Abstract
In this paper, an improved Fourier series method is presented for vibration analysis of moderately thick annular and circular sector plates subjected to general elastic boundary conditions along its edges. In literature, annular and circular sector plates subjected to classical boundary conditions have been studied in detail however in practical engineering applications the boundary conditions are not always classical in nature. Therefore, study of vibration response of these plates subjected to general elastic boundary conditions is far needed. In the method presented, artificial boundary spring technique has been employed to simulate the general elastic boundary conditions and first order shear deformation theory has been employed to formulate the theoretical model. Irrespective of the boundary conditions, each of the displacement function is expressed as a new form of trigonometric expansion with accelerated convergence. RayleighRitz method has been employed to determine the expansion coefficients. Unlike most of the studies on vibration analysis of moderately thick annular sector plates, the present method can be universally applied to a wide range of vibration problems involving different boundary conditions, varying material and geometric properties without modifying the solution algorithms and procedure. The effectiveness, reliability and accuracy of the present method is fully demonstrated and verified by several numerical examples. Bench mark solutions for moderately thick annular sector and circular plates under general elastic boundary conditions are also presented for future computational methods.
1. Introduction
Annular, circular and their sector parts are key structural components that are used in various engineering fields like civil, marine, aerospace and mechanical engineering. Due to different geometrical shapes of these structures, they have been analyzed separately using different solution techniques. Most of the initial research on these plates was done using Classical plate theory (CPT) in which the shear deformation and rotary inertia was neglected which in turn limited its application on moderately thick and thick plates. Later a lot of theories were proposed incorporating the shear deformation and rotary inertia which resulted in an increase in accuracy of the results for moderately thick and thick annular and circular plates. These theories’ have been well explained in Leissa’s book on vibration of plates.
Different methods have been employed by various researchers to study the vibration characteristics of annular and circular plates subjected to different boundary conditions. However, a few prominent studies related to these plates are highlighted here in this manuscript. Employing the Mindlin plate theory on thick sector plates, Guruswamy et al. [1] studied the dynamic response of these plates by proposing a sector finite element. Fully clamped boundary conditions were employed on all edges. Taking the effect of shear deformation in thickness direction, another study was performed by Soni et al. [2] on axisymmetric non uniform circular disks. In their research they employed Chebyshev collocation technique to study the vibration characteristics of these plates. In another study RayleighRitz method was employed by Liew et al. on circular plates with multiple internal ring supports. Later in another study they studied the characteristics of these plates subjected to inplane pressure [34].
Using three dimensional finite strip model, thick and thin sector plates subjected to various combinations of classical boundary condition were analyzed by Cheung et al. [5]. The integral equation technique and finite strip method was employed by Sirinivasan et al. and Misuzawa et al. respectively to study the vibration characteristics of Mindlin annular sector plates [67]. Employing the Mindlin plate theory and differential quadrature method, Liu et al. studied the effect of sector angle and thickness to radius ratio on the vibration characteristics of moderately thick sector plates [8]. Later they extended the same differential quadrature method to annular sector plates having shear deformation subjected to different combinations of classical boundary conditions [9]. In another study, the same differential quadrature method was employed on solid circular plates with variable thickness in radial direction and subjected to elastic boundary conditions by Wu et al. [10]. Similarly, Xiang et al. employed domain decomposition technique and studied the vibration response of circular plates with stepped thickness variation [11].
Using RayleighRitz method So et al. [12] studied the vibration characteristics of annular and circular plates by employing three dimensional elasticity theory. A similar three dimensional study was performed by Hashemi et al. on annular sector plates resting on elastic foundations. He employed polynomialRitz approach and studied the vibration characteristics for different sets of classical boundary conditions [13]. In another study similar polynomialRitz model was presented by Liew et al. to investigate the effect of boundary conditions and thickness on the vibration characteristics of annular plates [14]. In another prominent three dimensional study ChebyshevRitz technique was employed on circular and annular plates by Zhou et al. to study the vibration characteristics of these plates [15]. In plane vibrations of circular plates subjected to different boundary conditions were investigated by various researchers using different solution techniques [1617]
From the studies mentioned above it can be seen that most of the previous studies on annular and circular plates were limited to classical boundary conditions which includes free, simply supported, clamped or combination of these. However, in practical engineering applications the boundary conditions are not always classical in nature. Therefore, the development of an analytical method universally dealing with arbitrary boundary conditions was much needed. An improved Fourier series method was developed for vibration analysis of beams and plates by Li [1822]. Later Xianjie Shi et al. extended this method to thin annular plates to study its vibration characteristics [2326]. The main objective of this study is to realize and extend the same generalized Fourier series method to study the vibration analysis of Mindlin annular sector and circular sector plates under various boundary conditions including the general elastic restraints.
2. Theoretical formulation
2.1. Model description
Consider an annular sector plate of constant thickness $h$, inner radius $a$, outer radius $b$ and width $R$ in radial direction as shown in Fig. 1. The plate geometry and dimensions are defined in the cylindrical coordinate system $(r,\varphi ,z)$. A local coordinate system $(s,\varphi ,z)$ is also shown in Fig. 1. The radial and thickness coordinates $s$ and $z$ are measured normally from the inner edge and mid plane of the annular sector plate respectively whereas $\varphi $ is the circumferential angle. Four sets of distributed springs (one translational and two rotational) of arbitrary stiffness values are attached at each edge to simulate arbitrary boundary conditions. All the classical sets of boundary conditions can easily be achieved by varying the stiffness value of each spring from zero to an infinitely large number i.e. 10^{14}.
The domain of the annular sector plate can be defined as:
The relationship between local and global coordinate system can be expressed as:
The material of the plate is assumed to be isotropic with material density $\rho $, young’s modulus $E$ and Poisson ratio $\nu $. It should be noted that a circular sector plate can be defined as a special case of an annular sector plate if the inner radius ‘$a$’ is set either equal to 0 or to a very small number say 0.00001.
The displacement field of Mindlin annular sector plate in cylindrical coordinates is given by:
${u}_{\varphi}\left(r,\varphi ,z,t\right)={u}_{\varphi}\left(r,\varphi ,z\right)+z{\theta}_{\varphi}\left(r,\varphi ,t\right),$
$w\left(r,\varphi ,z,t\right)={w}_{o}\left(r,\varphi ,t\right),$
where $z$ is the thickness coordinate, ${u}_{r}$ and ${u}_{\varphi}$ are displacements of the mid plane in $r$ and $\varphi $ directions, respectively, $w$ is the transverse displacement. ${\theta}_{r}$ and ${\theta}_{\varphi}$ are the rotation functions of the middle surface and $t$ is the time. Assuming the plain stress distribution in accordance with Hooks law, the stress resultants are obtained for Mindlin annular plate by integrating the stresses as shown below:
${M}_{\varphi \varphi}=\underset{h/2}{\overset{h/2}{\int}}{\sigma}_{\varphi \varphi}zdz=\underset{h/2}{\overset{h/2}{\int}}\frac{E}{1{\nu}^{2}}\left({\epsilon}_{\varphi \varphi}+\nu {\epsilon}_{rr}\right)zdz=D\left[\frac{1}{r}\left({\theta}_{r}+\frac{\partial {\theta}_{\varphi}}{\partial \varphi}\right)+\nu \left(\frac{\partial {\theta}_{r}}{\partial r}\right)\right],$
${M}_{r\varphi}=\underset{h/2}{\overset{h/2}{\int}}{\tau}_{r\varphi}zdz=\underset{h/2}{\overset{h/2}{\int}}G{\gamma}_{r\varphi}zdz=D\left(\frac{1\nu}{2}\right)\left[\frac{1}{r}\left(\frac{\partial {\theta}_{r}}{\partial \varphi}{\theta}_{\varphi}\right)+\frac{\partial {\theta}_{\varphi}}{\partial r}\right],$
${Q}_{rr}={K}^{2}\underset{h/2}{\overset{h/2}{\int}}{\tau}_{rz}dz={K}^{2}Gh\left[{\theta}_{r}+\frac{\partial {w}_{o}}{\partial r}\right],$
${Q}_{\varphi \varphi}={K}^{2}\underset{h/2}{\overset{h/2}{\int}}{\tau}_{\varphi z}dz={K}^{2}Gh\left[{\theta}_{\varphi}+\frac{1}{r}\frac{\partial {w}_{o}}{\partial \varphi}\right],$
where ${M}_{rr}$, ${M}_{\varphi \varphi}$ and ${M}_{r\varphi}$ are the bending moments per unit length of the plate, ${Q}_{rr}$ and ${Q}_{\varphi \varphi}$ are the transverse shear forces per unit length of the plate, ${\sigma}_{rr}$, ${\sigma}_{\varphi \varphi}$ are the normal stresses, and ${\tau}_{r\varphi}$, ${\tau}_{rz}$ and ${\tau}_{rz}$ are the shear stresses, $h$ is the plate thickness, $E$ is the modulus of elasticity, $G$ is the shear modulus, $\nu $ is the Poisson ratio, $D=E{h}^{3}/12\left(1{\nu}^{2}\right)$ is the flexural rigidity and ${K}^{}={\pi}^{2}/12$ is the shear correction factor to compensate for the error in assuming the constant shear stress throughout the plate thickness.
The equation of motion of Mindlin plates in $(r,\varphi ,z)$ is given by:
$\frac{\partial {M}_{r\varphi}}{\partial r}+\frac{1}{r}\frac{\partial {M}_{\varphi}}{\partial \varphi}+\frac{2}{r}{M}_{r\varphi}{Q}_{\varphi \varphi}=\frac{\rho {h}^{3}}{12}\left(\frac{{\partial}^{2}{\theta}_{\varphi}}{\partial {t}^{2}}\right),$
$\frac{\partial {Q}_{rr}}{\partial r}+\frac{1}{r}\frac{\partial {Q}_{\varphi \varphi}}{\partial \varphi}+\frac{{Q}_{rr}}{r}=\rho h\frac{{\partial}^{2}{w}_{o}}{\partial {t}^{2}}.$
Fig. 1Mindlin sector plate geometry
2.2. Solution scheme
2.2.1. Selection of admissible displacement function
Assume that the displacement field of Mindlin annular sector plate in local coordinate system $(s,\varphi ,z)$ is defined by the following series:
${\theta}_{\varphi (s,\varphi )}=\sum _{m=n=2}^{\mathrm{\infty}}{B}_{mn}{\phi}_{m}\left(s\right){\phi}_{n}\left(\varphi \right),$
${w}_{o(s,\varphi )}=\sum _{m=n=2}^{\mathrm{\infty}}{C}_{mn}{\phi}_{m}\left(s\right){\phi}_{n}\left(\varphi \right),$
where:
and ${\lambda}_{m}=m\pi /R\text{,}$${\lambda}_{n}=n\pi /\alpha $ and ${A}_{mn}\text{,}$${B}_{mn}\text{,}$${C}_{mn}$ denotes the Fourier series expansion coefficients. The sine terms in the equations Eq. (6) are introduced to overcome the potential discontinuities (convergence problem) of the displacement function, along the edges of the plate, when it is periodically extended and sought in the form of trigonometric series expansion. The addition of these auxiliary functions in the admissible functions plays an important role in the convergence and accuracy of the present method or in other words the elimination of potential discontinuities at the ends or elimination of Gibbs effect.
In order to illustrate this, take a beam problem for example. The governing equations for free vibration of a general supported Euler beam is obtained as:
where $D$, $\rho $ and $A$ are, respectively, the flexural rigidity, the mass density and the cross sectional area of the beam, and $\omega $ is frequency in radian. From Eq. (7) it can be observed that the displacement solution $w\left(x\right)$ on a beam of length $L$ is required to have up to the fourth derivatives, that is, $w\left(x\right)\in {C}^{3}$. In general, the displacement function $w\left(x\right)$ defined over a domain [0, $L$] can be expanded into a Fourier series inside the domain excluding the boundary points:
where ${A}_{m}$ are the expansion coefficients. From the Eq. (8), we can see that the displacement function $w\left(x\right)$ can be viewed as a part of an even function defined over $\left[L,L\right]$, as shown in Fig. 2.
Fig. 2An illustration of the possible discontinuities of the displacement at the end points
Fig. 3An illustration of removal of possible discontinuities (convergence problem) at ends
Thus, the Fourier cosine series is able to correctly converge to $w\left(x\right)$ at any point over [0, $L$]. However, its firstderivative $w\mathrm{\text{'}}\left(x\right)$ is an odd function over $\left[L,L\right]$ leading to a jump at end locations. The corresponding Fourier expansion of $w\mathrm{\text{'}}\left(x\right)$ continue on [0, $L$] and can be differentiated termbyterm only if $w\left(0\right)=w\left(L\right)=0$. Thus, its Fourier series expansion (sine series) will accordingly have a convergence problem due to the discontinuity at end points (Gibbs phenomena) when $w\mathrm{\text{'}}\left(x\right)$ is required to have up to the firstderivative continuity.
To overcome this problem, this Improved Fourier Series technique was proposed by Li [18, 19]. In this technique a new function $P\left(x\right)$ is considered in the displacement function:
where the auxiliary function $P\left(x\right)$ in equation above represents an arbitrary continuous function that, regardless of boundary conditions, is always chosen to satisfy the following equations:
The actual values of the first and third derivatives (a sine series) at the boundaries need to be determined from the given boundary conditions. Essentially, $\stackrel{}{w}\left(x\right)$ represents a residual beam function which is continuous over [0, $L$] and has zero slopes at the both ends, as shown in Fig. 3. Apparently, the cosine series representation of $\stackrel{}{w}\left(x\right)$ is able to converge correctly to the function itself and its first derivative at every point on the beam.
Thus, based on the above analysis, $P\left(x\right)$ can be understood as a continuous function that satisfies Eq. (9), and its form is not a concern but must be a closedform and sufficiently smooth over a domain [0, $L$] of the beam in order to meet the requirements provided by the continuity conditions and boundary constraints. Furthermore, it is noticeable that the auxiliary function $P\left(x\right)$ can improve the convergent properties of Fourier series.
2.2.2. Determining the expansion coefficients
Once the proper admissible function for the displacement field is selected Eq. (6), the next step is to find the expansion coefficients in the assumed displacement field. In order to do so RayleighRitz method is employed which is an energybased method. To employ this method, it is necessary to state the potential and kinetic energies first in terms of displacement fields. The expression for the potential energy of the sector plate in local coordinates $(s,\varphi ,z)$ is derived from the constitutive laws and straindisplacement relations. According to the Mindlin plate theory. The strain energy of the annular sector plates can be expressed as:
the kinetic energy expression for annular sector plate is expressed as:
The potential energy stored in the boundary springs is given by:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\frac{1}{2}\underset{0}{\overset{R}{\int}}\left[{\left[{k}_{0}^{}{w}_{o}^{2}+{K}_{0}^{r}{{\theta}_{s}}^{2}+{K}_{0}^{t}{{\theta}_{\varphi}}^{2}\right]}_{\varphi =0}+{\left[{k}_{\alpha}^{}{w}_{o}^{2}+{K}_{\alpha}^{r}{{\theta}_{s}}^{2}+{K}_{\alpha}^{t}{{\theta}_{\varphi}}^{2}\right]}_{\varphi =\alpha}\right]ds,$
where ${k}_{a}^{}$, ${k}_{b}^{}$ (${k}_{0}^{}$ and ${k}_{\alpha}^{}$) are linear spring constants, ${K}_{a}^{r}$, ${K}_{b}^{r}$ (${K}_{0}^{r}$ and ${K}_{\alpha}^{r}$) are rotational spring constants in radial direction, ${K}_{a}^{t}$, ${K}_{b}^{t}$ (${K}_{0}^{t}$ and ${K}_{\alpha}^{t}$) are rotational spring constants in tangential direction at edges $s=$0 and $s=R$ and $\varphi =$ 0 and $\varphi =\alpha $ respectively. All the classical homogeneous boundary conditions can be simply considered as special cases when the spring constants are either extremely large or substantially small. The units for the translational and rotational springs are N/m and Nm/rad, respectively.
After the potential and kinetic energies are expressed, then all the assumed displacement functions are inserted in the potential and kinetic energy equations and these equations are then further minimized with respect to the expansion coefficients in the displacement field. Mathematically, the Lagrangian for the annular sector plate can be generally expressed as:
where ${U}_{p}$ is strain energy of the plate, ${U}_{sp}$ is strain energy stored in the boundary springs and ${T}_{p}$ is the kinetic energy of the plate. Substituting Eq. (6) in Eqs. (11)(13) and then minimizing Lagrangian Eq. (14) against all the unknown series expansion coefficients that is:
we can obtain a series of linear algebraic expressions in a matrix form as:
where $E$ is a vector which contains all the unknown series expansion coefficients and $K$ and $M$ are the stiffness and mass matrices, respectively. $E$, $K$ and $M$ can be expressed as:
where the subscripts $i$, $j$ and $k$ represents $w$, ${\theta}_{s}$ and ${\theta}_{\varphi}$ and the superscripts $p$ and $sp$ represents plate and boundary springs respectively. For conciseness, the detailed expressions for the stiffness and mass matrices are not shown here.
2.2.3. Determining the eigen values and eigen vectors
Once Eq. (16) is established, the eigenvalues (or natural frequencies) and eigenvectors of Mindlin annular sector plates can now be easily and directly determined from solving a standard matrix eigenvalue problem i.e. Eq. (16) using MATLAB. For a given natural frequency, the corresponding eigenvector actually contains the series expansion coefficients which can be used to construct the physical mode shape based on Eq. (6). Although this investigation is focused on the free vibration of Mindlin annular sector plate, the response of the annular sector plate to an applied load can be easily considered by simply including the work done by this load in the Lagrangian, eventually leading to a force term on the right side of Eq. (16).
3. Results and discussion
To check the accuracy and usefulness of the proposed technique, several numerical examples are presented in this section. It is important to mention here that the accuracy of the proposed method is greatly controlled by the number of truncation terms i.e. $M=N$, the more number of truncation terms we use we get more accurate results however the computational cost and time will increase with increasing number of truncation terms. Theoretically, there are infinite terms in the assumed displacement functions. However, the series is numerically truncated and finite terms are counted in actual calculations which will be further explained in the text to follow. Moreover, in identifying the boundary conditions in this section, letters C, S, and F have been used to indicate the clamped, simply supported and free boundary condition along an edge, respectively. Therefore, the boundary conditions for a plate are fully specified by using four alphabets with the first one indicating the B.C. along the first edge, $r=a$. The remaining (the second to the fourth) edges are ordered in the counterclockwise direction.
First of all, in order to check the accuracy and usefulness we first consider a fully clamped Mindlin annular sector plate. Fully clamped (CCCC) boundary conditions can easily be achieved by setting the stiffnesses of the restraining springs to an infinitely large number (10^{14}) in the numerical calculations. The first six nondimensional frequency parameter, $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$ are tabulated in Table 1 along with the reference results from [9] and [27].
Table 1First six non dimensional frequency parameter Ω=ωb2ρh/D1/2 for fully clamped (CCCC) Mindlin Annular sector plates (ϕ=2π/3, a/b= 0.25, h/b= 0.2)
$M=N$  Mode sequence  
1  2  3  4  5  6  
2  31.481  42.907  62.894  66.199  73.800  94.776 
4  31.084  41.872  56.108  62.435  72.906  75.775 
6  31.059  41.823  55.972  62.411  71.269  72.862 
8  31.054  41.813  55.951  62.406  71.148  72.852 
10  31.053  41.810  55.946  62.405  71.125  72.849 
12  31.053  41.809  55.943  62.404  71.118  72.847 
14  31.053  41.809  55.942  62.404  71.116  72.847 
Ref. [9]  31.056  41.814  55.951  62.420  71.127  72.862 
Ref. [27]  31.057  41.814  55.951  62.420  71.127  72.862 
Similarly, in Table 2, first six nondimensional frequency parameter for Mindlin annular sector plate having simply supported radial edges and clamped circumferential edges (CSCS) boundary conditions has been given along with the reference results from [9, 27]. A good agreement in the present values and reference values can be observed.
Table 2First six non dimensional frequency parameter Ω=ωb2ρh/D1/2 for Mindlin Annular sector plates (ϕ=π/3, a/b= 0.5, h/b= 0.1) having simply supported radial edges and clamped circumferential edges (CSCS)
$M=N$  Mode sequence  
1  2  3  4  5  6  
2  77.625  104.054  161.254  170.932  194.112  243.145 
4  76.567  102.955  149.795  166.952  190.467  215.603 
6  76.476  102.765  149.410  166.772  190.093  206.703 
8  76.449  102.696  149.323  166.726  189.961  206.315 
10  76.439  102.667  149.290  166.710  189.908  206.228 
12  76.435  102.654  149.275  166.703  189.883  206.196 
14  76.432  102.647  149.267  166.699  189.871  206.182 
Ref. [9]  76.902  103.682  150.413  167.327  191.593  207.276 
Ref. [27]  76.902  103.682  150.413  167.327  191.593  207.276 
Next we consider annular sector plate simply supported at radial edges and having different combination of boundary conditions (freeclamped, freesimply supported, simply supportedsimply supported, simply supportedfree and clampedfree) at the circumferential edges. The simply supported condition is simply produced by setting the stiffnesses of the translational and rotational springs to $\infty $ and 0, respectively, and the free edge condition by setting both stiffnesses to zero. The fundamental frequency parameters with different boundary conditions are shown in Table 3. The current results agree well with those taken from references [28, 29].
Next to illustrate the convergence and numerical stability of the current solution procedure, several sets of results for fully clamped Mindlin annular sector plates having different sector angles and using different truncation numbers ($M=N=$2, 4, 6, 8, 10, 12, 14) are presented in Tables 47. Furthermore, the fast convergence pattern can also be observed in Fig. 4.
Table 3Fundamental frequency parameter Ω=ωb2ρh/D1/2 for Mindlin Annular sector plates having simply supported radial edges and different boundary conditions at circumferential edges (a/b= 0.5)
Sector angle ($\varphi $)  Thickness to radius ratio ($h/b$)  Method  Boundary conditions at circumferential edges  
FC  FS  SS  SF  CF  
195  0.1  Present  19.998  10.224  38.365  4.560  12.696 
Ref. [28]  19.999  10.227  38.636  4.675  12.680  
Ref. [29]  20.097  10.239  38.764  –  –  
0.2  Present  17.503  9.130  32.508  4.005  11.413  
Ref. [28]  17.582  9.366  32.871  4.542  11.427  
Ref. [29]  17.764  9.396  33.190  –  –  
210  0.1  Present  19.620  9.685  38.222  4.507  12.678 
Ref. [28]  19.610  9.664  38.455  4.584  12.659  
Ref. [29]  19.706  9.675  38.582  –  –  
0.2  Present  17.235  8.681  32.419  3.997  11.417  
Ref. [28]  17.294  8.877  32.734  4.458  11.425  
Ref. [29]  17.294  8.904  33.050  –  –  
270  0.1  Present  18.654  8.213  37.868  4.392  12.639 
Ref. [28]  18.622  8.130  38.010  4.372  12.615  
Ref. [29]  18.715  8.139  38.134  –  –  
0..2  Present  16.548  7.450  32.200  3.999  11.433  
Ref. [28]  16.566  7.546  32.394  4.263  11.430  
Ref. [29]  16.739  7.567  32.704  –  – 
A fast convergence pattern can be observed in the tabulated results as well as Fig. 4, therefore it can be concluded that sufficiently accurate results can be obtained with a small number of terms in the series expansion and the solution is consistently refined as more and more terms are included in the series expansion.
Table 4First five nondimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)
$M=N$  Sector angle  Mode sequence  
1  2  3  4  5  
2  $\varphi =\frac{\pi}{6}$  145.584  240.451  251.723  331.721  391.843 
4  144.104  237.420  249.040  328.660  350.364  
6  144.032  237.301  248.941  328.487  349.816  
8  144.020  237.280  248.924  328.452  349.739  
10  144.017  237.274  248.920  328.442  349.720  
12  144.016  237.272  248.918  328.438  349.713  
14  144.015  237.271  248.917  328.436  349.711 
Table 5First five nondimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)
$M=N$  Sector angle  Mode number  
1  2  3  4  5  
2  $\varphi =\frac{\pi}{2}$  104.250  116.563  163.387  223.325  232.424 
4  102.977  112.649  130.467  173.504  220.564  
6  102.911  112.460  129.978  155.396  187.319  
8  102.900  112.420  129.896  154.985  185.972  
10  102.897  112.408  129.871  154.895  185.781  
12  102.896  112.403  129.862  154.865  185.726  
14  102.895  112.401  129.857  154.852  185.705 
Table 6First five nondimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)
$M=N$  Sector angle  Mode number  
1  2  3  4  5  
2  $\varphi =\frac{2\pi}{3}$  102.797  109.453  139.708  221.999  227.027 
4  101.566  106.400  115.457  143.396  194.321  
6  101.503  106.239  115.077  128.766  147.504  
8  101.492  106.203  115.006  128.447  146.310  
10  101.489  106.192  114.984  128.368  146.140  
12  101.488  106.188  114.975  128.339  146.089  
14  101.488  106.186  114.970  128.327  146.068 
Table 7First five nondimensional frequency parameters Ω=ωb2ρh/D1/2 for CCCC Mindlin annular sector plates (a/b= 0.6, h/b= 0.1)
$M=N$  Sector angle  Mode sequence  
1  2  3  4  5  
2  $\varphi =\frac{7\pi}{6}$  101.717  103.774  115.254  220.887  222.496 
4  100.537  101.885  104.287  114.756  138.302  
6  100.480  101.779  104.088  107.716  113.103  
8  100.470  101.753  104.042  107.560  112.462  
10  100.467  101.744  104.026  107.512  112.360  
12  100.467  101.741  104.019  107.492  112.325  
14  100.466  101.739  104.015  107.482  112.308 
From Tables 47 it can be seen that when the truncated numbers change from $M\times N=$ 10×10 to 12×12, the maximum difference of the frequency parameters does not exceed 0.003 for the worst case, which is acceptable. Furthermore, in modal analysis the natural frequencies for higher order modes tend to converge slower as compared to the lower order modes which can easily be observed in Fig. 4 that the 9th mode frequency converges slowly as compared to the 6th and 3rd mode. Thus a suitable truncation number should be used to achieve the accuracy of the largest desired natural frequency. In view of above and excellent numerical behavior of the current solution, the truncation number for all subsequent calculations in the present method is taken as $M=N=$12.
Next we study the effect of sector angle and thicknessradius ratio and cutout ratio on nondimensional frequency parameter. The effect has been graphically represented in Figs. 57, respectively.
Fig. 4Convergence pattern of frequency parameters with no. of terms (M=N)
Fig. 5Effect of sector angle on nondimensional frequency parameter ‘Ω’
Fig. 6Effect of thickness to radius ratio (h/b) on the frequency parameter
Fig. 7Effect of cutout ratio (a/b) on the frequency parameters
It can be seen in Fig. 5 that for smaller sector angles i.e. $\varphi \le 2\pi /3$, the decrease in the frequency parameters is more as compared to the sector angles greater than $2\pi /3$. Similarly, in figure 6, 1st, 3rd and 5th mode frequency parameters have been plotted against different thickness to radius ratios ($h/b$) for a fully clamped (CCCC) Mindlin annular sector plate having sector angle $=2\pi /3$ and cutout ratio $=a/b=$ 0.6. It can be observed that with the increase in thickness to radius ratio the frequency parameter always decreases. Similarly, the effect of cutout ratio i.e. inner radius to outer radius ($a/b$) on the frequency parameters for a FSFS mindlin annular sector plate having sector angle $=\pi /3$, and $h/b=$ 0.2 can be seen in Fig. 7.
As mentioned previously, using this method, a Mindlin circular sector plate can also be analyzed easily just by equating the inner radius of Mindlin annular sector plate to zero without modifying the equations or the solution algorithm. Table 8 shows first six nondimensional frequency parameter along with reference results for Mindlin circular sector plates having inner radius $=a=$0.0001, different thickness to radius ratio and sector angles and subjected to simply supported radial edges and clamped circular edge (SCS) boundary condition respectively. A close agreement can be observed in the present values and the reference results.
Table 8First six nondimensional frequency parameters for SCS Mindlin Circular sector plates (a/b= 0.0001)
Sector angle ($\varphi $)  Thickness to radius ratio $(h/b$)  Method  Mode sequence  
1  2  3  4  5  6  
$\frac{\pi}{6}$  0.1  Present  91.419  148.357  205.117  208.046  278.356  281.097 
Ref. [8]  93.450  152.630  206.900  213.080  274.650  283.190  
0.2  Present  66.272  98.992  131.454  131.981  161.648  163.749  
Ref. [8]  67.933  102.560  132.860  135.610  165.680  167.820  
$\frac{\pi}{2}$  0.1  Present  31.657  60.112  71.009  92.956  110.151  118.760 
Ref. [8]  32.205  60.637  72.221  93.450  111.320  120.450  
0.2  Present  26.298  46.323  53.227  67.436  77.370  82.512  
Ref. [8]  26.993  46.906  54.466  67.933  78.582  83.903  
$\pi $  0.1  Present  20.778  31.969  45.586  54.582  60.614  71.698 
Ref. [8]  20.223  32.205  45.773  53.859  60.637  72.221  
0..2  Present  17.752  26.630  36.470  42.169  46.726  53.813  
Ref. [8]  17.773  26.993  36.758  42.393  46.906  54.466  
$\frac{7\pi}{6}$  0.1  Present  19.915  28.398  39.635  51.986  52.898  65.557 
Ref. [8]  19.489  28.599  39.786  51.998  52.416  65.091  
0.2  Present  16.871  23.958  32.269  40.788  40.916  49.836  
Ref. [8]  17.092  24.293  32.519  41.067  41.195  49.859 
All the examples mentioned above are limited to different combinations of classical boundary conditions which are viewed as special case of elastically restrained edges. After verifying the convergence, accuracy and effectiveness of the proposed method for different combinations of classical boundary conditions, the method is further employed here to study the vibration characteristics of Mindlin annular sector and circular sector plates subjected to general elastic boundary conditions.
In order to simulate the elastic boundary conditions, it is important to study the effect of restraining springs first on the frequency parameters so that proper value to the restraining springs could be assigned. Fig. 8 shows the effect of restraining springs stiffness on the frequency parameter for Mindlin annular sector plate ($a/b=$ 0.6, $h/b=$ 0.2 and $\varphi =$120).
Fig. 8 shows 1st, 5th and 10th mode frequency parameters plotted against the spring stiffnesses by varying the stiffnesses of one group of boundary spring from 0 to 10^{16} while keeping the stiffnesses of the other group equal to infinite i.e 10^{16}. It can be seen in Fig. 8(a) that the frequency parameter almost remains at a level when the stiffness of the translational spring in $z$ direction is less than 10^{8} and greater than 10^{12} where as other than this range the frequency parameter increases with increasing stiffness values. Similar phenomena can be observed in case of rotational spring stiffness however a slight change in frequency parameter can be observed with in the stiffness range from 10^{8} to 10^{10}. Based on the analysis it can be concluded that stable frequency parameter can be obtained when the stiffnesses for all the restraining springs is more than 10^{12} or less than 10^{8} and also it is suitable and valid to use the stiffness value 10^{14} to simulate the infinite stiffness in the numerical calculations since the frequency parameter remain at the same level for values greater than equal to 10^{14}.It can also be concluded that the elastic stiffness range for translational spring is more than the two rotational springs.
Fig. 8a) Effect of translational spring stiffness (k) on Ω, and effect of rotational spring stiffness b) in tangential direction (Kt) on Ω, c) in radial direction (Kr) on Ω
a)
b)
c)
From Fig. 8, an elastic boundary condition can easily be defined with any stiffness value between 10^{8} to 10^{12}. To the author’s best knowledge, no reported results are available in literature for vibration analysis of Mindlin annular sector plates under general elastic boundary conditions. As mentioned earlier the present method can be used to obtain natural frequency parameters for Mindlin annular sector plates under general elastic boundary condition regardless of modifying solution algorithm and procedure.
In order to achieve valuable results for annular sector plates subjected to elastic boundary conditions, we define an elastic restraint ‘E^{1}’^{}having corresponding translational and rotational spring stiffness values $k=$1e8, ${K}_{r}=$0 and ${K}_{t}=$0. Tables 911 shows first five natural frequency parameters for Mindlin annular sector plates having different sector angles and thicknessradius ratio ($h/b$) subjected to E^{1}E^{1}E^{1}E^{1}, FE^{1 }FE^{1} and CE^{1}CE^{1 }boundary conditions where E^{1}E^{1}E^{1}E^{1}, FE^{1 }FE^{1} and CE^{1}CE^{1 }represents the combination of classic and elastic boundary conditions at edges $a$ and $b$ as well as $\varphi =$ 0 and $\alpha $.
Due to unavailability of results in previous literature for these types of boundary conditions (E^{1}E^{1}E^{1}E^{1}, FE^{1 }FE^{1} and CE^{1}CE^{1}), the comparison has been made with results obtained from ABAQUS. However, we know that the core algorithm of the ABAQUS software is based on the finite element method. Furthermore, the FEM computational accuracy strongly depends on the size of the mesh and the type of element selection. For more accuracy in the higher frequency region and for complex geometries, a highly refined mesh and a higher order finite element is needed. We know that the smaller the mesh size, the greater the number of elements we get for analysis which further requires more computer memory and subsequently a high computational cost. Since the geometry under investigation in this manuscript is a simpler geometry therefore a simple free meshing technique with mesh size 0.005 and S4R (Shell 4 node Reduced Integration) element type has been used which is computationally inexpensive and is considered suitable for this type of geometry and modal analysis. Keeping the mesh size 0.005, the number of elements used in the analysis for annular sector plate having sector angle $\pi $/3, $\pi $/2, 2$\pi $/3 and $\pi $ are 13040, 20080, 29727 and 40240 respectively.
A very good agreement can be observed in Tables 911 between the calculated results and the one obtained from ABAQUS. This shows that the present method can be easily applied to classical and elastic type boundary conditions as well as their combination without modifying the solution algorithm and procedure. The results tabulated in Tables 911 can be used as a bench mark for future computational methods.
Table 9First five natural frequency parameters for Mindlin Annular sector plates subjected to E1E1E1E1 type elastic boundary conditions (a/b= 0.6)
Sector angle ($\varphi $)  Thickness to radius ratio ($h/b$)  Method  Natural frequency modes  
1  2  3  4  5  
$\frac{\pi}{3}$  0.1  Present  337.934  435.140  506.867  1520.326  1966.903 
ABAQUS  337.690  437.350  507.130  1522.500  1969.600  
0.2  Present  241.050  302.391  330.646  2587.184  3074.622  
ABAQUS  240.970  303.960  330.420  2590.100  3080.200  
$\frac{\pi}{2}$  0.1  Present  314.276  389.282  479.851  870.662  1294.086 
ABAQUS  313.770  390.700  480.060  872.380  1295.400  
0.2  Present  225.655  274.476  318.276  1411.322  1991.271  
ABAQUS  225.400  275.570  318.090  1412.500  1994.500  
$\frac{2\pi}{3}$  0.1  Present  300.548  361.101  440.420  703.848  884.878 
ABAQUS  301.310  364.010  443.800  707.710  901.740  
0..2  Present  216.209  257.039  302.020  1032.654  1342.753  
ABAQUS  216.710  259.290  303.580  1037.500  1374.000  
$\pi $  0.1  Present  288.597  325.207  378.969  516.005  622.710 
ABAQUS  288.110  325.230  380.110  517.160  622.670  
0.2  Present  206.659  235.054  270.651  650.104  811.858  
ABAQUS  206.290  235.350  271.530  650.900  811.980 
Table 10First five natural frequency parameters for Mindlin Annular sector plates subjected to FE1FE1 type elastic boundary conditions (a/b= 0.4)
Sector angle ($\varphi $)  Thickness to radius ratio ($h/b$)  Method  Natural frequency  
1  2  3  4  5  
$\frac{\pi}{3}$  0.1  Present  119.832  264.818  319.661  1483.220  1916.028 
ABAQUS  119.940  264.970  319.570  1484.100  1917.400  
0.2  Present  83.397  180.756  222.669  2577.428  3060.712  
ABAQUS  83.395  180.750  222.710  2580.000  3066.200  
$\frac{\pi}{2}$  0.1  Present  57.023  251.087  267.380  786.925  1222.649 
ABAQUS  56.935  251.210  267.440  787.520  1223.700  
0.2  Present  41.696  179.512  185.090  1389.419  1971.191  
ABAQUS  40.802  179.550  184.680  1390.300  1974.300  
$\frac{2\pi}{3}$  0.1  Present  24.833  201.973  271.669  561.948  806.253 
ABAQUS  24.039  202.070  271.430  562.430  806.550  
0.2  Present  18.952  149.812  189.712  995.529  1318.058  
ABAQUS  17.523  149.840  189.290  995.930  1319.700  
$\pi $  0.1  Present  3.867  120.269  233.439  411.437  434.108 
ABAQUS  3.856  120.260  233.370  411.390  434.060  
0.2  Present  3.869  105.112  171.054  610.370  758.047  
ABAQUS  0.000  105.100  170.900  610.820  758.160 
4. Conclusion
An improved Fourier series method has been presented for vibration analysis of moderately thick annular and circular sector plates with classical and general elastic restraints along its edges. Regardless of the boundary conditions, the displacement function is invariantly expressed as an improved trigonometric series which converges uniformly at an accelerated rate. The efficiency, accuracy and reliability of the present method have been fully demonstrated by various numerical examples for moderately thick annular sector plates having different cutout ratios and sector angles.
Table 11First five natural frequency parameters for Mindlin Annular sector plates subjected to CE1CE1 type elastic boundary conditions (a/b= 0.4)
Sector angle ($\varphi $)  Thickness to radius ratio ($h/b$)  Method  Natural frequency modes  
1  2  3  4  5  
$\frac{\pi}{3}$  0.1  Present  5161.242  5329.372  6024.050  7500.647  9771.648 
ABAQUS  5182.600  5349.600  6044.000  7521.400  8046.600  
0.2  Present  6628.586  6823.656  7744.270  9710.654  12411.918  
ABAQUS  6665.600  6859.200  7778.100  9746.900  10565.000  
$\frac{\pi}{2}$  0.1  Present  5167.620  5242.262  5540.684  6140.785  7122.635 
ABAQUS  5188.700  5261.800  5560.000  6158.100  7141.200  
0.2  Present  6633.347  6724.016  7105.490  7936.324  9269.663  
ABAQUS  6673.500  6762.800  7142.300  7970.700  9306.600  
$\frac{2\pi}{3}$  0.1  Present  5171.158  5212.374  5378.060  5694.631  6205.673 
ABAQUS  5191.900  5231.300  5397.000  5711.500  6222.000  
0..2  Present  6636.080  6687.398  6895.770  7327.756  8047.005  
ABAQUS  6676.100  6726.100  6933.100  7362.200  8080.800  
$\pi $  0.1  Present  5175.030  5192.529  5265.762  5397.220  5599.831 
ABAQUS  5195.000  5210.600  5284.500  5414.600  5616.400  
0.2  Present  6639.126  6661.568  6752.481  6926.255  7207.997  
ABAQUS  6678.800  6699.900  6790.300  6961.700  7241.800 
The effect of sector angle, thickness to radius ratio and restraining springs on the frequency parameters has been discussed. The present method is also employed to study the vibration analysis of moderately thick circular sector plates without modifying the solution procedure. Results for moderately thick annular sector plates under general elastic boundary conditions for various thicknesses to radius ratio and sector angle are presented which can serve as a bench mark for future computational methods. The accuracy of the results has been verified by comparing it with those available in literature and with ABAQUS. An excellent agreement is observed between the results obtained using the present method and with those available in literature. Keeping in view the accuracy and fast convergence behavior this method can easily be further extended to study the vibration analysis of various built up structures without modifying the solution algorithm and procedure.
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About this article
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. U1430236) and Natural Science Foundation of Heilongjiang Province of China (No. E2016024).