Abstract
The vibrations of circular, annular and sector plates are different boundary value problems due to different edge conditions and thus have been treated separately using different solution algorithms and procedures. In this paper, a unified method is proposed for vibration analysis of moderately thick annular, circular plates and their sector counterparts with arbitrary boundary conditions. The unification of these plates is physically achieved by applying the coupling spring’s technique at the radial edges to ensure appropriate continuity conditions. Irrespective of the shape of the plate and the type of boundary conditions, each of the displacement function is expressed as a new form of trigonometric expansion with high convergence rate. Unlike most of the previous studies the current method can be universally applied to a wide range of vibration problems involving different shapes, boundary conditions, varying materials and geometric properties without modifying the solution algorithms and procedure. Furthermore, the current method can easily be applied to sector plates with an arbitrary inclusion angle of 2$\pi $. The accuracy, reliability and versatility of the proposed method are fully demonstrated with several numerical examples for different shapes of plates and under different boundary conditions.
1. Introduction
Circular, annular and their sectorial counterparts are important structural components widely used in many engineering fields like civil, mechanical and marine engineering. As far as previous literature is concerned different solution algorithms and procedures have been adopted to study their vibration characteristics. The main reason behind these different solution algorithms and procedures was difference in their geometries resulting in different edge conditions.
A lot of research work has been done to study their dynamic characteristics under different boundary conditions. The important and comprehensive review on this subject can be found in Leissa’s 1973 book. The initial study on vibrations of circular plates or disks was done by Deresiewicz and Mindlin [1]. Employing the classical thin plate theory and Mindlin plate theory, these two researchers studied the vibration characteristics of axially symmetric circular disks. This work was further extended by Soni et al. [2] to axisymmetric orthotropic non uniform circular discs. They carried out their research using the same Mindlin plate theory and Chebyshev collocation technique. This technique was later employed by Gupta et al. to polar orthotropic annular Mindlin plates with nonuniform thickness [3]. Using Finite Element Method and threedimensional finite strip model, Cheung et.al studied the vibration characteristics of thick and thin sector plates subjected to different types of classical boundary conditions [4, 5]. Investigation on vibration characteristics of annular sector plates having internal radial line and circumferential arc supports was carried out by Xiang et al. [67]. In another study Xiang et al. used first order shear deformation theory and studied the vibration response of thick circular and annular plates with internal ring stiffeners [8]. Later he extended his research to stepped circular Mindlin plates by employing domain decomposition technique to study the vibration characteristics [9]. Another similar study was performed by B. Singh and S. M. Hassan [10]. They studied the out of plane vibrations of a circular plate with different thickness variation. They approximated the thickness polynomial by interpolating the sample points along the thickness of the plate. In another study a combination of RayleighRitz method and Lagrange multiplier method was developed by S. Kitipornchai et al. to study the vibration characteristics of arbitrary shaped plates with corner supports [11]. Exact solution for annular sector plates subjected to simply supported radial edge conditions and general boundary conditions at circular edges was obtained by McGee et al. [12] employing the Mindlin plate theory and using ordinary and modified Bessel functions of the first and second kind.
Differential quadrature method was employed by various researchers to study the vibration characteristics of sector plates, annular sector plates and solid circular plates. Extensive results were reported for these plates subjected to various sets of classical boundary conditions [1315]. Huang et al. [16] employed Frobenius method on orthotropic sector plates and studied the effect of Young modulus and shear modulus on the vibration characteristics of these plates. In another important research on thick circular and annular plates with uniform, linear and quadratic change in thickness along the radial edge was performed by Jae Hoon Kang [17]. A similar threedimensional study of thick annular and circular plates was carried out by J. So et al. [18] employing RayleighRitz method. In their research they used trigonometric functions and algebraic polynomial as admissible displacement functions along the circumferential and radial and axial coordinates respectively. Another threedimensional study of annular and circular plates was performed by Zhou et.al. They employed ChebyshevRitz technique and used Chebyshev polynomial as admissible function. Later they extended the same ChebyshevRitz technique to annular sector plates [19, 20]. Another important threedimensional investigation on annular plates resting on elastic foundation was done by Hashemi et al. They used polynomialRitz approach and studied the effect of cutout ratio, thickness to radius ratio and elastic foundation on the vibration characteristics of annular plates subjected to various combinations of classical boundary conditions [21].
Discrete singular convolution method was used by Civalek et.al to investigate the vibration characteristics of Mindlin annular plates and thick circular plates [22, 23]. Similarly employing the Mindlin plate theory and first order shear deformation theory, Jomehzadeh et.al investigated the transverse vibrations of isotropic sector plate and moderately thick annular sector plates subjected to simply supported boundary conditions and arbitrary boundary conditions at radial and circular edges respectively [2425]. In plane free vibration analysis of isotropic homogeneous circular disks subjected to arbitrary boundary conditions at the inner and outer edges was investigated by Bashmal et al. by employing twodimensional linear plane stress theory. In another study he employed RayleighRitz method to study the vibration characteristics of annular disk with point elastic support [26, 27]. Similarly, Ravari et al. investigated the in plane vibrations of orthotropic circular annular plates by using Helmholtz decomposition technique and separation of variables method [28].
In other similar studies on circular, annular and sector plates, Sari et al. [29] used Chebyshev collocation method to study the vibration characteristics of Mindlin annular plates with damaged boundary conditions. Similarly, Reddy’s higher order shear deformation theory was employed by Bisadi et.al and Es’Haghi [30, 31] to investigate the vibration characteristics of thick circular and annular plates subjected to different combinations of classical boundary conditions at edges. Employing the boundary restraining springs technique Shi et.al proposed a generalized Fourier series method to study the annular sector plates subjected to elastic boundary conditions at each edge [3233]. Later X. Shi et al. [34] proposed a unified method for vibration analysis of circular, annular and their sector counterparts by employing coupling springs technique at the coupling edge. The same idea has been adopted here to develop a unified method to study the vibration characteristics of Mindlin circular, annular and their sector counter parts subjected to general elastic boundary conditions. The beauty of this method is that it does not require any modification in the procedure or solution algorithm to accommodate these different geometries and boundary conditions.
2. Theoretical formulation
2.1. Description of the model
Consider a moderately thick annular sector plate with internal radius $a$, outer radius $b$, thickness $h$ and width $R$ in the radial direction as shown in Fig. 1. The angle $\varphi $ represents the sector angle of the plate. The plate geometry and dimensions are defined in the cylindrical coordinate system $\left(r,\varphi ,z\right)$.
Fig. 1Geometry of moderately thick annular sector plate
a)
b)
The elastic boundary conditions along the edges are specified using boundary spring technique. One translational and two rotational springs 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}. It can be seen in Fig. 2 that an annular plate can be obtained by annular sector plate when the sector angle becomes equal to 2$\pi $, a circular sector plate can be obtained from annular sector plate if the inner radius $a$ becomes equal to 0. Similarly, a circular plate can be obtained when the inclusion angle of the annular sector plate becomes equal to 2$\pi $ and the inner radius $a$ also becomes equal to 0. Therefore, the solution algorithm and procedure will be developed in such a way that it can easily be applied to annular, circular and circular sector plates just by varying geometric parameters mentioned earlier.
Fig. 2a) Annular sector plate, b) annular plate, c) circular sector plate, d) circular plate
a)$0\le \varphi <2\pi $
b)$\varphi =2\pi $
c)$0\le \varphi <2\pi $, $\alpha =0$
d)$\varphi =2\pi $, $\alpha =0$
2.2. Formulation
In the framework of first order shear deformation plate theory, the displacement field in an arbitrary point of a moderately thick annular sector pate 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 ${\theta}_{r}$ and ${\theta}_{\varphi}$ represents the rotation of transverse normal with respect to $\varphi $ and $r$ directions, $z$ is the thickness coordinate, ${u}_{r}$ and ${u}_{\varphi}$ are displacements of the mid plane in $r$ and $\varphi $ directions, respectively, ${w}_{o}$ is the transverse displacement and $t$ is the time. Thus the corresponding strains at this point are defined in terms of middle surface strains, curvature and twist changes as:
${\gamma}_{r\varphi}={\gamma}_{r\varphi}^{o}+z{\chi}_{r\varphi},{\gamma}_{rz}={\gamma}_{rz}^{o},{\gamma}_{\varphi z}={\gamma}_{\varphi z}^{o},$
where the middle surface strains, curvature and twist changes are written as:
${\epsilon}_{\varphi}^{o}=\frac{\partial {u}_{\varphi}}{r\partial \varphi}+\frac{{u}_{r}}{r},{\chi}_{\varphi}=\frac{\partial {\theta}_{\varphi}}{r\partial \varphi}+\frac{{\theta}_{r}}{r},$
${\gamma}_{r\varphi}^{o}=\frac{\partial {u}_{\varphi}}{\partial r}+\frac{\partial {u}_{r}}{r\partial \varphi}\frac{{u}_{\varphi}}{r},{\chi}_{r\varphi}=\frac{\partial {\theta}_{\varphi}}{\partial r}+\frac{\partial {\theta}_{r}}{r\partial \varphi}\frac{{\theta}_{\varphi}}{r},$
${\gamma}_{rz}^{o}=\frac{\partial {w}_{o}}{\partial r}+{\theta}_{r},{\gamma}_{\varphi z}^{o}=\frac{\partial {w}_{o}}{r\partial \varphi}+{\theta}_{\varphi}.$
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:
${M}_{\varphi}=\underset{h/2}{\overset{h/2}{\int}}{\sigma}_{\varphi}zdz=D\left[\frac{1}{r}\left({\theta}_{r}+\frac{\partial {\theta}_{\varphi}}{\partial \varphi}\right)+\nu \left(\frac{\partial {\theta}_{r}}{\partial r}\right)\right],$
${M}_{r\varphi}=\underset{h/2}{\overset{h/2}{\int}}{\tau}_{r\varphi}zdz=D\left(\frac{1\nu}{2}\right)\left[\frac{1}{r}\left(\frac{\partial {\theta}_{r}}{\partial \varphi}{\theta}_{\varphi}\right)+\frac{\partial {\theta}_{\varphi}}{\partial r}\right],$
${Q}_{r}={K}^{2}\underset{h/2}{\overset{h/2}{\int}}{\tau}_{rz}dz={K}^{2}Gh\left[{\theta}_{r}+\frac{\partial {w}_{o}}{\partial r}\right],$
${Q}_{\varphi}={K}^{2}\underset{h/2}{\overset{h/2}{\int}}{\tau}_{\varphi z}dz={K}^{2}Gh\left[{\theta}_{\varphi}+\frac{1}{r}\frac{\partial {w}_{o}}{\partial \varphi}\right],$
where ${M}_{r}$, ${M}_{\varphi}$ and ${M}_{r\varphi}$ are the bending moments per unit length of the plate, ${Q}_{r}$ and ${Q}_{\varphi}$ are the transverse shear forces per unit length of the plate, ${\sigma}_{r}$, ${\sigma}_{\varphi}$ are the normal stresses, ${\tau}_{r\varphi}$, ${\tau}_{rz}$ and ${\tau}_{\varphi z}$ are the shear stresses, $h$ is the plate thickness, $E$ is the modulus of elasticity, $G=E/2\left(1+\nu \right)$ is the shear modulus, $\nu $ is the Poisson ratio, $D=E{h}^{3}/12\left(1{\nu}^{2}\right)$ is the flexural rigidity and ${K}^{2}={\pi}^{2}/12$ 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 the Mindlin annular sector plate is given by:
$\frac{\partial {M}_{r\varphi}}{\partial r}+\frac{1}{r}\frac{\partial {M}_{\varphi}}{\partial \varphi}+\frac{2}{r}{M}_{r\varphi}{Q}_{\varphi}=\frac{\rho {h}^{3}}{12}\left(\frac{{\partial}^{2}{\theta}_{\varphi}}{\partial {t}^{2}}\right),$
$\frac{\partial {Q}_{r}}{\partial r}+\frac{1}{r}\frac{\partial {Q}_{\varphi}}{\partial \varphi}+\frac{{Q}_{r}}{r}=\rho h\frac{{\partial}^{2}{w}_{o}}{\partial {t}^{2}}.$
The boundary conditions for an elastically restrained moderately thick annular sector plate are:
${k}_{b}{w}_{o}={Q}_{r},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{b}^{r}{\theta}_{r}={M}_{r},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{b}^{t}{\theta}_{\varphi}={M}_{r\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}r=b,$
${k}_{\varphi 0}{w}_{o}={Q}_{\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{\varphi 0}^{r}{\theta}_{\varphi}={M}_{\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{\varphi 0}^{t}{\theta}_{\varphi}={M}_{r\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\varphi =0,$
${k}_{\varphi 1}{w}_{o}={Q}_{\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{\varphi 1}^{r}{\theta}_{\varphi}={M}_{\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{\varphi 1}^{t}{\theta}_{\varphi}={M}_{r\varphi},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\varphi =\alpha ,$
where ${k}_{a}$, ${k}_{b}$ (${k}_{\varphi 0}$ and ${k}_{\varphi 1})$ are translational spring constants, ${K}_{a}^{r}$, ${K}_{b}^{r}$ (${K}_{\varphi 0}^{r}$ and ${K}_{\varphi 1}^{r})$ are rotational spring constants attached in radial direction and ${K}_{a}^{t}$, ${K}_{b}^{t}$ (${K}_{\varphi 0}^{t}$ and ${K}_{\varphi 1}^{t})$ are rotational spring constants attached in circumferential direction at $r=a$ and $b$ ($\varphi =0$ 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. For instance, a clamped boundary (C) is achieved by simply setting the stiffness of the entire springs equal to infinity (which is represented by a very large number, 10^{14}). Inversely, a free boundary (F) is gained by setting the stiffness of the entire springs equal to zero. The units for the translational and rotational springs are N/m and Nm/rad, respectively.
2.3. Trigonometric series representation for the displacement functions
Regardless of the plate shape and type of boundary conditions, the displacement and rotation functions are invariably expressed in the form of simple trigonometric series expansion as:
${\theta}_{\varphi (r,\varphi )}=\sum _{m=n=2}^{\mathrm{\infty}}{B}_{mn}{\phi}_{m}\left(r\right){\phi}_{n}\left(\varphi \right),$
${w}_{o(r,\varphi )}=\sum _{m=n=2}^{\mathrm{\infty}}{C}_{mn}{\phi}_{m}\left(r\right){\phi}_{n}\left(\varphi \right),$
where ${A}_{mn}$, ${B}_{mn}$, ${C}_{mn}$ denotes the expansion coefficients and:
A solution can be obtained either in strong form by letting the series satisfy the relevant equations exactly on a pointwise basis, or in weak form by solving the series coefficients approximately using, for instance, the RayleighRitz technique. The weak form of solution will be sought here since it will be more attractive in modeling complex structures. To employ this method for this analysis, it is necessary to state the potential and kinetic energy in terms of displacement fields. The total potential energy of the spring restrained plate which is composed of two parts, namely, the strain energy of the Mindlin annular sector plate is given by:
and the potential energy stored in the boundary springs, can be expressed as:
The kinetic energy expression for annular sector plate is expressed as:
As mentioned above, an annular plate can be mathematically viewed as a special case when the sector angle of an annular sector plate is set equal to 2$\pi $. However, this transition of annular sector plate into annular plate is not possible with this simple mathematical operation because the continuity of the displacement and its derivatives at this simple mathematical operation alone cannot automatically ensure a complete transition of the sector into an annular plate that is, the continuities of the displacements and their derivatives at $\varphi =0$ and $\varphi =2\pi $. To overcome this problem, a set of coupling springs will be used to enforce the continuity conditions for the displacements at the edges $\varphi =0$ and $\varphi =2\pi $. The potential energy stored in these coupling springs will be given by:
where ${k}_{cs}$, ${K}_{cs}^{r}$ and ${K}_{cs}^{t}$ are the stiffnesses for translational coupling spring, rotational coupling springs in radial direction and rotational coupling springs in tangential direction respectively.
The Lagrangian for the annular sector plate can be generally expressed as:
Substituting Eqs. (911) in (12) and minimizing Lagrangian against all the unknown series expansion coefficients 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 that is:
And $\mathrm{K}$and $\mathrm{M}$are the stiffness and mass matrices, respectively. For conciseness, the detailed expressions for the stiffness and mass matrices are not shown here. The eigenvalues (or natural frequencies) and eigenvectors of moderately thick annular sector plates can now be easily and directly determined from solving a standard matrix eigenvalue problem Eq. (13). 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 Eqs. (7).
3. Results and discussion
In order to verify the convergence, accuracy, reliability and applicability of the present method for moderately thick annular, circular plates and their sector counter parts, several numerical examples are presented here along with the reference results from literature and ABAQUS. First of all the convergence of the present method is studied. Using different truncation terms ($M=N=$2, 4, 6, 8, 10, 12, 14) several sets of results are obtained for fully clamped Mindlin annular sector plate having different sector angles and presented in Table 1 and 2 as shown.
Table 1First five nondimensional frequency parameter for fully clamped Mindlin annular sector plate having a/b= 0.6, h/b= 0.1
Sector angle Ø  $M=N$  Non dimensional frequency parameter $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$  
Mode sequence  
1  2  3  4  5  
2  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  
$\pi $/6  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  
ABAQUS  144.534  238.503  250.295  329.634  350.523 
Table 2First five nondimensional frequency parameter for CCCC Mindlin annular sector plate having a/b= 0.6, h/b= 0.1
Sector angle Ø  $M=N$  Non Dimensional frequency parameter $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$  
Mode sequence  
1  2  3  4  5  
2  104.250  116.563  163.387  223.325  232.424  
$\pi $/2  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  
ABAQUS  103.271  112.776  130.262  155.343  182.096 
A fast convergence can be observed in the tabulated results for different truncation numbers and also a good agreement can be observed between the present values and the ABAQUS results. Similarly figure 3 shows convergence pattern for the 1st, 3rd, 5th and 8th mode for a moderately thick circular sector plate having clamped circular edge and simply supported radial edges.
Fig. 3Convergence pattern for frequency parameters with no. of truncation terms
It can be seen that the results converge very quickly even with small number of truncation terms. Thus a suitable truncation number should be used to achieve the accuracy of the largest desired frequency. In view of above and excellent convergence behavior of the current solution, the truncation number for subsequent calculation in the present method is taken as $M=N=$12.
After verifying the fast convergence of the preset method, results for Mindlin annular and circular plates and their sector counterparts are obtained and tabulated for various sector angles and different boundary conditions along with the reference results from literature. Table 3 shows fundamental frequency parameters for Mindlin annular sector plates having different sector angles and thickness to radius ratio. The plate has simply supported radial edges and different boundary conditions at the circular edges. The results have been compared with ABAQUS software as well as those available in literature.
Table 3Fundamental frequency parameter Ω=ωb2ρh/D1/2 for Mindlin annular sector plates having SS radial edges and various boundary conditions at the inner and outer circumferential edges (a/b= 0.5)
Sector angle Ø  $h/b$  Method  Boundary conditions  
SS  SF  FS  FC  
195  0.1  Present  38.365  4.560  10.224  19.998 
Ref [12]  38.636  4.675  10.227  19.999  
ABAQUS  38.580  4.540  11.159  20.923  
0.2  Present  32.508  4.005  9.130  17.503  
Ref [12]  32.871  4.542  9.366  17.582  
ABAQUS  32.676  4.067  10.014  18.239  
210  0.1  Present  38.222  4.507  9.685  19.620 
Ref [12]  38.455  4.584  9.664  19.610  
ABAQUS  38.223  4.230  9.479  19.516  
0.2  Present  32.419  3.997  8.681  17.235  
Ref [12]  32.734  4.458  8.877  17.294  
ABAQUS  32.469  3.923  8.590  17.201  
270  0.1  Present  37.868  4.392  8.213  18.654 
Ref [12]  38.010  4.372  8.130  18.622  
ABAQUS  37.875  4.197  8.015  18.574  
0.2  Present  32.200  3.999  7.450  16.548  
Ref [12]  32.394  4.263  7.546  16.566  
ABAQUS  32.252  3.932  7.366  16.524 
Next we verify the applicability of this unified method for annular plates. As mentioned previously an annular plate can be viewed as a special case of annular sector plate if the sector angle becomes equal to 2$\pi $. Results for Mindlin annular plate for different combination of classical boundary conditions at the inner and outer edges for various cutout ratios are also calculated and presented in Table 4 along with those obtained from ABAQUS. A very close agreement can be observed in the calculated results. This close agreement verifies the applicability of the coupling spring technique for calculating frequency parameters for a complete annular plate without modifying the solution procedure.
Table 4Non dimensional frequency parameter Ω=ωb2ρh/D1/2 for Mindlin annular plates with various cutout ratio and boundary conditions (h/b= 0.2)
B.C  $a/b$  Method  Mode sequence  
1  2  3  4  5  
SC  0.2  Present  21.161  22.228  22.228  27.545  27.550 
ABAQUS  21.200  22.271  22.272  27.596  27.596  
0.4  Present  32.154  32.829  32.829  35.411  35.414  
ABAQUS  32.243  32.920  32.920  35.511  35.511  
0.6  Present  56.120  56.458  56.458  57.642  57.643  
ABAQUS  56.357  56.697  56.697  57.885  57.887  
SF  0.2  Present  2.082  2.082  3.225  4.980  4.998 
ABAQUS  2.084  2.084  3.224  4.975  4.975  
0.4  Present  3.280  3.280  3.581  5.464  5.471  
ABAQUS  3.282  3.282  3.578  5.465  5.465  
0.6  Present  4.703  5.089  5.089  7.402  7.407  
ABAQUS  4.699  5.090  5.090  7.411  7.411  
FC  0.2  Present  9.476  16.774  16.774  26.240  26.247 
ABAQUS  9.481  16.797  16.797  26.288  26.289  
0.4  Present  12.156  15.995  15.995  24.240  24.246  
ABAQUS  12.163  16.014  16.014  24.248  24.284  
0.6  Present  21.219  22.812  22.812  27.275  27.278  
ABAQUS  21.244  22.844  22.844  27.324  27.325  
CF  0.2  Present  4.191  4.191  4.809  5.750  5.756 
ABAQUS  4.193  4.193  4.810  5.746  5.748  
0.4  Present  8.017  8.017  8.175  8.865  8.868  
ABAQUS  8.022  8.022  8.179  8.870  8.870  
0.6  Present  17.148  17.198  17.198  17.802  17.803  
ABAQUS  17.168  17.220  17.220  17.826  17.826  
CC  0.2  Present  24.346  25.313  25.313  29.515  29.519 
ABAQUS  24.413  25.380  25.380  29.583  29.584  
0.4  Present  37.641  38.196  38.196  40.238  40.240  
ABAQUS  37.784  38.339  38.339  40.382  40.383  
0.6  Present  64.159  64.462  64.462  65.485  65.486  
ABAQUS  64.496  64.799  64.800  65.821  65.823 
As mentioned earlier when the inner radius of an annular sector plate is approximated to a very small number say $a=$0.00001, then the annular sector plate converges to circular sector plate. The same method has been applied to circular sector plates and results for circular sector plates having different sector angles and boundary conditions at the radial and circumferential edges It should be noted that the symbol S stands for simply supported, C stands for clamped and F stand for free boundary conditions. The edges are taken in the counter clock wise direction, so SCS boundary conditions means simply supported radial edges and clamped circumferential edge. First three nondimensional frequency parameters are calculated and presented in the Table 5 along with the reference results. It can be observed that the frequency parameters are in close agreement with the reference data.
Next we calculate the frequency parameter for various boundary conditions for a complete Mindlin circular plate having different thickness to radius ratio. In order to achieve this two simple modification needs to be done in the solution algorithm. First is equating the inner radius equal to a very small number say $a=$0.00001 and second is equating the sector angle equal to 2$\pi $ Table 6 presents first five non dimensional frequency parameter for a complete circular plate subjected to different boundary conditions at the circumferential edge and having different thickness to radius ratio. It should be noted that for the ‘F’; free boundary condition; the zero frequency parameters for the first six rigid body modes have not been taken into account in the Table 6. It can be observed that the frequency parameter decreases with increasing thickness to radius ratio in all the three types of boundary conditions listed. A good agreement between the presented results and those obtained through ABAQUS can also be observed which proves the applicability of the present method for calculating the frequency parameters for Mindlin circular plates also.
Table 5First three nondimensional frequency parameters Ω=ωb2ρh/D1/2 for circular sector plates having different combination of classical boundary conditions and sector angle (h/b= 0.2, a/b= 0.00001)
Sector angle Ø  BC  Mode sequence  Present  Ref. [13]  ABAQUS 
30  SCS  1  66.256  67.933  66.490 
2  98.936  102.560  99.373  
3  131.364  132.860  132.146  
90  SSS  1  21.006  21.977  21.030 
2  41.254  42.699  41.339  
3  48.863  50.307  48.981  
120  CCC  1  27.311  27.314  27.372 
2  40.977  40.983  41.105  
3  52.324  52.338  52.515 
Table 6First five nondimensional frequency parameter Ω=ωb2ρh/D1/2 for a Mindlin circular plate having different boundary conditions and thickness to radius ratio (a/b= 0.00001)
B.C  $h/b$  Method  Mode Sequence  
1  2  3  4  5  
C  0.1  Present  9.941  20.178  20.178  32.210  32.223 
ABAQUS  9.939  20.176  20.176  32.220  32.222  
0.2  Present  9.240  17.758  17.758  26.994  27.000  
ABAQUS  9.246  17.782  17.782  27.044  27.045  
0.25  Present  8.807  16.446  16.446  24.478  24.482  
ABAQUS  8.816  16.479  16.479  24.540  24.540  
S  0.1  Present  4.895  13.512  13.512  24.324  24.336 
ABAQUS  4.892  13.508  13.508  24.315  24.317  
0.2  Present  4.777  12.620  12.620  21.690  21.696  
ABAQUS  4.776  12.625  12.625  21.710  21.710  
0.25  Present  4.696  12.080  12.080  20.272  20.276  
ABAQUS  4.696  12.089  12.089  20.300  20.300  
F  0.1  Present  5.283  5.299  8.869  12.153  12.248 
ABAQUS  5.275  5.275  8.865  12.062  12.062  
0.2  Present  5.117  5.125  8.505  11.366  11.428  
ABAQUS  5.113  5.113  8.504  11.316  11.316  
0.25  Present  5.011  5.018  8.268  10.910  10.960  
ABAQUS  5.009  5.009  8.268  10.871  10.871 
All the results tabulated so far have been calculated for various combinations of classical boundary conditions which are treated as a special case of elastic boundary conditions in which the stiffness values for the restraining springs are set either equal to a very high value i.e. 10^{14} or a very low number zero. It is therefore necessary to study the effect of these restraining spring stiffnesses on the frequency characteristics for these plates. Figs. 46 shows the effect of boundary restraining springs on the frequency parameter ‘$\mathrm{\Omega}$’ for a fully clamped annular plate having $a/b=$ 0.6 and $h/b=$ 0.2.
Fig. 4 shows effect of translational spring stiffness on the second and sixth mode frequency parameter of annular plate in which the stiffness of the translational spring stiffness varies from 0 to 1e14 while the stiffnesses of the rotational spring in radial and tangential direction ( ) are kept constant i.e. 1e14.
Similarly, Figs. 5 and 6 have been obtained by assigning the corresponding boundary spring stiffness, a value ranging from 0 to 10^{14} and keeping the stiffnesses of other sets of spring equal to 10^{14}.
Fig. 4Effect of translational spring stiffness k on frequency parameter Ω
Fig. 5Effect of rotational spring stiffness K attached in tangential direction on Ω
Fig. 6Effect of rotational spring stiffness K attached in radial direction on Ω
Similarly, Fig. 7(a)(c) shows the effect of coupling springs on the frequency parameter $\mathrm{\Omega}$.
It can be seen that the translational and rotational boundary springs sufficiently affect the frequency parameters. More precisely the translational boundary restraining spring tend to be more influential when its stiffness varies from 10^{8} to 10^{13}. Similarly, the influential range for the rotational boundary spring in the radial direction is 10^{6} to 10^{12}. However, the influence of rotational boundary spring in the tangential direction is very small as seen in Fig. 5. Also it can be seen in Fig. 7(a)(c) that influential range for the coupling springs is much smaller as compared to the boundary restraining springs. This influential range is the elastic range and frequency parameters can easily be calculated for elastic boundary conditions by assigning the proper stiffness values to the boundary restraining springs without modifying the solution procedure or algorithms.
We know that in practical engineering, designing or development of any mechanical system or a product, structure vibration analysis and testing is an important part to assess the real behavior of the structure when subjected to static or dynamic loads. In other words, to better understand any structural vibration problem, the resonant frequencies of a structure need to be identified and quantified in order to avoid well known resonance phenomena which can result in catastrophe. Today, modal analysis has become a widespread means of finding the modes of vibration of a machine or a structure.
Fig. 7Effect of coupling springs on the frequency parameter Ω
a) Effect of translational coupling spring $kc$ on frequency parameter $\mathrm{\Omega}$
b) Effect of rotational coupling spring in radial direction $Kc$ on frequency parameter $\mathrm{\Omega}$
c) Effect of rotational coupling spring in tangential direction $Kc$ on frequency parameter $\mathrm{\Omega}$
Various analytical methods have been developed over the years to accurately estimate the resonant frequencies or modes of vibrations of any structure when subjected to different boundary conditions. Once these frequencies are calculated they are used to estimate the modes of vibrations of a structure which are determined by the material properties and boundary conditions. Each mode of vibration is defined by a natural (modal or resonant) frequency, modal damping, and a mode shape. If there is a slight change in material properties or boundary conditions of a structure, its modes of vibration will also change. Therefore, it is important to estimate these frequencies for any change in material properties as well as boundary conditions because in practical engineering applications, the material properties of a structure and boundary conditions may vary. Furthermore, most of the existing techniques available so far to estimate these natural or resonant frequencies are limited to classical boundary conditions (clamped, free, simply supported etc.), however in practical engineering applications the structures are not always subjected to classical boundary conditions rather they may be subjected to elastic boundary conditions.
In the present manuscript, the unified method presented not only helps to accurately estimate these natural frequencies of circular and annular plates and their sector counter parts when they are subjected to classical boundary conditions but also when they are subjected to general elastic boundary conditions. The presented results give an insight of the modes of vibration of these plates having different material properties and subjected to elastic boundary conditions. Moreover, another important contribution of this technique is that this method does not require any changes in procedure or solution algorithms to accommodate different geometries, material properties or boundary conditions. The same solution algorithm or procedure can be used to estimate natural frequencies for different materials and boundary conditions. Different boundary conditions (classical, elastic, uniform & nonuniform) can easily be achieved by simply changing the stiffnesses of the translational and rotational springs attached at the boundaries or edges of these plates”.
4. Conclusion
In this paper a unified method is presented for vibration analysis of Mindlin annular, circular and their sector counter parts with arbitrary boundary conditions at their edges. Coupling springs technique has been utilized to avoid inconvenient formulation or procedural modification to accommodate different boundary conditions and geometrical shapes of the plates. Irrespective of the shape of the plate and the type of boundary conditions, each of the displacement function is expressed as a new form of trigonometric expansion with high convergence rate. RayleighRitz method has been used to determine the expansion coefficients. The current method therefore can be universally applied to a wide range of vibration problems involving different shapes, boundary conditions, varying materials and geometric properties without modifying the solution algorithms and procedure. The unification, fast convergence, accuracy and reliability have been fully demonstrated through several numerical examples involving different shapes and boundary conditions. Furthermore, the effect of boundary restraining springs and coupling springs on the frequency parameter have also been studied.
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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)