RETRACTED ARTICLE: A novel approach for free vibration of circular/annular sector plates using Reddy’s third order shear deformation theory

Abstract

A new exact closed-form solution based on Reddy’ third-order shear deformation plate theory for free vibration of thick circular/annular sector plates is presented. Characteristic equations are given for sector plates having hard simply supported radial edges and all combinations of free, soft simply supported, hard simply supported and clamped boundary conditions along the circular edges. Based on the present solution, the governing equations of the vibrated thick circular/annular sector plates were exactly solved by introducing the new auxiliary and potential functions as well as using the separation method of variables. To clarify the efficiency and accuracy of the present solution, several comparison studies are examined with the available data in the literature. Also, natural frequencies of the circular/annular sector plates are presented for different thickness–radius ratios, inner–outer radius ratios, sector angle values and boundary conditions.

Appendices

Appendix 1

There exist closed-form exact solutions to the characteristic equations of annular sector plates with hard simply supported at the radial edges and all possible combinations of free, soft simply supported, hard simply supported and clamped boundary conditions along circular edges.

Substituting Eqs. (19), (40) and (42) into the boundary conditions, given by Eqs. (47)–(50), yields an eighth-order determinant for the frequency parameters β. For the sake of clarity, the determinant will not be expanded and the characteristic equations are represented in a matrix form. First four rows of the 8 × 8 matrix are related to the boundary conditions at the inner edge, while second ones are related to those at the outer edge. Thus, the 8 × 8 matrix is divided into two 4 × 8 sub-matrices corresponding to boundary conditions at the inner and outer edges of the plate. Following sub-matrices are given for clamped, hard simply supported, soft simply supported and free boundary conditions:

Case 1: Clamped annular sector plates

$$ A = \left| {\begin{array}{*{20}c} {w_{11} \left( {\chi_{1} r^{ * } } \right)} & {w_{12} \left( {\chi_{1} r^{ * } } \right)} & {w_{21} \left( {\chi_{2} r^{ * } } \right)} & {w_{22} \left( {\chi_{2} r^{ * } } \right)} & {w_{31} \left( {\chi_{3} r^{ * } } \right)} & {w_{32} \left( {\chi_{3} r^{ * } } \right)} & 0 & 0 \\ {w^{\prime}_{11} \left( {\chi_{1} r^{ * } } \right)} & {w^{\prime}_{12} \left( {\chi_{1} r^{ * } } \right)} & {w^{\prime}_{21} \left( {\chi_{2} r^{ * } } \right)} & {w^{\prime}_{22} \left( {\chi_{2} r^{ * } } \right)} & {w^{\prime}_{31} \left( {\chi_{3} r^{ * } } \right)} & {w^{\prime}_{32} \left( {\chi_{3} r^{ * } } \right)} & 0 & 0 \\ {a_{1} w^{\prime}_{11} \left( {\chi_{1} r^{ * } } \right)} & {a_{1} w^{\prime}_{12} \left( {\chi_{1} r^{ * } } \right)} & {a_{2} w^{\prime}_{21} \left( {\chi_{2} r^{ * } } \right)} & {a_{2} w^{\prime}_{22} \left( {\chi_{2} r^{ * } } \right)} & {a_{3} w^{\prime}_{31} \left( {\chi_{3} r^{ * } } \right)} & {a_{3} w^{\prime}_{32} \left( {\chi_{3} r^{ * } } \right)} & {\frac{{pw_{41} \left( {\chi_{4} r^{ * } } \right)}}{{r^{ * } }}} & {\frac{{pw_{42} \left( {\chi_{4} r^{ * } } \right)}}{{r^{ * } }}} \\ {\frac{p}{{r^{ * } }}a_{1} w_{11} \left( {\chi_{1} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{1} w_{12} \left( {\chi_{1} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{2} w_{21} \left( {\chi_{2} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{2} w_{22} \left( {\chi_{2} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{3} w_{31} \left( {\chi_{3} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{3} w_{32} \left( {\chi_{3} r^{ * } } \right)} & {w^{\prime}_{41} \left( {\chi_{4} r^{ * } } \right)} & {w^{\prime}_{42} \left( {\chi_{4} r^{ * } } \right)} \\ \end{array} } \right| $$

(75)

where the prime (′) indicates the derivative with respect to the $r^{*}$; $w_{ij} \left( {p,\chi_{i} r^{ * } } \right)$ is concisely expressed as $w_{ij} \left( {\chi_{i} r^{ * } } \right)$; $\cos \left( {p\theta } \right)e^{i\omega t}$ and $\sin \left( {p\theta } \right)e^{i\omega t}$ are eliminated for the brevity.

Case 2: Hard simply supported annular sector plates

$$ B = \left| {\begin{array}{*{20}c} {w_{11} \left( {\chi_{1} r^{ * } } \right)} & {w_{12} \left( {\chi_{1} r^{ * } } \right)} & {w_{21} \left( {\chi_{2} r^{ * } } \right)} & {w_{22} \left( {\chi_{2} r^{ * } } \right)} & {w_{31} \left( {\chi_{3} r^{ * } } \right)} & {w_{32} \left( {\chi_{3} r^{ * } } \right)} & 0 & 0 \\ {\mu_{1} \left( {r^{ * } } \right)} & {\mu \mu_{1} \left( {r^{ * } } \right)} & {\mu_{2} \left( {r^{ * } } \right)} & {\mu \mu_{2} \left( {r^{ * } } \right)} & {\mu_{3} \left( {r^{ * } } \right)} & {\mu \mu_{3} \left( {r^{ * } } \right)} & {\mu_{4} \left( {r^{ * } } \right)} & {\mu \mu_{4} \left( {r^{ * } } \right)} \\ {\alpha_{1} \left( {r^{ * } } \right)} & {\alpha \alpha_{1} \left( {r^{ * } } \right)} & {\alpha_{2} \left( {r^{ * } } \right)} & {\alpha \alpha_{2} \left( {r^{ * } } \right)} & {\alpha_{3} \left( {r^{ * } } \right)} & {\alpha \alpha_{3} \left( {r^{ * } } \right)} & {\alpha_{4} \left( {r^{ * } } \right)} & {\alpha \alpha_{4} \left( {r^{ * } } \right)} \\ {\frac{p}{{r^{ * } }}a_{1} w_{11} \left( {\chi_{1} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{1} w_{12} \left( {\chi_{1} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{2} w_{21} \left( {\chi_{2} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{2} w_{22} \left( {\chi_{2} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{3} w_{31} \left( {\chi_{3} r^{ * } } \right)} & {\frac{p}{{r^{ * } }}a_{3} w_{32} \left( {\chi_{3} r^{ * } } \right)} & {w^{\prime}_{41} \left( {\chi_{4} r^{ * } } \right)} & {w^{\prime}_{42} \left( {\chi_{4} r^{ * } } \right)} \\ \end{array} } \right| $$

(76)

where

$$ \mu_{i} \left( {r^{ * } } \right) = - \frac{{\left( {4F + \left( { - 3C + 4F} \right)a_{i} } \right)}}{{3r^{ * 2} }}\left( {\nu r^{ * } w_{i1}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) +\, r^{ * 2} w_{i1}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) - \nu p^{2} w_{i1} \left( {p,\chi_{i} r^{ * } } \right)} \right) $$

(77)

$$ \mu_{4} \left( {r^{ * } } \right) = - \frac{{p\left( {3C - 4F} \right)\left( {\nu - 1} \right)}}{{3r^{ * 2} }}\left( {r^{ * } w_{41}^{{\prime }} \left( {p,\chi_{4} r^{ * } } \right) -\, w_{41} \left( {p,\chi_{4} r^{ * } } \right)} \right) $$

(78)

$$ \mu \mu_{i} \left( {r^{ * } } \right) = - \frac{{\left( {4F + \left( { - 3C + 4F} \right)a_{i} } \right)}}{{3r^{ * 2} }}\left( {\nu r^{ * } w_{i2}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) +\,r^{ * 2} w_{i2}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) - \nu p^{2} w_{i2} \left( {p,\chi_{i} r^{ * } } \right)} \right) $$

(79)

$$ \mu \mu_{4} \left( {r^{ * } } \right) = - \frac{{p\left( {3C - 4F} \right)\left( {\nu - 1} \right)}}{{3r^{ * 2} }}\left( {r^{ * } w_{42}^{{\prime }} \left( {p,\chi_{4} r^{ * } } \right) - \,w_{42} \left( {p,\chi_{4} r^{ * } } \right)} \right) $$

(80)

$$ \alpha_{i} \left( {r^{ * } } \right) = - \frac{{\left( {4G + \left( { - 3F + 4G} \right)a_{i} } \right)}}{{3r^{ * 2} }}\left( {\nu r^{ * } w_{i1}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) + r^{ * 2} w_{i1}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) - \nu p^{2} w_{i1} \left( {p,\chi_{i} r^{ * } } \right)} \right) $$

(81)

$$ \alpha_{4} \left( {r^{ * } } \right) = - \frac{{p\left( {3F - 4G} \right)\left( {\nu - 1} \right)}}{{3r^{ * 2} }}\left( {r^{ * } w_{41}^{{\prime }} \left( {p,\chi_{4} r^{ * } } \right) - w_{41} \left( {p,\chi_{4} r^{ * } } \right)} \right) $$

(82)

$$ \alpha \alpha_{i} \left( {r^{ * } } \right) = - \frac{{\left( {4G + \left( { - 3F + 4G} \right)a_{i} } \right)}}{{3r^{ * 2} }}\left( {\nu r^{ * } w_{i2}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) + r^{ * 2} w_{i2}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) - \nu p^{2} w_{i2} \left( {p,\chi_{i} r^{ * } } \right)} \right) $$

(83)

$$ \alpha \alpha_{4} \left( {r^{ * } } \right) = - \frac{{p\left( {3F - 4G} \right)\left( {\nu - 1} \right)}}{{3r^{ * 2} }}\left( {r^{ * } w_{42}^{{\prime }} \left( {p,\chi_{4} r^{ * } } \right) - w_{42} \left( {p,\chi_{4} r^{ * } } \right)} \right) $$

(84)

and i = 1, 2, 3.

Case 3: Soft simply supported annular sector plates

$$ C = \left| {\begin{array}{*{20}c} {w_{11} \left( {\chi_{1} r^{ * } } \right)} & {w_{12} \left( {\chi_{1} r^{ * } } \right)} & {w_{21} \left( {\chi_{2} r^{ * } } \right)} & {w_{22} \left( {\chi_{2} r^{ * } } \right)} & {w_{31} \left( {\chi_{3} r^{ * } } \right)} & {w_{32} \left( {\chi_{3} r^{ * } } \right)} & 0 & 0 \\ {\mu_{1} \left( {r^{ * } } \right)} & {\mu \mu_{1} \left( {r^{ * } } \right)} & {\mu_{2} \left( {r^{ * } } \right)} & {\mu \mu_{2} \left( {r^{ * } } \right)} & {\mu_{3} \left( {r^{ * } } \right)} & {\mu \mu_{3} \left( {r^{ * } } \right)} & {\mu_{4} \left( {r^{ * } } \right)} & {\mu \mu_{4} \left( {r^{ * } } \right)} \\ {\alpha_{1} \left( {r^{ * } } \right)} & {\alpha \alpha_{1} \left( {r^{ * } } \right)} & {\alpha_{2} \left( {r^{ * } } \right)} & {\alpha \alpha_{2} \left( {r^{ * } } \right)} & {\alpha_{3} \left( {r^{ * } } \right)} & {\alpha \alpha_{3} \left( {r^{ * } } \right)} & {\alpha_{4} \left( {r^{ * } } \right)} & {\alpha \alpha_{4} \left( {r^{ * } } \right)} \\ {\eta_{1} \left( {r^{ * } } \right)} & {\eta \eta_{1} \left( {r^{ * } } \right)} & {\eta_{2} \left( {r^{ * } } \right)} & {\eta \eta_{2} \left( {r^{ * } } \right)} & {\eta_{3} \left( {r^{ * } } \right)} & {\eta \eta_{3} \left( {r^{ * } } \right)} & {\eta_{4} \left( {r^{ * } } \right)} & {\eta \eta_{4} \left( {r^{ * } } \right)} \\ \end{array} } \right| $$

(85)

where

$$ \eta_{i} \left( {r^{ * } } \right) = - \frac{{2p\left( {4\left( {3F - 4G} \right) + \left( { - 9C + 24F - 16G} \right)a_{i} } \right)}}{{9r^{ * 2} }}\left( {r^{ * } w_{i1}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) - w_{i1} \left( {p,\chi_{i} r^{ * } } \right)} \right) $$

(86)

$$ \eta_{4} \left( {r^{ * } } \right) = - \left( {C - \frac{8}{3}F + \frac{16}{9}G} \right)\left( { - \frac{{p^{2} }}{{r^{ * 2} }}w_{41} \left( {p,\chi_{i} r^{ * } } \right) - w_{41}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) + \frac{{w_{41}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right)}}{{r^{ * } }}} \right) $$

(87)

$$ \eta \eta_{i} \left( {r^{ * } } \right) = - \frac{{2p\left( {4\left( {3F - 4G} \right) + \left( { - 9C + 24F - 16G} \right)a_{i} } \right)}}{{9r^{ * 2} }}\left( {r^{ * } w_{i2}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) - w_{i2} \left( {p,\chi_{i} r^{ * } } \right)} \right) $$

(88)

$$ \eta \eta_{4} \left( {r^{ * } } \right) = - \left( {C - \frac{8}{3}F + \frac{16}{9}G} \right)\left( { - \frac{{p^{2} }}{{r^{ * 2} }}w_{42} \left( {p,\chi_{i} r^{ * } } \right) -\, w_{42}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) + \frac{{w_{42}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right)}}{{r^{ * } }}} \right) $$

(89)

Case 4: Free annular sector plates

$$ D = \left| {\begin{array}{*{20}c} {\varGamma_{1} \left( {r^{ * } } \right)} & {\varGamma \varGamma_{1} \left( {r^{ * } } \right)} & {\varGamma_{2} \left( {r^{ * } } \right)} & {\varGamma \varGamma_{2} \left( {r^{ * } } \right)} & {\varGamma_{3} \left( {r^{ * } } \right)} & {\varGamma \varGamma_{3} \left( {r^{ * } } \right)} & {\varGamma_{4} \left( {r^{ * } } \right)} & {\varGamma \varGamma_{4} \left( {r^{ * } } \right)} \\ {\mu_{1} \left( {r^{ * } } \right)} & {\mu \mu_{1} \left( {r^{ * } } \right)} & {\mu_{2} \left( {r^{ * } } \right)} & {\mu \mu_{2} \left( {r^{ * } } \right)} & {\mu_{3} \left( {r^{ * } } \right)} & {\mu \mu_{3} \left( {r^{ * } } \right)} & {\mu_{4} \left( {r^{ * } } \right)} & {\mu \mu_{4} \left( {r^{ * } } \right)} \\ {\alpha_{1} \left( {r^{ * } } \right)} & {\alpha \alpha_{1} \left( {r^{ * } } \right)} & {\alpha_{2} \left( {r^{ * } } \right)} & {\alpha \alpha_{2} \left( {r^{ * } } \right)} & {\alpha_{3} \left( {r^{ * } } \right)} & {\alpha \alpha_{3} \left( {r^{ * } } \right)} & {\alpha_{4} \left( {r^{ * } } \right)} & {\alpha \alpha_{4} \left( {r^{ * } } \right)} \\ {\eta_{1} \left( {r^{ * } } \right)} & {\eta \eta_{1} \left( {r^{ * } } \right)} & {\eta_{2} \left( {r^{ * } } \right)} & {\eta \eta_{2} \left( {r^{ * } } \right)} & {\eta_{3} \left( {r^{ * } } \right)} & {\eta \eta_{3} \left( {r^{ * } } \right)} & {\eta_{4} \left( {r^{ * } } \right)} & {\eta \eta_{4} \left( {r^{ * } } \right)} \\ \end{array} } \right| $$

(90)

where

$$ \begin{aligned} & J_{1} = \frac{{12\left( {1 - \nu } \right)}}{{\delta^{4} }}\left( {\frac{{\hat{I}_{1} }}{2} - 4\hat{I}_{3} + 8\hat{I}_{5} } \right) + \frac{4}{3}\tilde{I}_{5} \beta^{2} ,\,\,\,\,\,\,\,\,\,\,J_{2} = \frac{{12\left( {1 - \nu } \right)}}{{\delta^{4} }}\left( {\frac{{\hat{I}_{1} }}{2} - 4\hat{I}_{3} + 8\hat{I}_{5} } \right) - \frac{16}{9}\hat{I}_{7} \beta^{2} \\ & J_{3} = \frac{{16\left( {3\hat{I}_{5} - 4\hat{I}_{7} } \right)\left( {\nu - 2} \right)}}{{3\delta^{2} }},\,\,\,\,\,\,\,\,\,\,J_{4} = - \frac{{16\left( {3\hat{I}_{5} - 4\hat{I}_{7} } \right)\left( {\nu - 1} \right)}}{{3\delta^{2} }},\,\,\,\,\,\,\,\,\,\,J_{5} = - \frac{{64\hat{I}_{7} }}{{3\delta^{2} }} \\ & J_{6} = \frac{{64\hat{I}_{7} \left( {\nu - 2} \right)}}{{3\delta^{2} }},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,J_{7} = \frac{{16\left( {3\hat{I}_{5} - 4\hat{I}_{7} } \right)}}{{3\delta^{2} }},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,J_{8} = - \frac{{64\hat{I}_{7} \left( {\nu - 3} \right)}}{{3\delta^{2} }} \\ \end{aligned} $$

(91)

and

$$ \begin{aligned} & \varGamma_{i} \left( {r^{ * } } \right) = \frac{1}{{r^{ * 3} }}\left( {r^{ * } \left( { - J_{5} - J_{6} p^{2} + J_{2} r^{ * 2} - \left( {J_{7} + J_{4} p^{2} + J_{7} p^{2} - J_{1} r^{ * 2} } \right)a_{i} } \right)w_{i1}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right)} \right. \\ & \quad + \left. {r^{ * 2} \left( {J_{5} + J_{7} a_{i} } \right)\left( {w_{i1}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) + r^{ * } w_{i1}^{{{\prime \prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right)} \right) - p^{2} \left( {J_{8} + \left( {J_{3} - J_{7} } \right)a_{i} } \right)w_{i1} \left( {p,\chi_{i} r^{ * } } \right)} \right) \\ \end{aligned}. $$

(92)

$$ \varGamma_{4} \left( {r^{ * } } \right) = - \frac{{p\left( {\left( {J_{3} + J_{7} } \right)r^{ * } w_{41}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) + \left( {J_{4} p^{2} - J_{1} r^{ * 2} } \right)w_{41} \left( {p,\chi_{i} r^{ * } } \right)} \right)}}{{r^{ * 3} }} $$

(93)

$$ \begin{aligned} \varGamma \varGamma_{i} \left( {r^{ * } } \right) = \frac{1}{{r^{ * 3} }}\left( {r^{ * } \left( { - J_{5} - J_{6} p^{2} + J_{2} r^{ * 2} - \left( {J_{7} + J_{4} p^{2} + J_{7} p^{2} - J_{1} r^{ * 2} } \right)a_{i} } \right)w_{i2}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right)} \right. \hfill \\ + \left. {r^{ * 2} \left( {J_{5} + J_{7} a_{i} } \right)\left( {w_{i2}^{{{\prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right) + r^{ * } w_{i2}^{{{\prime \prime \prime }}} \left( {p,\chi_{i} r^{ * } } \right)} \right) - p^{2} \left( {J_{8} + \left( {J_{3} - J_{7} } \right)a_{i} } \right)w_{i2} \left( {p,\chi_{i} r^{ * } } \right)} \right) \hfill \\ \end{aligned} $$

(94)

$$ \varGamma \varGamma_{4} \left( {r^{ * } } \right) = - \frac{{p\left( {\left( {J_{3} + J_{7} } \right)r^{ * } w_{42}^{{\prime }} \left( {p,\chi_{i} r^{ * } } \right) + \left( {J_{4} p^{2} - J_{1} r^{ * 2} } \right)w_{42} \left( {p,\chi_{i} r^{ * } } \right)} \right)}}{{r^{ * 3} }} $$

(95)

For each case, an exact solution can be obtained by setting the determinant of the matrices in Table 11 equal to zero. Roots of the determinant are the natural frequencies of annular sector plates with specific boundary conditions at inner and outer edges of annular sector plates for a given wave number.

Table 11 Exact closed-form characteristic equations for annular sector plates with different combination of boundary conditions in matrix forms

Appendix 2

The closed-form exact solutions to the characteristic equations of circular sector plates under hard simply supported at the radial edge and free, soft simply supported, hard simply supported and clamped boundary conditions at the radial edge are presented for two cases.

Case 1: P ≥ 1

After expanding the determinant and performing mathematical manipulations, exact characteristic equations can be listed below for each individual case.

Case 1–1: Clamped circular sector plates

$$ \begin{aligned} & - x_{1} \left( {L_{3} \left( 1 \right)w_{21} \left( 1 \right) - L_{2} \left( 1 \right)w_{31} \left( 1 \right)} \right)\left( {L_{1} \left( 1 \right)L_{4} \left( 1 \right) - p^{2} w_{11} \left( 1 \right)w_{41} \left( 1 \right)} \right) \\ & + x_{2} \left( {L_{3} \left( 1 \right)w_{11} \left( 1 \right) - L_{1} \left( 1 \right)w_{31} \left( 1 \right)} \right)\left( {L_{2} \left( 1 \right)L_{4} \left( 1 \right) - p^{2} w_{21} \left( 1 \right)w_{41} \left( 1 \right)} \right) \\ & - x_{3} \left( {L_{2} \left( 1 \right)w_{11} \left( 1 \right) - L_{1} \left( 1 \right)w_{21} \left( 1 \right)} \right)\left( {L_{3} \left( 1 \right)L_{4} \left( 1 \right) - p^{2} w_{31} \left( 1 \right)w_{41} \left( 1 \right)} \right) = 0 \\ \end{aligned} $$

(96)

where

$$ w_{i1} \left( {r^{ * } } \right) = w_{i1} \left( {p,\chi_{i} r^{ * } } \right),\,\,\,\,\,\,\,\,\,\,L_{i} \left( {r^{ * } } \right) = \frac{\partial }{{\partial r^{ * } }}w_{i1} \left( {p,\chi_{i} r^{ * } } \right),\,\,\,\,\,i = 1,2,3,4 $$

(97)

Case 1–2: Hard simply supported circular sector plates

$$ \begin{aligned} & L_{4} \left( 1 \right)\left( {\left( { - \bar{\alpha }_{3} \varLambda_{2} + \bar{\alpha }_{2} \varLambda_{3} } \right)w_{11} \left( 1 \right) + \left( {\bar{\alpha }_{3} \varLambda_{1} - \bar{\alpha }_{1} \varLambda_{3} } \right)w_{21} \left( 1 \right) + \left( { - \bar{\alpha }_{2} \varLambda_{1} + \bar{\alpha }_{1} \varLambda_{2} } \right)w_{31} \left( 1 \right)} \right) \\ & \quad + p\left( {\left( {x_{2} - x_{3} } \right)\left( {\bar{\alpha }_{4} \varLambda_{1} - \bar{\alpha }_{1} \varLambda_{4} } \right)w_{21} \left( 1 \right)w_{31} \left( 1 \right)} \right. \\ & \quad + \left. {w_{11} \left( 1 \right)\left( {\left( {x_{1} - x_{2} } \right)\left( {\bar{\alpha }_{4} \varLambda_{3} - \bar{\alpha }_{3} \varLambda_{4} } \right)w_{21} \left( 1 \right) - \left( {x_{1} - x_{3} } \right)\left( {\bar{\alpha }_{4} \varLambda_{2} - \bar{\alpha }_{2} \varLambda_{4} } \right)w_{31} \left( 1 \right)} \right)} \right) = 0 \\ \end{aligned} $$

(98)

where

$$ \bar{\alpha }_{i} = - \frac{1}{3}\left( {4G + \left( { - 3F + 4G} \right)x_{i} } \right)\left( {q_{i} \left( 1 \right) + \nu \left( {L_{i} \left( 1 \right) - p^{2} w_{i1} \left( 1 \right)} \right)} \right),\,\,\,i = 1,2,3 $$

(99)

$$ \bar{\alpha }_{4} = - \frac{1}{3}\left( {3F - 4G} \right)p\left( {\nu - 1} \right)\left( {L_{4} \left( 1 \right) - w_{41} \left( 1 \right)} \right) $$

(100)

$$ \varLambda_{i} = - \frac{1}{3}\left( {4F + \left( { - 3C + 4F} \right)x_{i} } \right)\left( {q_{i} \left( 1 \right) + \nu \left( {L_{i} \left( 1 \right) - p^{2} w_{i1} \left( 1 \right)} \right)} \right),\,\,\,i = 1,2,3 $$

(101)

$$ \varLambda_{4} = - \frac{1}{3}\left( {3C - 4F} \right)p\left( {\nu - 1} \right)\left( {L_{4} \left( 1 \right) - w_{41} \left( 1 \right)} \right) $$

(102)

$$ q_{i} \left( {r^{ * } } \right) = \frac{\partial }{{\partial r^{ * } }}L_{i} \left( {r^{ * } } \right),\,\,\,\,\,\,i = 1,2,3,4 $$

(103)

Case 1–3: Soft simply supported circular sector plates

$$ \begin{aligned} & - \left( {\varLambda_{4} \varPsi_{3} + \varLambda_{3} \varPsi_{4} } \right)\left( {\bar{\alpha }_{2} w_{11} \left( 1 \right) - \bar{\alpha }_{1} w_{21} \left( 1 \right)} \right) \\ & \quad + \bar{\alpha }_{3} \left( {\left( {\varLambda_{4} \varPsi_{2} + \varLambda_{2} \varPsi_{4} } \right)w_{11} \left( 1 \right) - \left( {\varLambda_{4} \varPsi_{1} + \varLambda_{1} \varPsi_{4} } \right)w_{21} \left( 1 \right)} \right) \\ & \quad + \left( {\bar{\alpha }_{2} \left( {\varLambda_{4} \varPsi_{1} + \varLambda_{1} \varPsi_{4} } \right) - \bar{\alpha }_{1} \left( {\varLambda_{4} \varPsi_{2} + \varLambda_{2} \varPsi_{4} } \right)} \right)w_{31} \left( 1 \right) \\ & \quad + \bar{\alpha }_{4} \left( {\varPsi_{3} \left( {\varLambda_{2} w_{11} \left( 1 \right) - \varLambda_{1} w_{21} \left( 1 \right)} \right) + \varLambda_{3} \left( { - \varPsi_{2} w_{11} \left( 1 \right) + \varPsi_{1} w_{21} \left( 1 \right)} \right) + \left( { - \varLambda_{2} \varPsi_{1} + \varLambda_{1} \varPsi_{2} } \right)w_{31} \left( 1 \right)} \right) = 0 \\ \end{aligned} $$

(104)

where

$$ \varPsi_{i} = \frac{2}{9}p\left( { - 12F + 16G + \left( {9C - 24F + 16G} \right)x_{i} } \right)\left( {L_{i} \left( 1 \right) - w_{i1} \left( 1 \right)} \right),\,\,\,\,\,i = 1,2,3 $$

(105)

$$ \varPsi_{4} = \left( {C - \frac{8}{3}F + \frac{16}{9}G} \right)\left( {L_{4} \left( 1 \right) - q_{4} \left( 1 \right) - p^{2} w_{41} \left( 1 \right)} \right) $$

(106)

Case 1–4: Free circular sector plates

$$ \begin{aligned} & \bar{\alpha }_{4} \left( {\varLambda_{3} \left( {\varUpsilon_{2} \varPsi_{1} - \varUpsilon_{1} \varPsi_{2} } \right) + \varLambda_{2} \left( { - \varUpsilon_{3} \varPsi_{1} + \varUpsilon_{1} \varPsi_{3} } \right) + \varLambda_{1} \left( {\varUpsilon_{3} \varPsi_{2} - \varUpsilon_{2} \varPsi_{3} } \right)} \right) \\ & \quad + \bar{\alpha }_{3} \left( {\varLambda_{4} \left( { - \varUpsilon_{2} \varPsi_{1} + \varUpsilon_{1} \varPsi_{2} } \right) + \varLambda_{2} \left( {\varUpsilon_{4} \varPsi_{1} + \varUpsilon_{1} \varPsi_{4} } \right) - \varLambda_{1} \left( {\varUpsilon_{4} \varPsi_{2} + \varUpsilon_{2} \varPsi_{4} } \right)} \right) \\ & \quad + \bar{\alpha }_{2} \left( {\varLambda_{4} \left( {\varUpsilon_{3} \varPsi_{1} - \varUpsilon_{1} \varPsi_{3} } \right) - \varLambda_{3} \left( {\varUpsilon_{4} \varPsi_{1} + \varUpsilon_{1} \varPsi_{4} } \right) + \varLambda_{1} \left( {\varUpsilon_{4} \varPsi_{3} + \varUpsilon_{3} \varPsi_{4} } \right)} \right) \\ & \quad + \bar{\alpha }_{1} \left( {\varLambda_{4} \left( { - \varUpsilon_{3} \varPsi_{2} + \varUpsilon_{2} \varPsi_{3} } \right) + \varLambda_{3} \left( {\varUpsilon_{4} \varPsi_{2} + \varUpsilon_{2} \varPsi_{4} } \right) - \varLambda_{2} \left( {\varUpsilon_{4} \varPsi_{3} + \varUpsilon_{3} \varPsi_{4} } \right)} \right) = 0 \\ \end{aligned} $$

(107)

where

$$ \begin{aligned} & \varUpsilon_{i} = p^{2} \left( {J_{7} a_{i} - J_{3} a_{i} - J_{8} } \right)w_{i1} \left( 1 \right) + \left( {J_{1} a_{i} + J_{2} - J_{7} a_{i} p^{2} - J_{7} a_{i} - J_{4} a_{i} p^{2} - J_{6} p^{2} - J_{5} } \right)L_{i} \left( 1 \right) \\ & \quad + \left( {J_{7} a_{i} + J_{5} } \right)\left( {q_{i} \left( 1 \right) + s_{i} \left( 1 \right)} \right),\quad i = 1,2,3 \\ \end{aligned} $$

(108)

$$ \varUpsilon_{4} = p\left( {J_{1} - p^{2} J_{4} } \right)w_{41} \left( 1 \right) - p\left( {J_{7} + J_{3} } \right)L_{4} \left( 1 \right) $$

(109)

$$ s_{i} \left( {r^{ * } } \right) = \frac{\partial }{{\partial r^{ * } }}q_{i} \left( {r^{ * } } \right) $$

(110)

Case 2: 0 < p < 1

In this case, obtaining frequency parameter needs to determine the determinant of a 8 × 8 matrix. The first four rows of the 8 × 8 matrix are replaced by A, B, C or D sub matrices in Appendix 1, depending on boundary condition, and the last four rows is E sub matrix as follows

Subcase 1.x ₁ > 0, x ₂ < 0, x ₃ > 0, x ₄ > 0

$$ E = \left| {\begin{array}{*{20}c} 0 & {\varOmega_{12} } & 0 & {\varOmega_{14} } & 0 & {\varOmega_{16} } & 0 & 0 \\ 0 & {\varOmega_{22} } & 0 & {\varOmega_{24} } & 0 & {\varOmega_{26} } & 0 & {\varOmega_{28} } \\ {\varOmega_{31} } & {\varOmega_{32} } & {\varOmega_{33} } & {\varOmega_{34} } & {\varOmega_{35} } & {\varOmega_{36} } & {\varOmega_{37} } & {\varOmega_{38} } \\ {\varOmega_{41} } & {\varOmega_{42} } & {\varOmega_{43} } & {\varOmega_{44} } & {\varOmega_{45} } & {\varOmega_{46} } & 0 & 0 \\ \end{array} } \right| $$

(111)

where

$$ \begin{aligned} & \varOmega_{12} = \frac{\pi }{2}\chi_{1}^{ - p} ,\,\,\,\,\,\varOmega_{14} = - \chi_{2}^{ - p} ,\,\,\,\,\,\varOmega_{16} = \frac{\pi }{2}\chi_{3}^{ - p} \\ & \varOmega_{22} = - a_{1} \frac{\pi }{2}\chi_{1}^{ - p} ,\,\,\,\,\,\varOmega_{24} = a_{2} \chi_{2}^{ - p} ,\,\,\,\,\,\varOmega_{26} = - a_{3} \frac{\pi }{2}\chi_{3}^{ - p} ,\,\,\,\,\,\varOmega_{28} = - \frac{\pi }{2}\chi_{4}^{ - p} \\ & \varOmega_{31} = a_{1} \chi_{1}^{p} ,\,\,\,\,\,\varOmega_{32} = \frac{{ - \pi a_{1} \chi_{1}^{p} }}{{2\sin \left( {\pi p} \right)}},\,\,\,\,\,\varOmega_{33} = a_{2} \chi_{2}^{p} ,\,\,\,\,\,\varOmega_{34} = a_{2} \chi_{2}^{p} \cot \left( {\pi p} \right), \\ & \varOmega_{35} = a_{3} \chi_{3}^{p} ,\,\,\,\,\,\varOmega_{36} = \frac{{ - \pi a_{3} \chi_{3}^{p} }}{{2\sin \left( {\pi p} \right)}},\,\,\,\,\,\varOmega_{37} = - \chi_{4}^{p} ,\,\,\,\,\,\varOmega_{38} = \frac{{\pi \chi_{4}^{p} }}{{2\sin \left( {\pi p} \right)}} \\ & \varOmega_{41} = \chi_{1}^{p} ,\,\,\,\,\,\varOmega_{42} = \frac{{ - \pi \chi_{1}^{p} }}{{2\sin \left( {\pi p} \right)}},\,\,\,\,\,\varOmega_{43} = \chi_{2}^{p} ,\,\,\,\,\,\varOmega_{44} = \chi_{2}^{p} \cot \left( {\pi p} \right), \\ & \varOmega_{45} = \chi_{3}^{p} ,\,\,\,\,\,\varOmega_{46} = \frac{{ - \pi \chi_{3}^{p} }}{{2\sin \left( {\pi p} \right)}} \\ \end{aligned} $$

(112)

Subcase 2.x ₁ > 0, x ₁ < 0, x ₃ < 0, x ₄ < 0

$$ E = \left| {\begin{array}{*{20}c} 0 & {\varPi_{12} } & 0 & {\varPi_{14} } & 0 & {\varPi_{16} } & 0 & 0 \\ 0 & {\varPi_{22} } & 0 & {\varPi_{24} } & 0 & {\varPi_{26} } & 0 & {\varPi_{28} } \\ {\varPi_{31} } & {\varPi_{32} } & {\varPi_{33} } & {\varPi_{34} } & {\varPi_{35} } & {\varPi_{36} } & {\varPi_{37} } & {\varPi_{38} } \\ {\varPi_{41} } & {\varPi_{42} } & {\varPi_{43} } & {\varPi_{44} } & {\varPi_{45} } & {\varPi_{46} } & 0 & 0 \\ \end{array} } \right| $$

(113)

where

$$ \begin{aligned} & \varPi_{12} = \frac{\pi }{2}\chi_{1}^{ - p} ,\,\,\,\,\,\varPi_{14} = - \chi_{2}^{ - p} ,\,\,\,\,\,\varPi_{16} = - \chi_{3}^{ - p} \\ & \varPi_{22} = - a_{1} \frac{\pi }{2}\chi_{1}^{ - p} ,\,\,\,\,\,\varPi_{24} = a_{2} \chi_{2}^{ - p} ,\,\,\,\,\,\varPi_{26} = a_{3} \chi_{3}^{ - p} ,\,\,\,\,\,\varPi_{28} = \chi_{4}^{ - p} \\ & \varPi_{31} = a_{1} \chi_{1}^{p} ,\,\,\,\,\,\varPi_{32} = \frac{{ - \pi a_{1} \chi_{1}^{p} }}{{2\sin \left( {\pi p} \right)}},\,\,\,\,\,\varPi_{33} = a_{2} \chi_{2}^{p} ,\,\,\,\,\,\varPi_{34} = a_{2} \chi_{2}^{p} \cot \left( {\pi p} \right), \\ & \varPi_{35} = a_{3} \chi_{3}^{p} ,\,\,\,\,\,\varPi_{36} = a_{3} \chi_{3}^{p} \cot \left( {\pi p} \right),\,\,\,\,\,\varPi_{37} = - \chi_{4}^{p} ,\,\,\,\,\,\varPi_{38} = - \cot \left( {\pi p} \right)\chi_{4}^{p} \\ & \varPi_{41} = \chi_{1}^{p} ,\,\,\,\,\,\varPi_{42} = \frac{{ - \pi \chi_{1}^{p} }}{{2\sin \left( {\pi p} \right)}},\,\,\,\,\,\varPi_{43} = \chi_{2}^{p} ,\,\,\,\,\,\varPi_{44} = \chi_{2}^{p} \cot \left( {\pi p} \right), \\ & \varPi_{45} = \chi_{3}^{p} ,\,\,\,\,\,\varPi_{46} = \chi_{3}^{p} \cot \left( {\pi p} \right) \\ \end{aligned} $$

(114)