SV Shock and Vibration 1875-9203 1070-9622 Hindawi Publishing Corporation 517946 10.1155/2014/517946 517946 Research Article Vibration Analysis of Annular Sector Plates under Different Boundary Conditions http://orcid.org/0000-0002-0488-5358 Shi Dongyan 1 http://orcid.org/0000-0002-7674-1046 Shi Xianjie 1 Li Wen L. 2 http://orcid.org/0000-0003-0613-3229 Wang Qingshan 1 Han Jiashan 3 Haddadpour Hassan 1 College of Mechanical and Electrical Engineering Harbin Engineering University Harbin 150001 China hrbeu.edu.cn 2 Department of Mechanical Engineering Wayne State University Detroit, MI 48201 USA wayne.edu 3 Luoyang Sunrui Special Equipment Co., Ltd. Luoyang 471003 China 2014 962014 2014 18 10 2013 18 05 2014 9 6 2014 2014 Copyright © 2014 Dongyan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which 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 each edge of plates. Regardless of boundary conditions, the displacement solution is invariably expressed as a new form of trigonometric expansion with accelerated convergence. The expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. This work allows a capability of modeling annular sector plates under a variety of boundary conditions and changing the boundary conditions as easily as modifying the material properties or dimensions of the plates. Of equal importance, the proposed approach is universally applicable to annular sector plates of any inclusion angles up to 2π. The reliability and accuracy of the current method are adequately validated through numerical examples.

1. Introduction

Annular sector plates are one of the most important structural components used in industrial applications and civil engineering. The vibrational characteristics of sector plates are thus of great interest to engineers and designers. Although there is a vast pool of studies about vibrations of circular and rectangular plates , relatively few results are reported for annular sector plates.

It appears that the previous investigations on the annular sector plates are mostly limited to classical edge conditions. It is widely believed that an exact analytical solution is only possible for an annular sector plate which is simply supported along, at least, two radial edges. However, a variety of possible boundary conditions such as elastic restraints are usually encountered in many engineering applications [2, 8, 23]. Moreover, the existing solution procedures are often only customized for a specific kind of boundary conditions and thus typically require constant modifications of the trial functions and corresponding solution procedures to adapt to different boundary conditions. As a result, the use of the existing solution procedures will result in very tedious calculations and will be easily inundated with a variety of possible boundary conditions. Therefore, it is important to develop an analytical method which is capable of universally dealing with annular sector plates subjected to different boundary conditions. In addition, the results of annular sector plates with reentrant angle are scarce.

In this paper, an improved Fourier series method (IFSM) previously proposed for the vibration analysis of beams and plates  is extended to annular sector plates under different boundary conditions, including the general elastic restraints. The displacement solution of the annular sector plate, regardless of boundary conditions, is expressed as a new form of trigonometric expansion with accelerated convergence. The reliability and accuracy of the proposed solution technique are validated extensively through numerical examples.

2. Theoretical Formulations 2.1. Basic Equations for an Annular Sector Plate

An annular sector plate (consisted with two radial and two circumferential edges) and the coordinate systems used in this investigation are shown in Figure 1. This plate is of constant thickness h, inner radius a, outer radius b, width R of plate in radial direction, and sector angle ϕ. The plate geometry and dimensions are defined in a cylindrical coordinate system (r, θ, z). A local coordinate system (s, θ, z) is also shown in Figure 1, which will be used in the analysis. The boundary conditions for the bending motion can be generally specified in terms of two kinds of restraining springs (translational and rotational) along each edge, resulting in four sets of distributed springs of arbitrary stiffness values.

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

The governing differential equation for the free vibration of an annular sector plate is given by (1)Dr2r2w(r,θ)-ρhω2w(r,θ)=0, where r2=2/r2+/rr+2/r2θ2, w(r,θ) is the flexural displacement, ω is angular frequency, and D=Eh3/(12(1-μ2)), ρ, and h are the flexural bending rigidity, the mass density, and the thickness of the plate, respectively.

In terms of the flexural displacement, the bending and twisting moments and transverse shearing forces can be expressed as (2)Mr=-D[2wr2+μr(wr+1r2wθ2)],Mθ=-D[1r(wr+1r2wθ2)+μ2wr2],Mrθ=-(1-μ)Dr[2wrθ-1rwθ],Qr=-D(2w)r,Qθ=-D1r(2w)θ, where μ is Poisson’s ratio.

The boundary conditions for an elastically restrained annular sector plate are given as (3)kraw=Qr,Krawr=-Mrat  r=a,krbw=-Qr,Krbwr=Mrat  r=bkθ0w=Qθ,Kθ0wrr=-Mθat  θ=0,kθ1w=-Qθ,Kθ1wrr=Mθat  θ=ϕ, where kra and krb (kθ0 and kθ1) are translational spring constants, and Kra and Krb (kθ0 and kθ1) are the rotational spring constants at r=a and b (θ=0 and ϕ), respectively. All the classical homogeneous boundary conditions can be simply considered as special cases when the spring constants are either extremely large or substantially small. The units for the translational and rotational springs are N/m and Nm/rad, respectively.

The solution for the vibration problem of an annular sector plate can be generally written in the forms of Bessel functions : (4)w(r,θ)=n=0[AnJn(kr)+BnYn(kr)+CnIn(kr)+DnKn(kr)]cosnθ,+n=1[An*Jn(kr)+Bn*Yn(kr)+Cn*In(kr)m+Dn*Kn(kr)]sinnθ, where krλ, Jn and Yn are the Bessel functions of the first and second kinds, respectively, and In and Kn are modified Bessel functions of the first and second kinds, respectively. The coefficients, An,,Dn, An*,,Dn*, which determine the shape of a mode, are to be solved from the boundary conditions.

If the boundary conditions are symmetric with respect to one or more diameters of the plate, the terms involving sinnθ are not needed and the solution (4) is simplified to (5)w(r,θ)=n=0[AnJn(kr)+BnYn(kr)+CnIn(kr)+DnKn(kr)]cosnθ.

The characteristic equation is derived by substituting the solution into the boundary conditions and setting the determinant of the resulting coefficient matrix equal to zero. The eigenvalues are obtained as the roots of characteristic equation using an appropriate nonlinear root-searching algorithm. The eigenvalues can also be found approximately since the Bessel functions are tabulated in many mathematical books or handbooks. Regardless of what procedures are adopted, the results are understandably dependent on the specific set of boundary conditions involved. The modal properties for annular plates are comprehensively reviewed in  for various boundary conditions or complicating factors. However, annular sector plates are rarely dealt with in the literature. This is evident from the fact that reviewing the vibrations of sector plates occupies only less than two pages in the classical monograph .

2.2. An Accelerated Trigonometric Series Representation for the Displacement Function

In the previous papers [26, 29], each displacement component of a rectangular plate is expressed as a 2D Fourier cosine series supplemented by eight auxiliary terms, which are introduced to accelerate the convergence of the series expansion. In this study, a similar but much simpler and more concise form of series expansion is employed to expand the flexural displacement of an annular sector plate in local coordinate system (s, θ, z): (6)w(s,θ)=m,n=-4Amnφm(s)φn(θ)(s=r-a), where Amn denotes the series expansion coefficients and (7)φm(s)={cosλmsm0sinλmsm<0(λm=mπR).

The basis function φn(θ) in the θ-direction is also given by (7) except for λn=nπ/ϕ. The sine terms in the above equation are introduced to overcome the potential discontinuities, along the edges of the plate, of the displacement function when it is periodically extended and sought in the form of trigonometric series expansion. As a result, the Gibbs effect can be eliminated and the convergence of the series expansion can be substantially improved.

To clarify this point, consider a function f(x) having Cn-1 continuity on the interval [0,π] and the nth derivative is absolutely integrable (the nth derivative may not exist at certain points). Denote the partial sum of the trigonometric series as (8)FM,2P[f](x)=m=-2PMamφm(x).

It can then be mathematically proven that the series expansion coefficients satisfy (9)limmamm2P=0(for  2Pn) if the negatively indexed coefficients, am  (m<0), are calculated from (10)am=k=1P[(-1)mf(2k-1)(π)+f(2k-1)(0)]×1j1<<jP-kP,j1,,jP-kixj12xjP-k2xij=1,jiP(xj2-xi2),m=xi={2i-1if  m  is  odd2iif  m  is  even,(i=1,2,...,P).

More explicitly, the convergence estimate (9) can be expressed as (11)am=O(m-(2P+1))for  2Pn, which means (12)max0xπ|f(x)-FM,2P[f](x)|=O(M-2P).

It is seen that convergence can be drastically improved virtually at no extra cost. It should be pointed out that the convergence rate of the series expansion (8) can be controlled by setting P to any appropriate value. In reality, however, the smoothness of the solution required for a given boundary value problem is mathematically dictated by the highest order of derivatives that appeared in the governing differential equation. Take the current plate problem for example. The plate equation demands that the third-order derivatives are continuous and the fourth-order derivatives exist everywhere over the surface area of the plate. Accordingly, one needs to set P=2 in seeking for a strong C3 solution, or P=1 for C1 solution in a weak formulation. Because the smoothness (or, explicitly, the convergence rate) of the current series expansion can be managed, at will, over the solution domain, the unknown series expansion coefficients can be obtained from either a weak or strong formulation. In seeking for a strong form of solution, the series is required to simultaneously satisfy the governing equation and the boundary conditions exactly on a point-wise basis. As a consequence, the expansion coefficients are not totally independent; the negatively indexed coefficients are related to the others via the boundary conditions. In a weak formulation such as the Rayleigh-Ritz technique, however, all the expansion coefficients are considered as the generalized coordinates independent from each other. The strong and weak solutions are mathematically equivalent if they are constructed with the same degree of smoothness over the solution domain. The Rayleigh-Ritz technique will be adopted in this study since the solution can be obtained much easily. More importantly, such a solution process is more suitable for future modeling of built-up structures.

2.3. Final System for an Annular Sector Plate

For a purely bending plate, the total potential energy can be expressed as (13)Vp=D20ϕ0R[(2ws2+1s+aws+1(s+a)22wθ2)2+2(1-μ)-2(1-μ)2ws2(1s+aws+1(s+a)22wθ2)+2(1-μ)+2(1-μ){s(1s+awθ)2}2](s+a)dsdθ.

By neglecting rotary inertia, the kinetic energy of the annular sector plate is given by (14)T=12ρhω20R0ϕw2(s+a)dsdθ.

The potential energies stored in the boundary springs are calculated as (15)Vs=120ϕ{a[kraw2+Kra(ws)2]s=0m+b[krbw2+Krb(ws)2]s=R}dθ+120R{[kθ0w2+Kθ0(w(s+a)θ)2]θ=0mm+[kθ1w2+Kθ1(w(s+a)θ)2]θ=ϕ}ds.

The Lagrangian for the annular sector plate can be generally expressed as (16)L=Vp+Vs-T.

By substituting (6) into (16) and minimizing Lagrangian against all the unknown series expansion coefficients, one is able to obtain a system of linear algebraic equations, in a matrix form, as (17)(K-ω2M)E=0, where E is a vector which contains all the unknown series expansion coefficients, and K and M are the stiffness and mass matrices, respectively. For conciseness, the detailed expressions for the stiffness and mass matrices are not shown here.

The eigenvalues (or natural frequencies) and eigenvectors of annular sector plates can now be easily and directly determined from solving a standard matrix eigenvalue problem (17). For a given natural frequency, the corresponding eigenvector actually contains the series expansion coefficients which can be used to construct the physical mode shape based on (6). Although this investigation is focused on the free vibration of an annular sector plate, the response of the annular sector plate to an applied load can be easily considered by simply including the work done by this load in the Lagrangian, eventually leading to a force term on the right side of (17). Since the displacement is constructed with the same smoothness as required of a strong form of solution, other variables of interest such as shear forces and power flows can be calculated directly, and perhaps more accurately, by applying appropriate mathematical operations to the displacement function.

3. Result and Discussion

To demonstrate the accuracy and usefulness of the proposed technique, several numerical examples will be presented in this section. First, consider a completely clamped annular sector plate. A clamped B.C. can be viewed as a special case when the stiffness constants for both sets of restraining springs become infinitely large (represented by a very large number, 5.0 × 1013, in the numerical calculations). The first six nondimensional frequency parameters, Ω=ωb2(ρh/D)1/2, are tabulated in Table 1 together with the reference results from  and an FEM prediction.

Frequency parameters, Ω=ωb2(ρh/D)1/2, for a completely clamped annular sector plate (a/b=0.4,ϕ=π/3, and μ=0.33).

Mode number
1 2 3 4 5 6
M = N = 5 85.267 150.14 194.29 243.66 266.21 358.17
M = N = 6 85.253 150.13 194.27 243.61 266.20 358.07
M = N = 7 85.257 150.10 194.23 243.61 266.07 358.05
M = N = 8 85.251 150.10 194.23 243.59 266.07 358.03
M = N = 9 85.251 150.10 194.22 243.59 266.05 358.03
M = N = 10 85.250 150.10 194.22 243.59 266.05 358.03
M = N = 11 85.250 150.10 194.22 243.59 266.04 358.02
M = N = 12 85.250 150.10 194.22 243.59 266.04 358.02
FEM 85.230 150.08 194.29 243.69 265.92 358.56
Reference  85.250 150.10 194.22 243.59 266.04

Next, consider an annular sector plate with simply supported radial edges. Three different boundary conditions (free, simply supported, and clamped) are sequentially applied to the circumferential edges. The simply supported condition is simply produced by setting the stiffnesses of the translational and rotational springs to and 0, respectively, and the free edge condition by setting both stiffnesses to zero. The first six nondimensional frequency parameters are shown in Table 2. The current results compare well with those taken from [22, 30].

Frequency parameters, Ω=ωb2(ρh/D)1/2, for an annular sector plate with radial edges simply supported (a/b=0.5,ϕ=π/4, and μ=0.3).

Circumferential edges Source Mode number
1 2 3 4 5 6
Free M = N = 5 21.069 66.726 81.606 146.42 176.12 176.91
M = N = 7 21.067 66.723 81.604 146.41 176.12 176.90
M = N = 9 21.067 66.722 81.604 146.41 176.12 176.90
M = N = 11 21.067 66.722 81.604 146.41 176.12 176.90
M = N = 12 21.067 66.722 81.604 146.41 176.12 176.90
Reference  21.067 66.722 81.604 146.41 176.12 176.90
Reference  21.067 66.722 81.604 146.41 176.12 176.90

Simply supported Present 68.379 150.98 189.60 278.39 283.59 387.62
Reference  68.379 150.98 189.60 278.39 283.59 387.62
Reference  68.379 150.98 189.60 278.39 283.59 387.64

Clamped Present 107.56 178.82 269.48 305.84 346.44 476.29
Reference  107.57 178.82 269.49 305.84 346.46 476.30
Reference  107.58 178.82 269.49 305.84 346.46 476.30

To illustrate the convergence and numerical stability of the current solution, several sets of results in Tables 1 and 2 are presented for using different truncation numbers: M=N=5,6,7,,12. A highly desired convergence characteristic is observed in that (a) sufficiently accurate results can be obtained with only a small number of terms in the series expansions and (b) the solution is consistently refined as more terms are included in the expansions. While the convergence of the current solution is mathematically established via (11) and (12), the actual (truncation) error will be case-dependent and cannot be exactly determined a priori. However, this should not constitute a problem in practice because one can always verify the accuracy of the solution by increasing the truncation number until a desired numerical precision is achieved. As a matter of fact, this “quality control” scheme can be easily implemented automatically. In modal analysis, the natural frequencies for higher-order modes tend to converge slower (see Table 1). Thus, an adequate truncation number should be dictated by the desired accuracy of the largest natural frequencies of interest. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as M=N=12 in the following calculations.

In the very limited existing studies, the sector angles are typically assumed to be less than π, as specified in terms of m=π/ϕ being an integer. Although it is not clear whether ϕ=π inherently constitutes a pivoting point for mathematically solving sector plate problems, it has been a limit practically defining the previous investigations. However, the value of the sector angle appears to have no binding effect on the current solution procedures as described earlier. To verify this statement and illustrate the versatility of the proposed technique, the plates with a full range of sector angles are studied under various restraining conditions. Presented in Table 3 are the first six frequency parameters, Ω=ωb2(ρh/D)1/2, for annular sector plates (a/b=0.4) which are completely free along all of their edges. Due to a lack of analytical solutions, the numerical results calculated using an FEM (ABAQUS) model are given there for comparison. Since the reference solutions for annular sector plates are not readily available, the plates with other classical boundary conditions are also studied systematically and the corresponding results are listed in Tables 4 and 5 for a range of sector angles up to 2π. Such results can be particularly useful in benchmarking other solution methods. In identifying the boundary conditions, letters C, S, and F have been used to indicate the clamped, simply supported, and free boundary condition along an edge, respectively. Thus, the boundary conditions for a plate are fully specified by using four letters with the first one indicating the B.C. along the first edge, r=a. The remaining (the second to the fourth) edges are ordered in the counterclockwise direction. In all these cases, the current solutions are adequately validated by the FEM results obtained using ABAQUS models. Also included are the results previously given in  for smaller sector angles, ϕ=π/6 and π/2. The mode shapes for the first six modes are plotted in Figure 2 for the fully clamped annular sector plate with cutout ratio a/b=0.4 and sector angle ϕ=π. These modes are verified by the FEM results although they will not be shown here for conciseness.

Frequency parameters, Ω=ωb2(ρh/D)1/2, for completely free annular sector plates (a/b=0.4 and μ=0.3).

ϕ Source Mode number
1 2 3 4 5 6
π / 6 Present 61.515 67.249 113.97 149.51 171.91 244.95
FEM 61.516 67.230 113.97 149.46 171.94 244.92

π / 2 Present 15.647 23.576 38.428 53.649 63.390 70.739
FEM 15.646 23.572 38.425 53.645 63.389 70.740

2 π / 3 Present 10.148 16.348 24.588 35.673 44.649 59.663
FEM 10.148 16.345 24.584 35.672 44.645 59.663

π Present 7.0437 7.6871 15.378 17.404 28.449 28.559
FEM 7.0435 7.6858 15.374 17.404 28.445 28.558

7 π / 6 Present 5.4291 6.4773 13.094 13.136 21.512 24.262
FEM 5.4281 6.4769 13.093 13.131 21.511 24.258

3 π / 2 Present 2.8863 5.2324 8.8762 9.7849 14.001 17.608
FEM 2.8858 5.2320 8.8759 9.7822 13.999 17.606

16 π / 9 Present 1.8048 3.8657 7.3593 7.7069 11.229 13.075
FEM 1.8044 3.8653 7.3581 7.7066 11.227 13.074

2 π Present 1.2956 2.9318 5.7558 7.1511 9.9277 10.581
FEM 1.2963 2.9334 5.7582 7.1519 9.9283 10.584

Frequency parameters, Ω=ωb2(ρh/D)1/2, for fully clamped annular sector plates (μ=0.3).

ϕ a / b Source Mode number
1 2 3 4 5 6
π / 6 0.2 Present 188.21 300.11 417.61 429.57 577.84 599.88
0.4 Present 188.37 305.06 417.47 461.14 600.03 671.78
FEM 188.41 305.11 417.76 461.26 600.35 672.18
Reference  188.36 305.04 417.39 461.00 596.16 672.01
0.6 Present 216.12 422.67 454.32 662.94 728.63 820.91

π / 2 0.2 Present 50.283 87.826 113.99 136.90 165.36 195.50
0.4 Present 69.822 95.706 138.96 179.79 195.38 207.76
FEM 69.839 95.709 138.98 179.85 195.49 207.84
Reference  60.835 95.701 138.96 179.79 195.51 207.82
0.6 Present 144.27 159.26 187.24 228.34 285.34 351.81

2 π / 3 0.2 Present 41.835 63.382 93.964 104.76 129.64 133.58
0.4 Present 65.700 78.393 101.37 133.72 174.32 175.71
0.6 Present 142.05 149.60 163.44 184.16 212.92 249.07

π 0.2 Present 37.061 45.338 59.618 78.667 98.872 100.43
FEM 37.043 45.334 59.767 78.671 98.962 100.81
0.4 Present 63.331 68.008 76.617 89.647 107.22 128.75
FEM 63.329 68.006 76.606 89.639 107.14 128.75
0.6 Present 140.80 143.64 149.03 156.59 167.53 181.84
FEM 140.64 143.68 149.98 156.85 167.62 181.55

7 π / 6 0.2 Present 36.241 41.835 52.102 66.284 83.552 98.083
FEM 36.244 41.844 52.088 66.314 83.364 98.114
0.4 Present 62.904 66.110 72.109 80.996 93.532 108.83
FEM 62.900 66.142 72.044 81.035 93.320 108.84
0.6 Present 140.34 142.53 146.35 151.85 159.58 170.05
FEM 140.36 142.53 146.28 151.78 159.25 168.88

3 π / 2 0.2 Present 35.495 38.369 44.053 52.411 63.089 75.717
0.4 Present 62.412 64.212 67.419 72.312 79.187 87.876
0.6 Present 140.00 140.83 143.54 146.75 150.99 155.98

16 π / 9 0.2 Present 35.192 37.153 40.797 46.446 54.017 63.212
0.4 Present 62.298 63.555 65.696 69.042 73.579 79.273
0.6 Present 139.94 141.66 142.46 144.56 147.48 151.01

2 π 0.2 Present 35.061 36.520 39.252 43.464 49.307 56.495
FEM 35.056 36.502 39.208 43.417 49.199 56.438
0.4 Present 62.188 63.167 64.820 67.263 70.707 75.100
FEM 62.192 65.153 64.821 67.294 70.663 75.033
0.6 Present 139.72 140.68 141.75 143.44 145.58 148.37
FEM 139.88 140.57 141.74 143.41 145.61 148.38

Frequency parameters, Ω=ωb2(ρh/D)1/2, for CSCS annular sector plates (μ=0.3).

ϕ a / b Source Mode number
1 2 3 4 5 6
π / 4 0.2 Present 70.255 144.22 168.24 243.43 282.20 304.07
FEM 70.247 144.22 168.27 243.47 282.23 304.19
Reference  70.255 144.22 168.24 243.45 282.04 304.04
0.4 Present 84.594 169.65 199.19 296.82 304.11 366.32
FEM 84.589 169.65 199.21 296.82 304.20 366.51
Reference  84.592 169.65 199.17 296.76 304.08 366.23
0.6 Present 154.74 211.64 321.79 404.70 467.22 481.99

π / 2 0.2 Present 41.833 70.256 106.58 114.23 144.23 168.24
0.4 Present 66.678 86.611 119.45 169.43 177.64 199.21
FEM 66.678 84.593 119.45 169.66 177.68 199.22
0.6 Present 143.14 154.76 176.85 211.65 259.99 321.76

2 π / 3 0.2 Present 38.332 53.392 80.167 101.64 114.23 121.67
0.4 Present 64.480 73.627 91.703 119.46 155.88 174.66
0.6 Present 141.59 147.78 159.19 176.87 201.54 234.20

π 0.2 Present 36.108 41.828 53.387 70.236 90.850 98.272
FEM 36.105 41.820 53.389 70.255 90.859 98.294
0.4 Present 62.989 66.672 73.625 84.585 99.938 119.46
FEM 63.004 66.678 73.633 84.597 99.907 119.46
0.6 Present 140.48 143.13 147.79 154.74 164.32 176.83
FEM 140.51 143.17 147.82 154.76 164.34 176.87

7 π / 6 0.2 Present 35.684 39.622 47.703 60.066 75.868 94.024
FEM 35.684 39.629 47.699 60.063 75.821 94.041
0.4 Present 62.693 65.318 70.196 77.814 88.570 102.45
FEM 62.700 65.329 70.199 77.810 88.526 102.45
0.6 Present 140.25 142.27 145.54 150.61 157.21 165.96
FEM 140.29 142.22 145.56 150.49 157.22 165.94

3 π / 2 0.2 Present 35.241 37.429 41.823 48.904 58.484 70.372
0.4 Present 62.367 63.876 66.619 70.902 76.824 84.539
0.6 Present 140.12 141.09 143.12 145.99 149.67 154.81

16 π / 9 0.2 Present 35.036 36.539 39.462 44.137 50.791 59.178
0.4 Present 62.218 63.301 65.160 68.051 71.966 77.440
0.6 Present 139.89 140.63 142.11 144.11 146.83 149.99

2 π 0.2 Present 34.948 36.119 38.293 41.824 46.882 53.362
FEM 34.961 36.106 38.298 41.824 46.862 53.401
0.4 Present 62.148 63.172 64.429 66.683 69.767 73.579
FEM 62.156 63.004 64.481 66.682 69.699 73.641
0.6 Present 139.95 140.33 141.57 143.13 145.19 147.67
FEM 139.87 140.51 141.61 143.17 145.23 147.83

The first six mode shapes for a CCCC annular sector plate (a/b=0.4 and ϕ=π): the (a) first, (b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth mode shape.

All the above examples involve the classical homogeneous boundary conditions which are viewed as special cases (of elastically restrained edges) when the stiffness constants take extreme values. We now turn to annular sector plates with general elastically restrained edges. First consider an annular sector plate simply supported, but with uniform rotational restraint, along each edge. The first six frequency parameters are presented in Table 6 together with the results calculated using an ABAQUS model. The second example concerns a cantilever annular sector plate (clamped at θ=0) with identical elastic restraints at other edges. While the stiffness of the translational springs is fixed to k=104 N/m, the rotational springs will be specified to take different stiffness values: K=100, 104, 108, 1012 Nm/rad. The corresponding frequency parameters are shown in Table 7. In all the cases, a good agreement is observed between the current solution and the FEM results.

Frequency parameters, Ω=ωb2(ρh/D)1/2, for simply supported annular sector plates with uniform rotational restraint along each edge (a/b=0.4,ϕ=2π/3, and μ=0.3).

1 2 3 4 5 6
100 Present 32.618 46.318 68.813 98.903 115.80 131.64
FEM 32.578 46.212 68.677 98.751 115.82 131.52

104 Present 41.395 54.245 76.164 106.05 125.56 141.73
FEM 41.329 54.148 76.060 105.99 126.75 141.82

108 Present 65.688 78.383 101.29 133.72 173.81 175.66
FEM 65.762 78.429 101.34 133.83 174.11 176.29

1012 Present 65.698 78.394 101.30 133.74 173.83 175.69
FEM 65.772 78.439 101.34 133.85 174.13 176.32

Frequency parameters, Ω=ωb2(ρh/D)1/2, for an FCFF annular sector plate with identical elastic restraint at “free” edges (a/b=0.4,ϕ=π, and μ=0.3).

1 2 3 4 5 6
100 Present 4.4992 5.9227 9.6841 12.490 18.444 24.601
FEM 4.4898 5.9352 9.6797 12.491 18.448 24.576

104 Present 4.6998 7.3625 12.924 20.819 21.912 30.652
FEM 4.6961 7.3665 12.926 20.801 21.938 30.630

108 Present 4.7531 8.4457 15.344 24.898 30.587 36.816
FEM 4.7507 8.4435 15.345 24.909 30.605 36.842

1012 Present 4.7531 8.4460 15.345 24.899 30.589 36.817
FEM 4.7508 8.4435 15.346 24.909 30.606 36.842

Lastly, consider reentrant annular sector plates (ϕ=16π/9) elastically restrained along all the four edges. The stiffnesses for the translational and rotational restraints is chosen as k=105 N/m and K=107 Nm/rad, respectively. The first six frequency parameters are shown in Table 8 for three different cutout ratios. Plotted in Figure 3 are the mode shapes for the plate with a/b=0.4.

Frequency parameters, Ω=ωb2(ρh/D)1/2, for annular sector plates with elastic restraint at all four edges: k=105 N/m and K=107 Nm/rad (ϕ=16π/9 and μ=0.3).

a / b Source Mode number
1 2 3 4 5 6
0.2 Present 10.268 11.702 13.324 15.209 17.664 21.015
FEM 10.251 11.684 13.311 15.209 17.678 21.038

0.4 Present 11.884 12.593 13.802 15.471 17.763 20.988
FEM 11.862 12.575 13.795 15.477 17.780 21.009

0.6 Present 14.586 14.934 15.663 16.794 18.413 20.849
FEM 14.557 14.908 15.650 16.802 18.441 20.884

The first six mode shapes for an annular sector plate (a/b=0.4 and ϕ=16π/9) with elastic restraints, k=105 N/m and K=107 Nm/rad, at all the four edges: the (a) first, (b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth mode shape.

4. Conclusions

An analytical method has been presented for the vibration analysis of annular sector plates with general elastic restraints along each edge, which allows treating all the classical homogenous boundary conditions as the special cases when the stiffness for each of the restraining spring is equal to either zero or infinity. Regardless of boundary conditions, the displacement function is invariantly expressed as an improved trigonometric series which converges uniformly at an accelerated rate. Since the displacement solution is constructed to have C3 continuity, the current solution, although sought in a weak form from the Rayleigh-Ritz procedure, is mathematically equivalent to a strong solution which simultaneously satisfies both the governing differential equation and the boundary conditions on a point-wise basis.

The present method provides a unified means for predicting the free vibration characteristics of annular sector plates with a variety of boundary conditions and any sector angles. The efficiency, accuracy, and reliability of the proposed method are fully illustrated for free vibration analysis of annular sector plates with different boundary supports and model parameters such as radius ratio and sector angle. Numerical results obtained by the present approach are in excellent agreement with those available in the literature. Although the stiffness for each restraining spring is here assumed to be uniform, any nonuniform, discrete, or partial stiffness distribution can be readily considered by modifying potential energies accordingly.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous reviewers for their very valuable comments. This work was supported by the International S&T Cooperation Program of China (2011DFR90440) and the key project of the National Natural Science of Foundation of China (50939002). The second author is also grateful for the supports from China Scholarship Council (2011668004).