# Vibration Analysis of Annular Sector Plates under Different Boundary Conditions

Authors: Dongyan Shi; Xianjie Shi; Wen L. Li; Qingshan Wang; Jiashan Han
Journal: Shock and Vibration
 (2014)

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## Abstract

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.


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## Body

## 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 [1], relatively few results are reported for annular sector plates.Over the past decades, vibrations of annular sector plates have been investigated using various analytical or numerical methods, such as the energy method [2], spline element method [3, 4], finite element method [5], integral equation method [6], and so on [7]. In particular, a general technique was developed by Leissa [1] to obtain exact modal frequencies for plates which are simply supported along the radial edges and have arbitrary boundary conditions at the circumferential edges. This method utilizes the well-known Bessel function solutions for a circular plate by allowing the functions to have noninteger orders. His following work [8] using Ritz method advocated that the use of the ordinary Bessel functions solution is incorrect for solid sector thin plates having simply supported radial edges and sector angle larger than  π. Liew et al. [9] reviewed many investigations about the vibration of thick plates published before 1993. It is shown that a majority of them are focused on the classical boundary conditions (simply supported, clamped, or free edges). In comparison, other more complicated boundary conditions such as elastic boundary supports are rarely attempted. A closed-form solution is proposed by Kim and Yoo [10] in which the displacements are expressed in terms of trigonometric and exponential functions under the polar coordinate system. Ramakrishnan and Kunukkasseril [11] solved the vibration problem of an annular sector plate with simply supported radial edges and arbitrary conditions along the circumferential edges. Aghdam et al. [12] presented an approximate solution for bending deformation of thin sector plates using extended Kantorovich method in which the fourth-order governing equation is converted into two ordinary differential equations. Employing thepb-2 Rayleigh-Ritz method, Xiang et al. [13] tackled the vibration problem of annular sector Mindlin plates. Frequency parameters for annular sector plates with different geometry parameters and boundary conditions were presented. Liu and Chen [14] proposed an axisymmetric finite element for axisymmetric vibration analysis of annular and circular plates. Civalek and Ülker [15] utilized harmonic differential quadrature (HDQ) method to study the linear bending characteristics of circular plates. Civalek [16] compared the methods of differential quadrature (DQ) and harmonic differential quadrature (HDQ). These methods were utilized for buckling, bending, and free vibrations of thin isotropic plates. Accurate three-dimensional elasticity solutions of annular sector plates are presented under arbitrary boundary conditions by Liew et al. [17]. X. Wang and Y. Wang [18] extended the differential quadrature (DQ) method to analyze the free vibration problem of thin sector plates. Irie et al. [19] investigated the free vibrations of ring-shaped polar-orthotropic sector plates using a spline function as an admissible function for the deflection of the plates. In this approach, the flexural transverse deflection of sector plates is expressed as a series of the products of the deflection functions of a sectorial beam and a circular beam that satisfy the similar type of boundary conditions. Three-dimensional vibrations of annular sector plates with various boundary conditions were studied by Zhou et al. [20] using Chebyshev-Ritz method. Also the solutions of annular sector plates with reentrant angle are presented in Zhou et al. [20] investigation. Baferani et al. [21] presented an analytical solution for the free vibration of functionally graded (FG) thin annular sector plates resting on elastic foundations. The plates are considered to be simply supported along radial edges and arbitrarily supported at the circumferential edges. Mirtalaie and Hajabasi [22] studied the free vibration analysis of functionally graded (FG) thin annular sector plates with DQ method.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 [24–28] 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 Figure1. 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.
(a)
(b)The governing differential equation for the free vibration of an annular sector plate is given by(1)D∇r2∇r2w(r,θ)-ρhω2w(r,θ)=0,
where ∇r2=∂2/∂r2+∂/r∂r+∂2/r2∂θ2, 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[∂2w∂r2+μr(∂w∂r+1r∂2w∂θ2)],Mθ=-D[1r(∂w∂r+1r∂2w∂θ2)+μ∂2w∂r2],Mrθ=-(1-μ)Dr[∂2w∂r∂θ-1r∂w∂θ],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,Kra∂w∂r=-Mratr=a,krbw=-Qr,Krb∂w∂r=Mratr=bkθ0w=Qθ,Kθ0∂wr∂r=-Mθatθ=0,kθ1w=-Qθ,Kθ1∂wr∂r=Mθatθ=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 [1]:
(4)w(r,θ)=∑n=0∞[AnJn(kr)+BnYn(kr)+CnIn(kr)+DnKn(kr)]cos⁡nθ,+∑n=1∞[An*Jn(kr)+Bn*Yn(kr)+Cn*In(kr)m+Dn*Kn(kr)]sinnθ,
where kr≡λ, 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 involvingsinnθ are not needed and the solution (4) is simplified to
(5)w(r,θ)=∑n=0∞[AnJn(kr)+BnYn(kr)+CnIn(kr)+DnKn(kr)]cos⁡nθ.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 [1] 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 [1].


### 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=-4∞Amnφm(s)φn(θ)(s=r-a),
where Amn denotes the series expansion coefficients and
(7)φm(s)={cos⁡λmsm≥0sinλmsm<0(λm=mπR).The basis functionφn(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 functionf(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)lim⁡m→∞amm2P=0(for2P≤n)
if the negatively indexed coefficients, am(m<0), are calculated from
(10)am=∑k=1P[(-1)mf(2k-1)(π)+f(2k-1)(0)]×∑1≤j1<⋯<jP-k≤P,j1,…,jP-k≠ixj12⋯xjP-k2xi∏j=1,j≠iP(xj2-xi2),m=xi={2i-1ifmisodd2iifmiseven,(i=1,2,...,P).More explicitly, the convergence estimate (9) can be expressed as
(11)am=O(m-(2P+1))for2P≤n,
which means
(12)max⁡0≤x≤π|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 bemanaged,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 conditionsexactly 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=D2∫0ϕ∫0R[(∂2w∂s2+1s+a∂w∂s+1(s+a)2∂2w∂θ2)2+2(1-μ)-2(1-μ)∂2w∂s2(1s+a∂w∂s+1(s+a)2∂2w∂θ2)+2(1-μ)+2(1-μ){∂∂s(1s+a∂w∂θ)2}2](s+a)dsdθ.By neglecting rotary inertia, the kinetic energy of the annular sector plate is given by(14)T=12ρhω2∫0R∫0ϕw2(s+a)dsdθ.The potential energies stored in the boundary springs are calculated as(15)Vs=12∫0ϕ{a[kraw2+Kra(∂w∂s)2]s=0m+b[krbw2+Krb(∂w∂s)2]s=R}dθ+12∫0R{[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.


## 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 Figure1. 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.
(a)
(b)The governing differential equation for the free vibration of an annular sector plate is given by(1)D∇r2∇r2w(r,θ)-ρhω2w(r,θ)=0,
where ∇r2=∂2/∂r2+∂/r∂r+∂2/r2∂θ2, 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[∂2w∂r2+μr(∂w∂r+1r∂2w∂θ2)],Mθ=-D[1r(∂w∂r+1r∂2w∂θ2)+μ∂2w∂r2],Mrθ=-(1-μ)Dr[∂2w∂r∂θ-1r∂w∂θ],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,Kra∂w∂r=-Mratr=a,krbw=-Qr,Krb∂w∂r=Mratr=bkθ0w=Qθ,Kθ0∂wr∂r=-Mθatθ=0,kθ1w=-Qθ,Kθ1∂wr∂r=Mθatθ=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 [1]:
(4)w(r,θ)=∑n=0∞[AnJn(kr)+BnYn(kr)+CnIn(kr)+DnKn(kr)]cos⁡nθ,+∑n=1∞[An*Jn(kr)+Bn*Yn(kr)+Cn*In(kr)m+Dn*Kn(kr)]sinnθ,
where kr≡λ, 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 involvingsinnθ are not needed and the solution (4) is simplified to
(5)w(r,θ)=∑n=0∞[AnJn(kr)+BnYn(kr)+CnIn(kr)+DnKn(kr)]cos⁡nθ.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 [1] 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 [1].


## 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=-4∞Amnφm(s)φn(θ)(s=r-a),
where Amn denotes the series expansion coefficients and
(7)φm(s)={cos⁡λmsm≥0sinλmsm<0(λm=mπR).The basis functionφn(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 functionf(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)lim⁡m→∞amm2P=0(for2P≤n)
if the negatively indexed coefficients, am(m<0), are calculated from
(10)am=∑k=1P[(-1)mf(2k-1)(π)+f(2k-1)(0)]×∑1≤j1<⋯<jP-k≤P,j1,…,jP-k≠ixj12⋯xjP-k2xi∏j=1,j≠iP(xj2-xi2),m=xi={2i-1ifmisodd2iifmiseven,(i=1,2,...,P).More explicitly, the convergence estimate (9) can be expressed as
(11)am=O(m-(2P+1))for2P≤n,
which means
(12)max⁡0≤x≤π|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 bemanaged,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 conditionsexactly 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=D2∫0ϕ∫0R[(∂2w∂s2+1s+a∂w∂s+1(s+a)2∂2w∂θ2)2+2(1-μ)-2(1-μ)∂2w∂s2(1s+a∂w∂s+1(s+a)2∂2w∂θ2)+2(1-μ)+2(1-μ){∂∂s(1s+a∂w∂θ)2}2](s+a)dsdθ.By neglecting rotary inertia, the kinetic energy of the annular sector plate is given by(14)T=12ρhω2∫0R∫0ϕw2(s+a)dsdθ.The potential energies stored in the boundary springs are calculated as(15)Vs=12∫0ϕ{a[kraw2+Kra(∂w∂s)2]s=0m+b[krbw2+Krb(∂w∂s)2]s=R}dθ+12∫0R{[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 [22] and an FEM prediction.Table 1
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 [22]
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].Table 2
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 [22]
21.067
66.722
81.604
146.41
176.12
176.90
Reference [30]
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 [22]
68.379
150.98
189.60
278.39
283.59
387.62
Reference [30]
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 [22]
107.57
178.82
269.49
305.84
346.46
476.30
Reference [30]
107.58
178.82
269.49
305.84
346.46
476.30To illustrate the convergence and numerical stability of the current solution, several sets of results in Tables1 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 determineda 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 thanπ, 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 [13] 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.Table 3
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
10.584Table 4
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 [13]
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 [13]
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
148.38Table 5
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 [13]
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 [13]
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
147.83The 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.
(a)
(b)
(c)
(d)
(e)
(f)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 Table6 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.Table 6
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).
K(Nm/rad)
Source
Mode number
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
176.32Table 7
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).
K (Nm/rad)
Source
Mode number
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
36.842Lastly, 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.Table 8
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
20.884The 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.
(a)
(b)
(c)
(d)
(e)
(f)


## 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 haveC3 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.



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*Source: 517946-2014-06-09.xml*