Kaipeng Zhang^{1} , Dongyan Shi^{2} , Xiaoyan Teng^{3} , Yunke Zhao^{4} , Qian Liang^{5}
^{1, 2, 3, 4, 5}College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, China
^{2}Corresponding author
Journal of Vibroengineering, Vol. 17, Issue 2, 2015, p. 870882.
Received 15 September 2014; received in revised form 12 January 2015; accepted 20 January 2015; published 31 March 2015
Copyright © 2015 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this investigation, a SpectroGeometric Method (SGM) is presented for the inplane vibration analysis of sector plates with an arbitrary inclusion angle, and general boundary conditions along each of its edges. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures; that is, the inplane displacement functions are invariably expressed as an accelerated trigonometric series expansion and different boundary conditions can be easily dealt with through modifying the stiffness values for restraining springs in the same way as varying other model parameters such as Young’s modulus and mass density. The expansion coefficients are considered as the generalized coordinates, and determined using the RayleighRitz technique. The effectiveness and reliability of the current method are demonstrated by the calculated modal properties for sector plates with a range of different combinations of boundary conditions and inclusion angles up to $2\pi $.
Keywords: sector plates, inplane vibration, arbitrary inclusion angles, arbitrary boundary conditions, spectrogeometric method (SGM).
Sector plates are widely used in industrial applications as a standalone structure or a constituent structural component. The dynamic characteristics of sector plates are thus of great interest to engineers and designers. Although there is a vast pool of studies about the outofplane vibrations of sector plates, few inplane vibration results are available for sector plates [1]. The reason may be that the modes related to inplane vibrations typically fall outside the dominant frequency bands of excitations. In some practical engineering applications, however, the inplane vibrations need to be taken into consideration. Furthermore, some studies have shown that the inplane vibrations can play a prominent role in affecting the sound radiation and transmissions of vibration energies in buildup structures [2, 3].
Onoe [4] first obtained the inplane displacements and frequency equations of circular disks based on Love’s theory. In his study, he utilized trigonometric functions in the circumferential direction and Bessel functions in the radial direction. An error in the eigenfunctions for compound modes was identified in his latter works [5]. Holland [6] used trigonometric and Bessel functions to study the inplane vibration of a free circular plate. The frequency parameters for different Poisson’s ratios were presented and compared with the results previously given by Onoe [4, 5]. Chen, et al. [7] studied extensional (inplane) vibration of thin plates of Hookean material with different shapes. Based on plane stress condition, a general solution of the governing differential equation was derived for an arbitrarily shaped plate with the boundary conditions being satisfied only in a leastsquare sense. Irie et al. [8] used a transfer matrix method to study the inplane vibration of circular and annular plates with free and clamped boundary conditions specified at the inner and outer edges, respectively. The frequency parameters were presented for circumferential wave numbers from 0 to 4. The circular plates were treated as a limiting case of an annular plate when the radius of the inner edge tends to zero. Farag and Pan [9] examined the inplane vibration of circular plates clamped at the outer edges. The mode functions were expressed in terms of trigonometric functions in the circumferential direction and the series expansion of Bessel functions in the radial direction. The frequency parameters were compared with finite element results and the data published previously. Recently, Park [10] derived exact frequency equations for inplane vibration of a clamped circular plate using Hamilton’s principle. His results were validated by finite element results and those from Farag and Pan [9]. Bashmal et al. [11] derived a generalized formula for inplane vibration of circular annular plates under various combinations of classical boundary conditions. Bashmal, et al. [12] also studied the inplane vibration behavior of an annular disk with elastic constraints at the inner and outer boundaries. Ravari, et al. [13] derived the frequency equation for the inplane vibration of orthotropic circular annular plates. Kim et al. [14] studied the inplane vibration of a circular plate with its outer edge being restrained elastically. The mode shapes are expressed in terms of trigonometric functions in the circumferential direction and modal functions in the radial direction. The elastic boundary conditions at outer edge were represented by circumferentially distributed radial and tangential stiffness. Shi et al. [15] investigated the inplane vibration characteristics of annular sector plates with elastic boundary conditions.
It should be pointed out that all the above studies are focused on annular and circular plate problems to which the solutions can be reduced to solving two coupled ordinary differential equations with respect to the radial dimension. Using variational approach, Seok and Tiersten [16] investigated the inplane vibration of a cantilevered annular sector plate where the plate is fixed on one radial edge, and free on the rest. Singh, et al. [17] studied free inplane vibration of nonrectangular plate, including annular sector plates.
When an analytical solution is sought for a sector plate, its inclusion (or sector) angle is typically limited within a range; a sector plate with reentrant sector angle (larger than $\pi $) is rarely attempted in the literature. It is also widely accepted that an analytical solution is only possible for a sector plate which is simply supported along, at least, two radial edges. To the best authors’ knowledge, there is no reported study about the inplane vibration of a sector plate with elastically restrained edges. However, sector plates with general elastic boundary conditions have always been of research and application interests to many engineers [1820]. Recently, Shi et al. [21] presented a socalled SpectroGeometric Method (SGM) for the free inplane vibration analysis of orthotropic rectangular plates with arbitrary elastic supports. In this investigation, the spectrogeometric method (SGM) is extended to the inplane vibration analysis of sector plates with arbitrary inclusion angles and boundary conditions. Numerical results are presented to verify the accuracy and reliability of the current solution method.
Fig. 1 shows an annular sector plate with uniform thickness $h$, inner radius $a$, outer radius $b$, width $R$ in radial direction, and sector angle $\varphi $. The geometry and dimensions are defined in an orthogonal cylindrical coordinate system ($r$, $\theta $, $z$). A local coordinate system ($s$, $\theta $, $z$) is also shown in the Fig. 1 which will be used in the analysis. In this investigation, we deal with the inplane vibrations of sector plates with arbitrary boundary conditions in a unified manner. The material of the plate is assumed to be isotropic, having a mass density $\rho $, Young’s modulus $E$ and Poisson’s ratio $\mu $. The displacement fields at a material point are denoted as $u$ and $v$ in the radial and circumferential directions, respectively. Circular sector plates can be viewed as the special cases of an annular sector plate when its radius ratio $a/b$ is a very small number. The annular sector plate can be accordingly defined as:
The boundary conditions are generally specified in terms of elastic restraints with independent stiffness values. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness values become extremely large or small. For instance, a free edge condition is simply created by setting to zero the stiffness values for the two sets of springs.
Fig. 1. Sketch of an elastically restrained annular sector plate
a) An annular sector plate
b) Crosssection
Regardless of boundary conditions and sector angles, the displacement functions will be invariably sought as trigonometric series expansions in the form of:
and:
where ${\lambda}_{rm}=m\pi /R$, ${\lambda}_{\theta n}=n\pi /\varphi $; ${A}_{mn}$, ${B}_{mn}$, ${C}_{mn}$, ${D}_{mn}$, ${E}_{mn}$, ${F}_{mn}$ denote the coefficients of series expansions.
It should be pointed out that in the above series expansions the cosine terms (Fourier series) alone will constitute a complete set which is adequate to span a vector space of infinite dimensions. But this form of series representation of a sufficiently smooth function on a compact interval is usually slow converged due to the possible discontinuities at the endpoints of the interval after its periodic extension as implied by the Fourier representation. By including these auxiliary terms, the inplace displacements are governed by secondorder differential equations over the plate, their first derivatives are mathematically required to be continuous and second derivatives exist everywhere. Thus, these series are able to expand and uniformly converge to any function $f(s,\theta )\in {C}^{1}$ for $\forall (s,\theta )\in D$: ([0, $R$]×[0, $\varphi $]). According to the convergence theorem [22], the convergence (rate) for the series expansion of a periodic function is proportional to the degree of smoothness of the function. Therefore, the series expressions in Eqs. (2)(3) are guaranteed to converge at a substantially improved rate for any boundary conditions. The accompanying benefits were previously demonstrated for beams and plates with general boundary conditions [21, 23]. As a matter of fact, this subtraction techniques have been proposed by mathematicians as a means to accelerate the convergence of the Fourier series expansion for an explicitly given function [24].
The expansion coefficients for the sine terms are determined only by the first derivatives at the boundaries of the solution domain. Strictly speaking, the sine terms are “redundant” in view of the fact that the basis set (that is, the infinite cosine terms) is complete. More explicitly, the coefficients for the sine terms can be shown to be dependent on those for the cosine series under the condition that the (cosine) series converges fast enough so that it can be differentiated termbyterm. In such a case, for example, the relationships between the coefficients for sine and cosine terms can be established through the boundary conditions. A strong form of solution can be subsequently obtained by letting the series satisfy the governing differential equations on a pointwise basis (i.e., at each field point). Such a solution may be considered exact in the sense that the solution error can be controlled within any prespecified tolerance.
In seeking an approximate solution, the truncated version of the series expansions will have to be adopted. As a result, all the expansion coefficients, including those for the sine terms, will be treated equally as the generalized coordinates and solved from, such as, the RayleighRitz procedure. It shall be noted that since the trial solution is constructed with the same degree of smoothness as required for a strong solution, the approximate and exact solutions are equivalent mathematically (this is evident from integrating, by part, the appropriate energy equation). In terms of numerical implementations, however, the approximate solution is far more advantageous, especially when the modeling method is extended to structures involving many plates or other structural elements.
For small deformations, the straindisplacement relations in local cylindrical coordinate system ($s$, $\theta $, $z$) can be expressed as:
The strain energy of the sector plate is given as:
In terms of displacements, stresses can be calculated from:
Submitting Eqs. (4)(6) and (8) into Eq. (7) will lead to the following expression for the strain energy:
where $G(s,\theta )=E(s,\theta )h(s,\theta )/(1{\mu}_{2}(s,\theta \left)\right)$ is the extentional rigidity of the plate.
By neglecting rotary inertia, the kinetic energy of an annular sector plate can be written as:
where $\omega $ is the frequency in radians.
When an edge of the plate is elastically restrained, two sets of springs of arbitrary stiffness distributions will be specified accordingly, which directly describe the relationships between the forces and displacements along the edge. All the familiar homogeneous boundary conditions (i.e., simply supported, free and fixed) can be readily realized by setting the constants of restraint springs to either zero or an extremely large number. The stiffness for any of the restraining spring is allowed to vary continuously or disruptively.
In seeking a weak form of solution, the presence of the restraining springs can be easily accounted for in terms of potential energies stored in the boundary springs:
where ${k}_{ps0}$, ${k}_{ps1}$ (${k}_{p\theta 0}$, ${k}_{p\theta 1}$) are the stiffnesses for tangential springs, respectively, at $s=$ 0 and $R$ ($\theta =$ 0 and $\varphi $); ${k}_{ns0}$, ${k}_{ns1}$ (${k}_{n\theta 0}$, ${k}_{n\theta 1}$) represente the stiffnesses for normal springs, respectively, at $s=$ 0 and $R$ ($\theta =$ 0 and $\varphi $).
The Lagrangian function for an annular sector plate can be finally expressed as:
In the above derivations, the thickness of the plate, material properties (Young’s modulus, Poisson’s ratio, and mass density) and the stiffness for each elastic restraint are all generally specified as an arbitrary function of the spatial coordinates. To unify the descriptions and facilitate the analytical calculations of the involved integrals, all these distribution functions can be expanded into 1D or 2D Fourier cosine series.
By substituting the displacement expressions, Eqs. (2) and (3) into the Lagrangian, Eq. (12), and minimizing the result with respect to the series coefficients, one is able to yield a final system of equation as:
where $\mathbf{E}$ is a vector that contains the unknown series expansion coefficients, and $\mathbf{K}$ and $\mathbf{M}$ are the stiffness and mass matrices, respectively. For conciseness, the detailed expression for stiffness and mass matrices will not be shown here.
The natural frequencies and eigenvectors of sector plates can now be easily and directly determined from solving a standard matrix eigenvalue problem. Each of the eigenvectors actually contains the series coefficients for the corresponding mode; the mode shape can be simply obtained by using Eqs. (2) and (3). Although this investigation is focused on the free vibration of annular sector plates, the dynamic response of the plates to any applied load can be easily considered by including the work done by this load in the Lagrangian, which will eventually lead to a force term on the right side of Eq. (13).
It should also be noted that the current method is particularly advantageous in obtaining other variables of interest such as power flows. Since the displacements are constructed sufficiently smooth as required in a strong formulation, postprocessing the solution can be done easily through appropriate mathematical operations, including termbyterm differentiations.
Several numerical examples will be presented in this section to demonstrate the accuracy, reliability and versatility of the presented method. Throughout these examples, the mass density, Young’s modulus and Poisson’s ratio are specified as: $\rho =$ 7800 kg/m^{3}, $E=$ 2.0×10^{11} Pa and $\mu =$ 0.3. In identifying the boundary conditions, letters C and F have been used to indicate the clamped and free boundary condition along an edge, respectively. Thus, the boundary conditions for a plate are fully specified by using four letters with the first one indicating the B.C. along the first edge, $r=a\text{.}$ The remaining (the second to the fourth) edges are ordered in the counterclockwise direction.
We first consider an annular sector plate which is fully clamped along each edge (CCCC) to validate the current results. The geometry parameters of plate are as follows: cutout ratio $a/b=$ 0.5 and sector angle $\varphi =\pi /$3. A clamped edge refers to a special case of the elastic supports when the stiffnesses for both of the (normal and tangential) restraining springs become infinity (represented by a very large number, 10^{13}, in the actual calculations). The first six nondimensional frequency parameters, $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$, are presented in Table 1 for different series truncation numbers: $M=N=$ 4, 5,…, 12. Two sets of reference results taken from Ref. [17] and obtained using an FEM (ABAQUS) model are also given there. A good convergence behavior is observed. Of equal importance, the solution also shows an excellent numerical stability, meaning that the values essentially remain the same as the truncation numbers become increasingly large. Once the convergence behavior is understood, the series will be consistently truncated to $M=N=$ 10 in all the subsequent calculation.
In the very limited existing studies, the sector angles are typically assumed to be less than $\pi $. Although it is not clear whether $\varphi =\pi $ inherently constitutes a pivoting point for mathematically solving sector plate problems, it has been a limit point practically defining the previous investigation. However, the value of the sector angle appears to have no binding effect on the current solution procedures as described earlier. In Table 2, the first six frequency parameters, $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$, are shown for the completely clamped (CCCC) annular sector plates ($a/b=$ 0.5) with various sector angles. The results match very well with those from Ref. [17] and an FEM model.
Table 1. Frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for a CCCC annular sector plate ($a/b=$ 0.5 and $\varphi =\pi /$3)
Mode sequence


1

2

3

4

5

6


$M=N=$ 4

5.3397

6.5689

7.1412

8.2149

8.6793

8.8482

$M=N=$ 5

5.3384

6.5682

7.1402

8.2130

8.6776

8.8457

$M=N=$ 6

5.3382

6.5681

7.1401

8.2115

8.6744

8.8442

$M=N=$ 7

5.3379

6.5679

7.1399

8.2111

8.6741

8.8437

$M=N=$ 8

5.3378

6.5678

7.1398

8.2108

8.6736

8.8433

$M=N=$ 9

5.3377

6.5678

7.1398

8.2106

8.6735

8.8432

$M=N=$ 10

5.3377

6.5678

7.1398

8.2104

8.6734

8.8430

$M=N=$ 11

5.3377

6.5678

7.1398

8.2104

8.6734

8.8430

$M=N=$ 12

5.3377

6.5678

7.1398

8.2104

8.6734

8.8430

Ref. [17]

5.3385

6.6046

7.0882

8.2349

8.6217

8.8282

FEM

5.3391

6.5690

7.1431

8.2127

8.6778

8.8469

Table 2. Frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for a CCCC annular sector plate ($a/b=$ 0.5)
$\varphi $

Method

Mode sequence


1

2

3

4

5

6


$\pi /$2

Present

4.5320

6.0304

6.5406

6.5746

7.2092

7.7243

Ref. [17]

4.5385

6.0535

6.0535

6.6578

7.2260

7.7108


FEM

4.5327

6.0314

6.5425

6.5758

7.2115

7.7276


$\pi $

Present

3.9798

4.5089

5.2279

5.9389

6.3407

6.3519

FEM

3.9805

4.5091

5.2276

5.9384

6.3414

6.3524


3$\pi /$2

Present

3.8707

4.1223

4.5018

4.9647

5.4595

5.9162

FEM

3.8710

4.1223

4.5013

4.9635

5.4578

5.9132


16$\pi /$9

Present

3.8453

4.0274

4.3093

4.6657

5.0682

5.4855

FEM

3.8469

4.0286

4.3099

4.6655

5.0671

5.4822


2$\pi $

Present

3.8320

3.9769

4.2045

4.4977

4.8374

5.2025

FEM

3.8337

3.9786

4.2061

4.4992

4.8387

5.2040

For any given modal frequency, the corresponding mode shape can be readily determined from Eqs. (2) and (3). As an example, the first six mode shapes are plotted in Fig. 2 for the CCCC annular sector plate with $\varphi =\pi $. It is seen that although those are the lowerorder modes, they tend to exhibit unfamiliar and more complicated patterns than their counterparts in the flexural vibrations. For instance, the extensioncompression deformation in one mode (or region) can quickly turn into a shear state in another mode (or region). This characteristic, however, may have some useful implications to the nondestructive evaluation of material and structural parameters or monitoring of structural conditions or failures, as evidenced by the more distinctively different modal signatures and more probing natures of the inplane displacement fields. The complexity of mode shapes also graphically confirms the fact that the displacement fields can no longer be determined by the separation of variables for a plate under general boundary condition.
In this investigation, we deal with the inplane vibrations of thin sector plates with arbitrary boundary conditions in a unified manner. As the inner radius of the sector plate tends to zero, the annular sector plate coincides with the circular sector plate. In the previous paper [25], this configuration is practically realized by letting $a/b=$ 1×10^{5}. Table 3 lists the frequency parameters for fully free sector plates with various cutout ratios and sector angles. This problem is about a completely free sector plate, which represents a classical, but quite challenging, case for validating the solution. Under the current framework, the free edge condition is easily realized by setting all the stiffness constants to zero. Since the inplane vibration problems of circular sector plate are rarely investigated, the FEM results are used as the reference. A good agreement is observed between the current and the FEM solutions. The results have demonstrated that the radius ratio 10^{5} is considered adequately small to practically allow a decent prediction of the dynamic behaviors of the corresponding circular sector plate. A visual comparison of mode shape is shown in Fig. 3. The results convincingly demonstrate that the present series solution is accurate and reliable.
Fig. 2. The mode shapes for a CCCC annular sector plate with sector angle $\varphi =\pi $ and cutout ratio $a/b=$ 0.5. (The dashed background grid is the original undeformed configuration)
a) First mode
b) Second mode
c) Third mode
d) Fourth mode
e) Fifth mode
f) Sixth mode
Table 3. Comparison of frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for fully free circular and annular sector plates
$\varphi $

$a/b$

Method

Mode sequence


1

2

3

4

5

6


$\pi /$6

10^{5}

Present

1.5604

2.9922

3.5813

4.4336

5.4987

5.6205

FEM

1.5601

2.9922

3.5814

4.4336

5.4987

5.6221


0.3

Present

2.7882

4.3432

4.7680

5.6629

6.2355

6.7435


FEM

2.7882

4.3434

4.7687

5.6642

6.2368

6.7444


$\pi $

10^{5}

Present

1.1360

1.7546

1.8266

2.3500

2.5294

2.8974

FEM

1.1361

1.7546

1.8268

2.3503

2.5297

2.8978


0.3

Present

0.6947

1.5162

1.7022

2.4227

2.4317

2.9291


FEM

0.6947

1.5160

1.7023

2.4227

2.4316

2.9290


16$\pi /$9

10^{5}

Present

0.4581

1.0323

1.3985

1.6350

1.8257

1.9226

0.3

Present

0.2232

0.5060

0.9641

1.4736

1.5885

1.8013

Fig. 3. The mode shapes for a fully free circular sector plate with sector angle $\varphi =$ 16$\pi /$9. (The first row is calculated by the current approach, the second row is obtained using FEM)
a) First mode
b) Second mode
c) Third mode
In the literature, there are two distinct types of ‘simply supported’ boundary conditions for the inplane vibration of a plate [26]. For convenience, these two types of simply supported conditions are designated by SS1 and SS2. An edge associated with the first type, SS1, is characterized by the fact that the plate displacement parallel to the edge is specified to zero as well as normal stress perpendicular to the edge. In the numerical calculation, it can be physically realized by setting the stiffness of the tangential and normal springs to $\infty $ and 0, respectively. The second type is an exactly opposite scenario.
Table 4. Frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for CSS1CSS1 sector plates
$\varphi $

$a/b$

Method

Mode sequence


1

2

3

4

5

6


$\pi /$6

10^{5}

Present

2.2668

4.1503

6.0185

6.2555

7.8291

7.8821

0.5

Present

3.7821

6.8633

7.4685

8.2440

10.359

11.174


$\pi $

10^{5}

Present

1.9570

2.2668

3.0473

3.1784

3.9630

4.0858

FEM

1.9571

2.2671

3.0476

3.1790

3.9638

4.0870


0.5

Present

3.7821

3.9739

4.4861

5.1769

5.8617

6.2792


FEM

3.7838

3.9756

4.4879

5.1788

5.8641

6.2834


16$\pi /$9

10^{5}

Present

2.0238

2.2205

2.2668

2.7296

3.2526

3.2891

0.5

Present

3.7821

3.8441

4.0229

4.2994

4.6485

5.0415

The first six nondimensional frequency parameters, $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$, are given in Table 4 for sector plates with clamped and simply supported conditions (SS1) along the circumferential and radial edges, respectively. To understand the difference between the SS1 and SS2 boundary conditions, the problems are resolved by only replacing the SS1 conditions with SS2 while all the other parameters are kept the same. The results are shown in Table 5. A comparison of results in Tables 4 and 5 has revealed the noticeable difference between the SS1 and SS2 boundary conditions, as manifested in values of the frequency parameters. However, some frequency parameters appear to be unaffected even though the modal sequence is clearly changed. It is interesting to see that the first frequency parameters in Table 4 for annular sector plates ($a/b=$ 0.5) are not affected by the sector angles. The first three mode shapes for CSS1CSS1 sector plates ($a/b=$ 10^{5} and $\varphi =$ 16$\pi /$9) are plotted in Fig. 4. The results clearly show that the boundary springs have played a dominant role in these modes. In other words, one has to change the spring stiffness to effectively modify the modal properties.
Table 5. Frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for CSS2CSS2 sector plates
$\varphi $

$a/b$

Method

Mode sequence


1

2

3

4

5

6


$\pi /$6

10^{5}

Present

3.8313

6.2556

7.0148

7.8287

9.1762

10.172

0.5

Present

6.3925

6.8633

8.2435

10.359

11.192

11.880


$\pi $

10^{5}

Present

1.9570

3.0473

3.1784

3.8316

3.9630

4.0858

FEM

1.9571

3.0476

3.1790

3.8320

3.9638

4.0870


0.5

Present

3.9739

4.4861

5.1769

5.8617

6.2792

6.3322


FEM

3.9743

4.4865

5.1774

5.8623

6.2800

6.3330


16$\pi /$9

10^{5}

Present

2.0238

2.2205

2.7296

3.2526

3.2891

3.3577

0.5

Present

4.0229

4.2994

4.6485

5.0417

5.4466

5.8241

Fig. 4. The mode shapes for a CSS1CSS1 sector plate ($a/b=$ 10^{5} and $\varphi =$ 16$\pi /$9)
a) First mode
b) Second mode
c) Third mode
All the examples considered thus far have been limited to the classical boundary conditions which have been treated as the special cases of elastically restrained edges. Our attention will now be direct to elastically restrained plates. We first consider a simply supported (SS2SS2SS2) circular sector plate ($a/b=$ 10^{5}, $\varphi =$ 3$\pi /$2) with a uniform tangential restraint along each edge. This condition can be realized by setting ${k}_{ps1}={k}_{p\theta 0}={k}_{p\theta 1}=K$ and infinite stiffness for all other restraining springs. The first six frequency parameters, $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}\text{,}$ are presented in Table 6 for several different spring stiffness values. For a very large stiffness value, this edge condition will effectively degenerate to the familiar clamped condition. The second example concerns a sector plate ($\varphi =$16$\pi /$9) elastically supported along all edges. The stiffness values for the normal and tangential restraints are both set equal to 10^{8 }N/m^{2}. The first six frequency parameters are shown in Table 7 for sector plates with different radius ratios. Since the elastically restrained plates are rarely investigated, the FEM results are used as the reference. A good agreement is observed between the current and the FEM solutions. It can be seen that the increase of the radius ratio leads to increase of the frequency parameter of the plate. Fig. 5 shows the first three mode shapes for sector plates ($a/b=$ 10^{5} and $\varphi =$16$\pi /$9) with identical restraints at all edges $K=$10^{8} N/m^{2}.
In addition to its wide applicability to a range of boundary conditions, the current solution method is generally good for any inclusion angle. Although the results were not shown here, we have actually validated the model for an annular “sector” plate with a $2\pi $ inclusion angle. While making no attempt to determine how small a sector angle is allowed in this model, we can reasonably assume that the threshold will set up by the fact that the corresponding annular sector will have to behave like a plate, rather than a bar (of varying crosssection) as the sector angle becomes sufficiently small. Being applicable to any inclusion angle is one of the important advantages of the current solution method in view that the actual limitations of other solution techniques were often not stated clearly in the (limited) existing publications.
Table 6. Frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for SS2SS2SS2 sector plates ($\varphi =$3$\pi /$2) with uniform elastic restraint ${k}_{ps1}={k}_{p\theta 0}={k}_{p\theta 1}=K$
$K$
(N/m^{2})

Method

Mode sequence


1

2

3

4

5

6


1

Present

0.8793

1.7319

2.4129

2.4818

2.5076

3.0475

FEM

0.8776

1.7320

2.4132

2.4819

2.5077

3.0476


10^{4}

Present

0.8794

1.7320

2.4130

2.4818

2.5074

3.0475

FEM

0.8783

1.7321

2.4136

2.4821

2.5076

3.0477


10^{8}

Present

1.5094

2.1765

2.5713

2.6622

2.8940

3.1289

FEM

1.5063

2.1732

2.5706

2.6617

2.8932

3.1269


∞

Present

2.3609

2.7548

3.2007

3.6334

3.7276

3.7623

FEM

2.3605

2.7550

3.2003

3.6340

3.7278

3.7628

Table 7. Frequency parameters $\mathrm{\Omega}=\omega b\sqrt{\rho (1{\mu}^{2})/E}$ for sector plates ($\varphi =$16$\pi /$9) with identical restraints at all edges $K=$ 10^{8} N/m^{2}
$a/b$

Method

Mode sequence


1

2

3

4

5

6


10^{5}

Present

0.9592

0.9984

1.2594

1.4121

1.6924

1.9470

FEM

0.9611

0.9997

1.2577

1.4073

1.6878

1.9443


0.3

Present

1.1168

1.1357

1.2752

1.3300

1.5126

1.7824

FEM

1.1160

1.1351

1.2749

1.3297

1.5121

1.7820


0.5

Present

1.3206

1.3256

1.4053

1.4441

1.5432

1.6975

0.7

Present

1.7128

1.7129

1.7446

1.7790

1.8204

1.8716

Fig. 5. The mode shapes for a sector plate ($a/b=$ 10^{5} and $\varphi =$ 16$\pi /$9) with identical restraints at all edges $K=$ 10^{8}N/m^{2}
a) First mode
b) Second mode
c) Third mode
A general series solution has been developed for the inplane vibration analysis of sector plates with arbitrary boundary conditions and inclusion angles. The inplane displacement fields are invariably expressed as a new form of trigonometric series expansions with drastically improved convergence as compared with the conventional Fourier series. Unlike most of the existing methods in which each of the frequency parameters is calculated, repeatedly and iteratively, from a nonlinear characteristic equation, all the modal parameters can now be easily and simultaneously determined from solving a standard matrix eigenvalue problem. The proposed method provides a unified means for extracting the modal parameters and predicting the inplane vibrations of sector plates with different boundary conditions. Of equal importance, the current solution is generally applicable to sector plates with any inclusion angle up to a full 360 degree. The excellent accuracy and convergence of the present solutions have been validated through a number of examples.
The work was supported by the International S&T Cooperation Program of China (ISTCP, Grant No. 2011DFR90440) and the key project of the National Natural Science Foundation of China (Grant No. 50939002).