A Unified method for vibration analysis of moderately thick annular, circular plates and their sector counterparts subjected to arbitrary boundary conditions
Fazl e Ahad^{1} , Dongyan Shi^{2} , Anees Ur Rehman^{3} , Hafiz M. Waqas^{4}
^{1, 2, 3, 4}College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 18, Issue 8, 2016, p. 50485062.
https://doi.org/10.21595/jve.2016.16834
Received 14 January 2016; received in revised form 14 July 2016; accepted 17 August 2016; published 31 December 2016
JVE Conferences
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.
Keywords: vibrations, circular plates, annular plates, sector plates, natural frequency, mode shapes, arbitrary 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. 1. Geometry 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. 2. a) 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 1. First 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 2. First 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. 3. Convergence 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 3. Fundamental frequency parameter $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$ 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 4. Non dimensional frequency parameter $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$ 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 5. First three nondimensional frequency parameters $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$ 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 6. First five nondimensional frequency parameter $\mathrm{\Omega}=\omega {b}^{2}{\left(\rho h/D\right)}^{1/2}$ 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. 4. Effect of translational spring stiffness $k$ on frequency parameter $\mathrm{\Omega}$
Fig. 5. Effect of rotational spring stiffness $K$ attached in tangential direction on $\mathrm{\Omega}$
Fig. 6. Effect of rotational spring stiffness $K$ attached in radial direction on $\mathrm{\Omega}$
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. 7. Effect of coupling springs on the frequency parameter $\mathrm{\Omega}$
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.
Acknowledgements
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)
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