A unified solution for free vibration of orthotropic annular sector thin plates with general boundary conditions, internal radial line and circumferential arc supports

Dongyan Shi1 , Xiuhai Lv2 , Qingshan Wang3 , Qian Liang4

1, 2, 3, 4College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, P. R. China

1, 2, 3, 4Department of Electrical and Mechanical Engineering, Heilongjiang Agricultural Engineering Vocational College, Harbin, P. R. China

3Corresponding author

Journal of Vibroengineering, Vol. 18, Issue 1, 2016, p. 361-377.
Received 22 August 2015; received in revised form 21 October 2015; accepted 29 October 2015; published 15 February 2016

Copyright © 2016 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

In this paper, a modified Fourier-Ritz 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 closed-form 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.

Keywords: free vibration, orthotropic annular sector thin plates, general boundary conditions, internal circumferential arc supports, internal radial line supports, Ritz approach.

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 Rayleigh-Ritz 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 non-uniform 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 ring-shaped polar-orthotropic sector plates with classical boundary conditions. Singh et al. [9] used Rayleigh-Ritz 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 p-element 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., [22-30]. Therefore, the present work can be considered the combination of the modified Fourier series technique and Ritz method to present a modified Fourier-Ritz 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 closed-form 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 ϕ. The geometry and dimensions are defined in an orthogonal cylindrical coordinate system (r,θ,z) . A local coordinate system (s,θ,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 (kw) and one group of rotational springs (Kr) 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 mid-plane 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:

(1)
D 11 4 w s 4 + 2 D 12 1 s + a 2 4 w s 2 θ 2 + D 22 1 s + a 4 4 w θ 4 + 2 D 11 1 s + a 3 w s 3
            - 2 D 12 1 s + a 3 3 w s θ 2 - D 22 1 s + a 2 2 w s 2 + 2 D 22 + D 12 1 s + a 4 2 w θ 2
            + D 22 1 s + a 3 w s - ρ h ω 2 w 2 = 0 ,

where w is the sector plate deflection, Dij 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:

(2)
D 11 = E r h 3 12 1 - μ r μ θ ,
(3)
D 22 = E θ h 3 12 1 - μ r μ θ ,
(4)
D 12 = μ r E θ h 3 12 1 - μ r μ θ ,
(5)
D 66 = G r θ 12 h 3 .

In terms of the flexural displacement, the bending and twisting moment and transverse shearing forces can be expressed as:

(6)
M r = - D 11 2 w s 2 - D 12 1 s + a w s + 1 s + a 2 w θ 2 ,
(7)
M θ = - D 22 1 s + a w s + 1 s + a 2 w θ 2 - D 12 2 w s 2 ,
(8)
M r θ = - 2 D 66 1 s + a 2 w s θ - 1 s + a w θ ,
(9)
Q r = - D 11 3 w s 3 - D 12 + 4 D 66 1 s + a 2 3 w s θ 2 - 1 s + a 2 2 w θ 2 ,
(10)
Q θ = - D 22 1 s + a 3 2 w θ 3
            - D 12 + 4 D 66 1 s + a 3 w s 2 θ - 2 s + a 2 2 w s θ + 2 s + a 3 w θ .

Fig. 1. Schematic diagram of annular sector thin plate with arbitrary boundary condition

 Schematic 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 kγ and a rotational spring Kγ, where subscripts γ=s0, s1, θ0 and θ1 represent the springs at the boundary edges of the plate respectively, as shown in Fig. 1. Accordingly, the boundary conditions become:

(11)
s = 0 : Q r + k s 0 ,           w = 0 , M r + K s 0 ,       w s = 0 ,
(12)
s = R : Q r - k s 1 ,           w = 0 , M r - K s 1 ,       w s = 0 ,
(13)
θ = 0 : Q θ + k θ 0 ,                                         w = 0 , M θ + K θ 0 1 s + a ,       w θ = 0 ,
(14)
θ = ϕ : Q θ - k θ 1 ,                                         w = 0 , M θ - K θ 1 1 s + a ,       w θ = 0 .

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:

(15)
k s 0 w = D 11 3 w s 3 + D 12 + 4 D 66 1 s + a 2 3 w s θ 2 - 1 s + a 2 2 w θ 2 ,
(16)
K s 0 w s = D 11 2 w s 2 + D 12 1 s + a w s + 1 s + a 2 w θ 2 .

At s=R:

(17)
k s 1 w = - D 11 3 w s 3 - D 12 + 4 D 66 1 s + a 2 3 w s θ 2 - 1 s + a 2 2 w θ 2 ,
(18)
K s 1 w s = - D 11 2 w s 2 - D 12 1 s + a w s + 1 s + a 2 w θ 2 .

At θ= 0:

(19)
k θ 0 w = D 22 1 s + a 3 2 w θ 3
            + D 12 + 4 D 66 1 s + a 3 w s 2 θ - 2 s + a 2 2 w s θ + 2 s + a 3 w θ ,
(20)
K θ 0 1 s + a w θ = D 22 1 s + a w s + 1 s + a 2 w θ 2 + D 12 2 w s 2 .

At θ=ϕ:

(21)
k θ 1 w = - D 22 1 s + a 3 2 w θ 3
            - D 12 + 4 D 66 1 s + a 3 w s 2 θ - 2 s + a 2 2 w s θ + 2 s + a 3 w θ ,
(22)
K θ 1 1 s + a w θ = - D 22 1 s + a w s + 1 s + a 2 w θ 2 - D 12 2 w s 2 .

2.3. Admissible displacement functions

Mathematically, it is often desired to express the displacement, wr,θ, 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 term-by-term differentiation. To overcome these problems, the displacement function will be here expressed as a more robust form of Fourier series expansion:

(23)
w ( s , θ , t ) = m = 0 n = 0 A m n c o s λ R m s c o s λ ϕ n θ + l = 1 4 ζ l ( θ ) m = 0 a m l c o s λ R m s
            + l = 1 4 χ l ( s ) n = 0 b n l c o s λ ϕ n θ e j ω 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 ζ-functions in the θ direction and χ-functions in the s direction in Eq. (23) are here chosen as:

(24)
ζ 1 θ = ϕ 2 π s i n π θ 2 ϕ + ϕ 2 π s i n 3 π θ 2 ϕ ,
(25)
ζ 2 θ = - ϕ 2 π c o s π θ 2 ϕ + ϕ 2 π c o s 3 π θ 2 ϕ ,
(26)
ζ 3 θ = ϕ 3 π 3 s i n π θ 2 ϕ - ϕ 3 2 π 3 s i n 3 π θ 2 ϕ ,
(27)
ζ 4 θ = ϕ 3 π 3 c o s π θ 2 ϕ - ϕ 3 2 π 3 c o s 3 π θ 2 ϕ ,
(28)
χ 1 s = R 2 π s i n π s 2 R + R 2 π s i n 3 π s 2 R ,
(29)
χ 2 s = - R 2 π c o s π s 2 R + R 2 π c o s 3 π s 2 R ,
(30)
χ 3 s = R   3 π 3 s i n π s 2 R - R   3 2 π 3 s i n 3 π s 2 R ,
(31)
χ 4 s = R   3 π 3 c o s π s 2 R - R   3 2 π 3 c o s 3 π s 2 R .

It is easy to verify that their first and third derivatives are mostly equal to zero along the boundary edges except for:

(32)
ζ 1 ' 0 = ζ 2 ' ϕ = ζ 3 ' ' ' 0 = ζ 4 ' ' ' ϕ = 1 ,
(33)
χ 1 ' 0 = χ 3 ' R = χ 3 ' ' ' 0 = χ 4 ' ' ' R = 1 .

It can be proven mathematically that the series expression in Eq. (23) is able to expand and uniformly converge to any function Θr,θC3 for x,yD:0,R×0,ϕ. Also, this series can be simply differentiated, through term-by-term, to obtain the uniformly convergent series expansions for up to the fourth-order derivatives. Mathematically, an exact displacement (or classical) solution is a particular function wr,θC3 for x,yD 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:

(34)
U p = 1 2 o R 0 ϕ D 11 2 w s 2 2 + D 22 1 s + a w s + 1 s + a 2 2 w θ 2 2 + 2 D 12 2 w s 2 1 s + a w s
            + 1 s + a 2 2 w θ 2 + 4 D 66 1 s + a 2 w s θ - 1 s + a 2 w θ s + a d s d θ .

By neglecting the rotary inertia, the kinetic energy of an orthotropic annular sector thin plate can be written as:

(35)
T = 1 2 ρ h ω 2 o R 0 ϕ w 2 ( s + a ) d s d θ .

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 (kw) and one group of rotational springs (Kr) to simulate the given or typical boundary conditions. Therefore, the deformation strain energy (Ubs) stored in the boundary springs during vibration can be defined as:

(36)
U b s = 1 2 0 ϕ a k s 0 w 2 + K s 0 w s 2 s = 0 + b k s 1 w 2 + K s 1 w s 2 s = R d θ
            + 1 2 0 R a k θ 0 w 2 + K θ 0 1 s + a w θ 2 θ = 0 + b k θ 1 w 2 + K θ 1 1 s + a w θ 2 θ = ϕ d s .

The Lagrangian functional (L) of the plates during vibration can be expressed in terms of the energy expressions:

(37)
L = T - U p - U b s .

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:

(38)
K - ω 2 M G = 0 ,

where:

(39)
G u = A 00 , A 01 , , A m ' 0 , A m ' 1 , , A m ' n ' , , A M N , a 0 1 , , a M 1 , a 0 4 , , a M 4 , b 0 1 , , b N 1 , b 0 4 , b N 4 .

In Eq. (38), the K is the stiffness matrix of the plate, and the 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 non-dimensional Ω=ωb2/ρ1h1/D111/2 is used in the presentation, and the material and geometry properties of orthotropic annular sector thin plates under consideration are: ρ= 7800 kg/m3, Eθ= 70 GPa, Er= 40Eθ, Grθ= 7.3 GPa, μr= 0.3, ϕ= 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 non-dimensional frequency parameters Ω of orthotropic annular sector thin plates are studied. A frequency parameter ΔΩ which is defined as the difference of the non-dimensional frequency parameter Ω to those of the elastic restraint parameters Гλλ=w,  rto 10-2, i.e., ΔΩ=ΩГλ-ΩГλ= 10-2 is used in the calculation. The plates under consideration are completely clamped at boundaries s= 0, s=R, and free at boundary θ= 0, while at edge θ=ϕ, the plates are elastically supported by only one group of spring component with stiffnesses varying from 10-2 to 1016. In Fig. 2(a), the influences of the line springs kw on frequency parameters are given. It is shown that the frequency parameter almost stays unchanged when the non-dimensional stiffness of the liner springs k2 is larger than 109 or smaller than 104. In Fig. 2(b), the influences of the rotation springs kw on frequency parameters are given. It is shown that the frequency curve changes greatly within the stiffness range from 103 to 108. 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:

(40)
C : k s 0 = 1 0 14   N/m ,       K s 0 = 1 0 14   N m / r a d ;   S : k s 0 = 1 0 14   N / m ,       K s 0 = 0   N m / r a d , F : k s 0 = 0   N / m ,       K s 0 = 0   N m / r a d ;       E 1 : k s 0 = 1 0 6   N / m ,       K s 0 = 0   N m / r a d , E 2 : k s 0 = 0   N / m ,       K s 0 = 1 0 6   N m / r a d ;       E 3 : k s 0 = 1 0 6   N / m ,       K s 0 = 1 0 6   N m / r a d .

The appropriateness of defining the classical boundary conditions in terms of boundary spring parameters will be proved by several examples in later sub-sections. 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, θ= 0, s=R and θ=ϕ, respectively.

3.2. Convergence study

In this sub-section, the convergence of orthotropic annular sector plates with different boundary conditions is studied. The first eight frequency parameters Ω for CCCC and FFFF orthotropic annular sector plates with different truncated number M and N (i.e. M=N= 8-14) 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 Ω 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 higher-order 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. 2. Variation of the frequency parameters Ω versus the elastic boundary restraint parameters for annular sector plate: a) transverse spring stiffness; b) rotation spring stiffness

 Variation of the frequency parameters Ω versus the elastic boundary restraint parameters  for annular sector plate: a) transverse spring stiffness; b) rotation spring stiffness
 Variation of the frequency parameters Ω versus the elastic boundary restraint parameters  for annular sector plate: a) transverse spring stiffness; b) rotation spring stiffness

Fig. 3. Variations of frequency parameter Ω with respect to truncated number M and N: a) FFFF; b) CCCC

 Variations of frequency parameter Ω with respect to truncated number M and N: a) FFFF; b) CCCC
 Variations 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 sub-section 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 Ω 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 classical-elastic 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 1. Convergence 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 2. Frequency 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 (E1/E2) and sector angle (ϕ) 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 Ω of orthotropic annular sector thin plates along the variations of the stiffness ratio with CCCC and E3E3E3E3 boundary conditions. From the Fig. 4, the frequency parameters Ω 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 Ω 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 3. Frequency 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 4. Frequency 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
CE1CE1
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
SE2SE2
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
FE3FE3
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
E2E1E2E1
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
E3E2E3E2
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
E1E3E1E3
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
E1E1E1E1
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
E2E2E2E2
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
E3E3E3E3
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. ri and θj represent the position of the ith and jth internal radial line and circumferential arc supports along the r- and θ- directions, respectively. The displacement fields in the position of the line support satisfy wri,θ,t= 0 and wr,θi,t= 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, 1014). Thus, the potential energy (Prals) stored in these springs is:

(41)
P r a l s = 1 2 0 ϕ i = 1 M i k r i i w r i , θ , t 2 ( r i + a ) d θ + 1 2 0 R j = 1 N j k θ j j w r , θ j , t 2 d s ,

where Mi and Nj are the amount of circumferential arc supports and radial line in the θ and r directions. krii and kθjj denote the corresponding circumferential arc supported and radial line springs distributed at r=ri and θ=θj. By adding the potential energy Prals 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. 4. Variation of the frequency parameters Ω versus the stiffness rations (E1/E2) for annular sector plate: a) CCCC; b) E3E3E3E3

 Variation of the frequency parameters Ω versus the stiffness rations (E1/E2)  for annular sector plate: a) CCCC; b) E3E3E3E3
 Variation of the frequency parameters Ω versus the stiffness rations (E1/E2)  for annular sector plate: a) CCCC; b) E3E3E3E3

Fig. 5. Variation of the frequency parameters Ω versus the sector angle for annular sector plate: a) CCCC; b) E3E3E3E3

 Variation of the frequency parameters Ω versus the sector angle  for annular sector plate: a) CCCC; b) E3E3E3E3
 Variation 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 Ω 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 θ 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 Ω of the considered sector plate with against the radial line support location parameter θ1/ϕ and against the circumferential arc support location parameter R1/R are depicted. Six types of edge conditions used in the investigation are: C-C, S-S, F-C, C-F, F-S, S-F. 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. 6. Schematic diagram of an annular sector plate with arbitrary internal radial line and circumferential supports

 Schematic diagram of an annular sector plate with arbitrary internal radial line  and circumferential supports

Table 5. Comparison 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
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
R 2 ,   ϕ 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 Ω 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 CE1CE1 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. 7. Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions

 Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions
 Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions
 Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions
 Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions
 Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions
 Variation of the frequency parameters Ω versus the radial support locations and arc support locations for annular sector plate with different boundary conditions

Fig. 8. The 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

 The 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 Fourier-Ritz 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 6. The 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
CE1CE1
SE2SE2
FE3FE3
E2E1E2E1
E3E2E3E2
E1E3E1E3
E1E1E1E1
E2E2E2E2
E3E3E3E3
r 1 = 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 = R 3 ,  
r 2 = 2 R 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 = R 2 ,
θ 1 = ϕ 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   = R 3 ,
θ 1 = ϕ 2 ,
r 2 = 2 R 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 real-world 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.

Acknowledgements

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).

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