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Theoretical analyses and numerical simulation of flexural vibration based on Reddy and modified higher-order plate theories for a transversely isotropic circular plate

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Abstract

This paper is concerned with free vibration analyses of a circular transversely isotropic elastic plate based on Reddy plate theory and modified higher-order shear deformation plate theory (MHSDT). Hamilton’s principle is used to derive the equations of motion and boundary conditions of the circular plate. Natural frequencies are determined from the solution of the governing equations and boundary conditions along the circular edge. Comparisons of natural frequencies and mode shapes arising from the Mindlin plate theory, the Reddy plate theory, MHSDT, and finite element method are made for fully free and clamped circular plates. The comparison results show that MHSDT has better accuracy than the other analytical plate theories in terms of the natural frequencies and corresponding mode shapes.

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Acknowledgement

The authors gratefully acknowledge the financial supports of this research by the Ministry of Science and Technology (Republic of China) under Grant MOST 107-2221-E-002 -086 -MY3.

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Taiwan University, Taipei, 106, Taiwan, Republic of China

    Ming Ji & Chien-Ching Ma

  2. Department of Mechanical Engineering, School of Engineering, Tokyo Institute of Technology, Tokyo, Japan

    Ming Ji

  3. Advanced Institute of Manufacturing with High-tech Innovations & Department of Mechanical Engineering, National Chung Cheng University, Chiayi, 106, Taiwan, Republic of China

    Yi-Chuang Wu

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  1. Ming Ji
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  2. Yi-Chuang Wu
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  3. Chien-Ching Ma
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Correspondence to Chien-Ching Ma.

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Appendices

Appendix A

Case 1, clamped boundary condition:

$$ S_{1i}^{R} = \hat{w}_{in} (\alpha_{i} ){\kern 1pt} \quad {\text{for}}\;i = {1},{2},{3}\quad {\text{and}}\quad S_{14}^{R} = 0, $$
(49)
$$ S_{2i}^{R} = \partial \hat{w}_{in} (\alpha_{i} )/\partial \xi \quad {\text{for}}\;i = { 1},{2},{3}\quad {\text{and}}\quad S_{24}^{R} = 0, $$
(50)
$$ S_{3i}^{R} = \chi_{i} \partial \hat{w}_{in} (\alpha_{i} )/\partial \xi \quad {\text{for}}\;i = {1},{2},{3}\quad {\text{and}}\quad S_{34}^{R} = n\hat{w}_{4n} (\alpha_{4} ), $$
(51)
$$ S_{4i}^{R} = \chi_{i} n\hat{w}_{in} (\alpha_{i} )\quad {\text{for}}\;i = {1},{2},{3}\quad {\text{and}}\quad S_{44}^{R} = \partial \hat{w}_{4n} (\alpha_{4} )/\partial \xi. $$
(52)

Case 2, free boundary condition:

$$ \begin{aligned} S_{1i}^{R} &= \left( {\chi_{i} - \frac{5}{16}} \right)\left( {\frac{{\partial^{3} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{3} }} + \frac{{\partial^{2} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{2} }}} \right)\\ &\quad+ \left\{ {\left( {\frac{5}{16} - \chi_{i} } \right)\left[ {\left( {1 - \gamma_{12} } \right)n^{2} + \left( {1 + n^{2} } \right) - \lambda^{4} } \right]} \right.\\ &\quad + \left.\frac{21}{2}\frac{{\gamma_{55} }}{{\eta^{2} }}\left( {\chi_{i} + 1} \right) \right\}\frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi }\\ &\quad+ \left( {3 - \gamma_{12} } \right)\left( {\chi_{i} - \frac{5}{16}} \right)n^{2} \hat{w}_{in} (\alpha_{i} )\end{aligned}\quad {\text{for}}\;i = 1,2,3, $$
(53)
$$ S_{14}^{R} = \left( { - n^{3} \left( {1 - \gamma_{12} } \right) + \lambda^{4} n + \frac{21}{2}\frac{{\gamma_{55} n}}{{\eta^{2} }}} \right)\hat{w}_{4n} (\alpha_{4} ) + \left( {1 - \gamma_{12} } \right)n\frac{{\partial \hat{w}_{4n} (\alpha_{4} )}}{\partial \xi }, $$
(54)
$$ S_{2i}^{R} = \left( {\chi_{i} - \frac{5}{16}} \right)\left( {\frac{{\partial^{2} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{2} }} + \gamma_{12} \frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi } - \gamma_{12} n^{2} \hat{w}_{in} (\alpha_{i} )} \right)\quad {\text{for}}\;i = {1},{2},{3}, $$
(55)
$$ S_{24}^{R} = \left( {1 - \gamma_{12} } \right)n\left( {\frac{{\partial \hat{w}_{4n} (\alpha_{4} )}}{\partial \xi } - \hat{w}_{4n} (\alpha_{4} )} \right), $$
(56)
$$ S_{3i}^{R} = \left( {\chi_{i} - \frac{1}{4}} \right)\left( {\frac{{\partial^{2} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{2} }} + \gamma_{12} \frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi } - \gamma_{12} n^{2} \hat{w}_{in} (\alpha_{i} )} \right)\quad {\text{for}}\;i = {1},{2},{3}, $$
(57)
$$ S_{34}^{R} = \left( {1 - \gamma_{12} } \right)n\left( {\frac{{\partial \hat{w}_{4n} (\alpha_{4} )}}{\partial \xi } - \hat{w}_{4n} (\alpha_{4} )} \right), $$
(58)
$$ S_{4i}^{R} = n\left( {\frac{17}{4}\chi_{i} - 1} \right)\left( { - \frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi } + \hat{w}_{in} (\alpha_{i} )} \right)\quad {\text{for}}\;i = {1},{2},{3}, $$
(59)
$$ S_{44}^{R} = \frac{17}{8}\left( { - n^{2} \hat{w}_{4n} (\alpha_{4} ) + \frac{{\partial \hat{w}_{4n} (\alpha_{4} )}}{\partial \xi } - \frac{{\partial^{2} \hat{w}_{4n} (\alpha_{4} )}}{{\partial^{2} \xi }}} \right). $$
(60)

Appendix B

Case 1, clamped boundary condition:

$$ S_{1i}^{L} = \hat{w}_{in} (\alpha_{i} )\quad {\text{for}}\;i = {1},{2},{3},{4}\quad {\text{and}}\quad S_{15}^{L} = 0, $$
(61)
$$ S_{2i}^{L} = \chi_{i} \hat{w}_{in} (\alpha_{i} )\quad {\text{for}}\;i = { 1},{2},{3},{4}\quad {\text{and}}\quad S_{25}^{L} = 0, $$
(62)
$$ S_{3i}^{L} = \left( {1 + \eta^{2} \chi_{i} } \right)\partial \hat{w}_{in} (\alpha_{i} )/\partial \xi \quad {\text{for}}\;i = {1},{2},{3},{4}\quad {\text{and}}\quad S_{35}^{L} = 0, $$
(63)
$$ S_{4i}^{L} = \lambda_{i} \partial \hat{w}_{in} (\alpha_{i} )/\partial \xi \quad {\text{for}}\;i = { 1},{2},{3},{4}\quad {\text{and}}\quad S_{45}^{L} = n\hat{w}_{5n} (\alpha_{5} ), $$
(64)
$$ S_{5i}^{L} = \lambda_{i} n\hat{w}_{in} {\kern 1pt} {\kern 1pt} (\alpha_{i} ){\text{for}}\;i = {1},{2},{3},{4}\quad {\text{and}}\quad S_{55}^{L} = \partial \hat{w}_{5n} (\alpha_{5} )/\partial {\xi} .$$
(65)

Case 2, free boundary condition:

$$ S_{1i}^{L} = \left( {\lambda_{i} + 1} \right)\partial \hat{w}_{in} (\alpha_{i} )/\partial \xi \quad {\text{for}}\;i = {1},{2},{3},{4}\quad {\text{and}}\quad S_{15}^{L} = \hat{w}_{5n} (\alpha_{5} ), $$
(66)
$$ \begin{aligned} S_{2i}^{L} &= \left[ {\lambda_{i} - \frac{5}{16}\left( {1 + \eta^{2} \chi_{i} } \right)} \right]\left( {\frac{{\partial^{3} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{3} }} + \frac{{\partial^{2} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{2} }}} \right)\\ &\quad + \left\{ \left[ {\frac{5}{16}\left( {1 + \eta^{2} \chi_{i} } \right) - \lambda_{i} } \right]\left[ {\left( {n^{2} + 1} \right) + \left( {1 - \gamma_{12} } \right)n^{2} - \lambda^{4} } \right] \right. \\ &\quad +\left. \frac{21}{8}\gamma_{13} \chi \right\}\frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi }\\ &\quad + \left( {3 - \gamma_{12} } \right)\left[ {\lambda_{i} - \frac{5}{16}\left( {1 + \eta^{2} \chi_{i} } \right)} \right]n^{2} \hat{w}_{in} (\alpha_{i} )\\ \end{aligned}\quad {\text{for}}\;i = 1,2,3,4, $$
(67)
$$ S_{25}^{L} = \left( { - n^{3} \left( {1 - \gamma_{12} } \right) + \lambda^{4} n} \right)\hat{w}_{5n} (\alpha_{5} ) + \left( {1 - \gamma_{12} } \right)n\frac{{\partial \hat{w}_{5n} (\alpha_{5} )}}{\partial \xi }, $$
(68)
$$\begin{aligned} S_{3i}^{L} &= \left( {\lambda_{i} - \frac{5}{16} - \frac{5}{16}\eta^{2} \chi_{i} } \right)\left( {\frac{{\partial^{2} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{2} }} + \gamma_{12} \frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi } - \gamma_{12} n^{2} \hat{w}_{in} (\alpha_{i} )} \right)\\ &\quad + \frac{7}{8}\gamma_{13} \chi_{i} \hat{w}_{in} (\alpha_{i} )\\ \end{aligned}\quad {\text{for}}\;i = 1,2,3,4, $$
(69)
$$ S_{35}^{L} = \left( {1 - \gamma_{12} } \right)n\left( {\frac{{\partial \hat{w}_{5n} (\alpha_{5} )}}{\partial \xi } - \hat{w}_{5n} (\alpha_{5} )} \right), $$
(70)
$$ \begin{aligned} S_{4i}^{L} &= \left( {\frac{17}{4}\lambda_{i} - 1 - \eta^{2} \chi_{i} } \right)\left( {\frac{{\partial^{2} \hat{w}_{in} (\alpha_{i} )}}{{\partial \xi^{2} }} + \gamma_{12} \frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi } - \gamma_{12} n^{2} \hat{w}_{in} (\alpha_{i} )} \right)\\ &\quad+ \frac{21}{2}\gamma_{13} \chi_{i} \hat{w}_{in} (\alpha_{i} )\\ \end{aligned}\quad {\text{for}}\;i = 1,2,3,4, $$
(71)
$$ S_{45}^{L} = \frac{17}{4}\left( {1 - \gamma_{12} } \right)n\left( {\frac{{\partial \hat{w}_{5n} (\alpha_{5} )}}{\partial \xi } - \hat{w}_{5n} (\alpha_{5} )} \right), $$
(72)
$$ S_{5i}^{L} = n\left( {\frac{17}{4}\lambda_{i} - 1 - \eta^{2} \chi_{i} } \right)\left( { - \frac{{\partial \hat{w}_{in} (\alpha_{i} )}}{\partial \xi } + \hat{w}_{in} (\alpha_{i} )} \right)\quad {\text{for}}\;i = {1},{2},{3},{4}, $$
(73)
$$ S_{55}^{L} = \frac{17}{8}\left( { - n^{2} \hat{w}_{5n} (\alpha_{5} ) + \frac{{\partial \hat{w}_{5n} (\alpha_{5} )}}{\partial \xi } - \frac{{\partial^{2} \hat{w}_{5n} (\alpha_{5} )}}{{\partial^{2} \xi }}} \right). $$
(74)

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Ji, M., Wu, YC. & Ma, CC. Theoretical analyses and numerical simulation of flexural vibration based on Reddy and modified higher-order plate theories for a transversely isotropic circular plate. Acta Mech 232, 2825–2842 (2021). https://doi.org/10.1007/s00707-021-02973-y

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