[11989612057901199082+ 1198701205790(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=0

+[11989612057911199082+ 1198701205791(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=120601

119889119904

(15)

The Lagrangian for the annular sector plate can begenerally expressed as

119871 = 119881119901+ 119881119904minus 119879 (16)

By substituting (6) into (16) and minimizing Lagrangianagainst all the unknown series expansion coefficients oneis able to obtain a system of linear algebraic equations in amatrix form as

(K minus 1205962M)E = 0 (17)

where E is a vector which contains all the unknown seriesexpansion coefficients and K and M are the stiffness andmass matrices respectively For conciseness the detailed

Shock and Vibration 5

Table 1 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for a completely clamped annular sector plate (119886119887 = 04 120601 = 1205873 and 120583 = 033)

Mode number1 2 3 4 5 6

119872 = 119873 = 5 85267 15014 19429 24366 26621 35817119872 = 119873 = 6 85253 15013 19427 24361 26620 35807119872 = 119873 = 7 85257 15010 19423 24361 26607 35805119872 = 119873 = 8 85251 15010 19423 24359 26607 35803119872 = 119873 = 9 85251 15010 19422 24359 26605 35803119872 = 119873 = 10 85250 15010 19422 24359 26605 35803119872 = 119873 = 11 85250 15010 19422 24359 26604 35802119872 = 119873 = 12 85250 15010 19422 24359 26604 35802FEM 85230 15008 19429 24369 26592 35856Reference [22] 85250 15010 19422 24359 26604 mdash

Table 2 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an annular sector plate with radial edges simply supported (119886119887 = 05 120601 = 1205874 and120583 = 03)

Circumferential edges Source Mode number1 2 3 4 5 6

Free

119872 = 119873 = 5 21069 66726 81606 14642 17612 17691119872 = 119873 = 7 21067 66723 81604 14641 17612 17690119872 = 119873 = 9 21067 66722 81604 14641 17612 17690119872 = 119873 = 11 21067 66722 81604 14641 17612 17690119872 = 119873 = 12 21067 66722 81604 14641 17612 17690Reference [22] 21067 66722 81604 14641 17612 17690Reference [30] 21067 66722 81604 14641 17612 17690

Simply supportedPresent 68379 15098 18960 27839 28359 38762

Reference [22] 68379 15098 18960 27839 28359 38762Reference [30] 68379 15098 18960 27839 28359 38764

ClampedPresent 10756 17882 26948 30584 34644 47629

Reference [22] 10757 17882 26949 30584 34646 47630Reference [30] 10758 17882 26949 30584 34646 47630

Table 3 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for completely free annular sector plates (119886119887 = 04 and 120583 = 03)

120601 Source Mode number1 2 3 4 5 6

1205876Present 61515 67249 11397 14951 17191 24495FEM 61516 67230 11397 14946 17194 24492

1205872Present 15647 23576 38428 53649 63390 70739FEM 15646 23572 38425 53645 63389 70740

21205873Present 10148 16348 24588 35673 44649 59663FEM 10148 16345 24584 35672 44645 59663

120587Present 70437 76871 15378 17404 28449 28559FEM 70435 76858 15374 17404 28445 28558

71205876Present 54291 64773 13094 13136 21512 24262FEM 54281 64769 13093 13131 21511 24258

31205872Present 28863 52324 88762 97849 14001 17608FEM 28858 52320 88759 97822 13999 17606

161205879Present 18048 38657 73593 77069 11229 13075FEM 18044 38653 73581 77066 11227 13074

2120587Present 12956 29318 57558 71511 99277 10581FEM 12963 29334 57582 71519 99283 10584

6 Shock and Vibration

Table 4 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for fully clamped annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205876

02 Present 18821 30011 41761 42957 57784 59988

04Present 18837 30506 41747 46114 60003 67178FEM 18841 30511 41776 46126 60035 67218

Reference [13] 18836 30504 41739 46100 59616 6720106 Present 21612 42267 45432 66294 72863 82091

1205872

02 Present 50283 87826 11399 13690 16536 19550

04Present 69822 95706 13896 17979 19538 20776FEM 69839 95709 13898 17985 19549 20784

Reference [13] 60835 95701 13896 17979 19551 2078206 Present 14427 15926 18724 22834 28534 35181

21205873

02 Present 41835 63382 93964 10476 12964 1335804 Present 65700 78393 10137 13372 17432 1757106 Present 14205 14960 16344 18416 21292 24907

120587

02 Present 37061 45338 59618 78667 98872 10043FEM 37043 45334 59767 78671 98962 10081

04 Present 63331 68008 76617 89647 10722 12875FEM 63329 68006 76606 89639 10714 12875

06 Present 14080 14364 14903 15659 16753 18184FEM 14064 14368 14998 15685 16762 18155

71205876

02 Present 36241 41835 52102 66284 83552 98083FEM 36244 41844 52088 66314 83364 98114

04 Present 62904 66110 72109 80996 93532 10883FEM 62900 66142 72044 81035 93320 10884

06 Present 14034 14253 14635 15185 15958 17005FEM 14036 14253 14628 15178 15925 16888

31205872

02 Present 35495 38369 44053 52411 63089 7571704 Present 62412 64212 67419 72312 79187 8787606 Present 14000 14083 14354 14675 15099 15598

161205879

02 Present 35192 37153 40797 46446 54017 6321204 Present 62298 63555 65696 69042 73579 7927306 Present 13994 14166 14246 14456 14748 15101

2120587

02 Present 35061 36520 39252 43464 49307 56495FEM 35056 36502 39208 43417 49199 56438

04 Present 62188 63167 64820 67263 70707 75100FEM 62192 65153 64821 67294 70663 75033

06 Present 13972 14068 14175 14344 14558 14837FEM 13988 14057 14174 14341 14561 14838

expressions for the stiffness and mass matrices are not shownhere

The eigenvalues (or natural frequencies) and eigenvectorsof annular sector plates can now be easily and directlydetermined from solving a standard matrix eigenvalue prob-lem (17) For a given natural frequency the correspondingeigenvector actually contains the series expansion coefficientswhich can be used to construct the physical mode shapebased on (6) Although this investigation is focused on thefree vibration of an annular sector plate the response ofthe annular sector plate to an applied load can be easilyconsidered by simply including the work done by this load

in the Lagrangian eventually leading to a force term onthe right side of (17) Since the displacement is constructedwith the same smoothness as required of a strong form ofsolution other variables of interest such as shear forces andpower flows can be calculated directly and perhaps moreaccurately by applying appropriate mathematical operationsto the displacement function

3 Result and Discussion

To demonstrate the accuracy and usefulness of the proposedtechnique several numerical examples will be presented in

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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Shock and Vibration 3

a

b

Os

120579

r O998400

120601

(a)

r

z

O

R

h

(b)

Figure 1 Geometry and dimensions of an annular sector plate (a) Annular sector plate (b) Cross section of the annular sector plate

spring constants at 119903 = 119886 and 119887 (120579 = 0 and 120601) respectivelyAll the classical homogeneous boundary conditions can besimply considered as special cases when the spring constantsare either extremely large or substantially smallThe units forthe translational and rotational springs are Nm andNmradrespectively

The solution for the vibration problem of an annularsector plate can be generally written in the forms of Besselfunctions [1]

119908 (119903 120579) =

infin

sum

119899=0

[119860119899119869119899(119896119903) + 119861

119899119884119899(119896119903) + 119862

119899119868119899(119896119903)

+119863119899119870119899(119896119903)] cos 119899120579

+

infin

sum

119899=1

[119860lowast

119899119869119899(119896119903) + 119861

lowast

119899119884119899(119896119903) + 119862

lowast

119899119868119899(119896119903)

+ 119863lowast

119899119870119899(119896119903)] sin 119899120579

(4)

where 119896119903 equiv 120582 119869119899and 119884

119899are the Bessel functions of the first

and second kinds respectively and 119868119899and 119870

119899are modified

Bessel functions of the first and second kinds respectivelyThe coefficients 119860

119899 119863

119899 119860lowast119899 119863

lowast

119899 which determine

the shape of a mode are to be solved from the boundaryconditions

If the boundary conditions are symmetric with respect toone or more diameters of the plate the terms involving sin 119899120579are not needed and the solution (4) is simplified to

119908 (119903 120579) =

infin

sum

119899=0

[119860119899119869119899(119896119903) + 119861

119899119884119899(119896119903) + 119862

119899119868119899(119896119903)

+119863119899119870119899(119896119903)] cos 119899120579

(5)

The characteristic equation is derived by substitutingthe solution into the boundary conditions and setting thedeterminant of the resulting coefficient matrix equal to zeroThe eigenvalues are obtained as the roots of characteristicequation using an appropriate nonlinear root-searching algo-rithmThe eigenvalues can also be found approximately sincethe Bessel functions are tabulated in many mathematical

books or handbooks Regardless of what procedures areadopted the results are understandably dependent on thespecific set of boundary conditions involved The modalproperties for annular plates are comprehensively reviewed in[1] for various boundary conditions or complicating factorsHowever annular sector plates are rarely dealt with in theliterature This is evident from the fact that reviewing thevibrations of sector plates occupies only less than two pagesin the classical monograph [1]

22 An Accelerated Trigonometric Series Representation for theDisplacement Function In the previous papers [26 29] eachdisplacement component of a rectangular plate is expressedas a 2D Fourier cosine series supplemented by eight auxiliaryterms which are introduced to accelerate the convergence ofthe series expansion In this study a similar butmuch simplerand more concise form of series expansion is employed toexpand the flexural displacement of an annular sector platein local coordinate system (119904 120579 119911)

119908 (119904 120579) =

infin

sum

119898119899=minus4

119860119898119899120593119898(119904) 120593119899(120579) (119904 = 119903 minus 119886) (6)

where 119860119898119899

denotes the series expansion coefficients and

120593119898(119904) =

cos 120582119898119904 119898 ge 0

sin 120582119898119904 119898 lt 0

(120582119898=

119898120587

119877

) (7)

The basis function120593119899(120579) in the 120579-direction is also given by

(7) except for 120582119899= 119899120587120601 The sine terms in the above equa-

tion are introduced to overcome the potential discontinuitiesalong the edges of the plate of the displacement functionwhen it is periodically extended and sought in the form oftrigonometric series expansion As a result the Gibbs effectcan be eliminated and the convergence of the series expansioncan be substantially improved

To clarify this point consider a function 119891(119909) having119862119899minus1 continuity on the interval [0 120587] and the 119899th derivative

is absolutely integrable (the 119899th derivative may not exist at

4 Shock and Vibration

certain points) Denote the partial sum of the trigonometricseries as

F1198722119875[119891] (119909) =

119872

sum

119898=minus2119875

119886119898120593119898(119909) (8)

It can then be mathematically proven that the seriesexpansion coefficients satisfy

lim119898rarrinfin

1198861198981198982119875= 0 (for 2119875 le 119899) (9)

if the negatively indexed coefficients 119886119898(119898 lt 0) are

calculated from

119886119898=

119875

sum

119896=1

[(minus1)119898119891(2119896minus1)

(120587) + 119891(2119896minus1)

(0)]

times

sum1le1198951ltsdotsdotsdotlt119895

119875minus119896le1198751198951119895119875minus119896= 1198941199092

1198951

sdot sdot sdot 1199092

119895119875minus119896

119909119894prod119875

119895=1119895 = 119894(1199092

119895minus 1199092

119894)

119898 = 119909119894=

2119894 minus 1 if 119898 is odd2119894 if 119898 is even

(119894 = 1 2 119875)

(10)

More explicitly the convergence estimate (9) can beexpressed as

119886119898= O (119898

minus(2119875+1)) for 2119875 le 119899 (11)

which means

max0le119909le120587

1003816100381610038161003816119891 (119909) minus F

1198722119875[119891] (119909)

1003816100381610038161003816= O (119872

minus2119875) (12)

It is seen that convergence can be drastically improvedvirtually at no extra cost It should be pointed out thatthe convergence rate of the series expansion (8) can becontrolled by setting 119875 to any appropriate value In realityhowever the smoothness of the solution required for a givenboundary value problem is mathematically dictated by thehighest order of derivatives that appeared in the governingdifferential equation Take the current plate problem forexample The plate equation demands that the third-orderderivatives are continuous and the fourth-order derivativesexist everywhere over the surface area of the plate Accord-ingly one needs to set 119875 = 2 in seeking for a strong 1198623solution or 119875 = 1 for 1198621 solution in a weak formulationBecause the smoothness (or explicitly the convergence rate)of the current series expansion can be managed at willover the solution domain the unknown series expansioncoefficients can be obtained from either a weak or strongformulation In seeking for a strong form of solution theseries is required to simultaneously satisfy the governingequation and the boundary conditions exactly on a point-wise basis As a consequence the expansion coefficients arenot totally independent the negatively indexed coefficientsare related to the others via the boundary conditions Ina weak formulation such as the Rayleigh-Ritz techniquehowever all the expansion coefficients are considered asthe generalized coordinates independent from each other

The strong and weak solutions are mathematically equivalentif they are constructed with the same degree of smoothnessover the solution domain The Rayleigh-Ritz technique willbe adopted in this study since the solution can be obtainedmuch easily More importantly such a solution process ismore suitable for future modeling of built-up structures

23 Final System for an Annular Sector Plate For a purelybending plate the total potential energy can be expressed as

119881119901

=

119863

2

int

120601

0

int

119877

0

[(

1205972119908

1205971199042+

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

2

minus 2 (1 minus 120583)

1205972119908

1205971199042(

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

+2 (1 minus 120583)

120597

120597119904

(

1

119904 + 119886

120597119908

120597120579

)

2

2

](119904+119886) 119889119904 119889120579

(13)

By neglecting rotary inertia the kinetic energy of theannular sector plate is given by

119879 =

1

2

120588ℎ1205962int

119877

0

int

120601

0

1199082(119904 + 119886) 119889119904 119889120579 (14)

The potential energies stored in the boundary springs arecalculated as

119881119904=

1

2

int

120601

0

119886[1198961199031198861199082+ 119870119903119886(

120597119908

120597119904

)

2

]

119904=0

+119887[1198961199031198871199082+ 119870119903119887(

120597119908

120597119904

)

2

]

119904=119877

119889120579

+

1

2

int

119877

0

[11989612057901199082+ 1198701205790(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=0

+[11989612057911199082+ 1198701205791(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=120601

119889119904

(15)

The Lagrangian for the annular sector plate can begenerally expressed as

119871 = 119881119901+ 119881119904minus 119879 (16)

By substituting (6) into (16) and minimizing Lagrangianagainst all the unknown series expansion coefficients oneis able to obtain a system of linear algebraic equations in amatrix form as

(K minus 1205962M)E = 0 (17)

where E is a vector which contains all the unknown seriesexpansion coefficients and K and M are the stiffness andmass matrices respectively For conciseness the detailed

Shock and Vibration 5

Table 1 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for a completely clamped annular sector plate (119886119887 = 04 120601 = 1205873 and 120583 = 033)

Mode number1 2 3 4 5 6

119872 = 119873 = 5 85267 15014 19429 24366 26621 35817119872 = 119873 = 6 85253 15013 19427 24361 26620 35807119872 = 119873 = 7 85257 15010 19423 24361 26607 35805119872 = 119873 = 8 85251 15010 19423 24359 26607 35803119872 = 119873 = 9 85251 15010 19422 24359 26605 35803119872 = 119873 = 10 85250 15010 19422 24359 26605 35803119872 = 119873 = 11 85250 15010 19422 24359 26604 35802119872 = 119873 = 12 85250 15010 19422 24359 26604 35802FEM 85230 15008 19429 24369 26592 35856Reference [22] 85250 15010 19422 24359 26604 mdash

Table 2 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an annular sector plate with radial edges simply supported (119886119887 = 05 120601 = 1205874 and120583 = 03)

Circumferential edges Source Mode number1 2 3 4 5 6

Free

119872 = 119873 = 5 21069 66726 81606 14642 17612 17691119872 = 119873 = 7 21067 66723 81604 14641 17612 17690119872 = 119873 = 9 21067 66722 81604 14641 17612 17690119872 = 119873 = 11 21067 66722 81604 14641 17612 17690119872 = 119873 = 12 21067 66722 81604 14641 17612 17690Reference [22] 21067 66722 81604 14641 17612 17690Reference [30] 21067 66722 81604 14641 17612 17690

Simply supportedPresent 68379 15098 18960 27839 28359 38762

Reference [22] 68379 15098 18960 27839 28359 38762Reference [30] 68379 15098 18960 27839 28359 38764

ClampedPresent 10756 17882 26948 30584 34644 47629

Reference [22] 10757 17882 26949 30584 34646 47630Reference [30] 10758 17882 26949 30584 34646 47630

Table 3 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for completely free annular sector plates (119886119887 = 04 and 120583 = 03)

120601 Source Mode number1 2 3 4 5 6

1205876Present 61515 67249 11397 14951 17191 24495FEM 61516 67230 11397 14946 17194 24492

1205872Present 15647 23576 38428 53649 63390 70739FEM 15646 23572 38425 53645 63389 70740

21205873Present 10148 16348 24588 35673 44649 59663FEM 10148 16345 24584 35672 44645 59663

120587Present 70437 76871 15378 17404 28449 28559FEM 70435 76858 15374 17404 28445 28558

71205876Present 54291 64773 13094 13136 21512 24262FEM 54281 64769 13093 13131 21511 24258

31205872Present 28863 52324 88762 97849 14001 17608FEM 28858 52320 88759 97822 13999 17606

161205879Present 18048 38657 73593 77069 11229 13075FEM 18044 38653 73581 77066 11227 13074

2120587Present 12956 29318 57558 71511 99277 10581FEM 12963 29334 57582 71519 99283 10584

6 Shock and Vibration

Table 4 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for fully clamped annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205876

02 Present 18821 30011 41761 42957 57784 59988

04Present 18837 30506 41747 46114 60003 67178FEM 18841 30511 41776 46126 60035 67218

Reference [13] 18836 30504 41739 46100 59616 6720106 Present 21612 42267 45432 66294 72863 82091

1205872

02 Present 50283 87826 11399 13690 16536 19550

04Present 69822 95706 13896 17979 19538 20776FEM 69839 95709 13898 17985 19549 20784

Reference [13] 60835 95701 13896 17979 19551 2078206 Present 14427 15926 18724 22834 28534 35181

21205873

02 Present 41835 63382 93964 10476 12964 1335804 Present 65700 78393 10137 13372 17432 1757106 Present 14205 14960 16344 18416 21292 24907

120587

02 Present 37061 45338 59618 78667 98872 10043FEM 37043 45334 59767 78671 98962 10081

04 Present 63331 68008 76617 89647 10722 12875FEM 63329 68006 76606 89639 10714 12875

06 Present 14080 14364 14903 15659 16753 18184FEM 14064 14368 14998 15685 16762 18155

71205876

02 Present 36241 41835 52102 66284 83552 98083FEM 36244 41844 52088 66314 83364 98114

04 Present 62904 66110 72109 80996 93532 10883FEM 62900 66142 72044 81035 93320 10884

06 Present 14034 14253 14635 15185 15958 17005FEM 14036 14253 14628 15178 15925 16888

31205872

02 Present 35495 38369 44053 52411 63089 7571704 Present 62412 64212 67419 72312 79187 8787606 Present 14000 14083 14354 14675 15099 15598

161205879

02 Present 35192 37153 40797 46446 54017 6321204 Present 62298 63555 65696 69042 73579 7927306 Present 13994 14166 14246 14456 14748 15101

2120587

02 Present 35061 36520 39252 43464 49307 56495FEM 35056 36502 39208 43417 49199 56438

04 Present 62188 63167 64820 67263 70707 75100FEM 62192 65153 64821 67294 70663 75033

06 Present 13972 14068 14175 14344 14558 14837FEM 13988 14057 14174 14341 14561 14838

expressions for the stiffness and mass matrices are not shownhere

The eigenvalues (or natural frequencies) and eigenvectorsof annular sector plates can now be easily and directlydetermined from solving a standard matrix eigenvalue prob-lem (17) For a given natural frequency the correspondingeigenvector actually contains the series expansion coefficientswhich can be used to construct the physical mode shapebased on (6) Although this investigation is focused on thefree vibration of an annular sector plate the response ofthe annular sector plate to an applied load can be easilyconsidered by simply including the work done by this load

in the Lagrangian eventually leading to a force term onthe right side of (17) Since the displacement is constructedwith the same smoothness as required of a strong form ofsolution other variables of interest such as shear forces andpower flows can be calculated directly and perhaps moreaccurately by applying appropriate mathematical operationsto the displacement function

3 Result and Discussion

To demonstrate the accuracy and usefulness of the proposedtechnique several numerical examples will be presented in

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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4 Shock and Vibration

certain points) Denote the partial sum of the trigonometricseries as

F1198722119875[119891] (119909) =

119872

sum

119898=minus2119875

119886119898120593119898(119909) (8)

It can then be mathematically proven that the seriesexpansion coefficients satisfy

lim119898rarrinfin

1198861198981198982119875= 0 (for 2119875 le 119899) (9)

if the negatively indexed coefficients 119886119898(119898 lt 0) are

calculated from

119886119898=

119875

sum

119896=1

[(minus1)119898119891(2119896minus1)

(120587) + 119891(2119896minus1)

(0)]

times

sum1le1198951ltsdotsdotsdotlt119895

119875minus119896le1198751198951119895119875minus119896= 1198941199092

1198951

sdot sdot sdot 1199092

119895119875minus119896

119909119894prod119875

119895=1119895 = 119894(1199092

119895minus 1199092

119894)

119898 = 119909119894=

2119894 minus 1 if 119898 is odd2119894 if 119898 is even

(119894 = 1 2 119875)

(10)

More explicitly the convergence estimate (9) can beexpressed as

119886119898= O (119898

minus(2119875+1)) for 2119875 le 119899 (11)

which means

max0le119909le120587

1003816100381610038161003816119891 (119909) minus F

1198722119875[119891] (119909)

1003816100381610038161003816= O (119872

minus2119875) (12)

It is seen that convergence can be drastically improvedvirtually at no extra cost It should be pointed out thatthe convergence rate of the series expansion (8) can becontrolled by setting 119875 to any appropriate value In realityhowever the smoothness of the solution required for a givenboundary value problem is mathematically dictated by thehighest order of derivatives that appeared in the governingdifferential equation Take the current plate problem forexample The plate equation demands that the third-orderderivatives are continuous and the fourth-order derivativesexist everywhere over the surface area of the plate Accord-ingly one needs to set 119875 = 2 in seeking for a strong 1198623solution or 119875 = 1 for 1198621 solution in a weak formulationBecause the smoothness (or explicitly the convergence rate)of the current series expansion can be managed at willover the solution domain the unknown series expansioncoefficients can be obtained from either a weak or strongformulation In seeking for a strong form of solution theseries is required to simultaneously satisfy the governingequation and the boundary conditions exactly on a point-wise basis As a consequence the expansion coefficients arenot totally independent the negatively indexed coefficientsare related to the others via the boundary conditions Ina weak formulation such as the Rayleigh-Ritz techniquehowever all the expansion coefficients are considered asthe generalized coordinates independent from each other

The strong and weak solutions are mathematically equivalentif they are constructed with the same degree of smoothnessover the solution domain The Rayleigh-Ritz technique willbe adopted in this study since the solution can be obtainedmuch easily More importantly such a solution process ismore suitable for future modeling of built-up structures

23 Final System for an Annular Sector Plate For a purelybending plate the total potential energy can be expressed as

119881119901

=

119863

2

int

120601

0

int

119877

0

[(

1205972119908

1205971199042+

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

2

minus 2 (1 minus 120583)

1205972119908

1205971199042(

1

119904 + 119886

120597119908

120597119904

+

1

(119904 + 119886)2

1205972119908

1205971205792)

+2 (1 minus 120583)

120597

120597119904

(

1

119904 + 119886

120597119908

120597120579

)

2

2

](119904+119886) 119889119904 119889120579

(13)

By neglecting rotary inertia the kinetic energy of theannular sector plate is given by

119879 =

1

2

120588ℎ1205962int

119877

0

int

120601

0

1199082(119904 + 119886) 119889119904 119889120579 (14)

The potential energies stored in the boundary springs arecalculated as

119881119904=

1

2

int

120601

0

119886[1198961199031198861199082+ 119870119903119886(

120597119908

120597119904

)

2

]

119904=0

+119887[1198961199031198871199082+ 119870119903119887(

120597119908

120597119904

)

2

]

119904=119877

119889120579

+

1

2

int

119877

0

[11989612057901199082+ 1198701205790(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=0

+[11989612057911199082+ 1198701205791(

120597119908

(119904 + 119886) 120597120579

)

2

]

120579=120601

119889119904

(15)

The Lagrangian for the annular sector plate can begenerally expressed as

119871 = 119881119901+ 119881119904minus 119879 (16)

By substituting (6) into (16) and minimizing Lagrangianagainst all the unknown series expansion coefficients oneis able to obtain a system of linear algebraic equations in amatrix form as

(K minus 1205962M)E = 0 (17)

where E is a vector which contains all the unknown seriesexpansion coefficients and K and M are the stiffness andmass matrices respectively For conciseness the detailed

Shock and Vibration 5

Table 1 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for a completely clamped annular sector plate (119886119887 = 04 120601 = 1205873 and 120583 = 033)

Mode number1 2 3 4 5 6

119872 = 119873 = 5 85267 15014 19429 24366 26621 35817119872 = 119873 = 6 85253 15013 19427 24361 26620 35807119872 = 119873 = 7 85257 15010 19423 24361 26607 35805119872 = 119873 = 8 85251 15010 19423 24359 26607 35803119872 = 119873 = 9 85251 15010 19422 24359 26605 35803119872 = 119873 = 10 85250 15010 19422 24359 26605 35803119872 = 119873 = 11 85250 15010 19422 24359 26604 35802119872 = 119873 = 12 85250 15010 19422 24359 26604 35802FEM 85230 15008 19429 24369 26592 35856Reference [22] 85250 15010 19422 24359 26604 mdash

Table 2 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an annular sector plate with radial edges simply supported (119886119887 = 05 120601 = 1205874 and120583 = 03)

Circumferential edges Source Mode number1 2 3 4 5 6

Free

119872 = 119873 = 5 21069 66726 81606 14642 17612 17691119872 = 119873 = 7 21067 66723 81604 14641 17612 17690119872 = 119873 = 9 21067 66722 81604 14641 17612 17690119872 = 119873 = 11 21067 66722 81604 14641 17612 17690119872 = 119873 = 12 21067 66722 81604 14641 17612 17690Reference [22] 21067 66722 81604 14641 17612 17690Reference [30] 21067 66722 81604 14641 17612 17690

Simply supportedPresent 68379 15098 18960 27839 28359 38762

Reference [22] 68379 15098 18960 27839 28359 38762Reference [30] 68379 15098 18960 27839 28359 38764

ClampedPresent 10756 17882 26948 30584 34644 47629

Reference [22] 10757 17882 26949 30584 34646 47630Reference [30] 10758 17882 26949 30584 34646 47630

Table 3 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for completely free annular sector plates (119886119887 = 04 and 120583 = 03)

120601 Source Mode number1 2 3 4 5 6

1205876Present 61515 67249 11397 14951 17191 24495FEM 61516 67230 11397 14946 17194 24492

1205872Present 15647 23576 38428 53649 63390 70739FEM 15646 23572 38425 53645 63389 70740

21205873Present 10148 16348 24588 35673 44649 59663FEM 10148 16345 24584 35672 44645 59663

120587Present 70437 76871 15378 17404 28449 28559FEM 70435 76858 15374 17404 28445 28558

71205876Present 54291 64773 13094 13136 21512 24262FEM 54281 64769 13093 13131 21511 24258

31205872Present 28863 52324 88762 97849 14001 17608FEM 28858 52320 88759 97822 13999 17606

161205879Present 18048 38657 73593 77069 11229 13075FEM 18044 38653 73581 77066 11227 13074

2120587Present 12956 29318 57558 71511 99277 10581FEM 12963 29334 57582 71519 99283 10584

6 Shock and Vibration

Table 4 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for fully clamped annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205876

02 Present 18821 30011 41761 42957 57784 59988

04Present 18837 30506 41747 46114 60003 67178FEM 18841 30511 41776 46126 60035 67218

Reference [13] 18836 30504 41739 46100 59616 6720106 Present 21612 42267 45432 66294 72863 82091

1205872

02 Present 50283 87826 11399 13690 16536 19550

04Present 69822 95706 13896 17979 19538 20776FEM 69839 95709 13898 17985 19549 20784

Reference [13] 60835 95701 13896 17979 19551 2078206 Present 14427 15926 18724 22834 28534 35181

21205873

02 Present 41835 63382 93964 10476 12964 1335804 Present 65700 78393 10137 13372 17432 1757106 Present 14205 14960 16344 18416 21292 24907

120587

02 Present 37061 45338 59618 78667 98872 10043FEM 37043 45334 59767 78671 98962 10081

04 Present 63331 68008 76617 89647 10722 12875FEM 63329 68006 76606 89639 10714 12875

06 Present 14080 14364 14903 15659 16753 18184FEM 14064 14368 14998 15685 16762 18155

71205876

02 Present 36241 41835 52102 66284 83552 98083FEM 36244 41844 52088 66314 83364 98114

04 Present 62904 66110 72109 80996 93532 10883FEM 62900 66142 72044 81035 93320 10884

06 Present 14034 14253 14635 15185 15958 17005FEM 14036 14253 14628 15178 15925 16888

31205872

02 Present 35495 38369 44053 52411 63089 7571704 Present 62412 64212 67419 72312 79187 8787606 Present 14000 14083 14354 14675 15099 15598

161205879

02 Present 35192 37153 40797 46446 54017 6321204 Present 62298 63555 65696 69042 73579 7927306 Present 13994 14166 14246 14456 14748 15101

2120587

02 Present 35061 36520 39252 43464 49307 56495FEM 35056 36502 39208 43417 49199 56438

04 Present 62188 63167 64820 67263 70707 75100FEM 62192 65153 64821 67294 70663 75033

06 Present 13972 14068 14175 14344 14558 14837FEM 13988 14057 14174 14341 14561 14838

expressions for the stiffness and mass matrices are not shownhere

The eigenvalues (or natural frequencies) and eigenvectorsof annular sector plates can now be easily and directlydetermined from solving a standard matrix eigenvalue prob-lem (17) For a given natural frequency the correspondingeigenvector actually contains the series expansion coefficientswhich can be used to construct the physical mode shapebased on (6) Although this investigation is focused on thefree vibration of an annular sector plate the response ofthe annular sector plate to an applied load can be easilyconsidered by simply including the work done by this load

in the Lagrangian eventually leading to a force term onthe right side of (17) Since the displacement is constructedwith the same smoothness as required of a strong form ofsolution other variables of interest such as shear forces andpower flows can be calculated directly and perhaps moreaccurately by applying appropriate mathematical operationsto the displacement function

3 Result and Discussion

To demonstrate the accuracy and usefulness of the proposedtechnique several numerical examples will be presented in

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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

Shock and Vibration 5

Table 1 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for a completely clamped annular sector plate (119886119887 = 04 120601 = 1205873 and 120583 = 033)

Mode number1 2 3 4 5 6

119872 = 119873 = 5 85267 15014 19429 24366 26621 35817119872 = 119873 = 6 85253 15013 19427 24361 26620 35807119872 = 119873 = 7 85257 15010 19423 24361 26607 35805119872 = 119873 = 8 85251 15010 19423 24359 26607 35803119872 = 119873 = 9 85251 15010 19422 24359 26605 35803119872 = 119873 = 10 85250 15010 19422 24359 26605 35803119872 = 119873 = 11 85250 15010 19422 24359 26604 35802119872 = 119873 = 12 85250 15010 19422 24359 26604 35802FEM 85230 15008 19429 24369 26592 35856Reference [22] 85250 15010 19422 24359 26604 mdash

Table 2 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an annular sector plate with radial edges simply supported (119886119887 = 05 120601 = 1205874 and120583 = 03)

Circumferential edges Source Mode number1 2 3 4 5 6

Free

119872 = 119873 = 5 21069 66726 81606 14642 17612 17691119872 = 119873 = 7 21067 66723 81604 14641 17612 17690119872 = 119873 = 9 21067 66722 81604 14641 17612 17690119872 = 119873 = 11 21067 66722 81604 14641 17612 17690119872 = 119873 = 12 21067 66722 81604 14641 17612 17690Reference [22] 21067 66722 81604 14641 17612 17690Reference [30] 21067 66722 81604 14641 17612 17690

Simply supportedPresent 68379 15098 18960 27839 28359 38762

Reference [22] 68379 15098 18960 27839 28359 38762Reference [30] 68379 15098 18960 27839 28359 38764

ClampedPresent 10756 17882 26948 30584 34644 47629

Reference [22] 10757 17882 26949 30584 34646 47630Reference [30] 10758 17882 26949 30584 34646 47630

Table 3 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for completely free annular sector plates (119886119887 = 04 and 120583 = 03)

120601 Source Mode number1 2 3 4 5 6

1205876Present 61515 67249 11397 14951 17191 24495FEM 61516 67230 11397 14946 17194 24492

1205872Present 15647 23576 38428 53649 63390 70739FEM 15646 23572 38425 53645 63389 70740

21205873Present 10148 16348 24588 35673 44649 59663FEM 10148 16345 24584 35672 44645 59663

120587Present 70437 76871 15378 17404 28449 28559FEM 70435 76858 15374 17404 28445 28558

71205876Present 54291 64773 13094 13136 21512 24262FEM 54281 64769 13093 13131 21511 24258

31205872Present 28863 52324 88762 97849 14001 17608FEM 28858 52320 88759 97822 13999 17606

161205879Present 18048 38657 73593 77069 11229 13075FEM 18044 38653 73581 77066 11227 13074

2120587Present 12956 29318 57558 71511 99277 10581FEM 12963 29334 57582 71519 99283 10584

6 Shock and Vibration

Table 4 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for fully clamped annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205876

02 Present 18821 30011 41761 42957 57784 59988

04Present 18837 30506 41747 46114 60003 67178FEM 18841 30511 41776 46126 60035 67218

Reference [13] 18836 30504 41739 46100 59616 6720106 Present 21612 42267 45432 66294 72863 82091

1205872

02 Present 50283 87826 11399 13690 16536 19550

04Present 69822 95706 13896 17979 19538 20776FEM 69839 95709 13898 17985 19549 20784

Reference [13] 60835 95701 13896 17979 19551 2078206 Present 14427 15926 18724 22834 28534 35181

21205873

02 Present 41835 63382 93964 10476 12964 1335804 Present 65700 78393 10137 13372 17432 1757106 Present 14205 14960 16344 18416 21292 24907

120587

02 Present 37061 45338 59618 78667 98872 10043FEM 37043 45334 59767 78671 98962 10081

04 Present 63331 68008 76617 89647 10722 12875FEM 63329 68006 76606 89639 10714 12875

06 Present 14080 14364 14903 15659 16753 18184FEM 14064 14368 14998 15685 16762 18155

71205876

02 Present 36241 41835 52102 66284 83552 98083FEM 36244 41844 52088 66314 83364 98114

04 Present 62904 66110 72109 80996 93532 10883FEM 62900 66142 72044 81035 93320 10884

06 Present 14034 14253 14635 15185 15958 17005FEM 14036 14253 14628 15178 15925 16888

31205872

02 Present 35495 38369 44053 52411 63089 7571704 Present 62412 64212 67419 72312 79187 8787606 Present 14000 14083 14354 14675 15099 15598

161205879

02 Present 35192 37153 40797 46446 54017 6321204 Present 62298 63555 65696 69042 73579 7927306 Present 13994 14166 14246 14456 14748 15101

2120587

02 Present 35061 36520 39252 43464 49307 56495FEM 35056 36502 39208 43417 49199 56438

04 Present 62188 63167 64820 67263 70707 75100FEM 62192 65153 64821 67294 70663 75033

06 Present 13972 14068 14175 14344 14558 14837FEM 13988 14057 14174 14341 14561 14838

expressions for the stiffness and mass matrices are not shownhere

The eigenvalues (or natural frequencies) and eigenvectorsof annular sector plates can now be easily and directlydetermined from solving a standard matrix eigenvalue prob-lem (17) For a given natural frequency the correspondingeigenvector actually contains the series expansion coefficientswhich can be used to construct the physical mode shapebased on (6) Although this investigation is focused on thefree vibration of an annular sector plate the response ofthe annular sector plate to an applied load can be easilyconsidered by simply including the work done by this load

in the Lagrangian eventually leading to a force term onthe right side of (17) Since the displacement is constructedwith the same smoothness as required of a strong form ofsolution other variables of interest such as shear forces andpower flows can be calculated directly and perhaps moreaccurately by applying appropriate mathematical operationsto the displacement function

3 Result and Discussion

To demonstrate the accuracy and usefulness of the proposedtechnique several numerical examples will be presented in

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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

6 Shock and Vibration

Table 4 Frequency parametersΩ = 1205961198872(120588ℎ119863)12 for fully clamped annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205876

02 Present 18821 30011 41761 42957 57784 59988

04Present 18837 30506 41747 46114 60003 67178FEM 18841 30511 41776 46126 60035 67218

Reference [13] 18836 30504 41739 46100 59616 6720106 Present 21612 42267 45432 66294 72863 82091

1205872

02 Present 50283 87826 11399 13690 16536 19550

04Present 69822 95706 13896 17979 19538 20776FEM 69839 95709 13898 17985 19549 20784

Reference [13] 60835 95701 13896 17979 19551 2078206 Present 14427 15926 18724 22834 28534 35181

21205873

02 Present 41835 63382 93964 10476 12964 1335804 Present 65700 78393 10137 13372 17432 1757106 Present 14205 14960 16344 18416 21292 24907

120587

02 Present 37061 45338 59618 78667 98872 10043FEM 37043 45334 59767 78671 98962 10081

04 Present 63331 68008 76617 89647 10722 12875FEM 63329 68006 76606 89639 10714 12875

06 Present 14080 14364 14903 15659 16753 18184FEM 14064 14368 14998 15685 16762 18155

71205876

02 Present 36241 41835 52102 66284 83552 98083FEM 36244 41844 52088 66314 83364 98114

04 Present 62904 66110 72109 80996 93532 10883FEM 62900 66142 72044 81035 93320 10884

06 Present 14034 14253 14635 15185 15958 17005FEM 14036 14253 14628 15178 15925 16888

31205872

02 Present 35495 38369 44053 52411 63089 7571704 Present 62412 64212 67419 72312 79187 8787606 Present 14000 14083 14354 14675 15099 15598

161205879

02 Present 35192 37153 40797 46446 54017 6321204 Present 62298 63555 65696 69042 73579 7927306 Present 13994 14166 14246 14456 14748 15101

2120587

02 Present 35061 36520 39252 43464 49307 56495FEM 35056 36502 39208 43417 49199 56438

04 Present 62188 63167 64820 67263 70707 75100FEM 62192 65153 64821 67294 70663 75033

06 Present 13972 14068 14175 14344 14558 14837FEM 13988 14057 14174 14341 14561 14838

expressions for the stiffness and mass matrices are not shownhere

The eigenvalues (or natural frequencies) and eigenvectorsof annular sector plates can now be easily and directlydetermined from solving a standard matrix eigenvalue prob-lem (17) For a given natural frequency the correspondingeigenvector actually contains the series expansion coefficientswhich can be used to construct the physical mode shapebased on (6) Although this investigation is focused on thefree vibration of an annular sector plate the response ofthe annular sector plate to an applied load can be easilyconsidered by simply including the work done by this load

in the Lagrangian eventually leading to a force term onthe right side of (17) Since the displacement is constructedwith the same smoothness as required of a strong form ofsolution other variables of interest such as shear forces andpower flows can be calculated directly and perhaps moreaccurately by applying appropriate mathematical operationsto the displacement function

3 Result and Discussion

To demonstrate the accuracy and usefulness of the proposedtechnique several numerical examples will be presented in

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Shock and Vibration 7

Table 5 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for CSCS annular sector plates (120583 = 03)

120601 119886119887 Source Mode number1 2 3 4 5 6

1205874

02Present 70255 14422 16824 24343 28220 30407FEM 70247 14422 16827 24347 28223 30419

Reference [13] 70255 14422 16824 24345 28204 30404

04Present 84594 16965 19919 29682 30411 36632FEM 84589 16965 19921 29682 30420 36651

Reference [13] 84592 16965 19917 29676 30408 3662306 Present 15474 21164 32179 40470 46722 48199

1205872

02 Present 41833 70256 10658 11423 14423 16824

04 Present 66678 86611 11945 16943 17764 19921FEM 66678 84593 11945 16966 17768 19922

06 Present 14314 15476 17685 21165 25999 32176

21205873

02 Present 38332 53392 80167 10164 11423 1216704 Present 64480 73627 91703 11946 15588 1746606 Present 14159 14778 15919 17687 20154 23420

120587

02 Present 36108 41828 53387 70236 90850 98272FEM 36105 41820 53389 70255 90859 98294

04 Present 62989 66672 73625 84585 99938 11946FEM 63004 66678 73633 84597 99907 11946

06 Present 14048 14313 14779 15474 16432 17683FEM 14051 14317 14782 15476 16434 17687

71205876

02 Present 35684 39622 47703 60066 75868 94024FEM 35684 39629 47699 60063 75821 94041

04 Present 62693 65318 70196 77814 88570 10245FEM 62700 65329 70199 77810 88526 10245

06 Present 14025 14227 14554 15061 15721 16596FEM 14029 14222 14556 15049 15722 16594

31205872

02 Present 35241 37429 41823 48904 58484 7037204 Present 62367 63876 66619 70902 76824 8453906 Present 14012 14109 14312 14599 14967 15481

161205879

02 Present 35036 36539 39462 44137 50791 5917804 Present 62218 63301 65160 68051 71966 7744006 Present 13989 14063 14211 14411 14683 14999

2120587

02 Present 34948 36119 38293 41824 46882 53362FEM 34961 36106 38298 41824 46862 53401

04 Present 62148 63172 64429 66683 69767 73579FEM 62156 63004 64481 66682 69699 73641

06 Present 13995 14033 14157 14313 14519 14767FEM 13987 14051 14161 14317 14523 14783

this section First consider a completely clamped annularsector plate A clamped BC can be viewed as a specialcase when the stiffness constants for both sets of restrainingsprings become infinitely large (represented by a very largenumber 50times 1013 in the numerical calculations)Thefirst sixnondimensional frequency parameters Ω = 1205961198872(120588ℎ119863)12are tabulated in Table 1 together with the reference resultsfrom [22] and an FEM prediction

Next consider an annular sector plate with simplysupported radial edges Three different boundary condi-tions (free simply supported and clamped) are sequentially

applied to the circumferential edges The simply supportedcondition is simply produced by setting the stiffnesses of thetranslational and rotational springs toinfin and 0 respectivelyand the free edge condition by setting both stiffnesses to zeroThefirst six nondimensional frequency parameters are shownin Table 2 The current results compare well with those takenfrom [22 30]

To illustrate the convergence and numerical stabilityof the current solution several sets of results in Tables 1and 2 are presented for using different truncation numbers119872 = 119873 = 5 6 7 12 A highly desired convergence

8 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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 Shock and Vibration

Table 6 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for simply supported annular sector plates with uniform rotational restraint along eachedge (119886119887 = 04 120601 = 21205873 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 32618 46318 68813 98903 11580 13164FEM 32578 46212 68677 98751 11582 13152

104 Present 41395 54245 76164 10605 12556 14173FEM 41329 54148 76060 10599 12675 14182

108 Present 65688 78383 10129 13372 17381 17566FEM 65762 78429 10134 13383 17411 17629

1012 Present 65698 78394 10130 13374 17383 17569FEM 65772 78439 10134 13385 17413 17632

Table 7 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for an FCFF annular sector plate with identical elastic restraint at ldquofreerdquo edges (119886119887 =04 120601 = 120587 and 120583 = 03)

119870 (Nmrad) Source Mode number1 2 3 4 5 6

100 Present 44992 59227 96841 12490 18444 24601FEM 44898 59352 96797 12491 18448 24576

104 Present 46998 73625 12924 20819 21912 30652FEM 46961 73665 12926 20801 21938 30630

108 Present 47531 84457 15344 24898 30587 36816FEM 47507 84435 15345 24909 30605 36842

1012 Present 47531 84460 15345 24899 30589 36817FEM 47508 84435 15346 24909 30606 36842

characteristic is observed in that (a) sufficiently accurateresults can be obtained with only a small number of termsin the series expansions and (b) the solution is consistentlyrefined as more terms are included in the expansions Whilethe convergence of the current solution is mathematicallyestablished via (11) and (12) the actual (truncation) error willbe case-dependent and cannot be exactly determined a prioriHowever this should not constitute a problem in practicebecause one can always verify the accuracy of the solution byincreasing the truncation number until a desired numericalprecision is achieved As amatter of fact this ldquoquality controlrdquoscheme can be easily implemented automatically In modalanalysis the natural frequencies for higher-order modes tendto converge slower (see Table 1)Thus an adequate truncationnumber should be dictated by the desired accuracy of thelargest natural frequencies of interest In view of the excellentnumerical behavior of the current solution the truncationnumbers will be simply set as119872 = 119873 = 12 in the followingcalculations

In the very limited existing studies the sector anglesare typically assumed to be less than 120587 as specified interms of 119898 = 120587120601 being an integer Although it is notclear whether 120601 = 120587 inherently constitutes a pivotingpoint for mathematically solving sector plate problems ithas been a limit practically defining the previous investi-gations However the value of the sector angle appears tohave no binding effect on the current solution proceduresas described earlier To verify this statement and illustrate

the versatility of the proposed technique the plates with a fullrange of sector angles are studied under various restrainingconditions Presented in Table 3 are the first six frequencyparameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates(119886119887 = 04) which are completely free along all of their edgesDue to a lack of analytical solutions the numerical resultscalculated using an FEM (ABAQUS) model are given therefor comparison Since the reference solutions for annularsector plates are not readily available the plates with otherclassical boundary conditions are also studied systematicallyand the corresponding results are listed in Tables 4 and 5for a range of sector angles up to 2120587 Such results can beparticularly useful in benchmarking other solution methodsIn identifying the boundary conditions letters C S and Fhave been used to indicate the clamped simply supportedand free boundary condition along an edge respectivelyThus the boundary conditions for a plate are fully specified byusing four letters with the first one indicating the BC alongthe first edge 119903 = 119886The remaining (the second to the fourth)edges are ordered in the counterclockwise direction In allthese cases the current solutions are adequately validatedby the FEM results obtained using ABAQUS models Alsoincluded are the results previously given in [13] for smallersector angles 120601 = 1205876 and 1205872 The mode shapes for thefirst six modes are plotted in Figure 2 for the fully clampedannular sector plate with cutout ratio 119886119887 = 04 and sectorangle 120601 = 120587 These modes are verified by the FEM resultsalthough they will not be shown here for conciseness

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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

Shock and Vibration 9

Table 8 Frequency parameters Ω = 1205961198872(120588ℎ119863)12 for annular sector plates with elastic restraint at all four edges 119896 = 105Nm and 119870 =107 Nmrad (120601 = 161205879 and 120583 = 03)

119886119887 Source Mode number1 2 3 4 5 6

02 Present 10268 11702 13324 15209 17664 21015FEM 10251 11684 13311 15209 17678 21038

04 Present 11884 12593 13802 15471 17763 20988FEM 11862 12575 13795 15477 17780 21009

06 Present 14586 14934 15663 16794 18413 20849FEM 14557 14908 15650 16802 18441 20884

(a) (b) (c)

(d) (e) (f)

Figure 2 The first six mode shapes for a CCCC annular sector plate (119886119887 = 04 and 120601 = 120587) the (a) first (b) second (c) third (d) fourth (e)fifth and (f) sixth mode shape

All the above examples involve the classical homogeneousboundary conditions which are viewed as special cases (ofelastically restrained edges) when the stiffness constants takeextreme values We now turn to annular sector plates withgeneral elastically restrained edges First consider an annularsector plate simply supported but with uniform rotationalrestraint along each edge The first six frequency parametersare presented in Table 6 together with the results calculatedusing an ABAQUS model The second example concerns acantilever annular sector plate (clamped at 120579 = 0) withidentical elastic restraints at other edges While the stiffnessof the translational springs is fixed to 119896 = 104Nm therotational springs will be specified to take different stiffnessvalues 119870 = 100 104 108 1012Nmrad The correspondingfrequency parameters are shown in Table 7 In all the casesa good agreement is observed between the current solutionand the FEM results

Lastly consider reentrant annular sector plates (120601 =161205879) elastically restrained along all the four edges Thestiffnesses for the translational and rotational restraintsis chosen as 119896 = 10

5Nm and 119870 = 107Nmrad

respectively The first six frequency parameters are shownin Table 8 for three different cutout ratios Plotted inFigure 3 are the mode shapes for the plate with 119886119887 =04

4 Conclusions

An analytical method has been presented for the vibrationanalysis of annular sector plates with general elastic restraintsalong each edge which allows treating all the classicalhomogenous boundary conditions as the special cases whenthe stiffness for each of the restraining spring is equal toeither zero or infinity Regardless of boundary conditionsthe displacement function is invariantly expressed as animproved trigonometric series which converges uniformlyat an accelerated rate Since the displacement solution isconstructed to have 1198623 continuity the current solutionalthough sought in a weak form from the Rayleigh-Ritzprocedure is mathematically equivalent to a strong solutionwhich simultaneously satisfies both the governing differentialequation and the boundary conditions on a point-wise basis

The presentmethod provides a unifiedmeans for predict-ing the free vibration characteristics of annular sector plateswith a variety of boundary conditions and any sector anglesThe efficiency accuracy and reliability of the proposedmethod are fully illustrated for free vibration analysis ofannular sector plates with different boundary supports andmodel parameters such as radius ratio and sector angleNumerical results obtained by the present approach are inexcellent agreement with those available in the literature

10 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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 Shock and Vibration

(a) (b) (c)

(d) (e) (f)

Figure 3 The first six mode shapes for an annular sector plate (119886119887 = 04 and 120601 = 161205879) with elastic restraints 119896 = 105Nm and 119870 =107 Nmrad at all the four edges the (a) first (b) second (c) third (d) fourth (e) fifth and (f) sixth mode shape

Although the stiffness for each restraining spring is hereassumed to be uniform any nonuniform discrete or partialstiffness distribution can be readily considered by modifyingpotential energies accordingly

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous reviewersfor their very valuable comments This work was supportedby the International SampT Cooperation Program of China(2011DFR90440) and the key project of the National NaturalScience of Foundation of China (50939002) The secondauthor is also grateful for the supports from China Scholar-ship Council (2011668004)

References

[1] A W Leissa Vibration of Plates U S Government PrintingOffice Washington DC USA 1969

[2] G K Ramaiah and K Vijayakumar ldquoNatural frequencies ofcircumferentially truncated sector plates with simply supportedstraight edgesrdquo Journal of Sound and Vibration vol 34 no 1 pp53ndash61 1974

[3] T Mizusawa and H Ushijima ldquoVibration of annular sectorMindlin plates with intermediate arc supports by the spline stripmethodrdquo Computers and Structures vol 61 no 5 pp 819ndash8291996

[4] TMizusawa H Kito and T Kajita ldquoVibration of annular sectormindlin plates by the spline strip methodrdquo Computers andStructures vol 53 no 5 pp 1205ndash1215 1994

[5] M N Bapu Rao P Guruswamy and K S SampathkumaranldquoFinite element analysis of thick annular and sector platesrdquoNuclear Engineering and Design vol 41 no 2 pp 247ndash255 1977

[6] R S Srinivasan and V Thiruvenkatachari ldquoFree vibration ofannular sector plates by an integral equation techniquerdquo Journalof Sound and Vibration vol 89 no 3 pp 425ndash432 1983

[7] A Houmat ldquoA sector Fourier p-element applied to free vibra-tion analysis of sectorial platesrdquo Journal of Sound and Vibrationvol 243 no 2 pp 269ndash282 2001

[8] A W Leissa O G McGee and C S Huang ldquoVibrations ofsectorial plates having corner stress singularitiesrdquo Journal ofApplied Mechanics vol 60 no 1 pp 134ndash140 1993

[9] K M Liew Y Xiang and S Kitipornchai ldquoResearch onthick plate vibration a literature surveyrdquo Journal of Sound andVibration vol 180 no 1 pp 163ndash176 1995

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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

Shock and Vibration 11

[10] K Kim and C H Yoo ldquoAnalytical solution to flexural responsesof annular sector thin-platesrdquo Thin-Walled Structures vol 48no 12 pp 879ndash887 2010

[11] R Ramakrishnan and V X Kunukkasseril ldquoFree vibration ofannular sector platesrdquo Journal of Sound and Vibration vol 30no 1 pp 127ndash129 1973

[12] M M Aghdam M Mohammadi and V Erfanian ldquoBendinganalysis of thin annular sector plates using extended Kan-torovich methodrdquo Thin-Walled Structures vol 45 no 12 pp983ndash990 2007

[13] Y Xiang KM Liew and S Kitipornchai ldquoTransverse vibrationof thick annular sector platesrdquo Journal of EngineeringMechanicsvol 119 no 8 pp 1579ndash1599 1993

[14] C F Liu and G T Chen ldquoA simple finite element analysisof axisymmetric vibration of annular and circular platesrdquoInternational Journal of Mechanical Sciences vol 37 no 8 pp861ndash871 1995

[15] O Civalek and M Ulker ldquoHarmonic differential quadrature(HDQ) for axisymmetric bending analysis of thin isotropiccircular platesrdquo Structural Engineering and Mechanics vol 17no 1 pp 1ndash14 2004

[16] O Civalek ldquoApplication of differential quadrature (DQ) andharmonic differential quadrature (HDQ) for buckling analysisof thin isotropic plates and elastic columnsrdquo Engineering Struc-tures vol 26 no 2 pp 171ndash186 2004

[17] K M Liew T Y Ng and B P Wang ldquoVibration of annularsector plates from three-dimensional analysisrdquo Journal of theAcoustical Society of America vol 110 no 1 pp 233ndash242 2001

[18] X Wang and Y Wang ldquoFree vibration analyses of thin sectorplates by the new version of differential quadrature methodrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 36ndash38 pp 3957ndash3971 2004

[19] T Irie G Yamada and F Ito ldquoFree vibration of polar-orthotropic sector platesrdquo Journal of Sound and Vibration vol67 no 1 pp 89ndash100 1979

[20] D Zhou S H Lo and Y K Cheung ldquo3-D vibration analysisof annular sector plates using the Chebyshev-Ritz methodrdquoJournal of Sound and Vibration vol 320 no 1-2 pp 421ndash4372009

[21] AH Baferani A R Saidi and E Jomehzadeh ldquoExact analyticalsolution for free vibration of functionally graded thin annularsector plates resting on elastic foundationrdquo Journal of Vibrationand Control vol 18 no 2 pp 246ndash267 2012

[22] S H Mirtalaie and M A Hajabasi ldquoFree vibration analysisof functionally graded thin annular sector plates using thedifferential quadrature methodrdquo Proceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 225 no 3 pp 568ndash583 2011

[23] E Jomehzadeh and A R Saidi ldquoAnalytical solution for freevibration of transversely isotropic sector plates using a bound-ary layer functionrdquoThin-Walled Structures vol 47 no 1 pp 82ndash88 2009

[24] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[25] W L Li M W Bonilha and J Xiao ldquoVibrations of twobeams elastically coupled together at an arbitrary anglerdquo ActaMechanica Solida Sinica vol 25 no 1 pp 61ndash72 2012

[26] W L Li ldquoVibration analysis of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol273 no 3 pp 619ndash635 2004

[27] W L Li X Zhang J Du and Z Liu ldquoAn exact series solutionfor the transverse vibration of rectangular plates with generalelastic boundary supportsrdquo Journal of Sound and Vibration vol321 no 1-2 pp 254ndash269 2009

[28] D Y Shi X J Shi W L Li and Q S Wang ldquoFree transversevibrations of orthotropic thin rectangular plates with arbitraryelastic edge supportsrdquo Journal of Vibroengineering vol 16 no 1pp 389ndash398 2014

[29] X Zhang and W L Li ldquoVibrations of rectangular plates witharbitrary non-uniform elastic edge restraintsrdquo Journal of Soundand Vibration vol 326 no 1-2 pp 221ndash234 2009

[30] C S Kim and S M Dickinson ldquoOn the free transversevibration of annular and circular thin sectorial plates subjectto certain complicating effectsrdquo Journal of Sound and Vibrationvol 134 no 3 pp 407ndash421 1989

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