Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

Abstract

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Keywords

weak form quadrature element method free vibration thin sectorial plates arbitrary boundary conditions corner functions

1. Introduction

Sectorial plates are widely used in special structures in civil, ocean, nuclear, and aerospace engineering. Therefore, research on the vibration characteristics of sectorial plates with various boundary conditions is very meaningful. However, due to the existence of a sharp vertex, singularity problems must be considered when conducting sectorial plate vibration analysis, which is a severe challenge for many numerical methods. Williams was the first to study the singularities of thin plates with a stationary crack (Williams, 1952, 1961). In his work, characteristic equations for various boundary condition combinations were given explicitly, which indicate the order of stress singularity around the vertex. Thereafter, Ojikutu analyzed the stress singularities in laminated composite plates (Ojikutu et al., 1984), which was further extended to the anisotropic case by Chue and Liu (Chue and Liu, 2002). Leissa studied the singularity effect on membrane, plate, and shell behaviors (Leissa, 2001). Several typical examples of concentrated forces, concentrated moments, and sharp corners were discussed to show the effect of stress singularities. Huang conducted extensive research on corner stress singularities in thick plates (Huang et al., 1994), bi-material Mindlin plates (Huang, 2002), high-order plates (Huang, 2004), and functionally graded material (FGM) thin plates (Huang and Chang, 2007). McGee et al. derived the sharp corner functions for Mindlin plates (McGee et al., 2005). Saidi et al. introduced a boundary layer function to investigate the singularities near the vertex (Saidi et al., 2010).

Along with the development of singularity analysis, several analytical solutions for sectorial plate vibration analysis were derived. Huang et al. introduced the Bessel functions to obtain the analytical solutions for vibration analysis of thin and thick sectorial plates with simply-supported radial edges (Huang et al., 1993, 1994). Huang and Ho gave the analytical solution for vibration analysis of a polarly orthotropic sectorial Mindlin plate with simply-supported radial edges for the first time (Huang and Ho, 2004). Jomehzadeh and Saidi provided an analytical solution for free vibration of sectorial Mindlin plates using a boundary layer function (Jomehzadeh and Saidi, 2009). Es’haghi derived a closed-form solution for free vibration of thick sectorial plates based on Reddy’s third-order shear deformation plate theory (Es’haghi, 2014).

Compared to the limited analytical methods, numerous numerical methods have been proposed for free vibration analysis of sectorial plates. Kim and Dickson applied the Rayleigh–Ritz method with orthogonally generated polynomials to obtain the lowest six frequency parameters (Kim and Dickinson, 1989). Leissa et al. provided the first six frequency parameters of sectorial plates using two sets of admissible functions, which were chosen for accelerating convergence (Leissa et al., 1993a, 1993b; McGee et al., 1993). Liu and Liew conducted a free vibration analysis of sectorial Mindlin plates by the differential quadrature method (DQM) (Liu and Liew, 1999). Houmat applied a sector Fourier p-element to free vibration analysis of sectorial plates (Houmat, 2001). The triangular differential quadrature method (TDQM) proposed by Zhong (Zhong, 2000) was used for free vibration analysis of sectorial Mindlin plates (Li et al., 2004). Wang and Wang extended the differential quadrature method to free vibration analysis of thin sectorial plates (Wang and Wang, 2004). Gürses et al. analyzed the free vibration of thin sectorial plates using the discrete singular convolution (DSC) (Gürses et al., 2010). McGee et al. discussed the influence of stress singularities on the vibration of sectorial plates with arbitrary radial edge conditions (McGee et al., 2010). Cheng et al. applied the element-free Galerkin method and its variants for two-dimensional elastodynamics problems and Kirchhoff plate bending problems (Chen et al., 2015; Cheng et al., 2012; Wang et al., 2019; Zhang et al., 2013). Shi et al. proposed a unified method for free vibration analysis of sectorial plates with arbitrary boundary conditions (Shi et al., 2014). Su et al. applied a modified Fourier series to study the free vibration of laminated composite and FGM sectorial plates (Su et al., 2015). Zhang et al. established a unified model for vibration analysis of composite laminated annular, circular, and sectorial plates (Zhang et al., 2019).

In this paper, the weak form quadrature element method (QEM) is extended to free vibration analysis of thin sectorial plates. The QEM proposed by Zhong and Yu (Zhong and Yu, 2007) is an effective tool for solving structural problems. It starts from dividing the problem domain into several integrable subdomains. Geometric mapping to standard domains is adopted afterward. The relative integrals in functionals are estimated by a numerical quadrature rule, usually Lobatto quadrature. The derivatives at quadrature points are evaluated by the differential quadrature analog (C0 problem) (Quan and Chang, 1989) or the generalized differential quadrature analog (C1 problem) (Wu and Liu, 2000). After introducing the corresponding variational principles, results can be obtained by solving the algebraic equations. One can refer to the recent comprehensive review of the QEM for more details (Liao, 2023). For the free vibration analysis of thin sectorial plates, the generalized differential quadrature analog is adopted, and the final generalized eigenvalue formulation is solved by the QR algorithm.

Prior to this paper, a rotation-free quadrature element formulation was proposed by Cai et al. for free vibration analysis of thin sectorial plates (Cai et al., 2021). Though the approach in their work was entitled the QEM, their work is essentially different from the present QEM. As a tiny cutout at the vertex of a sectorial plate was assumed, the rotation-free quadrature element formulation encountered enormous difficulties in dealing with free vertex. Their results for sectorial plates with a free vertex were rather sensitive to the cutout size. Moreover, the results for sectorial plates with a larger sector angle were not accurate enough due to negligence of the singularity around the vertex. To address these issues, the displacement-based weak form quadrature element, which has been successfully applied to various problems with singularities (Liu and Zhong, 2022a, 2022b, 2023a, 2023b), is adopted in this paper. The present formulation has no restrictions on vertex angles or boundary conditions. In addition, the results obtained by the present formulation agree well with those credible analytical and numerical results, demonstrating the superiority of the present formulation.

2. Corner functions

Consider an isotropic elastic sectorial thin plate, as shown in Figure 1. The x-axis bisects $∠ AOB$ . The deflection of the plate is denoted as $w$ . $α$ is the sector angle. $E$ and $ν$ are Young’s modulus and Poisson’s ratio. For simplicity, simply supported, clamped, and free boundary conditions are abbreviated as S, C, and F, respectively. Then, the boundary conditions of a sectorial plate are identified by the combination of the three letters. For instance, CSS means that edge 1 is clamped, edges 2 and 3 are simply supported; SCF indicates that edge 1 is simply supported, edge 2 is clamped and edge 3 is free.

The differential equation for the free vibration of thin plates can be written as

\nabla^{4} w (r, θ, t) + \frac{ρ h}{D} \frac{\partial^{2} w (r, θ, t)}{\partial t^{2}} = 0 .

(1)

Ignoring the inertial term, the general solution of the homogeneous flexural equilibrium equation of thin plates

\nabla^{4} w (r, θ, t) = 0

(2)

can be expressed as

w (r, θ, t) = w_{s} (r, θ, t) + w_{a} (r, θ, t) = \sum_{k = 1}^{\infty} w_{k} (r, θ, t),

(3)

\begin{array}{l} w_{k} (r, θ, t) = r^{λ_{k} + 1} [a_{k} \sin (λ_{k} + 1) θ + b_{k} \cos (λ_{k} + 1) θ \\ + c_{k} \sin (λ_{k} - 1) θ + d_{k} \cos (λ_{k} - 1) θ], \end{array}

(4)

where

λ_{k}

is the kth eigenvalue;

a_{k}

b_{k}

c_{k}

, and

d_{k}

are undetermined coefficients;

w_{s}

and

w_{a}

are the symmetric and anti-symmetric corner functions, respectively. For various radial boundary conditions, the corresponding characteristic equations and corner functions (Leissa et al., 1993b; McGee et al., 2010) are given in Figure 2. For free vertex boundary conditions (F-F), the rigid body deflection term should be added. It should be noted that for specific vertex angles, some of the following four terms,

\sin (λ_{k} + 1) \frac{α}{2}

\cos (λ_{k} + 1) \frac{α}{2}

\sin (λ_{k} - 1) \frac{α}{2}

and

\cos (λ_{k} - 1) \frac{α}{2}

, may vanish. In that case, the corner functions need to be modified. Taking sectorial plates with C-C radial boundary conditions (

α = 2 π

) as an example, the corner functions should be chosen as

{\begin{cases} w_{s} = \sum_{k = 1}^{\infty} a_{k} r^{λ_{k} + 1} {- \frac{λ_{k} - 1}{λ_{k} + 1} \cos (λ_{k} + 1) θ + \cos (λ_{k} - 1) θ} \\ w_{a} = \sum_{k = 1}^{\infty} b_{k} r^{λ_{k} + 1} {- \sin (λ_{k} + 1) θ + \sin (λ_{k} - 1) θ} \end{cases}

(5)

{\begin{cases} w_{s} = \sum_{k = 1}^{\infty} a_{k} r^{λ_{k} + 1} {- \cos (λ_{k} + 1) θ + \cos (λ_{k} - 1) θ} \\ w_{a} = \sum_{k = 1}^{\infty} b_{k} r^{λ_{k} + 1} {- \frac{λ_{k} - 1}{λ_{k} + 1} \sin (λ_{k} + 1) θ + \sin (λ_{k} - 1) θ} \end{cases}

(6)

for different eigenvalues.

According to the characteristic equations in Figure 2, one can easily obtain the eigenvalues for various radial boundary conditions. The real parts of the first eigenvalue for different radial boundary conditions and vertex angles $(ν = 0.3)$ are plotted in Figure 3. The moment singularities exist when the minimum $Re (λ_{k})$ is smaller than unity. As seen from Figure 3, singularities arise when $α > 90^{\circ}$ for S-S and S-F boundary conditions, $α > 180^{\circ}$ for F-F and C-C boundary conditions, $α > 130^{\circ}$ for S-C boundary condition and $α > 95^{\circ}$ for F-C boundary condition. The singularity problems are so common for sectorial plates that corner singularities must be considered when conducting free vibration analysis of sectorial plates.

3. Formulation

The overall strain energy and kinetic energy of a sectorial plate can be expressed by

U = U_{S D} + U_{A D},

(7)

T = T_{S D} + T_{A D},

(8)

where

U_{S D}

and

U_{A D}

are the strain energy terms of the sectorial subdomain and the annular subdomain, respectively;

T_{S D}

and

T_{A D}

are the corresponding kinetic energy terms. The problem domain is divided into a sectorial subdomain and an annular subdomain, regardless of the value of the vertex angle. The energy terms of the two subdomains are computed separately in the following analysis.

3.1. Energy terms of sectorial subdomain

Using the analytical corner functions given in Figure 2, the deflection of a sectorial plate can be expressed as

w = C A,

(9)

where

C

is a corner function matrix,

A

is composed of undetermined coefficients. Taking the corner function for S-S radial boundary condition (

α < π

) as an example, the relevant matrices take the form

C = {[\begin{array}{l} r^{\frac{π}{α}} \sin (\frac{π}{α} θ + \frac{π}{2}) \\ ⋮ \\ r^{\frac{n π}{α}} \sin (\frac{k π}{α} θ + \frac{k π}{2}) \\ r^{\frac{π}{α} + 2} \sin (\frac{π}{α} θ + \frac{π}{2}) \\ ⋮ \\ r^{\frac{n π}{α} + 2} \sin (\frac{k π}{α} θ + \frac{k π}{2}) \end{array}]}^{T},

(10)

A = {[\begin{array}{l} a_{1} & \dots & a_{n} & b_{1} & \dots & b_{n} \end{array}]}^{T} .

(11)

Notably, the corresponding coefficients are also imaginary numbers for imaginary eigenvalues. Usually, the real and imaginary parts of corner functions and undetermined coefficients are separated and computed individually.

The strain energy of the sectorial subdomain is given by

U_{S D} = \iint \frac{1}{2} κ_{0}^{T} E κ_{0} r d r d θ,

(12)

where the curvature vector and the elasticity matrix are expressed as

κ_{0} = {[\begin{array}{l} κ_{r r} & κ_{θ θ} & κ_{r θ} \end{array}]}^{T} = {[\begin{array}{l} \frac{\partial^{2} w}{\partial r^{2}} \frac{1}{r} \frac{\partial w}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} w}{\partial θ^{2}} - 2 \frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial w}{\partial θ}) \end{array}]}^{T} = F A,

(13)

E = D [\begin{array}{l} 1 & ν & 0 \\ ν & 1 & 0 \\ 0 & 0 & (1 - ν) / 2 \end{array}] .

(14)

The strain matrix $F$ can be analytically derived from equation (9); the flexural rigidity of the sectorial plate is

D = \frac{E h^{3}}{12 (1 - ν^{2})},

(15)

where

h

, the thickness of the plate, is usually chosen as unity for simplicity.

Substituting equation (13) into equation (12) yields

U_{S D} = \frac{1}{2} \int_{- \frac{α}{2}}^{\frac{α}{2}} \int_{0}^{r_{s}} A^{T} F^{T} E F A r d r d θ = \frac{1}{2} A^{T} G A,

(16)

where

r_{s}

is the radius of the sectorial subdomain. As corner functions are seperated into functions of variables

r

and

θ

, the double integral can be decomposed into two individual single integrals about variables

r

and

θ

, which can be computed either analytically or numerically.

The kinetic energy of the sectorial subdomain is given by

T_{S D} = \iint \frac{1}{2} ρ h {\dot{w}}^{2} r d r d θ + \iint \frac{ρ h^{3}}{12} [{(\frac{\partial \dot{w}}{\partial r})}^{2} + \frac{1}{r^{2}} {(\frac{\partial \dot{w}}{\partial θ})}^{2}] r d r d θ,

(17)

where

ρ

is the mass density of the sectorial plate. For thin sectorial plates, the second integral in equation (17), referring to the influence of rotational inertia, is usually negligible. Thus, the final form of the kinetic energy term of the sectorial subdomain is expressed as

T_{S D} = \frac{1}{2} ρ h \int_{- \frac{α}{2}}^{\frac{α}{2}} \int_{0}^{r_{s}} {\dot{A}}^{T} C^{T} C \dot{A} r d r d θ = \frac{1}{2} {\dot{A}}^{T} H \dot{A},

(18)

which can be evaluated following the same procedures of computing

U_{S D}

3.2. Energy terms of annular subdomain

The QEM implementation procedures for free vibration analysis of annular plates were clarified in a previous paper (Zhong and Yue, 2012). Thus, only a brief introduction is given in this section. The strain energy and kinetic energy terms of the annular subdomain are expressed by

U_{A D} = \iint \frac{1}{2} κ^{T} E κ d x d y = \int_{- 1}^{1} \int_{- 1}^{1} Q^{(e)} d ξ d η, κ = - b ε, Q^{(e)} = \frac{1}{2} ε^{T} b^{T} E b ε | J |,

(19)

T_{A D} = \iint \frac{1}{2} ρ h {\dot{w}}^{2} d x d y = \int_{- 1}^{1} \int_{- 1}^{1} V^{(e)} d ξ d η, V^{(e)} = \frac{1}{2} ρ h {\dot{w}}^{2} | J |,

(20)

where the curvature vector under the Cartesian coordinate system is defined as

κ = {[\begin{array}{l} κ_{11} & κ_{22} & κ_{12} \end{array}]}^{T} = {[\begin{array}{l} \begin{matrix} - \frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial^{2} w}{\partial y^{2}} - 2 \frac{\partial^{2} w}{\partial x \partial y} \end{matrix} \end{array}]}^{T},

(21)

the auxiliary strain vector is expressed as

ε^{T} = [\begin{array}{l} \frac{\partial w}{\partial ξ} & \frac{\partial w}{\partial η} & \frac{\partial^{2} w}{\partial ξ \partial η} & \frac{\partial^{2} w}{\partial ξ^{2}} & \frac{\partial^{2} w}{\partial η^{2}} \end{array}],

(22)

the strain transform matrix is given by

b = [\begin{array}{lcccc} \frac{\partial^{2} ξ}{\partial x^{2}} & \frac{\partial^{2} η}{\partial x^{2}} & 2 \frac{\partial ξ}{\partial x} \frac{\partial η}{\partial x} & {(\frac{\partial ξ}{\partial x})}^{2} & {(\frac{\partial η}{\partial x})}^{2} \\ \frac{\partial^{2} ξ}{\partial y^{2}} & \frac{\partial^{2} η}{\partial y^{2}} & 2 \frac{\partial ξ}{\partial y} \frac{\partial η}{\partial y} & {(\frac{\partial ξ}{\partial y})}^{2} & {(\frac{\partial η}{\partial y})}^{2} \\ 2 \frac{\partial^{2} ξ}{\partial x \partial y} & 2 \frac{\partial^{2} η}{\partial x \partial y} & 2 (\frac{\partial ξ}{\partial x} \frac{\partial η}{\partial y} + \frac{\partial ξ}{\partial y} \frac{\partial η}{\partial x}) & 2 \frac{\partial ξ}{\partial x} \frac{\partial ξ}{\partial y} & 2 \frac{\partial η}{\partial x} \frac{\partial η}{\partial y} \end{array}],

(23)

and the Jacobian matrix is expressed as

J = [\begin{array}{l} \frac{\partial x}{\partial ξ} & \frac{\partial y}{\partial ξ} \\ \frac{\partial x}{\partial η} & \frac{\partial y}{\partial η} \end{array}] .

(24)

Estimating the integrals in equations (19) and (20) by Lobatto quadrature yields

U_{A D} = \sum_{i = 1}^{N_{ξ}} \sum_{j = 1}^{N_{η}} W_{i} W_{j} Q_{i j}^{(e)},

(25)

T_{A D} = \sum_{i = 1}^{N_{ξ}} \sum_{j = 1}^{N_{η}} W_{i} W_{j} V_{i j}^{(e)},

(26)

where

W_{i}

and

W_{j}

are Lobatto quadrature coefficients;

N_{ξ}

and

N_{η}

are the number of quadrature points in the two directions, which are usually identical. Significantly, the partial derivatives in equation (22) are approximated by the generalized differential quadrature analog (Wu and Liu, 2000)

ε_{i j} = B_{i j} d_{s},

(27)

where

B_{i j}

is the differential coefficient matrix at quadrature point

i j

;

d_{s}^{(e)}

is the displacement vector defined on the standard domain. The displacement vector defined on the original domain is denoted as

d

, and the transformation between the two displacement vectors can be expressed as

d_{s} = T d,

(28)

where

T = [\begin{array}{c} \underset{N^{2} \times N^{2}}{\underset{⏟}{I}} & 0 \\ 0 & \underset{4 N \times 4 N}{\underset{⏟}{T_{s}}} \end{array}],

(29)

T _s being the slope transform matrix. For slopes at four corner nodes (11, 1N, N1, NN), the relationship is given as

{\begin{cases} w_{ξ} \\ w_{η} \end{cases}} = J {\begin{cases} w_{x} \\ w_{y} \end{cases}},

(30)

where

J

is given in equation (24). For other slope variables (slopes at white nodes in Figure 1), the transformation is expressed by

\begin{array}{l} w_{ξ} = [| J | w_{n} - w_{η} (\frac{\partial x}{\partial ξ} n_{y} - \frac{\partial y}{\partial ξ} n_{x})] / (\frac{\partial y}{\partial η} n_{x} - \frac{\partial x}{\partial η} n_{y}), \\ for ξ = \pm 1; \\ w_{η} = [| J | w_{n} - w_{ξ} (\frac{\partial y}{\partial η} n_{x} - \frac{\partial x}{\partial η} n_{y})] / (\frac{\partial x}{\partial ξ} n_{y} - \frac{\partial y}{\partial ξ} n_{x}), \\ for η = \pm 1, \end{array}

(31)

where

n_{x}

and

n_{y}

are directional cosines of the normal,

w_{ξ}

and

w_{η}

on the right side of equation (31) are evaluated by the differential quadrature analog (Quan and Chang, 1989). Substituting equations (19), (27), and (28) into equation (25) yields

\begin{array}{l} U_{A D} = \sum_{i = 1}^{N_{ξ}} \sum_{j = 1}^{N_{η}} \frac{1}{2} W_{i} W_{j} d^{T} T^{T} B_{i j}^{T} b_{i j}^{T} {E b}_{i j} B_{i j} T d {| J |}_{i j} \\ = \frac{1}{2} d^{T} K d . \end{array}

(32)

Similarly, the kinetic energy term is given as

T_{A D} = \sum_{i = 1}^{N_{ξ}} \sum_{j = 1}^{N_{η}} \frac{1}{2} W_{i} W_{j} {(ρ h {\dot{w}}^{2})}_{i j} {| J |}_{i j} = \frac{1}{2} {\dot{d}}^{T} M \dot{d} .

(33)

3.3. Continuity on the interface

In order to enforce displacement and slope continuity on the interface, the original displacement vector $d$ is decomposed into two parts

d = d_{1} \cup d_{2},

(34)

where

d_{1}

, marked by half-white and half-black dots in Figure 1, is composed of displacement variables and slope variables at quadrature points outside the inner arc boundary;

d_{2}

, marked by dots with a cross in Figure 1, consists of those on the inner arc boundary. According to the decomposition rule, equations (32) and (33) can be rewritten as

\begin{array}{l} U_{A D} = \frac{1}{2} d^{T} K d = \frac{1}{2} [d_{1}^{T} d_{2}^{T}] [\begin{array}{l} K_{11} & K_{12} \\ K_{21} & K_{22} \end{array}] {[d_{1}^{T} d_{2}^{T}]}^{T} \\ = \frac{1}{2} d_{1}^{T} K_{11} d_{1} + \frac{1}{2} d_{1}^{T} K_{12} d_{2} + \frac{1}{2} d_{2}^{T} K_{21} d_{1} \end{array} + \frac{1}{2} d_{2}^{T} K_{22} d_{2},

(35)

\begin{array}{l} T_{A D} = \frac{1}{2} {\dot{d}}^{T} M \dot{d} = \frac{1}{2} [{\dot{d}}_{1}^{T} {\dot{d}}_{2}^{T}] [\begin{array}{l} M_{11} & 0 \\ 0 & M_{22} \end{array}] {[{\dot{d}}_{1}^{T} {\dot{d}}_{2}^{T}]}^{T} \\ = \frac{1}{2} {\dot{d}}_{1}^{T} M_{11} {\dot{d}}_{1} + \frac{1}{2} {\dot{d}}_{2}^{T} M_{22} {\dot{d}}_{2} . \end{array}

(36)

It is noteworthy that $d_{2}$ can be expressed by the analytical corner functions given in Figure 2, namely

d_{2} = L A,

(37)

where

L

is a transformation matrix consisting of corner function values for given radius and angle. Substitution of equations (16), (35), and (37) into equation (7) yields

U = \frac{1}{2} d_{1}^{T} K_{11} d_{1} + \frac{1}{2} d_{1}^{T} K_{12} L A + \frac{1}{2} A^{T} L^{T} K_{21} d_{1} + \frac{1}{2} A^{T} L^{T} K_{22} L A + \frac{1}{2} A^{T} G A .

(38)

Similarly, the kinetic energy of the sectorial plate can be expressed as

T = \frac{1}{2} {\dot{d}}_{1}^{T} M_{11} {\dot{d}}_{1} + \frac{1}{2} {\dot{A}}^{T} L^{T} M_{22} L \dot{A} + \frac{1}{2} {\dot{A}}^{T} H \dot{A} .

(39)

Introduction of Hamilton’s principle gives

[\begin{array}{l} K_{11} & K_{12} L \\ L^{T} K_{21} & L^{T} K_{22} L + G \end{array}] \{\begin{array}{l} d_{1} \\ A \end{array}\} + [\begin{array}{l} M_{11} & 0 \\ 0 & L^{T} M_{22} L + H \end{array}] \{\begin{array}{l} {\ddot{d}}_{1} \\ \ddot{A} \end{array}\} = \{0\} .

(40)

Assume that the undetermined vector takes the form

{\begin{cases} d_{1} \\ A \end{cases}} = {\begin{cases} d_{1}^{*} \\ A^{*} \end{cases}} \exp (i ω t)

(41)

where

i

is the imaginary unit. Substituting equation (41) into equation (40) leads to a generalized eigenvalue problem

[\begin{array}{l} K_{11} & K_{12} L \\ L^{T} K_{21} & L^{T} K_{22} L + G \end{array}] \{\begin{array}{l} d_{1}^{*} \\ A^{*} \end{array}\} = ω^{2} [\begin{array}{l} M_{11} & 0 \\ 0 & L^{T} M_{22} L + H \end{array}] \{\begin{array}{l} d_{1}^{*} \\ A^{*} \end{array}\},

(42)

which can be solved by various algorithms, such as the QR algorithm.

4. Numerical examples

Sectorial plates of various boundary conditions and vertex angles are analyzed in this section. Non-dimensional frequency parameters are computed and compared with those analytical and numerical results. $ν = 0.3$ is used in all examples. The influence of three major parameters, the number of quadrature points $N$ , the number of truncated terms $n$ , and the radius ratio of the sectorial subdomain $r_{s} / r_{l}$ , is discussed at length.

4.1. S-S sectorial plates

Free vibration analysis of radially simply supported sectorial plates is conducted first. A non-dimensional frequency parameter is defined as

\bar{ω} = ω r_{l}^{2} \sqrt{ρ h / D} .

(43)

With the exact solutions for the frequencies of sectorial plates with simply supported radial edges given by Huang et al. (Huang et al., 1993), a normalized frequency parameter $ω_{i} / ω_{i}^{e x a c t}$ is defined to examine the accuracy of the present formulation.

4.1.1. Convergence analysis

Taking $α = 240^{\circ}$ , the present results of SSS sectorial plates are compared with analytical solutions. Firstly, the relationship between the normalized frequency parameters and the radius ratio of the sectorial subdomain for $N = 15$ and $n = 10$ , is shown in Figure 4. It is seen that the computed frequency parameters are not sensitive to the radius ratio. With the increase of $r_{s} / r_{l}$ , more truncated terms are needed. The decrease in accuracy is mainly due to insufficient truncated terms. To ensure the accuracy of subsequent analysis, $r_{s} / r_{l} = 0.2$ is chosen in the following examples.

The relationship between the normalized frequency parameters and the number of truncated terms $n$ is plotted in Figure 5 ( $N = 15$ ). Only five truncated terms are needed to obtain accurate first six frequencies. For $n = 10$ , the relationship between the normalized frequency parameters and the number of quadrature points $N$ is plotted in Figure 6. The first two frequencies tend to be stable when $N > 8$ , while the fifth and sixth normalized frequencies are approaching unity for $N > 12$ . As seen from Figures 5 and 6, the results converge rapidly with the increase of $n$ and $N$ , implying that only a small number of truncated terms and quadrature points are needed to obtain accurate frequency parameters. To ensure the accuracy of computed results, $N = 15$ and $n = 10$ are chosen in the following examples.

4.1.2. Results and discussion

The first six non-dimensional frequency parameters for plates with various vertex angles under three different boundary conditions, SSS, SCS, and SFS, are computed and compared with analytical solutions given by Huang et al. (Huang et al., 1993) in Table 1. It is seen that the present results agree well with the analytical solutions, demonstrating the high accuracy of the present formulation.

Table 1.

Non-dimensional frequency parameters of sectorial plates with simply-supported radial edges.

Boundary conditions	Vertex angle (°)	Solution	Frequency parameter
Boundary conditions	Vertex angle (°)	Solution	1st	2nd	3rd	4th	5th	6th
SSS	90	Present	25.6133	56.8416	70.1183	97.9946	121.7024	134.3327
	90	Analytical (Huang et al., 1993)	25.6133	56.8416	70.1170	97.9946	121.7024	134.2978
	180	Present	13.8982	25.6133	39.9573	48.4859	56.8416	70.1183
	180	Analytical (Huang et al., 1993)	13.8982	25.6133	39.9573	48.4789	56.8416	70.1170
	270	Present	14.4129	17.5042	25.6133	34.8900	45.3065	49.0395
	270	Analytical (Huang et al., 1993)	14.4129	17.5042	25.6133	34.8900	45.3067	49.0344
	330	Present	14.5467	14.8515	21.0433	28.0288	35.7900	44.3166
	330	Analytical (Huang et al., 1993)	14.5467	14.8515	21.0433	28.0288	35.7902	44.3132
	355	Present	14.0444	14.5833	19.6684	25.9819	32.9696	40.6228
	355	Analytical (Huang et al., 1993)	14.0444	14.5833	19.6684	25.9819	32.9698	40.6199
SFS	90	Present	5.3583	21.8352	35.2601	47.3780	73.5426	81.7043
	90	Analytical (Huang et al., 1993)	5.3583	21.8352	35.2601	47.3780	73.5426	81.7043
	180	Present	5.3583	12.4390	20.4747	21.8352	33.4948	35.2601
	180	Analytical (Huang et al., 1993)	5.3583	12.4390	20.4746	21.8352	33.4949	35.2601
	270	Present	1.9096	2.7586	5.3583	9.8189	15.3164	20.7234
	270	Analytical (Huang et al., 1993)	1.9096	2.7586	5.3583	9.8189	15.3165	20.7233
	330	Present	0.7890	3.1164	3.3611	6.4715	10.2760	14.7761
	330	Analytical (Huang et al., 1993)	0.7890	3.1164	3.3611	6.4715	10.2760	14.7742
	355	Present	0.2854	2.7890	3.2089	5.5257	8.8568	12.7912
	355	Analytical (Huang et al., 1993)	0.2854	2.7890	3.2089	5.5257	8.8569	12.7895
SCS	90	Present	34.8771	69.6658	84.5860	114.2125	140.1080	153.8832
	90	Analytical (Huang et al., 1993)	34.8770	69.6658	84.5826	114.2125	140.1079	153.8151
	180	Present	21.2606	34.8771	51.0300	60.8468	69.6658	84.5860
	180	Analytical (Huang et al., 1993)	21.2604	34.8770	51.0300	60.8287	69.6658	84.5826
	270	Present	22.0882	25.5142	34.8771	45.3673	56.9684	61.4687
	270	Analytical (Huang et al., 1993)	22.0881	25.5141	34.8770	45.3673	56.9685	61.4550
	330	Present	22.3039	22.3922	29.6292	37.6270	46.3761	55.8721
	330	Analytical (Huang et al., 1993)	22.3038	22.3921	29.6291	37.6269	46.3763	55.8684
	355	Present	21.4345	22.3630	28.0366	35.2976	43.2102	51.7709
	355	Analytical (Huang et al., 1993)	21.4343	22.3629	28.0365	35.2976	43.2104	51.7676

4.2. Sectorial plates with free circumferential edge

Free vibration analysis of CFC, FFF, SFF, SFC, and FFC plates is conducted in this section. The computed results are compared with those available results obtained by the Ritz method (Leissa et al., 1993a; McGee et al., 1993, 1995, 2003) in Table 2. It is noteworthy that the zero frequency parameters, which correspond to rigid body modes, are weeded out from the computed results. It is seen that excellent agreement is achieved, whether the vertex is free or constrained. About 10 to 20 corner functions and 42 to 72 polynomial terms are needed for the Ritz method to obtain convergent results with five significant digits. In contrast, the present formulation uses ten corner functions and about 250 nodal deflection and slope variables. Besides, the differential quadrature coefficients and weight coefficients are usually computed and stored beforehand. Thus, the computational cost of the potential energy and its variation for the present formulation, is much smaller than that for the Ritz method.

Table 2.

Non-dimensional frequency parameters of sectorial plates with free circumferential edge.

Boundary conditions	Vertex angle (°)	Solution	Frequency parameter
Boundary conditions	Vertex angle (°)	Solution	1st	2nd	3rd	4th	5th	6th
CFC	90	Present	13.6212	33.7692	48.5127	63.4944	91.0841	102.0945
	90	Ritz method (McGee et al., 1995)	13.623	33.772	48.514	63.508	91.090	102.11
	180	Present	4.5407	9.3536	17.2302	27.2999	27.5703	40.2359
	180	Ritz method (McGee et al., 1995)	4.5399	9.3519	17.222	27.037	27.565	40.235
	270	Present	3.2613	4.7814	8.0399	12.8127	18.6734	22.3779
	270	Ritz method (McGee et al., 1995)	3.2613	4.5413	8.0384	12.802	18.671	22.378
	330	Present	3.1651	3.3321	5.5397	8.7286	12.7390	17.5046
	330	Ritz method (McGee et al., 1995)	3.1651	3.3318	5.5390	8.7267	12.738	17.487
	355	Present	3.0396	3.1590	4.8626	7.5878	11.0720	15.2476
	355	Ritz method (McGee et al., 1995)	3.0389	3.1589	4.8620	7.5867	11.072	15.208
FFF	90	Present	16.1625	23.1779	30.6858	38.4212	57.6383	67.3302
	90	Ritz method (McGee et al., 1993)	16.16	23.18	30.59	38.42	57.64	67.32
	180	Present	6.9321	9.4966	18.0556	18.1889	29.1793	29.2173
	180	Ritz method (Leissa et al., 1993a)	6.9318	9.4966	18.056	18.189	29.179	29.204
	270	Present	4.6061	5.9952	9.2778	12.8859	17.2848	18.6998
	270	Ritz method (Leissa et al., 1993a)	4.6061	5.9951	9.2776	12.886	17.284	18.700
	330	Present	3.1981	4.9147	7.9052	9.0001	13.1798	16.4320
	330	Ritz method (Leissa et al., 1993a)	3.1981	4.9147	7.9048	9.0000	13.181	16.428
	355	Present	2.8221	4.3832	7.7462	7.8394	11.5818	15.1872
	355	Ritz method (Leissa et al., 1993a)	2.8220	4.3831	7.7458	7.8393	11.583	15.177
SFF	90	Present	9.4966	18.0556	29.1792	46.9128	54.8250	59.0096
	90	Ritz method (McGee et al., 2003)	9.4966	18.055	29.179	—	—	—
	180	Present	2.7585	7.6326	14.8452	17.6400	24.0344	26.8468
	180	Ritz method (McGee et al., 2003)	2.7664	7.6352	14.852	—	—	—
	270	Present	2.2092	3.4803	6.8753	11.4580	16.7991	19.2696
	270	Ritz method (McGee et al., 2003)	2.2165	3.5006	6.8796	—	—	—
	330	Present	1.9728	2.9943	4.5824	7.7434	11.6243	16.0052
	330	Ritz method (McGee et al., 2003)	1.9728	2.9943	4.5823	—	—	—
	355	Present	1.7520	3.0543	3.9549	6.6905	10.0947	14.0546
	355	Ritz method (McGee et al., 2003)	1.7520	3.0543	3.9548	—	—	—
SFC	90	Present	9.3517	27.5635	41.9128	55.1449	81.9115	91.5739
	90	Ritz method (McGee et al., 2003)	9.3413	27.458	41.896	—	—	—
	180	Present	2.9912	7.3853	14.8038	24.2137	24.7623	36.7491
	180	Ritz method (McGee et al., 2003)	2.9913	7.3860	14.806	—	—	—
	270	Present	2.5161	3.5591	6.7272	11.3041	16.9808	21.2593
	270	Ritz method (McGee et al., 2003)	2.5161	3.5591	6.7273	—	—	—
	330	Present	2.1315	3.1620	4.5371	7.6217	11.5083	16.1238
	330	Ritz method (McGee et al., 2003)	2.1315	3.1594	4.5347	—	—	—
	355	Present	1.9748	3.1509	3.9604	6.5768	9.9641	13.9945
	355	Ritz method (McGee et al., 2003)	1.9748	3.1509	3.9605	—	—	—
FFC	90	Present	4.4926	13.5687	24.8951	35.2392	54.3081	64.0022
	90	Ritz method (McGee et al., 2003)	4.4927	13.571	24.895	—	—	—
	180	Present	2.4769	4.1540	9.6384	17.0164	20.0795	27.0152
	180	Ritz method (McGee et al., 2003)	2.4769	4.1541	9.6393	—	—	—
	270	Present	2.1656	2.4793	4.6904	8.2150	12.9393	18.1416
	270	Ritz method (McGee et al., 2003)	2.1656	2.4793	4.6903	—	—	—
	330	Present	1.8198	2.3452	3.5899	5.6267	8.8654	12.8589
	330	Ritz method (McGee et al., 2003)	1.8198	2.3452	3.5900	—	—	—
	355	Present	1.7534	2.2099	3.4262	4.9073	7.7097	11.2005
	355	Ritz method (McGee et al., 2003)	1.7534	2.2100	3.4263	—	—	—

4.3. Sectorial plates with simply supported circumferential edge

The first six non-dimensional frequency parameters of CSC, FSF, SSF, SSC, and FSC plates are computed in this example. The computed results are compared with those obtained using the differential quadrature method (Wang and Wang, 2004) and the Ritz method (McGee et al., 2010). As seen from Table 3, the present results agree well with the available results. Wang and Wang used a

15 \times 15

grid to obtain acceptable results. The differential quadrature method exhibits high efficiency in analyzing sectorial plates with small vertex angles. However, a sharp decline in its computational accuracy occurs for plates with reentrant corners. For the Ritz method adopted by McGee et al., about ten corner functions and 84 polynomials were used to achieve convergence of the lowest frequency with three significant digits. As an ill-conditioning problem occurs for large trial sets of polynomials and corner functions, two special numerical techniques, a Householder reduction of the dynamical matrices and a singular value decomposition technique, were employed in the work of McGee et al. (McGee et al., 2010). In contrast, only ten corner functions and a quadrature element are needed for the present formulation, avoiding the ill-conditioning problem.

Table 3.

Non-dimensional frequency parameters of sectorial plates with simply supported circumferential edge.

Boundary conditions	Vertex angle (°)	Solution	Frequency parameter
Boundary conditions	Vertex angle (°)	Solution	1st	2nd	3rd	4th	5th	6th
CSC	90	Present	37.4568	72.9511	88.7115	118.7063	144.4490	159.5321
		Ritz method (McGee et al., 2010)	37.457	72.951	88.711	118.71	144.45	159.53
		DQM (Wang and Wang, 2004)	37.4568	72.9511	88.7113	118.706	144.449	159.528
	180	Present	19.4969	31.4313	46.6004	58.5117	64.5544	79.6371
		Ritz method (McGee et al., 2010)	19.504	31.431	46.601	58.548	64.544	79.602
		DQM (Wang and Wang, 2004)	19.497	—	—	—	—	—
	270	Present	15.7766	21.2515	29.4423	39.0917	49.8532	51.6725
		Ritz method (McGee et al., 2010)	15.777	21.252	29.442	39.092	49.854	51.669
		DQM (Wang and Wang, 2004)	15.778	—	—	—	—	—
	330	Present	15.0932	18.1751	24.0917	31.2285	39.2488	48.0621
		Ritz method (McGee et al., 2010)	15.093	18.175	24.092	31.229	39.247	48.046
		DQM (Wang and Wang, 2004)	15.094	—	—	—	—	—
	355	Present	14.9633	17.3140	22.5082	28.8992	36.0882	43.9858
	355	Ritz method (McGee et al., 2010)	14.963	17.314	22.508	28.900	36.093	43.977
FSF	90	Present	2.4419	17.1660	23.0856	45.3869	54.9265	64.0456
	90	Ritz method (McGee et al., 2010)	2.4426	17.167	23.046	45.387	54.921	63.235
	180	Present	3.5146	12.4982	20.5959	26.7388	34.8714	45.6157
	180	Ritz method (McGee et al., 2010)	3.5149	12.498	20.593	26.707	34.871	45.608
	270	Present	4.1460	11.7078	15.2635	23.2213	27.4160	31.9985
	270	Ritz method (McGee et al., 2010)	4.1463	11.708	15.264	23.221	27.380	31.993
	330	Present	4.3899	11.8180	13.6255	19.3946	25.5283	28.5374
	330	Ritz method (McGee et al., 2010)	4.3902	11.818	13.626	19.392	25.512	28.500
	355	Present	4.4606	11.9448	13.1982	18.2291	23.9071	28.5660
	355	Ritz method (McGee et al., 2010)	4.4610	11.945	13.198	18.223	23.888	28.525
SSF	90	Present	12.4982	34.8714	45.6158	68.8681	86.6158	98.6275
	90	Ritz method (McGee et al., 2010)	12.498	34.872	45.608	68.868	86.609	98.448
	180	Present	11.9737	18.0096	30.2678	44.2201	45.2535	56.2517
	180	Ritz method (McGee et al., 2010)	11.989	18.022	30.269	44.265	45.254	56.285
	270	Present	13.3522	13.4648	20.3474	28.6968	38.2283	46.9626
	270	Ritz method (McGee et al., 2010)	13.358	13.672	20.368	28.696	38.228	47.212
	330	Present	12.4470	14.1463	17.1312	23.4789	30.6190	38.5425
	330	Ritz method (McGee et al., 2010)	12.447	14.146	17.131	23.457	30.610	38.540
	355	Present	12.3932	14.3211	16.1447	21.9131	28.3494	35.4779
	355	Ritz method (McGee et al., 2010)	12.393	14.321	16.144	21.883	28.337	35.474
SSC	90	Present	31.4306	64.5444	79.6021	107.9690	132.6463	147.5014
		Ritz method (McGee et al., 2010)	31.431	64.545	79.607	107.97	132.65	147.55
		DQM (Wang and Wang, 2004)	31.4306	64.5444	79.6017	107.969	132.646	147.488
	180	Present	17.1642	28.4812	43.2471	54.3894	60.6751	74.7583
		Ritz method (McGee et al., 2010)	17.164	28.481	43.247	54.386	60.675	74.758
		DQM (Wang and Wang, 2004)	17.165	—	—	—	—	—
	270	Present	14.8297	19.4461	27.5257	36.9803	47.5725	49.8806
		Ritz method (McGee et al., 2010)	14.830	19.454	27.527	36.973	47.591	49.861
		DQM (Wang and Wang, 2004)	14.807	—	—	—	—	—
	330	Present	14.6302	16.7036	22.5611	29.6167	37.5050	46.1735
		Ritz method (McGee et al., 2010)	14.631	16.737	22.585	29.624	37.506	46.167
		DQM (Wang and Wang, 2004)	14.612	—	—	—	—	—
	355	Present	14.6065	15.9906	21.1183	27.4502	34.5298	42.3082
	355	Ritz method (McGee et al., 2010)	14.606	16.010	21.115	27.441	34.522	42.288
FSC	90	Present	18.0487	40.9643	55.9172	76.8947	96.4330	113.5252
	90	Ritz method (McGee et al., 2010)	18.049	40.965	55.916	76.896	96.430	113.46
	180	Present	13.8280	20.7069	33.1795	47.3619	49.0229	60.8087
	180	Ritz method (McGee et al., 2010)	13.828	20.707	33.180	47.359	49.023	60.807
	270	Present	13.5040	15.7477	22.1607	30.6466	40.3470	47.0450
	270	Ritz method (McGee et al., 2010)	13.504	15.748	22.161	30.648	40.349	47.040
	330	Present	13.4014	14.7332	18.7023	24.9799	32.2221	40.2852
	330	Ritz method (McGee et al., 2010)	13.402	14.733	18.703	24.982	32.224	40.293
	355	Present	13.3180	14.6006	17.6937	23.2891	29.8040	37.0422
	355	Ritz method (McGee et al., 2010)	13.319	14.601	17.695	23.292	29.807	37.057

4.4. Sectorial plates with clamped circumferential edge

CCC, FCF, SCF, SCC, and FCC plates are investigated in this example. The computed results are given and compared with those computed by the DQM (Wang and Wang, 2004) and Ritz method (McGee et al., 2010) in Table 4. It is seen that the present results agree well with those computed by the Ritz method. On the other hand, the present formulation shows higher precision than the DQM for sectorial plates with reflex angle.

Table 4.

Non-dimensional frequency parameters of sectorial plates with clamped circumferential edge.

Boundary conditions	Vertex angle (°)	Solution	Frequency parameter
Boundary conditions	Vertex angle (°)	Solution	1st	2nd	3rd	4th	5th	6th
CCC	90	Present	48.7856	87.7788	104.8848	136.9268	164.5724	180.7181
		Ritz method (McGee et al., 2010)	48.786	87.774	104.88	136.93	164.57	180.71
		DQM (Wang and Wang, 2004)	48.7856	87.7786	104.884	136.926	164.570	180.709
	180	Present	28.1153	41.7285	58.6764	71.9154	78.3971	94.9960
		Ritz method (McGee et al., 2010)	28.125	41.726	58.677	71.950	78.375	94.924
		DQM (Wang and Wang, 2004)	28.115	—	—	—	—	—
	270	Present	23.7435	31.9766	39.3995	50.3133	62.1914	64.3623
		Ritz method (McGee et al., 2010)	23.743	30.041	39.399	50.241	62.192	64.352
		DQM (Wang and Wang, 2004)	23.703	—	—	—	—	—
	330	Present	22.9559	26.4545	33.2767	41.3891	50.3886	60.1677
		Ritz method (McGee et al., 2010)	22.956	26.455	33.277	41.389	50.383	60.147
		DQM (Wang and Wang, 2004)	22.898	—	—	—	—	—
	355	Present	22.8112	25.6260	31.4979	38.8018	46.8814	56.0492
	355	Ritz method (McGee et al., 2010)	22.810	25.447	31.450	38.745	46.848	55.634
FCF	90	Present	7.5627	24.7550	32.0904	56.5524	67.9317	78.5807
	90	Ritz method (McGee et al., 2010)	7.5632	24.760	31.991	56.561	67.925	77.195
	180	Present	8.6016	19.6598	28.8665	36.4154	45.0875	57.6879
	180	Ritz method (McGee et al., 2010)	8.6013	19.660	28.857	36.329	45.088	57.665
	270	Present	9.3288	18.8146	22.7800	32.0376	37.1204	42.0325
	270	Ritz method (McGee et al., 2010)	9.3280	18.814	22.799	32.036	37.029	42.016
	330	Present	9.6148	18.9745	20.9330	27.6857	34.6821	38.4729
	330	Ritz method (McGee et al., 2010)	9.6137	18.972	20.932	27.675	34.638	38.378
	355	Present	9.6967	19.1338	20.4508	26.3574	32.8494	38.5082
	355	Ritz method (McGee et al., 2010)	9.6955	19.131	20.449	26.334	32.795	38.406
SCF	90	Present	19.6584	45.0857	57.6813	82.4726	102.4281	115.7795
	90	Ritz method (McGee et al., 2010)	19.660	45.088	57.666	82.477	102.42	115.45
	180	Present	19.1667	26.0878	40.0519	56.0459	56.8496	69.4096
	180	Ritz method (McGee et al., 2010)	19.185	26.103	40.054	56.078	56.853	69.434
	270	Present	20.6245	21.0726	28.7916	38.3177	49.0177	59.1727
	270	Ritz method (McGee et al., 2010)	20.629	21.420	28.825	38.315	49.013	59.368
	330	Present	19.6712	21.8420	25.0779	32.4267	40.5144	49.3848
	330	Ritz method (McGee et al., 2010)	19.669	21.842	25.075	32.367	40.488	49.375
	355	Present	19.6312	22.0540	23.9352	30.6478	37.9598	45.9706
	355	Ritz method (McGee et al., 2010)	19.629	22.054	23.930	30.566	37.924	45.956
SCC	90	Present	41.7260	78.3748	94.9229	125.2009	151.8772	167.8920
		Ritz method (McGee et al., 2010)	41.726	78.375	94.922	125.20	151.87	167.87
		DQM (Wang and Wang, 2004)	41.7260	78.3748	94.9218	125.201	151.876	167.865
	180	Present	25.2713	38.2689	54.8213	67.3767	74.0008	89.6583
		Ritz method (McGee et al., 2010)	25.271	38.269	54.821	67.367	74.000	89.655
		DQM (Wang and Wang, 2004)	25.273	—	—	—	—	—
	270	Present	22.5751	27.8846	37.1762	47.8094	59.6023	62.3815
		Ritz method (McGee et al., 2010)	22.575	27.884	37.143	47.787	59.588	62.368
		DQM (Wang and Wang, 2004)	22.544	—	—	—	—	—
	330	Present	22.3761	24.6714	31.4417	39.4922	48.3719	58.0067
		Ritz method (McGee et al., 2010)	22.376	24.718	31.481	39.506	48.368	57.995
		DQM (Wang and Wang, 2004)	22.350	—	—	—	—	—
	355	Present	22.3547	23.8922	29.8526	37.0634	45.0656	53.8706
	355	Ritz method (McGee et al., 2010)	22.355	23.875	29.783	37.025	45.020	53.691
FCC	90	Present	26.4739	52.1072	69.0787	91.4999	113.0026	131.7202
	90	Ritz method (McGee et al., 2010)	26.476	52.109	69.078	91.503	113.00	131.60
	180	Present	21.5001	29.2943	43.4567	59.4946	61.1946	74.4248
	180	Ritz method (McGee et al., 2010)	21.501	29.295	43.458	59.486	61.193	74.420
	270	Present	21.1219	23.5659	30.9638	40.6024	51.4602	59.2703
	270	Ritz method (McGee et al., 2010)	21.122	23.566	30.964	40.603	51.460	59.255
	330	Present	20.9736	22.4345	26.9793	34.1790	42.3750	51.3962
	330	Ritz method (McGee et al., 2010)	20.973	22.435	26.980	34.180	42.377	51.400
	355	Present	20.8577	22.3087	25.8121	32.2470	39.6489	47.7805
	355	Ritz method (McGee et al., 2010)	20.858	22.309	25.813	32.251	39.653	47.795

4.5. Discussion

In subsections 4.1 to 4.4, sectorial plates with 18 combinations of boundary conditions are analyzed. All computed results agree well with those available data, demonstrating the accuracy of the present formulation.

Up to now, there exist analytical solutions only when two radial edges are simply supported. For other boundary conditions, it is almost impossible to find a series of admissible displacement functions. The Ritz method is an effective tool for the examples discussed in the present paper. However, it may be invalid when mixed boundary conditions or plates of irregular shape are present. The differential quadrature method (DQM) (Wang and Wang, 2004), an algorithm for strong-form solutions, suffers from application limitations. Furthermore, it is only capable of analyzing plates with simply supported or clamped boundary conditions. The present formulation overcomes these shortcomings. The free vertex, intractable for some numerical methods, is dealt with satisfactorily in the present formulation. Moreover, the weak form quadrature method has no restriction on plate shapes or boundary conditions, implying it can be extended to vibration analysis of arbitrary-shaped plates with arbitrary boundary conditions.

In addition to the numerical methods discussed above, some other numerical methods, such as the finite element method (FEM), the boundary element method (BEM) and meshless methods, are also applicable to singularity problems. In general, the total degree of freedom of the finite element analysis by ABAQUS is about ten times than that of the present formulation to achieve the same accuracy. The BEM enjoys high solution accuracy in analysis of crack problems, but unknowns are usually solved for from algebraic equations of asymmetric and full coefficient matrix which may limit the problem scale. For meshless methods including those combined with the phase field model (Amiri et al., 2014), the accuracy of the results depends heavily on the selection of the enrichment functions. General enrichment functions, such as $\sqrt{r} \sin θ$ , $\sqrt{r} \cos θ$ , are effective in dealing with crack problems. However, when the singularity order changes, the convergence speed may decrease. In contrast, series displacement descriptions for the crack tip are adopted in the present formulation, which ensures solution accuracy and convergence. As for the computational cost, the total number of degrees of freedom in the present formulation is greatly reduced. It is noteworthy that the weighting coefficients, the entries in $C$ , $H$ for specific angles, can be computed and stored beforehand. Thus, the CPU time consumption decreases significantly.

5. Discussion on vibration frequencies and modes

For sectorial plates with the same vertex angle, it is obvious that plates with more rigid constraints usually have larger vibration frequencies. However, the vibration frequencies show no correlation with the vertex angle. For SFS, SSS, and SCS sectorial plates, the relationship between the non-dimensional fundamental frequency parameter and the vertex angle is shown in Figure 7. For the same vertex angle, the lowest fundamental frequency parameter occurs in SFS plates, while the highest arises in SCS plates. The curves simultaneously reach the minimum at $α = 180^{\circ}$ , namely when the sectorial plate turns into a semicircle plate. However, different extremum points of the curves occur for $180^{\circ} < α < 360^{\circ}$ , specifically about $α = 245^{\circ}$ for SFS plates, and $α = 330^{\circ}$ for SCS and SSS plates.

The major factors influencing the vibration modes are the boundary conditions of the circumferential and the radial edges, and the vertex angle. The first six vibration modes of SFS, SSS, and SCS sectorial plates are plotted in Figure 8 to illustrate the influence of the circumferential edge. As seen from Figure 8, the free edge shows a significant influence on the vibration modes. The maximum amplitude usually appears at the free edge, which is an extremely unstable vibration mode for structures. In contrast, the maximum amplitude of SSS and SCS plates can only appear inside the sector domain since the boundary deflection is constrained. Besides, the vibration modes of SSS and SCS plates are similar. The influence of the vertex angle on the vibration frequencies and modes of sectorial plates ( $α \leq 180^{\circ}$ ) is noticeable. However, the influence is relatively small for sectorial plates with reentrant corners ( $α > 180^{\circ}$ ).

The first six vibration modes of CSC and FSF sectorial plates are plotted in Figure 9. As seen from Figures 8 and 9, the vibration modes of CSC plates are similar to those of SSS plates. However, the vibration modes of FSF plates are totally different. For sectorial plates with symmetrical radial boundary conditions, such as SSS, CSC, and FSF, some vibration modes can be regarded as adding additional radial simply-supported edges. For example, the third mode of SSS plates with $α = 270^{\circ}$ can be viewed as the combination of the first mode of three individual SSS plates with $α = 90^{\circ}$ , which is confirmed by the vibration frequencies given in Table 1. Another example is that the second vibration frequency of CSC plates with $α = 180^{\circ}$ is the same as the first vibration frequency of SSC plates with $α = 90^{\circ}$ , as shown in Table 3. In this case, the CSC plate is divided into two individual SSC plates by adding a radial simply-supported edge at its bisector.

All the aforementioned examples have symmetric boundary conditions. Some vibration modes of sectorial plates with asymmetric boundary conditions are plotted in Figure 10. Apparently, the vibration modes of these sectorial plates show no symmetry. Additionally, nodal lines are not easy to recognize.

6. Conclusions

Transverse free vibration analysis of thin sectorial plates has been conducted. Sectorial plates with 18 kinds of boundary conditions are investigated by the weak form quadrature element method. Different corner functions are introduced into the sectorial subdomain for various radial boundary conditions and vertex angles, while the outer annular subdomain is modeled by a single quadrilateral weak form quadrature element. Excellent agreement is reached between the present results and those computed by the Ritz method. Though corner functions are introduced in both the present formulation and the Ritz method, the latter is inflicted by an ill-conditioning problem which is avoided in the present formulation by cutting down the number of polynomial terms. The free vertex problem, which is challenging for many numerical methods, is tackled satisfactorily using the present formulation. With the high efficiency of the weak form quadrature element method, the total number of degrees of freedom is prominently reduced. Moreover, as the weak form quadrature element method has no restrictions on plate shapes or boundary conditions, the present formulation can be easily extended to vibration analysis of irregular plates or plates with complex boundary conditions.

Footnotes

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hongzhi Zhong

Data availability statement

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

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