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http://jvc.sagepub.com/

http://jvc.sagepub.com/content/12/6/635

The online version of this article can be found at: DOI: 10.1177/1077546306065396

2006 12: 635 Journal of Vibration and Control

C. S. Huang, M. J. Chang and A. W. Leissa

Vibrations of Mindlin Sectorial Plates Using the Ritz Method Considering Stress Singularities

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Method Considering Stress Singularities

C. S. HUANG M. J. CHANG

Department of Civil Engineering, National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu, Taiwan 30050 (cshuang@mail.nctu.edu.tw)

A. W. LEISSA

Colorado State University, Fort Collins, Colorado, USA

(Received 24 October 20051 accepted 14 February 2006)

Abstract: This paper reports very accurate vibration frequencies of moderately thick sectorial plates with

var-ious boundary conditions and vertex angles (1 1 902, 1802, 2702, 3002, 3302, and 3552) based on Mindlin plate theory, and provides the nodal patterns of their vibration modes for the first time in the published lit-erature. Most of the extensive frequencies presented are exact to the four digits shown. The classical Ritz method is employed, using corner functions and algebraic trigonometric functions as the admissible func-tions. Because the corner functions properly describe the singularity behaviors of moments and shear forces in the vicinity of the vertex of a sectorial plate, they substantially enhance the convergence and accuracy of the numerical results, which is shown by convergence studies.

Keywords: Mindlin sectorial plates, Ritz method, corner functions, vibration

1. INTRODUCTION

Plates are fundamental structural components, which are widely used in practical engineering projects and have caught many researchers’ interests. Leissa (1969, 1977a, 1977b, 1981a, 1981b, 1987a, 1987b) reviewed more than one thousand technical publications on vibrations of thin or thick plates published before 1985, while Liew et al. (1995) concentrated their review on the vibrations of thick plates on pre-1994 publications. These reviews show that there are far fewer studies on vibrations of sectorial plates than for circular, rectangular, or even annular sector plates. The existence of moment and shear force singularities at the vertex of a sectorial plate considerably increases the difficulty of numerically obtaining accurate frequencies and mode shapes for this configuration.

A number of studies have been carried out on vibrations of thin sectorial plates. Based on classical thin plate theory, Huang et al. (1993) provided the first known exact analytical solu-tion for sectorial plates with simply supported radial edges. No exact analytical solusolu-tions are possible with other boundary conditions along the radial edges. Various numerical solutions were developed, such as those based on energy methods (Rubin, 19751 Bhattacharya and Bhowmic, 1975), finite element method (Houmat, 2001), finite strip method (Cheung and Chan, 1981), differential quadrature method (Wang and Wang, 2004), and the Ritz method

Journal of Vibration and Control, 12(6): 635–657, 2006 DOI: 10.1177/1077546306065396

1

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(Leissa et al., 19931 McGee et al., 2003). Among these numerical solutions for thin plates, the solutions developed by Leissa et al. (1993) and McGee et al. (2003) are the most ac-curate because the admissible functions in the Ritz method include the corner functions, which appropriately describe the singular behavior of moments in the neighborhood of the vertex of a sectorial plate. Experimental studies were also performed by Waller (1952) and Maruyama and Ichinomiya (1981) into the vibration behavior of free and clamped sectorial plates, respectively.

Only a few investigations have been carried out into the vibrations of sectorial plates based on Mindlin or Reissner plate theory, even though shear deformation and rotary inertia are known to be important to any analysis of moderately thick plates, and in determining the higher vibration frequencies of thin plates. Huang et al. (1994) obtained exact analytical solutions for sectorial plates with simply supported radial edges, which involve ordinary and modified Bessel functions. Liu and Liew (1999) applied the differential quadrature method to analyze the free vibration of sectorial plates with edges either clamped or simply sup-ported. They considered annular sector plates with an inner to outer radius ratio of 0.00001 and with free boundary conditions along the inner circular edge, so that no moment and shear force singularities need to be taken into account. This was the same procedure used earlier by Leissa et al. (1993) for thin annular plates, presented there in addition to the method em-ploying corner functions. The accuracy of this approach depends on the inner to outer radius ratio chosen and the boundary conditions along the inner circular edge. While a number of researchers have investigated vibrations of thick annular sector plates using various methods (e.g., Guruswamy and Yang, 19791 Cheung and Chan, 19811 Srinivasan and Thiruvenkat-achari, 19851 Mizusawa, 19911 Xiang et al., 19931 Mizusawa et al., 19941 McGee et al., 1995a1 Liew and Liu, 2000), only Xiang et al. (1993) demonstrated results for plates with an inner to outer radius ratio of 0.00001 and with sector angles not larger than 902.

The studies cited above reveal that there is a need to develop accurate solutions for vibra-tion frequencies of thick sectorial plates with various boundary condivibra-tions and vertex angles, and which may have moment and shear force singularities at the neighborhood of the vertex of the sectorial plate. In the present work, a procedure recently developed for the analysis of skewed plates with re-entrant corners (Huang et al., 2005) is extended to sectorial plates. The Ritz method is used with displacement components, which are represented by a math-ematically complete set of admissible algebraic-trigonometric polynomials in conjunction with corner functions that appropriately represent the singular behaviors of moments and shear forces in the neighborhood of the vertex. The corner functions significantly accelerate the convergence of the numerical solutions. Accurate non-dimensional frequencies are pre-sented for sectorial plates with various boundary conditions, vertex angles (1 1 9022 18022 27022 30022 33022 and 35523, and thickness-to-radius ratios (h4a 1 051 or 0.2). The nodal patterns are also shown.

2. METHODOLOGY

In the Ritz method, the vibration frequencies of plates are obtained by minimizing the energy function

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Figure 1. Geometry and coordinate system of a sectorial plate.

where Vmax and Tmax are maximum strain energy and maximum kinetic energy during a vibratory cycle, respectively. In terms of polar coordinates (see Figure 1), Vmaxand Tmaxare expressed as Tmax 1 7 28 2 1 h3 12 22 92 r 5 2 3d A 5 h 22 2d A 3 2 (2) Vmax 1 1 2 22 A 4 D 56 r2r5 1 rr5 1 r 2 72 4 291 4 31 rr2r9 2 5 r3 5 1 291 4 3 6 2r5r1r2 4 1r 728 5 2Gh 5 9 r 5 2r 2 5 6 5 1 r 2 728 d A2 (3)

where w is the transverse displacement of the mid-plane,rand are the bending rotations

of the mid-plane normal in the r and directions respectively, h is the thickness of the plate, D 1 Eh341291 4 23 is the flexural rigidity, E is the modulus of elasticity, is Poisson’s ratio,2is the shear correction factor, G is the shear modulus,8 is the density of the plate, and7 is a free vibration frequency.

The admissible functions for displacement components are assumed as the sum of two sets of functions,

r9r2 3 1 r p9r2 3 5 r c9r2 32 (4a)

9r2 3 1  p9r2 3 5  c9r2 32 (4b)

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wherer p, p and Wp consist of algebraic-trigonometric polynomials, andr c, c and

Wcare three sets of corner functions accounting for the singular behaviors of moments and shear forces at the vertex. The algebraic-trigonometric polynomials are expressed as

r p9r2 3 1 g19r3 4 f19 3 5 I 1 i1224 i j1224 Bi jri41cos j 5 I2 i1325 i j1123 Bi jri41cos j 8 5 f29 3 5 I3 i1224 i j1224 6Bi jri41sin j 5 I4 i1325 i j1123 6Bi jri41sin j 8 2 (5a)  p9r2 3 1 g29r3 4 f39 3 5 I1 i1224 i j1224 Ci jri41cos j 5 I2 i1325 i j1123 Ci jri41cos j 8 5 f49 3 5 I3 i1224 i j1224 6Ci jri41sin j 5 I4 i1325 i j1123 6Ci jri41sin j 8 2 (5b) Wp9r2 3 1 g39r3 4 f59 3 5 I5 i102224 i j102224 Ai jricos j 5 I6 i1123 i j1123 Ai jricos j 8 5 f69 3 5 I7 i1224 i j1224 6Ai jrisin j 5 I8 i1123 i j1123 6Ai jrisin j 8 2 (5c)

where Ai j, Bi j, Ci j, 6Ai j, 6Bi j and 6Ci j are coefficients to be determined by minimizing6. In

equations (5), Ik can be different for different k. However, for simplicity, they are set to be

I1 1 I3 1 I5 1 I7 and I2 1 I4 1 I6 1 I8 in the following. Functions gi9r3 and fj9 3 are

chosen to make the admissible functions satisfy the geometric boundary conditions along the circular and radial edges, respectively.

For different boundary conditions along the circular edge (r 1 a), gi9r3 are chosen as

follows:

Clamped : gi9r3 1 91 4 r4a3 for i 1 12 22 and 37

Simply supported : g29r3 1 g39r3 1 91 4 r4a32 and g19r3 1 17 Free : gi9r3 1 1 for i 1 12 22 and 35

Functions fj9 3 9 j 1 12 22 5 5 5 2 63 are expressed as

fj9 3 1 mj91 4 413nj2 (6)

where mj and nj are either zero or one depending on the boundary conditions along the

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Clamped : m1 1 m3 1 m5 1 12 the rest all equal to zero7 Simply supported : m1 1 m5 1 12 the rest all equal to zero7 Free : all equal to zero5

The same rule is applied to nj for different boundary conditions along 1 1. Notably, the

simply supported conditions given above simulate the mechanical support of a line hinge along an edge.

When the problems under consideration are symmetric (i.e., have the same boundary conditions along the two radial edges), one can take advantage of the symmetry and set 1 0 as the symmetry axis. Then functions fj9 3 ( j 1 12 22 5 5 5 2 6) can expressed as

fj9 3 1 91 5 2 413kj91 4 2 413kj2 (7)

where the rule for determining kjis the same as that for mj given above.

The sets of corner functions are written as r c9r2 3 1 g19r3 K k11 BkRe9r k9r2 2 k33 5 8BkIm9r k9r2 2 k33  2 (8a)  c9r2 3 1 g29r3 K k11 9CkRe9 k9r2 2 k33 5 8CkIm9 k9r2 2 k33  2 (8b) Wc9r2 3 1 g39r3 L l11 AlWl9r2 2 k32 (8c)

wherer k2  k and Wlare established from the asymptotic solutions presented by Huang

(2003) and McGee et al. (2005). The expressions forr k2  k and Wl used in this work

are listed in Table 1. Whenk in Table 1 is a complex number, the correspondingr k and

 k are complex functions. Because 9k is always real, Wk is a real function. To meet the

regularity conditions at r 1 0 the real parts of k and 9k have to exceed zero, andkand 9k

are ranked by increasing order of magnitude of the real part.r kand k are usually more

complicated than the algebraic-trigonometric polynomials used inr pand p.

Using the Ritz method, the free vibration problem is solved by substituting equations (4), (5) and (8) into equations (2) and (3), and minimizing the energy functional6 given in equation (1) with respect to the coefficients Ai j, Bi j, Ci j, 6Ai j, 6Bi j, 6Ci j, Al, Bk, 9Ck, 8Bk, and

8Ckby partial differentiation. This yields a set of homogeneous linear algebraic equations in

terms of these coefficients, which lead to a standard eigenvalue problem. A similar formu-lation for these equations in matrix form can be found in Xiang et al. (1993). The resulting eigenvalues correspond to the vibration frequencies, and the corresponding eigenfunctions describe the mode shapes.

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Table 1. Corner functions corresponding to various boundary conditions along radial edges.

Boundary con- Corner functions ditions along

radial edges

Clamped- r k9r2 3 1 rkcos9k5 13 4 9k11sin9k5 13 4 cos9k4 13

Free 51sin9k4 13 2  k9r2 3 1 rk4 sin9 k5 13 4 9k11cos9k5 13 5 9k1sin9k4 13 59k11cos9k4 13 2 Wk9r2 3 1 r9ksin 9k , where 11 4 k91 4 3 cos9k5 131 4 99k19k4 13 4 k 4 13 cos9k4 131 99k19k4 13 4 k 4 13 sin9k4 131 4 9k1k91 4 3 sin9k5 131 9k11 4[291 4 3 5 91 5 39k5 13] [291 4 3 4 91 5 39k4 13]

, andkand 9kare the kth root of

sin21 1 44 

291 5 32sin21

93 4 391 5 3 and cos 91 1 0, respectively. Simply r k9r2 3 1 rk2sin9k5 13 5 sin9k4 13 ,

supported-  k9r2 3 1 rk2cos9 k5 13 5 9k1cos9k4 13 , Free Wk9r2 3 1 r9ksin 9 k , where21 4 91 5 391 5 3 43 5  5 5  sin9 4 131

sin9 5 131, andkand 9k are the kth root of sin 21 1  sin 21 and cos 91 1 0, respectively.

Simply r k9r2 3 1 rk  4sin9k4 131 sin9k5 131 sin9k5 13 5 sin9k4 13  , supported-  k9r2 3 1 rk  4sin9k4 131 sin9k5 131 cos9k5 13 5 9k1cos9k4 13  , Clamped Wk9r2 3 1 r9ksin 9k ,

wherekand 9kare the kt hroot of

sin 21 1 91 5 3

43 5 sin 21 and sin 91 1 0, respectively. Free- (1) Symmetric case

Free r k9r2 3 1 rk3cos9k5 13 5 cos9k4 13

 k9r2 3 1 rk43sin9 k5 13 4 9k1sin9k4 13 Wk9r2 3 1 r9kcos 9k where31 49k19k4 13 4 k 4 1 k91 4 3 cos9k4 13142 cos9k5 13142 , andkand 9k

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Table 1. Corner functions corresponding to various boundary conditions along radial edges. (Continued)

Boundary con- Corner functions ditions along radial edges (2) Antisymmetric case r k9r2 3 1 rk4sin9k5 13 5 sin9k4 13  k9r2 3 1 rk4cos9 k5 13 5 9k1cos9k4 13 Wk9r2 3 1 r9ksin 9k where 41 49k19k4 13 4 k 4 1 k91 4 3 sin9k4 13142 sin9k5 13142

, andkand 9kare the kt h

root of sin1 1  sin 1 and cos 9142 1 0, respectively. Clamped- (1) Symmetric case:

Clamped r k9r2 3 1 rk  cos9k5 13 4 6 cos9k5 13142 cos9k4 13142 7 cos9k4 13   k9r2 3 1 rk  4 sin9k5 13 5 9k1 6 cos9k5 13142 cos9k4 13142 7 sin9k4 13  Wk9r2 3 1 r9kcos 9k

wherekand 9kare the kt hroot of

sin1 1 491 5 3

43 5 sin1 and cos 9142 1 0, respectively. (2) Antisymmetric case r k9r2 3 1 rk  sin9k5 13 4 6 sin9k5 13142 sin9k4 13142 7 sin9k4 13   k9r2 3 1 rk  cos9k5 13 4 9k1 6 sin9k5 13142 sin9k4 13142 7 cos9k4 13  Wk9r2 3 1 r9ksin 9k

wherekand 9kare the kt hroot of

sin1 1 91 5 3

43 5 sin1 and sin 9k142 1 0, respectively.

3. CONVERGENCE STUDIES

The Ritz method always gives upper-bound solutions for vibration frequencies. Because the sets of polynomials (equations (5)) are mathematically complete, the numerical solutions will converge to exact solutions when the number of admissible functions is sufficiently large. The purpose of the corner functions (equations (8)) is to accelerate the convergence. Here, convergence studies were conducted to verify the accuracy of the present solutions and

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Table 2. Convergence of7a2 8h4D for a sectorial plate with C-F-F boundary conditions and1 1 902.

Mode No. of corner (Ieven,Iodd) in Eqs.(5)

no. functions (16,15) (18,17) (20,19) (22,21) (22,23) 9r c2  c2 Wc3 437# 546# 667# 800# 872# 1 (0,0,0) 4.466 4.432 4.419 4.415 4.412 (10,10,10) 4.427 4.416 4.409 4.403 4.402 (20,20,20) 4.414 4.407 4.402 4.400 4.400 2 (0,0,0) 12.88 12.78 12.73 12.73 12.72 (10,10,10) 12.76 12.74 12.72 12.71 12.71 (20,20,20) 12.73 12.72 12.71 12.71 12.71 3 (0,0,0) 23.22 23.17 23.12 23.08 23.08 (10,10,10) 23.15 23.11 23.07 23.07 23.06 (20,20,20) 23.10 23.07 23.06 23.06 23.06 4 (0,0,0) 32.42 32.27 32.23 32.22 32.21 (10,10,10) 32.27 32.23 32.21 32.20 32.20 (20,20,20) 32.22 32.21 32.20 32.20 32.20 5 (0,0,0) 48.01 47.87 47.81 47.80 47.78 (10,10,10) 47.89 47.85 47.80 47.77 47.77 (20,20,20) 47.81 47.79 47.77 47.77 47.77

Note: “#” denotes the total number of terms inr p, pand Wp.

to show the effects of corner functions on the numerical solutions. The results given here are for plates with 1 053 and 2(shear correction factor)1 2412.

The different combinations of boundary conditions along radial edges and circular edge are considered in this section and the next are F-F-F, C-F-F, S-F-F, C-C-F, S-C-F, and C-C-C (where S-C-F, for instance, denotes simply supported, clamped, and free boundary conditions along edges 1, 2, and 3, respectively, on the sectorial plate shown in Figure 1).

Tables 2–4 list the nondimensional frequencies7a2 8h4D of plates (h4a 1 051), pro-duced using different numbers of admissible functions, with C-F-F boundary conditions and having1 1 902, 2702 and 3552, respectively. Notably, increasing the vertex angle leads to more severe stress singularities at the neighborhood of r 1 0. It can be observed that the results for the plate with 1 1 902 converge well even using only algebraic-trigonometric polynomials, but that this is not the case for the plates with1 1 2702 and 3552. Adding corner functions to the admissible functions accelerates the convergence of the numerical results considerably, especially for larger1, where the stress singularities are more severe. Using 20 corner functions for each ofr c2  c and Wcand setting I2 1 I4 1 I6 1 I8 1 22

and I1 1 I3 1 I5 1 I7 1 21 in equations (4) gives results that are accurate to at least three significant figures. Table 4 (1 1 3552) shows that using 800 polynomial terms without any corner functions, the fundamental frequency obtained is 39 percent higher than the accurate value (1.712) obtained when 60 corner functions are added. It is worth noting that using Ik

larger than those given in Tables 2–4 may cause numerical difficulties through ill-conditioned matrices.

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Table 3. Convergence of7a2 8h4D for a sectorial plate with C-F-F boundary conditions and1 1 2702.

Mode No. of corner (Ieven,Iodd) in Eqs.(5)

no. functions (16,15) (18,17) (20,19) (22,21) (24,23) 9r c2  c2 Wc3 437# 546# 667# 800# 945# 1 (0,0,0) 2.482 2.425 2.388 2.359 2.335 (10,10,10) 2.191 2.036 1.983 1.966 1.965 (20,20,20) 2.032 2.012 1.966 1.964 1.962 2 (0,0,0) 2.974 2.906 2.832 2.804 2.774 (10,10,10) 2.484 2.296 2.225 2.204 2.201 (20,20,20) 2.299 2.244 2.204 2.200 2.198 3 (0,0,0) 4.684 4.635 4.629 4.617 4.596 (10,10,10) 4.535 4.468 4.373 4.370 4.368 (20,20,20) 4.465 4.397 4.369 4.366 4.366 4 (0,0,0) 8.157 8.063 8.046 8.030 8.009 (10,10,10) 7.868 7.808 7.803 7.795 7.791 (20,20,20) 7.808 7.804 7.794 7.789 7.787 5 (0,0,0) 12.54 12.50 12.48 12.47 12.45 (10,10,10) 12.27 12.18 12.16 12.16 12.15 (20,20,20) 12.19 12.16 12.16 12.15 12.15

Note: “#” denotes the total number of terms inr p, pand Wp.

Table 4. Convergence of7a2 8h4D for a sectorial plate with C-F-F boundary conditions and1 1 3552.

Mode No. of corner (Ieven,Iodd) in Eqs.(5)

no. functions (16,15) (18,17) (20,19) (22,21) 9r c2  c2 Wc3 437# 546# 667# 800# 1 (0,0,0) 2.504 2.448 2.375 2.319 (10,10,10) 1.808 1.745 1.717 1.714 (20,20,20) 1.745 1.714 1.712 1.712 2 (0,0,0) 3.029 2.918 2.839 2.765 (10,10,10) 2.401 2.286 2.184 2.182 (20,20,20) 2.304 2.183 2.180 2.177 3 (0,0,0) 3.717 3.654 3.624 3.597 (10,10,10) 3.421 3.338 3.275 3.272 (20,20,20) 3.344 3.296 3.267 3.266 4 (0,0,0) 5.452 5.335 5.281 5.237 (10,10,10) 5.096 4.782 4.735 4.725 (20,20,20) 5.007 4.733 4.725 4.722 5 (0,0,0) 8.041 7.979 7.927 7.880 (10,10,10) 7.595 7.383 7.343 7.339 (20,20,20) 7.385 7.340 7.337 7.336

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Tables 2 to 4 show corner functions accelerating the convergence of the numerical so-lution, but the corner functions themselves are rather too complicated. Can one use some simplified corner functions having the same singular order as the original corner functions and get accurate results? To answer this question, a simple convergence study was conducted for F-F-F and C-F-F sectorial plates with1 1 3552 and h4a 1 051. Tables 5 and 6 show the results obtained by using the corner functions given in Table 1 and by using simplified corner functions defined as follows:

For clamped-free radial edges,

r c9r2 3 1 g19r3 K

k11

rRe[k]B

ksin9Re[k] 3 5 8Bk cos9Re[k] 3 2 (9a)

 c9r2 3 1 g29r3 K

k11

rRe[k] 9C

ksin9Re[k] 3 5 8Ck cos9Re[k] 3 2 (9b)

Wc9r2 3 1 g39r3 K

k11

Akr9ksin99k 35 (9c)

For free-free radial edges,

r c2S9r2 3 1 g19r3

K

k11

8BkrRe[k]cos9Re[k] 32 (10a)

 c2S9r2 3 1 g29r3 K

k11

9CkrRe[k]sin9Re[k] 32 (10b)

Wc2S9r2 3 1 g39r3 K k11 8Akr9kcos99k 32 (10c) and r c2A9r2 3 1 g19r3 K k11

BkrRe[k]sin9Re[k] 32 (11a)

 c2A9r2 3 1 g29r3 K

k11

8CkrRe[k]cos9Re[k] 3 2 (11b)

Wc2A9r2 3 1 g39r3 K

k11

Akr9ksin99k 32 (11c)

where the subscripts “,S” and “,A” denote the symmetric and antisymmetric modes, respec-tively. These simplified corner functions give the correct singularity orders of moments and shear forces at r 1 0. The simplified corner functions for r c and c are much simpler

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Table 5. Comparison of7a2 8h4D for a completely free sectorial plate with 1 1 3552 andh4a 1 051 by using different corner functions.

Mode No. of corner (Ieven,Iodd) in Eqs.(5)

no. functions (16, 15) (18,17) (20,19) (22,21) (24,23) 9r c2  c2 Wc3 1 (0,0,0) 5.261 5.244 5.242 5.240 5.237 (A) (1,1,1) 4.875 4.814 4.779 4.755 4.750 [4.890] [4.819] [4.783] [4.758] [4.752] (5,5,5) 2.782 2.760 2.747 2.740 2.736 [2.795] [2.763] [2.752] [2.746] [2.740] 2 (0,0,0) 5.345 5.301 5.297 5.296 5.294 (S) (1,1,1) 4.713 4.690 4.642 4.601 4.579 [4.721] [4.695] [4.649] [4.607] [4.596] (5,5,5) 4.268 4.241 4.234 4.232 4.231 [4.277] [4.246] [4.239] [4.237] [4.235] 3 (0,0,0) 8.881 8.854 8.849 8.843 8.839 (S) (1,1,1) 8.010 7.980 7.936 7.894 7.845 [8.024] [7.988] [7.940] [7.899] [7.850] (5,5,5) 7.645 7.561 7.556 7.551 7.547 [7.659] [7.566] [7.560] [7.556] [7.551] 4 (0,0,0) 12.02 12.00 11.99 11.98 11.98 (A) (1,1,1) 10.25 10.18 10.09 10.02 10.00 [10.27] [10.19] [10.10] [10.03] [10.02] (5,5,5) 7.653 7.609 7.596 7.592 7.589 [7.667] [7.616] [7.603] [7.599] [7.594] 5 (0,0,0) 12.12 12.12 12.12 12.11 12.11 (S) (1,1,1) 11.51 11.51 11.50 11.49 11.49 [11.52] [11.52] [11.51] [11.51] [11.51] (5,5,5) 11.26 11.24 11.23 11.23 11.22 [11.27] [11.25] [11.24] [11.23] [11.23]

Note: (S) and (A) denote symmetric and antisymmetric modes, respectively. [] represents the results obtained by using simplified corner functions.

In Tables 5 and 6, using one corner function in eachr c2  cand Wcmeans that only

the functions leading to the correct singularity orders of moments and shear forces are added into the admissible functions. Whenk9k 1 13 is complex, the real part of the corresponding

complex corner function is used if only one corner function is used in the admissible func-tions. In the case of completely free sectorial plates (Table 5), the simplified corner functions give results very close (within 0.5%) to those produced by the original corner functions (Ta-ble 1), although the latter give slightly better solutions. In the case of sectorial plates with C-F-F boundary conditions (see Table 6), the simplified corner functions improve the con-vergence of the numerical results when compared with the model using just the polynomials. However, the improvement achieved by using the original corner functions is much better1 the difference can be more than 4%. The reason for the simplified corner functions having

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Table 6. Comparison of7a2 8h4D for a C-F-F sectorial plate with 1 1 3552 andh4a 1 051 by using different corner functions.

Mode No. of corner (Ieven,Iodd) in Eqs.(5)

no. functions (16,15) (18,17) (20,19) (22,21) (24,23) 9r c2  c2 Wc3 1 (0,0,0) 2.504 2.448 2.375 2.319 2.263 (1,1,1) 2.388 2.237 2.187 1.932 1.895 [2.428] [2.302] [2.200] [1.993] [1.955] (5,5,5) 2.216 2.186 1.884 1.851 1.811 [2.264] [2.201] [1.958] [1.929] [1.885] 2 (0,0,0) 3.029 2.918 2.839 2.765 2.744 (1,1,1) 2.857 2.571 2.473 2.416 2.380 [2.914] [2.689] [2.598] [2.518] [2.490] (5,5,5) 2.513 2.422 2.409 2.377 2.294 [2.607] [2.455] [2.429] [2.416] [2.372] 3 (0,0,0) 3.717 3.654 3.624 3.597 3.474 (1,1,1) 3.625 3.526 3.485 3.463 3.352 [3.641] [3.604] [3.564] [3.518] [3.431] (5,5,5) 3.510 3.489 3.464 3.345 3.319 [3.595] [3.537] [3.521] [3.412] [3.370] 4 (0,0,0) 5.452 5.335 5.281 5.237 5.196 (1,1,1) 5.298 5.223 5.149 5.112 5.048 [5.305] [5.278] [5.191] [5.166] [5.139] (5,5,5) 5.185 5.139 5.116 4.988 4.961 [5.252] [5.207] [5.139] [5.078] [5.005] 5 (0,0,0) 8.041 7.979 7.927 7.880 7.822 (1,1,1) 7.918 7.839 7.810 7.761 7.717 [7.950] [7.926] [7.875] [7.839] [7.794] (5,5,5) 7.762 7.633 7.536 7.498 7.463 [7.816] [7.701] [7.623] [7.568] [7.541]

Note: [] represents the results obtained by using simplified corner functions.

different effects on improving the accuracy of the results given in Tables 5 and 6 is that the first few values of k for clamped-free radial edges are complex numbers, so that the

simplified corner functions do not correctly portray the singular behaviors of moments in the vicinity of the vertex. Complexk 9k 1 13 yields bending moments approaching infinity

in an oscillatory manner as r approaches zero, while the simplified corner functions yield the moments approaching infinity monotonically. The results in Tables 5 and 6 imply that the ability of the corner functions to accurately describe the singular behaviors of moments and shear forces can considerably accelerate the convergence of numerical results. This observation also suggests a need for caution in the use of r-type singular elements in a finite element approach, in which is assigned to be real. Notably, the corner functions can cooperate with a finite element approaches as Gifford and Hilton (1978) developed enriched finite elements for determining stress intensity factor in plane crack problems.

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Table 7. Frequency parameters7a2 8h4D for completely clamped sectorial plates.

h/a 1 Mode Number

(degrees) 1 2 3 4 5 0.1 90 42.78 72.01 83.94 105.2 122.4 [42.75] [71.96] [83.86] [105.1] [122.2] 180 25.94 37.46 51.20 61.10 66.32 [25.91] [37.43] [51.16] [61.06] [66.26] 270 21.64 27.63 35.71 44.70 53.90 [21.62] [27.60] [35.68] [44.66] [53.87] 330 20.37 24.43 30.53 37.45 44.88 [20.35] [24.40] [30.51] [37.43] [44.85] 0.2 90 33.31 52.06 59.20 72.17 81.72 [33.28] [52.04] [59.16] [72.14] [81.71] 180 21.66 30.08 39.63 45.57 49.60 [21.63] [30.06] [39.61] [45.53] [49.55] 270 18.18 23.09 29.10 35.46 40.91 [18.14] [23.07] [29.07] [35.43] [40.88] 330 16.97 20.75 25.38 30.46 35.69 [16.95] [20.71] [25.36] [30.42] [35.66]

Note: [ ] represents results of Liu and Liew (1999).

4. FREQUENCIES AND MODE SHAPES

Extensive convergence studies were performed to give the accurate nondimensional fre-quency parameters 7a2 8h4D listed in Tables 7–9. All frequency results are guaranteed upper bounds to the exact values and exact (i.e., converged) to at least three significant figures. Tables 7–9 list the nondimensional frequencies of the first five modes for sectorial plates with different boundary conditions (C-C-C, F-F-F, C-F-F, S-F-F, C-C-F, and S-C-F), vertex angles (1 1 902, 1802, 2702 3002, 3302 and 3552) and ratios of thickness to radius (h4a 1 051 and 0.2). Since there are no published results for thick sectorial plates with 1 1802and boundary conditions other than simply supported along the radial edges, the results given in Tables 7–9 emphasize cases where1 1802 to fill the gap in existing fre-quency data in the published literature. Tables 8 and 9, together with the results given in Huang et al. (1994) represent all the possible cases when the circular edge is free.

As mentioned in the introduction, some researchers have investigated the vibrations of annular sector plates and used the free boundary conditions along the inner circular edge and a very small inner to outer radius ratio (0.00001) to approximate the solutions of sectorial plates. The accuracy of these approximate solutions depends on the chosen inner to outer radius ratio and the boundary conditions used along the inner circular edge. It is interest-ing to compare those published results with the present results, as Leissa et al. (1993) did with their results. Table 7 compares the present results with those of Liu and Liew (1999) for completely clamped sectorial plates. The agreement between the results is excellent, al-though the results produced by Liu and Liew are always smaller than the present results, as

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Table 8. Frequency parameters7a2 8h4D for sectorial plates with h4a 1 051 and having various boundary conditions.

Boundary 1 Mode Number

conditions (degrees) 1 2 3 4 5 F-F-F 90 15.37 22.00 28.66 35.25 51.66 180 6.813 9.146 17.30 17.37 27.35 270 4.454 5.859 9.063 12.29 16.40 300 3.675 5.378 8.155 10.29 15.01 330 3.101 4.757 7.737 8.679 12.73 355 2.730 4.228 7.548 7.577 11.23 C-F-F 90 4.400 12.71 23.06 32.20 47.77 180 2.388 3.817 8.971 15.53 18.50 270 1.962 2.198 4.366 7.787 12.15 300 1.937 2.198 3.839 6.411 10.05 330 1.802 2.180 3.317 5.373 8.432 355 1.712 2.177 3.266 4.722 7.336 S-F-F 90 9.055 17.13 27.09 42.39 48.79 180 2.726 7.342 14.02 16.59 22.63 270 2.161 3.350 6.681 11.01 15.99 300 2.156 2.744 5.423 9.059 13.34 330 1.891 2.868 4.268 7.589 11.33 355 1.613 2.987 3.742 6.605 10.93 C-C-F 90 13.36 29.81 41.53 56.75 72.98 180 4.462 8.789 16.81 24.64 24.93 270 3.066 4.074 7.851 11.12 17.63 300 2.920 3.461 6.109 10.08 14.13 330 2.839 3.062 5.113 8.306 11.76 355 2.688 2.858 3.947 7.240 10.83 S-C-F 90 8.881 25.45 37.66 48.54 69.31 180 2.899 7.155 14.14 22.99 23.40 270 2.403 3.281 6.591 10.90 16.17 300 2.238 3.156 5.415 8.909 13.30 330 2.027 3.055 4.116 7.497 11.23 355 1.815 2.847 3.794 7.232 10.81 they used a free boundary condition along the inner circular edge, which makes the plate less constrained than a completely clamped sectorial plate.

Notably, Tables 8 and 9 do not list the zero frequencies of the three rigid body modes and one rigid body mode for plates with F-F-F and S-F-F boundary conditions, respectively. If these rigid body modes are taken into account, as the boundary conditions change from F-F-F to S-F-F-F, C-F-F-F, S-C-F, and to C-C-F, the plates become stiffer, so that their frequencies increase for each mode. As the vertex angle (1) increases, the nondimensional frequency parameters generally decrease (except for the second modes of S-F-F plates). This trend was also observed in the solutions based on thin plate theory (McGee et al., 2003).

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Table 9. Frequency parameters7a2 8h4D for sectorial plates with h4a 1 052 and having various boundary conditions.

Boundary 1 Mode Number

conditions (degrees) 1 2 3 4 5 F-F-F 90 14.82 21.31 27.45 33.10 47.30 180 6.724 8.997 16.39 16.51 25.81 270 4.376 5.765 8.927 11.99 15.65 300 3.590 5.286 8.021 10.05 14.47 330 3.057 4.637 7.645 8.481 11.98 355 2.632 4.152 7.430 7.458 10.98 C-F-F 90 4.326 12.38 22.14 30.06 44.59 180 2.296 3.713 8.801 15.19 18.10 270 1.905 2.134 4.258 7.556 11.85 300 1.868 2.130 3.822 6.202 9.700 330 1.758 2.127 3.230 5.261 8.246 355 1.659 2.121 3.184 4.497 6.725 S-F-F 90 8.573 15.79 24.11 36.75 40.74 180 2.624 6.835 13.01 15.58 20.60 270 1.993 3.147 6.475 10.00 14.97 300 1.954 2.541 5.218 8.691 12.32 330 1.710 2.656 4.064 7.312 10.81 355 1.431 2.774 3.538 6.298 10.42 C-C-F 90 11.35 24.77 32.49 40.70 52.90 180 4.198 8.479 14.80 20.62 21.90 270 2.823 3.779 7.144 10.11 15.66 300 2.717 3.187 5.530 9.469 13.11 330 2.657 2.750 4.601 7.898 10.74 355 2.474 2.566 3.454 6.930 10.09 S-C-F 90 8.289 23.32 31.60 40.04 52.72 180 2.649 6.685 12.99 20.57 20.96 270 2.271 3.078 6.284 9.887 13.16 300 2.056 2.953 5.210 8.671 12.28 330 1.829 2.747 3.821 7.247 10.70 355 1.613 2.561 3.374 6.929 9.738 The effect of thickness on the vibration frequencies can be seen from the results in Ta-bles 8 and 9 and those for thin plates given by McGee et al. (1995b, 2003). The nondi-mensional frequency parameters decrease as the thickness increases if the radius (a) remains fixed. This is expected, because the effects of shear deformation and rotary inertia are in-cluded here. However, if h increases, with all other parameters (82 E, ) remaining the same, the frequencies (7) also increase.

Figures 2 to 4 show the nodal patterns of the first five modes for sectorial plates having h4a 1 051 and the same vertex angles (1 1 902, 1802, 2702 3002, 3302, and 3552) as in Tables 8 and 9 and various boundary conditions (F-F-F, C-F-F, and C-C-F). The parenthe-sized numbers are the nondimensional frequencies for the corresponding modes. The nodal

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Figure 2. Nodal patterns for F-F-F sectorial plates withh4a 1 051and various vertex angles.(Continued)

patterns of the F-F-F and C-F-F sectorial plates with1 1 3552shown in Figures 2 and 3 are very similar to those for thin plates given by Leissa et al. (1993) and McGee et al. (1995), respectively. The existence of the sharp V-notch (1 1 3552) severely distorts the nodal patterns of a completely free circular plate. When1 changes from 3302 to 3552, the nodal patterns for F-F-F plates remain very similar, and the nodal patterns for C-F-F plates are also similar except for the third mode, but the nodal patterns of the first three modes for C-C-F plates change dramatically.

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Figure 4. Nodal patterns for C-C-F sectorial plates with h4a 1 051 and various vertex angles.

(Continued)

The increase of vertex angle changes the sequence of the mode shapes for the symmetric and antisymmetric modes of F-F-F and C-C-F plates. For example, for F-F-F boundary conditions, the first modes of the plates with1 1 902and 1802are symmetric modes while the first modes for1 1 2702, 3002, 3302and 3552 are antisymmetric modes. The opposite sequence of symmetric and antisymmetric modes with changing1 is observed for the second modes. For C-C-F boundary conditions, the first modes of the plates with1 1 902, 1802,

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2702, 3002, and 3302 are symmetric modes with no nodal line, while the mode without a nodal line disappears for the case of1 1 3552, the first mode of which is an antisymmetric mode. These changes do not alter the trend of frequencies decreasing with the increase of the vertex angle.

5. CONCLUSION

This paper has demonstrated that adding corner functions to the algebraic-trigonometric polynomials as admissible functions of the Ritz method can significantly accelerate the con-vergence of the numerical results for sectorial plates. The main effect of the corner functions is to accurately present the singular behaviors of moments and shear forces in the vicinity of the vertex. When the moments or shear forces approach infinity in an oscillatory manner in the neighborhood of the vertex (i.e., whenkis complex), the admissible functions must not

only have the correct singularity order for the moments or shear forces, but also show the correct oscillation behavior.

Detailed numerical tables have been presented for frequencies of moderately thick sec-torial plates (h4a 1 051 and 0.2) with various boundary conditions and vertex angles. The presented frequencies are all exact to at least three significant figures and are, in most cases, the first such results to be shown in the published literature, as are the nodal patterns also given here. Generally speaking, the vibration frequencies decrease as the vertex angle in-creases. The nondimensional frequencies7a2 8h4D decrease as the thickness of the plate increases. This reliable information serves not only to improve the understanding of how sectorial plates can vibrate, but also as benchmark data against which other computational methods (such as finite element, boundary element or differential quadrature methods) may be checked.

Although this paper only considers the vibrations of sectorial plates, the method used here can be easily extended to study the vibration behaviors of a circular plate with V-notches (or cracks), where the notch vertex is not at the circle center.

Acknowledgements. This work was supported by the National Science Council, R.O.C. through research grant no. NSC93-2211-E-009-043. This support is gratefully acknowledged. Much appreciation is extended to National Center for High-performance Computing for granting the supercomputer resources.

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數據

Figure 1. Geometry and coordinate system of a sectorial plate.
Table 1. Corner functions corresponding to various boundary conditions along radial edges.
Table 1. Corner functions corresponding to various boundary conditions along radial edges
Table 2. Convergence of 7a 2  8h4D for a sectorial plate with C-F-F boundary conditions and 1 1 90 2 .
+7

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