Journal of Sound and < ibration (2000) 230(2), 335 } 356 doi:10.1006/jsvi.1999.2623, available online at http://www.idealibrary.com on DIFFERENTIAL QUADRATURE METHOD FOR VIBRATION ANALYSIS OF SHEAR DEFORMABLE ANNULAR SECTOR PLATES K. M. LIEW AND F.-L. LIU Centre for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang ¹ echnological ;niversity, Singapore 639798, Singapore (Received 10 September 1997, and in ,nal form 23 July 1999) This paper presents di!erential quadrature solutions for free vibration analysis of moderately thick annular sector plates based on the Mindlin "rst-order shear deformation theory. Numerical characteristics of the di!erential quadrature method are illustrated through solving selected annular sector plates with di!erent boundary conditions, relative thickness ratios, inner-to-outer radius ratios and various sector angles. Parametric studies in terms of the vibration frequency parameters are thoroughly investigated. ( 2000 Academic Press 1. INTRODUCTION The annular sector plate forms one of the most widely used structural components in engineering applications. The vibration analysis of annular sector plates is, therefore, of paramount importance in practical design. In the past few decades, many researches have been done on the solution of vibration problems of thin annular sector plates by analytical methods [1}5] and numerical methods such as the energy method [6}11], the integral equation method [12], the "nite strip method [13] and the spline element method [14, 15], and other methods [16]. In the mean time, the solution of vibration problems of thick annular sector plates has also attracted the attention of many researchers. Kobayashi et al. [17] obtained an analytical solution to the vibration of a Mindlin annular sector plate with two radial edges simply supported and two other circular edges free. Huang et al. obtained analytical solutions to the sectorial plates having simply supported radial edges and arbitrarily bounded circular edges [18]. Tanaka et al. [19] reported solutions to the free vibration of a cantilever annular sector plate with curved radial edges and varying thicknesses. Other researchers obtained numerical solutions for free vibration problems of Reissner or Mindlin annular sector plates by using the "nite element method [20, 21], the boundary element method [22], the "nite strip method [23, 24] and the Rayleigh}Ritz method [25]. The DQ method was "rst introduced by Bellman and Casti [26] and Bellman et al. [27] and developed further by Quan and Chang [28] and Shu and Richards 0022-460X/00/070335#22 $35.00/0 ( 2000 Academic Press
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Differential Quadrature Method for Vibration Analysis of Shear Deformable Annular Sector Plates
Differential Quadrature Method for Vibration Analysis of Shear Deformable Annular Sector Plates
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Journal of Sound and <ibration (2000) 230(2), 335}356doi:10.1006/jsvi.1999.2623, available online at http://www.idealibrary.com on
DIFFERENTIAL QUADRATURE METHOD FORVIBRATION ANALYSIS OF SHEAR DEFORMABLE
ANNULAR SECTOR PLATES
K. M. LIEW AND F.-L. LIU
Centre for Advanced Numerical Engineering Simulations, School of Mechanical andProduction Engineering, Nanyang ¹echnological ;niversity, Singapore 639798, Singapore
(Received 10 September 1997, and in ,nal form 23 July 1999)
This paper presents di!erential quadrature solutions for free vibration analysisof moderately thick annular sector plates based on the Mindlin "rst-order sheardeformation theory. Numerical characteristics of the di!erential quadraturemethod are illustrated through solving selected annular sector plates with di!erentboundary conditions, relative thickness ratios, inner-to-outer radius ratios andvarious sector angles. Parametric studies in terms of the vibration frequencyparameters are thoroughly investigated.
( 2000 Academic Press
1. INTRODUCTION
The annular sector plate forms one of the most widely used structural componentsin engineering applications. The vibration analysis of annular sector plates is,therefore, of paramount importance in practical design. In the past few decades,many researches have been done on the solution of vibration problems of thinannular sector plates by analytical methods [1}5] and numerical methods such asthe energy method [6}11], the integral equation method [12], the "nite stripmethod [13] and the spline element method [14, 15], and other methods [16]. Inthe mean time, the solution of vibration problems of thick annular sector plates hasalso attracted the attention of many researchers. Kobayashi et al. [17] obtained ananalytical solution to the vibration of a Mindlin annular sector plate with tworadial edges simply supported and two other circular edges free. Huang et al.obtained analytical solutions to the sectorial plates having simply supported radialedges and arbitrarily bounded circular edges [18]. Tanaka et al. [19] reportedsolutions to the free vibration of a cantilever annular sector plate with curved radialedges and varying thicknesses. Other researchers obtained numerical solutions forfree vibration problems of Reissner or Mindlin annular sector plates by using the"nite element method [20, 21], the boundary element method [22], the "nite stripmethod [23, 24] and the Rayleigh}Ritz method [25].
The DQ method was "rst introduced by Bellman and Casti [26] and Bellmanet al. [27] and developed further by Quan and Chang [28] and Shu and Richards
[29] into the generalized DQ method through introducing a simple algebraicformula to calculate the weighting coe$cients of di!erent derivatives. Manyprevious studies [30}35] have shown that the DQ method is capable of yieldinghighly accurate solutions to the initial boundary value problems with much lesscomputational e!ort. Therefore, it appears that the method has the potential tobecome an alternative to the conventional numerical methods. However,this powerful method has not been tested to solve the vibration analysis of sectorplates.
In this paper, the DQ method is thus applied to the problems of free vibrations ofthick Mindlin annular sector plates which are described by three di!erentialequations in a two-dimensional polar co-ordinate system. The accuracy and theconvergence characteristics of the DQ method for the free vibration analysis ofseveral thick annular plates of di!erent inner-to-outer radius ratios, relativethickness ratios and boundary conditions are investigated through directlycomparing the present results with the existing exact or other numerical solutions.The applicability and the simplicity of the DQ method for the vibration analysis ofMindlin annular sector plates have been demonstrated through solving examples inthe parameter studies.
2. METHOD OF DIFFERENTIAL QUADRATURE
The two-dimensional polar co-ordinate system can be treated in a similar way tothe two-dimensional Cartesian co-ordinate system in using the di!erentialquadrature rule. Suppose that there are N
Rgrid points in the R-direction and
NH grid points in the H direction with R1, R
2,2, R
Nxand H
1, H
2,2, H
NHas the
co-ordinates, the nth order partial derivative of f (R, H ) with respect to R, the mthorder partial derivative of f (R, H) with respect to H and the (n#m)th order partialderivative of f (R, H) with respect to both R and H can be expressed discretely at thepoint (R
i, H
j) as
f (n)R
(Ri, H
j)"
NR
+k/1
C(n)ik
f (Rk, H
j), n"1, 2,2, N
R!1, (1a)
f (m)H (Ri, H
j)"
NH
+k/1
CM (m)jk
f (Ri, H
k), m"1, 2,2, NH!1, (1b)
f (n`m)RH (R
i, H
j)"
NR
+k/1
C(n)ik
NH
+l/1
CM (m)jl
f (Rk, H
l)
for i"1, 2,2, NR
and j"1, 2,2, NH , (1c)
where C(n)ij
and CM (m)ij
are weighting coe$cients associated with nth order partialderivative of f (R, H) with respect to R at the discrete point R
iand mth order
derivative with respect to H at Hj.
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 337
According to Quan and Chang [28] and Shu and Richards [29], the weightingcoe$cients in equations (1a}c) can be determined as follows:
C(1)ij"
M(1)(Ri)
(Ri!R
j)M(1)(R
j), i, j"1, 2,2,N
R, but jOi, (2)
where
M(1) (Ri)"
NR
<j/1, jOi
(Ri!R
j) (3)
and
C(n)ij"nAC(n~1)
iiC(1)
ij!
C(n~1)ij
Ri!R
jB (4)
for i, j"1, 2,2, NR, but jOi; and n"2, 3,2, N
R!1,
C(n)il"!
NR
+j/1, jOi
C(n)ij
, i"1, 2,2,NR, and n"1, 2,2, N
R!1. (5)
CM (n)ij
can be determined using equations (1)}(5) simply by replacing all R with H.
3. MODELLING OF PROBLEMS BY DQ METHOD
3.1. GOVERNING EQUATIONS
The problem concerned here is the transverse free vibration of a thick isotropicannular sector plate with uniform thickness h, sector angle a, inner radius b andouter radius a as shown in Figure 1. According to Mindlin's plate theory, theequilibrium equations in terms of moment and shear resultants in polarco-ordinates are [36]
LMr
Lr#
1r
LMrh
Lh#
1r
(Mr!Mh )!Q
r"
oh3
12L2t
rLt2
, (6a)
LMrh
Lr#
1r
LMhLh
#
2r
Mrh!Qh"
oh3
12L2thLt2
, (6b)
LQr
Lr#
1r
LQhLh
#
1r
Qr"oh
L2wLt2
, (6c)
where o is the density of the plate. The moment resultants Mr, Mh , and M
rh and theforce resultants Q
rand Qh are expressed by the transverse de#ection w and the
Figure 1. Geometry and co-ordinate system of annular sector plate.
338 K. M. LIEW AND F.-L. LIU
bending rotations tr
in the radial plane and th in the circumferential plane asfollows:
Mr"DA
Ltr
Lr#l
1r Atr
#
LthLh BB , (7a)
Mh"DA1r Atr
#
LthLh B#l
Ltr
Lr B , (7b)
Mrh"
1!l2
DA1r A
Ltr
Lh!thB#
LthLr B (7c)
and
Qr"iGh Atr
#
LwLrB , (8a)
Qh"iGhAth#1r
LwLhB , (8b)
where
D"
Eh3
12(1!l2)(9)
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 339
and E, G and l are Young's modulus, shear modulus and the Poisson ratio of theplate respectively, and i is the shear correction factor.
Using the following dimensionless parameters,
R"r/a, H"h/a,="w/a, WR"t
r, WH"th , d"h/a, (10a)
¹"t/t0, t
0"S
oa2 (1!l2)E
, (10b)
and substituting equations (7) and (8) into equation (6), one can normalize thegoverning equations as follows:
R2L2W
RLR2
#RLW
RLR
!(1#mR2)WR#
(1!l)2a2
L2WR
LH2#
(1#l)2a
RL2WH
LRLH
!
(3!l)2a
LWhLH
!mR2L=LR
"R2L2W
RL¹2
, (11a)
(1#l)2
RL2W
RLRLH
#
(3!l)2a
LWR
LH#
1a2
L2WH
LH2#
(1!l)2
R2L2WH
LR2
#
(1!l)2
RLWH
LR!C
(1!l)2
#mR2D WH!ma
RL=LH
"R2L2WH
L¹2, (11b)
AR2L2=LR2
#RL=LR
#
1a2
L2=LH2B#AR2
LWR
LR#RW
RB#
Ra
LWH
LH"
2R2
i (1!l)L2=L¹2
, (11c)
where
m"6i(1!l)
d2(12)
and the stress-displacement relationships are given by
MR"
LWR
LR#l
1R AWR
#
1a
LWH
LHB , (13a)
MH"1R AWR
#
1a
LWH
LH B#lLW
RLR
, (13b)
MRH"
1!l2 C
1R A
1a
LWR
LH!WHB#
LWH
LR D , (13c)
where MR"M
r/(D/a), MH"Mh/(D/a) and M
RH"Mrh/(D/a).
340 K. M. LIEW AND F.-L. LIU
For free vibration, the solutions of motion in time can be assumed as
= (R, H, ¹ )"=j(R, H)e*XjT, W
R(R, H, ¹ )"W
Rj(R, H)e*XjT,
WH (R, H, ¹ )"WHj(R, H)e*XjT, (14a, b, c)
where Xjis the eigenvalue of the jth mode of vibration.
Substitution of equation (14) into equation (11) leads to
R2L2W
RLR2
#RLW
RLR
!(1#mR2)WR#
(1!l)2a2
L2WR
LH2#
(1#l)2a
RL2WH
LRLH
!
(3!l)2a
LWH
LH!mR2
L=LR
"!R2X2WR, (15a)
(1#l)2a
RL2W
RLRLH
#
(3!l)2a
LWR
LH#
1a2
L2WH
LH2#
(1!l)2
R2L2WH
LR2
#
(1!l)2
RLWH
LR!C
(1!l)2
#mR2DWH!ma
RL=LH
"!R2X2WH , (15b)
AR2L2=LR2
#RL=LR
#
1a2
L2=LH2B#AR2
LWR
LR#RW
RB#
Ra
LWH
LH"!
2R2
i (1!l)X2= (15c)
in which and also in the following, =, WR, WH and X should have been taken as
=j(R, H), W
Rj(R, H), WHj
(R, H) and Xjrespectively for the jth mode of vibration,
but the su$x j is omitted for the sake of convenience.According to the di!erential quadrature procedure, the normalized governing
equations (15) will be transformed into the following discrete forms:
CNR
+k/1
(C(2)ik
R2i#C(1)
ikR
i)W
R(k, j )D!(1#mR2
i)W
R(i, j )
#
1!l2
b2CNH
+m/1
CM (2)jm
WR(i, m)D#
1#l2
bRiC
NR
+k/1
C(1)ik
NH
+m/1
CM (1)jm
WH (k, m)D!
(3!l)2
bCNH
+m/1
CMjm
WH (i, m)D!mR2i C
NR
+k/1
C(1)ik=(k, j )D"!X2R2
iWR(i, j ),
(16a)
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 341
1#l2
bRiC
NR
+k/1
C(1)ik
NH
+m/1
CM (1)jm
WR(k, m)D#
3!l2
bCNH
+m/1
CM (1)jm
WR(i, m)D
#b2CNH
+m/1
CM (2)jm
WH (i, m)D#1!l
2 CNR
+k/1
(C(2)ik
R2i#C(1)
ikR
i)WH(k, j )D
!A1!l
2#mR2
i BWH (i, j )!mbRi C
NH
+m/1
CM (1)jm= (i, m)D"!)2R2
iWH(i, j ),
(16b)
CNR
+k/1
(C(2)ik
R2i#C(1)
ikR
i)=(k, j )D#b2C
NH
+m/1
CM (2)jm= (i, m)D#R2
i CNR
+k/1
C(1)ik
WR(k, j )D
#RiWR(i, j )#bR
iCNH
+m/1
CM (1)jm
WH(i, m)D"!
2i (1!l)
X2R2i=(i, j ), (16c)
where i"2,2, NR!1 and j"2,2, NH!1. C(n)
rsand CM (n)
rsare the weighting
coe$cients for the nth order partial derivatives of =, WR
and WH with respect toR and H respectively.
It should be noticed that the domain [b/a, 1] of dimensionless variable R is notthe often used [0, 1] or [!1, 1]. Therefore, the DQ weighting coe$cients, C(n)
rsand
CM (n)rs
, are di!erent from the standard ones corresponding to the [0, 1] or [!1, 1]domain.
3.2. BOUNDARY CONDITIONS
The boundary conditions considered herein are divided into four kinds. Takingthe radial edge with h"constant, for example, we have
Simply supported edge (S) : w"Mh"tr"0, (17)
(S@) : w"Mh"Mrh"0, (18)
Clamped edge (C): w"tr"th"0, (19)
Free edge (F): Qh"Mh"Mrh"0. (20)
Substituting equations (7) and (8) into equations (17)}(20) and normalizing themlead to
Simply supported edge (S) :
="0, lRLW
RLR
#WR#
1a
LWH
LH"0, W
R"0. (21)
342 K. M. LIEW AND F.-L. LIU
Simply supported edge (S@) :
="0, lRLW
RLR
#WR#
1a
LWH
LH"0,
1a
LWR
LH!WH#R
LWH
LR"0. (22)
Clamped edge (C):
="0, WR"0, WH"0 (23)
Free edge (F):
1a
L=LH
#RWH"0, lRLW
RLR
#WR#
1a
LWH
LH"0,
1a
LWR
LH!WH#R
LWH
LR"0. (24)
Using the DQ procedure, the normalized boundary conditions presented byequations (21)} (24) for an edge of H"constant, can then be described in thefollowing discrete forms. For an example, at the edge H"0:
(S) =i1"0, (25a)
lRi
NR
+k/1
C(1)k1
WR(k, 1)#W
R#
1a
NH
+m/1
CM (1)1m
WH (i, m)"0, (25b)
WR(i, 1)"0. (25c)
(S@) =1j"0, (26a)
lRi
NR
+k/1
C(1)k1
WR(k, 1)#W
R#
1a
NH
+m/1
CM (1)1m
WH (i, m)"0, (26b)
1a
NH
+m/1
CM (1)1m
WR(i, m)!WH (i, 1)#R
i
NR
+k/1
C(1)ik
WH (k, 1)"0, (26c)
(C) =i1"0, (27a)
WR(i, 1)"0, (27b)
WH(i, 1)"0, (27c)
(F)1a
NH
+k/1
CM (1)1k=(i, m)#R
iWH (i, 1)"0, (28a)
lRi
NR
+k/1
C(1)k1
WR(k, 1)#W
R#
1a
NH
+m/1
CM (1)1m
WH (i, m)"0, (28b)
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 343
1a
NH
+m/1
CM (1)1m
WR(i, m)!WH(i, 1)#R
i
NR
+k/1
C(1)ik
WH (k, 1)"0,
i"1, 2,2, NR
for equations (25a)}(28c). (28c)
For the edge of H"1, the discrete boundary conditions can be simply obtained bysubstituting all the subscripts of 1 into equations (25)}(28) with NH . The boundaryconditions for the circular edge with R"constant can also be written in the samemanner.
4. NUMERICAL RESULTS AND DISCUSSION
Based on the formulas presented in the previous section, a programme has beenbuilt up to solve the eigenvalues of the plate. For all the calculations here, thePoisson ratio and the shear correction factor i have been taken as l"0)3 andn2/12. The grid points employed in computation are designated by
Ri"Gb#
12 C1!cos A
(i!1)nN
R!1BD (a!b)HNa, i"1, 2,2, N
R, (29a)
Hj"
12 C1!cosA
( j!1)nNH!1 BD , j"1, 2,2, NH . (29b)
The moderately thick isotropic plates with six di!erent boundary conditions ofSSSS, CCCC, CSCS, CFCF, FCSC and SCFC have been considered here. Thesymbol FCSC, for instance, represents the free, clamped, simply supported andclamped boundary conditions of edges 1, 2, 3 and 4 on the plate shown in Figure 1respectively. The eigenvalues are expressed in terms of non-dimensional frequencyparameter j2 which is de"ned as follows:
j2"ua2SohD
and u"XSE
oa2 (1!l2). (30)
4.1. CONVERGENCE AND ACCURACY STUDIES
The convergence studies of the DQ method for free vibration of annular sectorMindlin plates should be carried out "rst to reveal the convergence characteristicsof this numerical method for the problem concerned and also to ensure theaccuracy of the present results. In the mean time, the e!ects of boundary conditions,relative thickness, inner-to-outer cut-out ratio and sector angle on the convergenceproperties should also be investigated so that the number of grid points required foran e!ective solution of the problem can be determined.
Figures 2}4 show the convergence patterns of an annular sector plate with SSSS,CCCC and CSCS boundary conditions respectively. The normalized frequency
Figure 2. Convergence pattern of normalized frequency parameter j2/j2%9!#5
of modes 1, 3, 4 and5 for a simply supported annular sector plate.
Figure 3. Convergence pattern of normalized frequency parameter j2/j2%9!#5
of "rst "ve modes fora fully clamped annular sector plate.
344 K. M. LIEW AND F.-L. LIU
parameters j2/j2%9!#5
of the "rst "ve mode sequences are presented in these "gures,and the values of j2
%9!#5are the exact solutions taken from Ramakrishnan and
Kunikkasseril [2]. The convergence pattern of CFCF annular sector plate is shown
Figure 4. Convergence pattern of normalized frequency parameter j2/j2%9!#5
of "rst "ve modes foran annular sector plate with CSCS boundary conditions.
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 345
in Figure 5 and the parameter j2#
stands for the completely converged DQ results(with "ve signi"cant digits). In Figures 2, 4 and 5, the sector angle a and theinner-to-outer cut-out ratio b/a are taken to be 453 and 0)5 respectively, whereas inFigure 3, the value of a and b/a are 903 and 0)00001 respectively so that the directcomparison can be made between the present DQ results and the existing exactsolutions [2]. For all the four cases, the relative thickness h/a is taken to be 0)005.From these "gures, it is found that (1) for all the four kinds of boundary conditionsconsidered here, the DQ results of the annular sector plates converge to the exactsolutions (Figures 2}4) or the corresponding converged values with the increase ofthe grid points (Figure 5); (2) among the DQ results of these four cases, only thefully clamped plate (CCCC) demonstrates the monotonic convergence pattern,while all other cases (SSSS, CSCS and CFCF) show the #uctuating characteristicsin the convergence patterns; (3) for di!erent mode sequences, the convergent speedsare di!erent. Normally, the higher the mode sequence, the slower the convergentspeed; (4) for all the mode sequences, the boundary condition plays the mostimportant part in the convergent speed of DQ solutions for the free vibrations ofannular sector plates. For example, for the fully clamped, simply supportedboundary condition and their combinations, all the frequency parameterscompletely converge to their corresponding converged values when the number ofthe grid points for each co-ordinate variable is equal to or greater than 11, whereasfor the CFCF plate, even when the number of grid points for each co-ordinate isequal to 25, the results are still #uctuating slightly.
Figure 5. Convergence pattern of normalized frequency parameter j2/j2%9!#5
of "rst "ve modes foran annular sector plate with CFCF boundary conditions.
346 K. M. LIEW AND F.-L. LIU
In order to further reveal the e!ects of other parameters such as relative thicknessh/a, sector angle a, and the inner-to-outer cut-out ratio b/a, the most di$cultconvergent case CFCF among the six cases concerned in this paper is selected.Figure 6 shows the e!ects of plate thickness h/a on the convergence of the frequencyparameter j2 of the "rst and the fourth modes. It is observed that the relativethickness h/a has the signi"cant e!ects on the convergent speed of the DQsolutions. The thicker a plate is (h/a"0)005}0)2), the faster the convergent speedfor both the fundamental frequency and the higher mode of frequencies. In otherwords, increasing the relative thickness h/a from 0)005 to 0)2 can greatly improvethe convergence ratio of the DQ results with the re"nement of the grid. The e!ectsof the sector angle on the convergence of frequency parameter j2 of the "rst and thethird modes for the annular sector plate are illustrated in Figure 7. It is very clear tosee that for the fundamental frequency (Figure 7(a)), with the increase of the sectorangle (in the range of 30}1203), the convergent rate increases, but for the highermode such as the third mode (Figure 7(b)), this conclusion is only true for the gridpoints of each co-ordinate variable between 5 and 12; if the grid points along eachco-ordinate direction are over 13, the value of the sector angle will almost bear noe!ects on the convergent rate. Figure 8 shows the e!ects of the inner-to-outercut-out ratio on the convergence of frequency parameter j2 of the "rst and the thirdmodes of the annular sector plate. It is also found that the inner-to-outer cut-outratio b/a can only have the e!ect on the convergence patterns when the grid pointsfor each co-ordinate variable are smaller than 14; when the grid points are larger
Figure 6. E!ects of plate thickness h/a on convergence of normalized frequency parameter j2/j2cof
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 347
than 14, the convergent rates for both of the "rst and the "fth modes of thefrequency parameter j2 are completely dominated by the number of the grid points.
To examine the accuracy of the converged DQ results, comparisons with theearlier results obtained by using other methods such as the analytical method [2], theMindlin "nite-strip method [15] and the Rayleigh}Ritz method [25] are made forfour boundary conditions (SSSS, CCCC, CSCS and CFCF) in Table 1. It is observedthat close agreement has been obtained for all the cases presented in the table.
4.2. PARAMETRIC STUDIES
The "rst six natural frequencies of the annular sector plates with six boundaryconditions, di!erent relative thicknesses, di!erent sector angles and inner-to-outerradius ratios are computed by using the DQ method and presented in Tables 2}7.The values of sector angle, relative thickness and inner-to-outer radius ratio aretaken as a"30, 60, and 1203, h/a"0)01, 0)1 and 0)2 and b/a"0)1, 0)25 and 0)5respectively in the calculation. All the results shown in these tables are completelyconverged ones with "ve signi"cant digits for the thick plates and four signi"cantdigits for the thin plates. Based on the results in all these tables, the followingconclusion remarks can be made:
(1) As the sector angle increases, for the annular sector plate with SSSS, CCCC,CSCS, FCSC and SCFC boundary conditions, he "rst six frequencyparameters decrease signi"cantly for any given relative thickness h/a and
Figure 7. E!ects of sector angle a on convergence of normalized frequency parameter j2/j2c
inner-to-outer radius ratio b/a, but for the annular sector plate with CFCFboundary condition, the frequency parameters may increase for some modessuch as the "rst and third modes in the case of b/a"0)1. This means thatincreasing the sector angles will lead to the decrease of the #exural sti!nessfor the annular sector plates with SSSS, CCCC, CSCS, FCSC and SCFCboundary conditions, but may not necessarily reduce the #exural sti!ness forthe CFCF plates.
TABLE 1
Comparison study of frequency parameters, j2"ua2Joh/D, for annular sector plates with di+erent boundary conditions
(2) As the relative thickness h/a increases, the frequency parameters for theannular sector plate with any boundary conditions will decrease greatly forany given sector angle, mode sequence and inner-to-outer radius ratio.
(3) For the CFCF annular sector plate, increasing the inner-to-outer radius ratiob/a from 0)1 to 0)5 will increase the values of frequency parameters regardlessof the sector angle, mode sequences and relative thickness, but for other casesconsidered in this paper, the e!ect of inner-to-outer radius ratio b/a on thefrequency parameter is not signi"cant in the range of the values of b/abetween 0)1 and 0)25; the e!ect will become signi"cant in the range of thevalues of b/a from 0)25 to 0)5, and in this range, the frequency parameters formost vibration modes will increase whereas some modes of frequencyparameters may decrease as the value of b/a increases.
For all the natural frequencies considered in these tables, except for the thinannular sector plate (h/a"0)01) with at least one free edge, using 17 grid pointsalong each co-ordinate variable will achieve the completely converged DQ resultswith "ve signi"cant digits. But for the thin annular sector plates (h/a"0)01) with atleast one free edge such as CFCF, FCSC, or SCFC plates, using 17 grid pointsalong each co-ordinate variable can only obtain the converged DQ results withthree to four signi"cant digits for some vibration modes. However, this has beenaccurate enough for the engineering applications.
DQM FOR VIBRATION OF ANNULAR SECTOR MINDLIN PLATES 355
5. CONCLUDING REMARKS
In this paper, the di!erential quadrature method has been applied to solve thefree vibration problem of thick annular sector plates based on the Mindlin "rstorder shear deformation theory. The "rst six natural frequencies have beencalculated for the plates with arbitrary combinations of free, clamped and simplysupported boundary conditions and with various relative thickness, sector angleand inner-to-outer radius ratios. The convergence characteristics of the DQmethod have been carefully investigated for di!erent boundary conditions, relativethicknesses, sector angles and inner-to-outer radius ratios. The numerical resultsshow that the DQ method can yield accurate results for the title problem witha relatively small number of grid points.
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