Abstract
In this paper, a modified FourierRitz approach is adopted to analyze the free vibration of orthotropic annular sector thin plates with general boundary conditions, internal radial line and circumferential arc supports. In the present method, regardless of boundary conditions, the displacements of the sector plates are invariantly expressed as a standard Fourier cosine series and several auxiliary closedform functions. These auxiliary functions are introduced to eliminate any potential discontinuities of the original displacement function and its derivatives throughout the whole domain including its edges, and then to effectively enhance the convergence of the results. Since the displacement field is constructed to be adequately smooth in the whole solution domain, an accurate solution can be obtained by using Ritz procedure based on the energy functions of the sector plates. The excellent accuracy and reliability of the current solutions are compared with the results found in the literature, and numerous new results for annular sector plates with various boundary conditions are presented. New results are obtained for annular sector plates subjected to elastic boundary restraints and arbitrary internal radial line and circumferential arc supports in both directions, and they may be served as benchmark solutions for future researches.
1. Introduction
Orthotropic annular sector thin plates are widely used in many engineering applications such as ships, curved bridge decks, aeronautical and space structures and other industrial applications due to their excellent engineering features. The orthotropic annular sector thin plates in these applications can be subjected to various boundary conditions, such as classical restraints, elastic supports and their combinations. In addition, the internal radial line and circumferential arc supports may be placed to reduce the magnitude of dynamic and static stresses and displacements of the plates as well as satisfy special functional requirements. Therefore a thorough understanding of the vibration behaviors of orthotropic annular sector thin plates with general boundary restraints, internal radial line and circumferential arc supports is of great interest for the designers to realize proper and comparatively accurate design of machines and structures.
In recent decades, the wide use of annular sector thin plate structures has motivated a huge amount of research efforts in developing the more accurate and applicable model and methods for analyzing their dynamic behaviors. Onoe [1] presented a mathematical method on basis of Love’s theory for contour vibrations of isotropic circular plates. Mcgee et al. [2] used a novel Ritz method to analyze free vibration of the sectorial plate with complete free boundary conditions. Wang and Thevendran [3] employed the RayleighRitz method to solve the free vibration problem of annular plates with internal axisymmetric supports. Wang et al. [4] extended the differential quadrature method to study the free vibration analysis of annular plates with classical boundary conditions. Later, Wang et al. [5] developed the differential quadrature method to analyze the free vibration of circular annular plates with classical boundary conditions and nonuniform thickness. Furthermore, Wang [6, 7] extended the differential quadrature method to analyze the free vibration of thin sector plates with various sector angles and six combinations of classical boundary conditions. Irie et al. [8] employed Ritz method to the free vibration of ringshaped polarorthotropic sector plates with classical boundary conditions. Singh et al. [9] used RayleighRitz method to analyze the transverse vibrations of circular plates with variable thickness and classical boundary conditions. Wong et al. [10] investigated the sensitivity of changes in displacement mode shapes of annular plates relative to the hole size and obtained approximations to frequencies and mode shapes of circular plates with variable thickness by using mode subtraction method. Houmat [11] presented a sector Fourier pelement on basis of finite element method for free vibration analysis of sectorial plates with classical boundary conditions. Chen et al. [12] applied a meshless method for free vibration analysis of circular and rectangular clamped plates with clamped boundary condition. Seok and Tiersten [13, 14] presented a variational approximation procedure for free vibration analysis of annular sector cantilever plates. Aghdam et al. [15] performed bending analysis of thin annular sector plates with clamped boundary condition by extended Kantorovich method. Li [16] employed finite strip method to study the free vibration of circular and annular sectorial thin plates subject to classical boundary conditions. Kim and Yoo [17] utilized a novel analytical solution to flexural responses of annular sector thin plates with classical boundary conditions. Mirtalaie and Hajabasi [18] studied the free vibration of annular sector thin plates with classical boundary conditions by using differential quadrature method.
A review of the scientific literature in this field reveals that the majority of the existing free vibration investigation mainly focused on the isotropic annular sector plate which has the same material property along different directions, while the reported work on the free vibration of the orthotropic annular sector thin plate is little. Most of the contributions on free vibration analysis of orthotropic annular sector thin plates with classical boundary supports are confined. In addition, orthotropic annular sector thin plates with internal radial line and circumferential arc supports are widely encountered in the engineering practices. Without these intermediate supports, the plates may undergo large deformation and acute shaking and eventually lead to structural failure. The only work focused on this subject is that Liew et al. [19] presented the vibrations of Thick isotropic annular sector plates with classical boundary conditions. However, a variety of possible elastic boundary condition cases which may not always be classical in nature can be encountered in practice. The existing solution procedures are often only customized for a specific set of different boundary conditions, and thus typically require constant modifications of the trial functions and corresponding solution procedures to adapt to different boundary cases. Therefore, the use of the existing solution procedures will result in very tedious calculations and be easily inundated with various classical boundary conditions and their combinations. To the best of authors’ knowledge, there are no reported solutions on the free vibration of orthotropic annular sector thin plates with general boundary conditions, internal radial line and circumferential arc supports in the literature. Therefore, it is necessary and of great significance to develop a unified, efficient and accurate formulation which is capable of universally dealing with orthotropic annular sector thin plates subjected to general boundary conditions, internal radial line and circumferential arc supports.
The purpose of the present study is to develop an efficient and accurate solution for free vibration analysis of orthotropic annular sector thin plates subjected to general boundary restraints, internal radial line supports, circumferential arc supports. In a previous study, a modified Fourier series technique proposed by Li [20, 21] is widely used in the vibrations of plates and shells with general boundary conditions by Ritz method, e.g., [2230]. Therefore, the present work can be considered the combination of the modified Fourier series technique and Ritz method to present a modified FourierRitz approach for free vibration of orthotropic annular sector thin plates subjected to general boundary conditions, internal radial line and circumferential arc supports. Under the current framework, regardless of boundary conditions, the displacements of the sector plates are invariantly expressed as a standard Fourier cosine series and several auxiliary closedform functions. These auxiliary functions are introduced to eliminate any potential discontinuities of the original displacement function and its derivatives, throughout the whole domain including its edges, and then to effectively enhance the convergence of the results. Since the displacement field is constructed to be adequately smooth in the whole solution domain, an accurate solution can be obtained by using Ritz procedure based on the energy functions of the sector plates. The excellent accuracy and reliability of the current solutions are compared with the results found in the literature, and numerous new results for annular sector plates with various boundary conditions are presented. New results are obtained for annular sector plates subjected to elastic boundary restraints and arbitrary internal radial line and circumferential arc supports in both directions, and they may serve as benchmark solutions for future researches.
2. Theoretical formulations
2.1. Description of the model
Fig. 1 shows a orthotropic annular sector thin plate with uniform thickness $h$, inner radius $a$, outer radius $b$, width $R$ of plate in the radial direction and sector angle $\varphi $. The geometry and dimensions are defined in an orthogonal cylindrical coordinate system $(r,\theta ,z)$ . A local coordinate system $(s,\theta ,z)$ is also shown in the Fig. 1, which will be used in the analysis. Since the main focus of this paper is to develop a unified solution for the vibration analysis of orthotropic annular sector thin plates with general boundary conditions, thus, in order to satisfy the request, the artificial spring boundary technique is adopted here, in which each boundary of a plate is assumed to be restrained by one group of linear springs (${k}_{w}$) and one group of rotational springs (${K}_{r}$) to simulate the given or typical boundary conditions. The stiffness of the boundary springs can take any value from zero to infinity. By assigning the stiffness of the boundary springs with various values, it is equivalent to impose different boundary forces on the midplane of the plate. For example, the clamped boundary conditions are essentially obtained by setting the spring stiffness substantially larger than the bending rigidity of the plate.
2.2. Governing equations and boundary conditions
The governing equation of motion for the free vibration of orthotropic annular sector thin plates can be written as:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}2{D}_{12}\frac{1}{{\left(s+a\right)}^{3}}\frac{{\partial}^{3}w}{\partial s\partial {\theta}^{2}}{D}_{22}\frac{1}{{\left(s+a\right)}^{2}}\frac{{\partial}^{2}w}{\partial {s}^{2}}+2\left({D}_{22}+{D}_{12}\right)\frac{1}{{\left(s+a\right)}^{4}}\frac{{\partial}^{2}w}{\partial {\theta}^{2}}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+{D}_{22}\frac{1}{{\left(s+a\right)}^{3}}\frac{\partial w}{\partial s}\rho h{\omega}^{2}{w}^{2}=0,$
where $w$ is the sector plate deflection, ${D}_{ij}$ are the standard bending rigidities in the classical lamination theory. For an orthotropic annular sector thin plate, the stiffness constants are related to the lamina engineering constants and the plate thickness as:
In terms of the flexural displacement, the bending and twisting moment and transverse shearing forces can be expressed as:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left({D}_{12}+4{D}_{66}\right)\left(\frac{1}{\left(s+a\right)}\frac{{\partial}^{3}w}{\partial {s}^{2}\partial \theta}\frac{2}{{\left(s+a\right)}^{2}}\frac{{\partial}^{2}w}{\partial s\partial \theta}+\frac{2}{{\left(s+a\right)}^{3}}\frac{\partial w}{\partial \theta}\right).$
Fig. 1Schematic diagram of annular sector thin plate with arbitrary boundary condition
In this study, the general boundary conditions along each edge will be described in terms of two restraining springs, a linear spring ${k}_{\gamma}$ and a rotational spring ${K}_{\gamma}$, where subscripts $\gamma ={s}_{0}$, ${s}_{1}$, ${\theta}_{0}$ and ${\theta}_{1}$ represent the springs at the boundary edges of the plate respectively, as shown in Fig. 1. Accordingly, the boundary conditions become:
Eqs. (11)(14) represent a set of general boundary conditions. By setting the spring stiffnesses to appropriate values, all the classical homogeneous boundary conditions can be readily simulated.
From Eqs. (6)(10), the boundary conditions can be finally written as:
At $s=$0:
At $s=R$:
At $\theta =$0:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\left({D}_{12}+4{D}_{66}\right)\left(\frac{1}{\left(s+a\right)}\frac{{\partial}^{3}w}{\partial {s}^{2}\partial \theta}\frac{2}{{\left(s+a\right)}^{2}}\frac{{\partial}^{2}w}{\partial s\partial \theta}+\frac{2}{{\left(s+a\right)}^{3}}\frac{\partial w}{\partial \theta}\right),$
At $\theta =\varphi $:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left({D}_{12}+4{D}_{66}\right)\left(\frac{1}{\left(s+a\right)}\frac{{\partial}^{3}w}{\partial {s}^{2}\partial \theta}\frac{2}{{\left(s+a\right)}^{2}}\frac{{\partial}^{2}w}{\partial s\partial \theta}+\frac{2}{{\left(s+a\right)}^{3}}\frac{\partial w}{\partial \theta}\right),$
2.3. Admissible displacement functions
Mathematically, it is often desired to express the displacement, $w\left(r,\theta \right)$, in the form of a Fourier series expansion because Fourier functions constitute a complete set and exhibit an excellent numerical stability. Unfortunately, the conventional Fourier series expression will generally have a convergence problem along the boundary edges except for a few simple boundary conditions. In addition, without being uniformly convergent, the derivatives of a Fourier series cannot be obtained simply through termbyterm differentiation. To overcome these problems, the displacement function will be here expressed as a more robust form of Fourier series expansion:
$\left.+\sum _{l=1}^{4}{\chi}_{l}\left(s\right)\sum _{n=0}^{\mathrm{\infty}}{b}_{n}^{l}\mathrm{c}\mathrm{o}\mathrm{s}{\lambda}_{\varphi n}\theta \right\}{e}^{j\omega t},$
where the eight supplementary terms are introduced to deal with any possible discontinuities or jumps at the boundaries which are potentially associated with the displacement function and its derivatives when they are periodically extended onto the entire solution domain. It should be noted that these discontinuities are not inherently related to the displacement function over the solution domain; instead they are the artifact resulting from the Fourier series representation of the displacement solution.
The four $\zeta $functions in the $\theta $ direction and $\chi $functions in the $s$ direction in Eq. (23) are here chosen as:
It is easy to verify that their first and third derivatives are mostly equal to zero along the boundary edges except for:
It can be proven mathematically that the series expression in Eq. (23) is able to expand and uniformly converge to any function $\Theta \left(r,\theta \right)\in {\mathbf{C}}^{3}$ for $\forall \left(x,y\right)\in \mathbf{D}:\left(\left[0,R\right]\times \left[0,\varphi \right]\right)$. Also, this series can be simply differentiated, through termbyterm, to obtain the uniformly convergent series expansions for up to the fourthorder derivatives. Mathematically, an exact displacement (or classical) solution is a particular function $w\left(r,\theta \right)\in {\mathbf{C}}^{3}$ for $\forall \left(x,y\right)\in \mathbf{D}$ which satisfies the governing equation at each field point and the boundary conditions at every boundary point.
2.4. Solution procedure
Once the admissible displacement functions and energy functions of the sector plate are established, the following task is to determine the coefficients in the admissible functions. Because of its simplicity and high accuracy, Ritz method is widely used in the vibration analysis of structural elements as a very powerful tool. In the Ritz method, the solutions can be obtained by minimizing the energy functional with respect to the coefficients of the admissible functions.
The strain energy of the sector plate is given as:
$\left.\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left.+\frac{1}{{\left(s+a\right)}^{2}}\frac{{\partial}^{2}w}{\partial {\theta}^{2}}\right)+4{D}_{66}\left(\frac{1}{s+a}\frac{{\partial}^{2}w}{\partial s\partial \theta}\frac{1}{{\left(s+a\right)}^{2}}\frac{\partial w}{\partial \theta}\right)\right\}\left(s+a\right)dsd\theta .$
By neglecting the rotary inertia, the kinetic energy of an orthotropic annular sector thin plate can be written as:
As mentioned in Sections 2.1 and 2.2, each boundary of a sector plate is assumed to be restrained by one group of linear springs (${k}_{w}$) and one group of rotational springs (${K}_{r}$) to simulate the given or typical boundary conditions. Therefore, the deformation strain energy (${U}_{bs}$) stored in the boundary springs during vibration can be defined as:
$+\frac{1}{2}{\int}_{0}^{R}\left[a{\left\{{k}_{{\theta}_{0}}{w}^{2}+{K}_{{\theta}_{0}}{\left(\frac{1}{s+a}\frac{\partial w}{\partial \theta}\right)}^{2}\right\}}_{\theta =0}+b{\left\{{k}_{{\theta}_{1}}{w}^{2}+{K}_{{\theta}_{1}}{\left(\frac{1}{s+a}\frac{\partial w}{\partial \theta}\right)}^{2}\right\}}_{\theta =\varphi}\right]ds.$
The Lagrangian functional ($L$) of the plates during vibration can be expressed in terms of the energy expressions:
Substituting Eqs. (34)(36) and Eq. (23) into Eq. (37) and performing the Ritz procedure with respect to each unknown coefficient, the equations of motion for plates can be yielded and are given in the matrix form:
where:
In Eq. (38), the $\mathbf{K}$ is the stiffness matrix of the plate, and the $\mathbf{M}$ is the mass matrix. For conciseness, the detailed expression for stiffness and mass matrices will not be shown here. By solving the Eq. (38), the frequencies (or eigenvalues) of orthotropic annular sector thin plates can be readily obtained and the mode shapes can be yielded by substituting the corresponding eigenvectors into series representations of displacement.
3. Numerical results and discussion
In this section, a systematic comparison between the current solutions and other methods is carried out to validate the excellent accuracy, reliability and feasibility of the present method. Unless otherwise stated, the nondimensional $\mathrm{\Omega}=\omega {b}^{2}/{\left({\rho}_{1}{h}_{1}/{D}_{11}\right)}^{1/2}$ is used in the presentation, and the material and geometry properties of orthotropic annular sector thin plates under consideration are: $\rho =$7800 kg/m^{3}, ${E}_{\theta}=$70 GPa, ${E}_{r}=$40${E}_{\theta}$, ${G}_{r\theta}=$7.3 GPa, ${\mu}_{r}=$0.3, $\varphi =$90, $b=$ 1 m, $b/a=$2 and $h/b=$0.005.
3.1. Determination of the boundary spring stiffness
In the present work, the general boundary conditions of the structure are implemented by introducing artificial spring boundary technique to separately simulate the boundary forces and displacements. As previously mentioned, the case of general boundary conditions of the plates can be easily simulated by assigning proper stiffness values to the boundary springs, for instance, a clamped boundary (C) can be readily achieved by simply setting the stiffness of the entire springs to be infinitely large. However, the “infinitely large” is represented by a sufficiently large number in actual calculations. Thus, effects of the spring stiffness of boundary springs on the modal characteristics should be investigated.
Effects of elastic boundary and coupling stiffness parameters on the nondimensional frequency parameters $\mathrm{\Omega}$ of orthotropic annular sector thin plates are studied. A frequency parameter $\mathrm{\Delta}\mathrm{\Omega}$ which is defined as the difference of the nondimensional frequency parameter $\mathrm{\Omega}$ to those of the elastic restraint parameters ${\u0413}_{\lambda \left(\lambda =w,r\right)}$_{}to 10^{2}, i.e., $\mathrm{\Delta}\mathrm{\Omega}={\mathrm{\Omega}}_{{\u0413}_{\lambda}}{\mathrm{\Omega}}_{{\u0413}_{\lambda}}=$10^{2} is used in the calculation. The plates under consideration are completely clamped at boundaries $s=$0, $s=R$, and free at boundary $\theta =$ 0, while at edge $\theta =\varphi $, the plates are elastically supported by only one group of spring component with stiffnesses varying from 10^{2} to 10^{16}. In Fig. 2(a), the influences of the line springs ${k}_{w}$ on frequency parameters are given. It is shown that the frequency parameter almost stays unchanged when the nondimensional stiffness of the liner springs ${k}_{2}$ is larger than 10^{9} or smaller than 10^{4}. In Fig. 2(b), the influences of the rotation springs ${k}_{w}$ on frequency parameters are given. It is shown that the frequency curve changes greatly within the stiffness range from 10^{3} to 10^{8}. Based on the analysis, it can be found the frequency parameters exist the large change as the stiffness parameters increase in the certain range.
In the following discussion, vibration frequencies and modal shapes of annular sector plates with arbitrary classical boundary conditions, general elastic boundary conditions and their combinations will be presented. Taking edge $s=$0 for example, the corresponding spring stiffness parameters for three types of classical boundary conditions and three types of elastic boundary conditions which are commonly encountered in engineering practices are given as follows:
The appropriateness of defining the classical boundary conditions in terms of boundary spring parameters will be proved by several examples in later subsections. For the sake of simplicity, a simple letter string is employed to represent the boundary condition of the annular sector plate, circular sector plate, annular plate and circular plate, such as the FCSE identifies the annular sector plate with $F$, $C$, $S$ and $E$ boundary conditions at boundaries $s=$ 0, $\theta =$ 0, $s=R$ and $\theta =\varphi $, respectively.
3.2. Convergence study
In this subsection, the convergence of orthotropic annular sector plates with different boundary conditions is studied. The first eight frequency parameters $\mathrm{\Omega}$ for CCCC and FFFF orthotropic annular sector plates with different truncated number $M$ and $N$ (i.e. $M=N=$ 814) are given in Table 1. The Table shows the proposed method has fast convergence. The maximum discrepancy for the worst case between the truncated configuration $M=$12 and $M=$14 is less than 0.004 %. In order to fully illustrate the convergence of the present method, the frequency parameters $\mathrm{\Omega}$ of the higher mode (10th, 15th and 20th) with various truncated numbers $M$, $N$ subjected to CCCC and FFFF boundary conditions are shown in Fig. 3. The highly desired convergence characteristics are observed: (a) sufficiently accurate results can be obtained with only a small number of terms in the series expansions; (b) the solution is consistently refined as more terms are included in the expansions; (c) the frequency parameters for higherorder modes tend to converge slower. Thus, an adequate truncation number should be dictated by the desired accuracy of the interesting largest natural frequencies. 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. To further validate the accuracy and reliability of the current solution, more numerical examples will be presented.
Fig. 2Variation of the frequency parameters Ω versus the elastic boundary restraint parameters for annular sector plate: a) transverse spring stiffness; b) rotation spring stiffness
Fig. 3Variations of frequency parameter Ω with respect to truncated number M and N: a) FFFF; b) CCCC
3.3. Orthotropic annular sector thin plates with general boundary conditions
The target of this subsection is to validate whether the present method can fit to solve the free vibration of orthotropic annular sector thin plates with general boundary conditions. First, a verification study about the classical boundary conditions is carried out to validate the accuracy and reliability of present method. In Tables 2 and 3, the first eight frequency parameters $\mathrm{\Omega}$ with different classical boundary conditions for isotropic annular sector thin plates and orthotropic annular sector thin plates are presented, respectively. In order to compare, the reference results taken from Ref. [18] and obtained using an FEM (ABAQUS) model are also given there. A great agreement can be obtained from the comparison. Next, we will focus on the free vibration of orthotropic annular sector thin plates with general elastic restraints. In Table 4, the detail comparisons between results obtained by the present method and those provided by FEM solutions (ABAQUS) are presented, in which nine types of elastic boundary conditions including classicalelastic case and complete elastic case are included. It is obvious that the current results match very well with the referential data. Based on the above analysis, it implies that the current method is able to make correct predictions for the modal characteristics of orthotropic annular sector thin plates with not only classical boundary but also elastically restrained boundary.
Table 1Convergence of frequencies parameters Ω for annular sector plate with CCCC and FFFF boundary conditions
Boundary conditions  $M=N$  Mode number  
1  2  3  4  5  6  7  8  
CCCC  8  88.838  89.265  90.382  92.702  96.822  103.30  112.57  128.41 
9  88.838  89.265  90.382  92.702  96.823  103.30  112.58  124.78  
10  88.838  89.265  90.381  92.702  96.821  103.30  112.55  124.78  
11  88.838  89.265  90.381  92.702  96.820  103.30  112.55  124.76  
12  88.838  89.265  90.381  92.702  96.820  103.30  112.55  124.76  
13  88.838  89.265  90.381  92.702  96.820  103.30  112.55  124.76  
14  88.838  89.265  90.381  92.702  96.820  103.30  112.55  124.76  
FFFF  8  2.3039  2.3588  5.5209  6.6546  10.370  13.820  16.793  24.198 
9  2.3017  2.3530  5.5201  6.6527  10.370  13.819  16.792  24.198  
10  2.3017  2.3530  5.5201  6.6526  10.370  13.819  16.792  24.197  
11  2.3013  2.3517  5.5199  6.6523  10.370  13.819  16.791  24.197  
12  2.3013  2.3517  5.5199  6.6522  10.370  13.819  16.791  24.197  
13  2.3012  2.3516  5.5199  6.6521  10.370  13.819  16.791  24.197  
14  2.3012  2.3516  5.5199  6.6521  10.370  13.819  16.791  24.197 
Table 2Frequency parameters Ω for isotropic annular sector plate with different classical boundary conditions
Boundary conditions  Methods  Mode number  
1  2  3  4  5  6  7  8  
FSFS  Present  21.067  66.722  81.604  146.41  176.12  176.9  274.57  298.37 
DQM  21.067  66.722  81.604  146.41  176.12  176.9  –  –  
FEM  21.032  66.498  81.46  145.66  175.83  176.51  273.51  297.25  
SSSS  Present  68.379  150.98  189.6  278.39  283.59  387.62  438.96  443.89 
DQM  68.379  150.98  189.6  278.39  283.59  387.62  –  –  
FEM  68.112  150.55  189.12  277.75  282.4  386.99  437.32  442.9  
CSCS  Present  107.57  178.82  269.49  305.84  346.46  476.3  487.39  508.57 
DQM  107.57  178.82  269.49  305.84  346.46  476.3  –  –  
FEM  107.41  178.48  269.11  305.35  345.51  475.56  485.84  507.83 
On the basis of verification of the presented method, next, the authors will study the influence of the stiffness ratio (${E}_{1}/{E}_{2}$) and sector angle ($\varphi $) on the vibration characteristics of orthotropic annular sector thin plates with classical boundary and elastically restrained conditions. Fig. 4 depicts the variations of the first four frequency parameters $\mathrm{\Omega}$ of orthotropic annular sector thin plates along the variations of the stiffness ratio with CCCC and E^{3}E^{3}E^{3}E^{3} boundary conditions. From the Fig. 4, the frequency parameters $\mathrm{\Omega}$ decrease rapidly and may reach their crest around a critical stiffness ratio, and beyond this range, the frequency parameters almost remain unchanged. The variations of the first four frequency parameters $\mathrm{\Omega}$ of orthotropic annular sector thin plates relative to the sector angle are given in Fig. 5. It is evident that the variations of the vibration characteristics of the orthotropic annular sector thin plate have the similar tendency with the variations relative to the stiffness ratio.
Table 3Frequency parameters Ω for orthotropic annular sector plate with different classical boundary conditions
Boundary conditions  Methods  Mode number  
1  2  3  4  5  6  7  8  
CCCC  Present  88.838  89.265  90.381  92.702  96.820  103.30  112.55  124.76 
FEM  89.033  89.443  90.530  92.823  96.935  103.46  112.85  125.32  
SSSS  Present  39.424  39.945  41.486  44.882  50.885  59.861  71.742  86.191 
FEM  39.457  39.971  41.504  44.905  50.941  60.012  72.072  86.798  
FFFF  Present  2.3013  2.3519  5.5199  6.6522  10.370  13.819  16.791  24.197 
FEM  2.3014  2.3514  5.5231  6.6528  10.384  13.828  16.834  24.257  
CSCF  Present  88.732  88.842  89.202  90.143  92.145  95.79  101.66  110.21 
FEM  88.930  89.029  89.365  90.270  92.234  95.85  101.74  110.37 
Table 4Frequency parameters Ω for orthotropic annular sector plate with various elastic boundary conditions
Boundary conditions  Methods  Mode number  
1  2  3  4  5  6  7  8  
CE^{1}CE^{1}  Present  88.791  89.014  89.566  90.561  92.087  94.513  98.550  104.90 
FEM  88.990  89.199  89.733  90.700  92.176  94.540  98.542  104.91  
SE^{2}SE^{2}  Present  39.321  39.423  39.944  41.481  44.870  50.861  59.820  71.681 
FEM  39.356  39.455  39.971  41.499  44.891  50.916  59.971  72.010  
FE^{3}FE^{3}  Present  1.8326  4.7181  4.7284  8.5757  9.8222  12.718  15.337  17.275 
FEM  1.8327  4.7354  4.7400  8.6054  9.9099  12.785  15.489  17.384  
E^{2}E^{1}E^{2}E^{1}  Present  1.3032  4.8025  9.8634  15.543  21.385  27.578  34.504  36.093 
FEM  1.3037  4.8063  9.8763  15.567  21.422  27.642  34.611  36.077  
E^{3}E^{2}E^{3}E^{2}  Present  11.561  11.643  12.666  16.101  22.051  29.423  37.753  39.744 
FEM  11.553  11.707  12.739  16.172  22.141  29.555  37.958  39.747  
E^{1}E^{3}E^{1}E^{3}  Present  11.338  13.077  15.856  19.141  20.533  21.884  22.887  24.534 
FEM  11.284  13.045  15.850  19.164  20.427  21.868  22.921  24.759  
E^{1}E^{1}E^{1}E^{1}  Present  11.032  12.212  15.039  18.793  20.373  21.338  22.881  24.137 
FEM  10.975  12.168  15.015  18.793  20.261  21.259  22.913  24.234  
E^{2}E^{2}E^{2}E^{2}  Present  1.3226  5.0470  10.944  18.286  26.321  35.077  35.935  36.106 
FEM  1.3229  5.0506  10.960  18.328  26.408  35.239  35.942  36.135  
E^{3}E^{3}E^{3}E^{3}  Present  11.892  13.416  16.195  19.872  24.922  31.443  39.178  39.986 
FEM  11.830  13.376  16.204  19.932  25.019  31.583  39.368  39.950 
3.4. Orthotropic annular sector thin plates with line/arc supports
In the engineering practices, the plate structures are often restrained by internal line supports to reduce the magnitude of dynamic and static stresses and displacements of the structure or to satisfy special architectural and functional requirements. However, the research work on plates with internal line supports is scanty. Thus, in this paper, except the classical and general elastic boundary conditions, the authors also investigate the free vibration behaviors of orthotropic annular sector plates with internal radial line and circumferential arc supports. As shown in Fig. 4, the orthotropic annular sector plate is restrained by arbitrary internal radial line and circumferential arc supports. ${r}_{i}$ and ${\theta}_{j}$ represent the position of the $i$th and $j$th internal radial line and circumferential arc supports along the ${r}^{}$ and ${\theta}^{}$ directions, respectively. The displacement fields in the position of the line support satisfy $w\left({r}_{i},\theta ,t\right)=$0 and $w\left(r,{\theta}_{i},t\right)=$0. This condition can be readily obtained by introducing a group of continuously distributed linear springs at the location of each line/arc support and setting the stiffnesses of these springs equal to be infinite (which is represented by a very large number, 10^{14}). Thus, the potential energy (${P}_{rals}$) stored in these springs is:
where ${M}_{i}$ and ${N}_{j}$ are the amount of circumferential arc supports and radial line in the $\theta $ and $r$ directions. ${k}_{ri}^{i}$ and ${k}_{\theta j}^{j}$ denote the corresponding circumferential arc supported and radial line springs distributed at $r={r}_{i}$ and $\theta ={\theta}_{j}$. By adding the potential energy ${P}_{rals}$ stored in the line/arc supported springs in the Lagrangian energy function (Eq. (37)) and carrying out the Ritz procedure, the characteristic equation for a orthotropic annular sector thin plates with arbitrary boundary conditions and internal radial line and circumferential arc supports is readily obtained.
Fig. 4Variation of the frequency parameters Ω versus the stiffness rations (E1/E2) for annular sector plate: a) CCCC; b) E3E3E3E3
Fig. 5Variation of the frequency parameters Ω versus the sector angle for annular sector plate: a) CCCC; b) E3E3E3E3
In order to prove the validity of the present formulations for the vibration of orthotropic annular sector thin plates with internal radial line and circumferential arc supports, Table 5 presents the comparison of the first eight frequency parameters $\mathrm{\Omega}$ for sector plates with three classical boundary conditions, i.e. CCCC, SSSS, FFFF. For the purpose of stressing the effects of the line/arc supports, corresponding results for the considered annular sector plate without line/arc supports are also presented in the table. The benchmark results are provided by ABAQUS based on FEA method. From the table a consistent agreement of present results and referential date is seen. The discrepancy is very small and doesn’t exceed 0.61 % for the worst case. In addition, the table shows that the line/arc supports can increase the frequencies of the sector plate. Then, the influence of the locations of internal radial line and circumferential arc supports on the frequency of orthotropic annular sector plates is investigated. For simplicity of this research, when investigating the influence of the radial line support along $\theta $ direction, the circumferential edges of the plates are under clamped boundary conditions and we only change the radial boundary conditions; on the contrary, when studying the effect of the circumferential arc supports along $s$ direction, the radial edges are with clamped boundary condition and only the boundary conditions of the circumferential edges vary. In Fig. 7, the variations of the fundamental frequency parameters $\mathrm{\Omega}$ of the considered sector plate with against the radial line support location parameter ${\theta}_{1}/\varphi $ and against the circumferential arc support location parameter ${R}_{1}/R$ are depicted. Six types of edge conditions used in the investigation are: CC, SS, FC, CF, FS, SF. It is obvious that the frequency parameters of the sector plate are significantly affected by the position of the radial line and circumferential arc support, and this effect varies with the edge conditions.
Fig. 6Schematic diagram of an annular sector plate with arbitrary internal radial line and circumferential supports
Table 5Comparison of the first eight frequency parameters Ω for orthotropic annular sector thin plate with different line supports
Line supports  Boundary conditions  Methods  Mode number  
1  2  3  4  5  6  7  8  
None  CCCC  20.453  20.552  20.809  21.343  22.291  23.783  25.913  28.724  
SSSS  9.0767  9.1966  9.5514  10.333  11.715  13.782  16.517  19.844  
FFFF  0.5298  0.5415  1.2709  1.5316  2.3875  3.1816  3.8658  5.5709  
$\frac{R}{2}$  CCCC  Present  245.75  246.09  246.83  248.18  250.43  253.90  258.92  265.72 
FEM  247.25  247.52  248.17  249.41  251.53  254.84  259.73  266.47  
SSSS  Present  157.85  158.20  158.99  160.50  163.08  167.11  172.92  180.65  
FEM  158.40  158.73  159.46  160.90  163.41  167.38  173.17  180.95  
FFFF  Present  1.3847  2.6689  5.9102  11.2125  18.490  26.815  35.130  43.346  
FEM  1.3822  2.6663  5.9101  11.2192  18.518  26.875  35.226  43.498  
$\frac{R}{2}\text{,}\text{}\frac{\varphi}{2}$  CCCC  Present  246.09  246.26  248.18  248.97  253.90  256.03  265.72  270.00 
FEM  247.52  247.68  249.41  250.17  254.84  256.85  266.47  270.36  
SSSS  Present  158.20  158.37  160.50  161.39  167.11  169.55  180.65  185.30  
FEM  158.73  158.89  160.90  161.76  167.38  169.73  180.95  185.25  
FFFF  Present  2.6690  3.5863  11.212  14.306  26.815  30.980  43.346  47.782  
FEM  2.6663  3.5829  11.219  14.304  26.875  30.992  43.498  47.812 
Since the vibration results for internal radial line and circumferential arc supported orthotropic annular sector thin plates with arbitrary boundary conditions are very limited in the literature, some new results are calculated here, which can be used for benchmark results by researchers as well as reference datum for practicing engineers. In Table 6, the first six frequency parameters $\mathrm{\Omega}$ of the considered orthotropic annular sector thin plate subjected to as many as nine possible boundary conditions are presented. And four different line/arc support conditions are considered in the calculation. In addition, the lowest four mode shapes for the sector plate with CE^{1}CE^{1} boundary condition presented in Table 6 are given in Fig. 8. These view mode shapes are served to enhance our understanding of the vibratory characteristics of the orthotropic annular sector thin plates with internal radial line and circumferential arc supports.
Fig. 7Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions
Fig. 8The lowest three modes shape for a CE1CE1 annular sector plate with various line supports: a) None; b) r1=R/2; c) r1=R/3, r2=2R/3; d) r1=R/2, θ1=ϕ/2; e) r1=R/3, θ1=ϕ/2, r2=2R/3
4. Conclusions
A modified FourierRitz approach is presented for the free vibration analysis of orthotropic annular sector thin plates with general boundary conditions, internal radial line and circumferential arc supports. Under the current framework, the admissible displacement function of the plate, regardless of the boundary conditions, is expressed as a modified Fourier series, which is constructed as the linear superposition of a standard Fourier cosine series supplemented with auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations.
Table 6The first six frequency parameters Ω for orthotropic annular sector thin plates with different boundary conditions and various arc and radial line supports
Line supports  Mode  Boundary conditions  
CE^{1}CE^{1}  SE^{2}SE^{2}  FE^{3}FE^{3}  E^{2}E^{1}E^{2}E^{1}  E^{3}E^{2}E^{3}E^{2}  E^{1}E^{3}E^{1}E^{3}  E^{1}E^{1}E^{1}E^{1}  E^{2}E^{2}E^{2}E^{2}  E^{3}E^{3}E^{3}E^{3}  
${r}_{1}=\frac{R}{2}$  1  245.70  157.75  3.9791  35.594  39.409  20.539  20.376  35.461  39.610 
2  245.88  157.85  8.1532  36.210  39.541  21.763  21.291  35.606  40.312  
3  246.21  158.20  12.735  37.691  40.215  23.898  23.511  36.347  41.593  
4  246.74  158.99  18.006  40.130  42.167  27.154  26.993  38.479  43.746  
5  247.60  160.49  24.718  43.639  46.157  32.158  31.553  42.773  47.563  
6  249.04  163.06  32.362  48.710  52.248  38.581  37.234  49.212  53.392  
${r}_{1}=\frac{R}{3}\text{,}\text{}$ ${r}_{2}=\frac{2R}{3}$  1  454.35  355.14  71.101  115.53  117.55  75.70  75.66  115.47  117.63 
2  454.47  355.24  71.448  115.79  117.63  76.03  75.92  115.55  117.91  
3  454.72  355.56  72.082  116.31  117.95  76.64  76.52  115.87  118.40  
4  455.16  356.18  73.134  117.16  118.74  77.64  77.57  116.67  119.26  
5  455.87  357.23  74.895  118.48  120.29  79.34  79.17  118.24  120.80  
6  456.97  358.90  77.555  120.52  122.78  81.91  81.46  120.74  123.24  
${r}_{1}=\frac{R}{2}\text{,}$ ${\theta}_{1}=\frac{\varphi}{2}$  1  245.88  157.85  8.1532  36.210  39.541  21.763  21.291  35.606  40.312 
2  245.94  157.89  10.185  36.694  39.664  22.614  22.070  35.743  40.779  
3  246.74  158.99  18.006  40.130  42.167  27.154  26.993  38.479  43.746  
4  247.00  159.43  21.022  41.583  43.696  29.301  29.013  40.134  45.253  
5  249.03  163.06  32.362  48.710  52.248  38.581  37.234  49.212  53.392  
6  249.88  164.62  36.274  51.791  55.762  42.035  40.395  52.880  56.802  
${r}_{1}=\frac{R}{3},$ ${\theta}_{1}=\frac{\varphi}{2},$ ${r}_{2}=\frac{2R}{3}$  1  454.47  355.24  71.448  115.79  117.63  76.030  75.922  115.55  117.91 
2  454.50  355.26  71.657  115.92  117.67  76.228  76.095  115.59  118.04  
3  455.16  356.18  73.134  117.16  118.74  77.643  77.570  116.67  119.26  
4  455.31  356.42  73.818  117.63  119.26  78.300  78.205  117.20  119.80  
5  456.97  358.90  77.555  120.52  122.78  81.906  81.458  120.74  123.24  
6  457.49  359.74  79.145  121.74  124.22  83.437  82.829  122.20  124.67 
The general boundary conditions of the sector plate are accounted for by using the artificial spring boundary technique, in which the elastic restraint stiffnesses can take any value from zero to infinity to better simulate many realworld boundary conditions. Ritz procedure is used to obtain the exact solution based on the energy functions of those structures. The convergence of the present solution is checked and the excellent accuracy is validated by the comparison with the existing results published in the literature and FEM solutions. Excellent agreements are obtained from these comparisons. The effects of elastic restraint parameters and locations of radial line and circumferential arc supports are also investigated and reported. New results for free vibration of orthotropic annular sector thin plates with various edge conditions and internal radial line and circumferential arc supports are presented, which may be used for benchmarking of researchers in the field.
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About this article
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 51209052), Heilongjiang Province Youth Science Fund Project (No. QC2011C013) and Harbin Science and Technology Development Innovation Foundation of Youth (No. 2011RFQXG021).