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Modeling and vibration analysis of a spinning assembled beam–plate structure reinforced by graphene nanoplatelets

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Abstract

The theoretical modeling of a functionally graded (FG) graphene nanoplatelet (GPL)-reinforced assembled beam–plate structure resting on elastic supports is presented for the first time, and its free vibration analysis is performed. Herein, the assembled structure is modeled according to the Kirchhoff plate theory and the Rayleigh beam theory. The graphene nanoplatelets (GPLs) gradiently distribute in the beam’s radial direction and in the plate’s thickness direction, respectively. By adopting the rule of mixture and the Halpin–Tsai model, the effective material properties can be obtained. By employing the Lagrange’s equation and considering the effects of Coriolis force and centrifugal force, the coupled governing equations of the assembled structure are determined. Furthermore, the assumed modes method and substructure modal synthesis method are applied to obtain the frequencies of the assembled beam–plate structure. A comprehensive numerical investigation is carried out to discuss the influence of the structural and material parameters on the vibration behavior of the beam–plate structure.

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Acknowledgements

This project is supported by the National Science Foundation of China (No. 51805076, No. U1708255, No. 51775093 and No. 11922205) and the Natural Science Foundation of Hebei Province (No. B2019501073).

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Correspondence to Yan Qing Wang.

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Appendix

Appendix

$$\begin{aligned} {\mathbf{M}}_{{{\text{11}}}} & = {\text{2}}\pi \int_{0}^{{r_{S} }} {\rho _{S} r{\text{d}}r} \int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi}}_{S} ^{{\text{T}}} {\varvec{\Upphi}}_{S} {\text{d}}x_{{\text{1}}} } + \pi \int_{0}^{{r_{S} }} {\rho _{S} r^{3} } {\text{d}}r\int_{{\text{0}}}^{{l_{S} }} {\varvec{\Upphi}}_{S}^{{\prime}{\text{T}}} {\varvec{\Upphi}}_{S}^{\prime } {\text{d}}x_{{\text{1}}} \\ & \left.\quad + \left[ \frac{{a^{3} b + b^{3} a}}{3}{\varvec{\Upphi}}_{S}^{{\prime}{\text{T}}} {\varvec{\Upphi}}_{S}^{\prime } + ab^{2} \frac{{{\varvec{\Upphi}}_{S} ^{{\text{T}}} {\varvec{\Upphi}} _{S}^{\prime}}} + {{\varvec{\Upphi}} _{S}^{{\prime}{\text{T}}} {\varvec{\Upphi}}_{S} }{2} +ab{\varvec{\Upphi}}_{S} ^{{\text{T}}} {\varvec{\Upphi}}_{S} \right] \right|_{{x_{{\text{1}}} = d}} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } , \\ \end{aligned}$$
$$\begin{aligned} {\mathbf{M}}_{{{\text{22}}}} & = {\text{2}}\pi \int_{0}^{{r_{S} }} {\rho _{S} r{\text{d}}r} \int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} {\text{d}}x_{{\text{1}}} } + \pi \int_{0}^{{r_{S} }} {\rho _{S} r^{3} } {\text{d}}r\int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S}^{{\prime}{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime } {\text{d}}x_{{\text{1}}} } \\ & \quad + \left. {\left[ {\frac{{b^{3} a}}{3}{\varvec{\Upphi }}_{S}^{{\prime}{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime } + ab^{2} \frac{{{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi}}_{S}^{\prime} + {\varvec{\Upphi}}_{S}^{{\prime}{\text{T}}} {\varvec{\Upphi }}_{S} }}{2}+ ab{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right]} \right|_{{x_{{\text{1}}} = d}} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } , \\ \end{aligned}$$
$${\mathbf{M}}_{{{\text{23}}}} = \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left. {{\varvec{\Upphi }}_{S} ^{{\text{T}}} } \right|_{{x_{{\text{1}}} = d}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } + \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left. {{\varvec{\Upphi ^{\prime}}}_{S} ^{{\text{T}}} } \right|_{{x_{{\text{1}}} = d}} x_{{\text{2}}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } ,$$
$${\mathbf{M}}_{{{\text{32}}}} = \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{B} ^{{\text{T}}} \left. {{\varvec{\Upphi }}_{S} } \right|_{{x_{{\text{1}}} = d}} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } + \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {x_{{\text{2}}} {\varvec{\Upphi}}_{B} ^{{\text{T}}} \left. {{\varvec{\Upphi }}_{S}^{\prime } } \right|_{{x_{{\text{1}}} = d}} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } ,$$
$${\mathbf{M}}_{{{\text{33}}}} = \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{B} ^{{\text{T}}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } ,$$
$${\mathbf{C}}_{{{\text{12}}}} = - {\text{4}}\pi \Omega \int_{0}^{{r_{S} }} {\rho _{S} r{\text{d}}r} \int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} {\text{d}}x_{{\text{1}}} } - \left. {\Omega \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \left( \begin{gathered} \frac{{2b^{3} a}}{{\text{3}}}{\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime } +ab^{2} {\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} {\varvec{\Upphi }}_{S} \hfill \\ + ab^{2} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime } + 2ab{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} \hfill \\ \end{gathered} \right)} \right|_{{x_{{\text{1}}} = d}} ,$$
$${\mathbf{C}}_{{{\text{13}}}} = - 2\Omega \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left. {{\varvec{\Upphi }}_{S} ^{{\text{T}}} } \right|_{{x_{{\text{1}}} = d}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } - 2\Omega \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left. {{\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} } \right|_{{x_{{\text{1}}} = d}} x_{{\text{2}}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } ,$$
$${\mathbf{C}}_{{{\text{21}}}} = {\text{4}}\pi \Omega \int_{0}^{{r_{S} }} {\rho _{S} r{\text{d}}r} \int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} {\text{d}}x_{{\text{1}}} } {\kern 1pt} +\left. {\Omega \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \left( \begin{gathered} \frac{{2b^{3} a}}{{\text{3}}}{\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime } +ab^{2} {\varvec{\Upphi ^{\prime}}}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} \hfill \\ + ab^{2} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime } + 2ab{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} \hfill \\ \end{gathered} \right)} \right|_{{x_{{\text{1}}} = d}} ,$$
$${\mathbf{C}}_{{{\text{31}}}} = 2\Omega \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{B} ^{{\text{T}}} \left. {{\varvec{\Upphi }}_{S} } \right|_{{x_{{\text{1}}} = d}} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } + 2\Omega \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {x_{{\text{2}}} {\varvec{\Upphi}}_{B} ^{{\text{T}}} \left. {{\varvec{\Upphi }}_{S}^{\prime } } \right|_{{x_{{\text{1}}} = d}} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} ,} }$$
$$\begin{aligned} {\mathbf{K}}_{{{\text{11}}}} & = - {\text{2}}\Omega ^{{\text{2}}} \pi \int_{0}^{{r_{S} }} {\rho _{S} r{\text{d}}r} \int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} {\text{d}}x_{{\text{1}}} } + \Omega ^{{\text{2}}} \pi \int_{0}^{{r_{S} }} {\rho _{S} r^{3} } {\text{d}}r\int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime } {\text{d}}x_{{\text{1}}} } \\ & \quad + \Omega ^{{\text{2}}} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \left. {\left[ {\frac{{a^{3} b - b^{3} a}}{3}{\varvec{\Upphi }}_{S}^{{\prime{\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime} - ab^{2} \frac{{{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime} + {\varvec{\Upphi }}_{S}^{{\prime{\text{T}}}} {\varvec{\Upphi }}_{S} }}{2}{\mathbf{ - }}ab{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right]} \right|_{{x_{{\text{1}}} = d}} , \\ & \quad +\pi \int_{0}^{{r_{S} }} \left( {\frac{{E_{S} r^{3} }}{{{\text{1}} - \mu _{S} ^{{\text{2}}} }}} \right){\text{d}}r\int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S}^{{\prime\prime}{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime\prime} {\text{d}}x_{{\text{1}}} }\left. + \left( {k_{{11}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right) \right|_{{x_{{\text{1}}} = 0}} + \left. \left( {k_{{21}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right) \right|_{{x_{{\text{1}}} = l_{S} }} \\ \end{aligned}$$
$$\begin{aligned} & {\mathbf{K}}_{{{\text{12}}}} = \left. {\left( {k_{{{\text{12}}}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right)} \right|_{{x_{{\text{1}}} = 0}} + \left. {\left( {k_{{{\text{22}}}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right)} \right|_{{x_{{\text{1}}} = l_{S} }} , \\ & {\mathbf{K}}_{{{\text{21}}}} = \left. { - \left( {k_{{{\text{12}}}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right)} \right|_{{x_{{\text{1}}} = 0}} - \left. {\left( {k_{{{\text{22}}}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right)} \right|_{{x_{{\text{1}}} = l_{S} }} , \\ \end{aligned}$$
$$\begin{aligned} {\mathbf{K}}_{{{\text{22}}}} & = - {\text{2}}\Omega ^{{\text{2}}} \pi \int_{0}^{{r_{S} }} {\rho _{S} r{\text{d}}r} \int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} {\text{d}}x_{{\text{1}}} } + \Omega ^{{\text{2}}} \pi \int_{0}^{{r_{S} }} {\rho _{S} r^{3} } {\text{d}}r\int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S}^{{\prime{\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime } {\text{d}}x_{{\text{1}}} } \\ & \quad - \Omega ^{{\text{2}}} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \left. {\left[ {\frac{{b^{3} a}}{3}{\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime } + ab^{2} \frac{{{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S}^{\prime } + {\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} {\varvec{\Upphi }}_{S} }}{2}+ab{\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right]} \right|_{{x_{{\text{1}}} = d}} , \\ & \quad +\pi \int_{0}^{{r_{S} }} {\left( {\frac{{E_{S} r^{3} }}{{{\text{1}} - \mu _{S} ^{{\text{2}}} }}} \right){\text{d}}r\int_{{\text{0}}}^{{l_{S} }} {{\varvec{\Upphi }}_{S}^{{\prime\prime {\text{T}}}} {\varvec{\Upphi }}_{S}^{\prime\prime } {\text{d}}x_{{\text{1}}} } } + \left. {\left( {k_{{11}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right)} \right|_{{x_{{\text{1}}} = 0}} + \left. {\left( {k_{{21}} {\varvec{\Upphi }}_{S} ^{{\text{T}}} {\varvec{\Upphi }}_{S} } \right)} \right|_{{x_{{\text{1}}} = l_{S} }} \\ \end{aligned}$$
$${\mathbf{K}}_{{{\text{23}}}} = - \Omega ^{2} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left. {{\varvec{\Upphi }}_{S} ^{{\text{T}}} } \right|_{{x_{{\text{1}}} = d}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } - \Omega ^{2} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left. {{\varvec{\Upphi }}_{S}^{{\prime {\text{T}}}} } \right|_{{x_{{\text{1}}} = d}} x_{{\text{2}}} {\varvec{\Upphi}}_{B} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } ,$$
$${\mathbf{K}}_{{{\text{32}}}} = - \Omega ^{2} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{B} ^{{\text{T}}} \left. {{\varvec{\Upphi }}_{S} } \right|_{{x_{{\text{1}}} = d}} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } - \Omega ^{2} \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {x_{{\text{2}}} {\varvec{\Upphi}}_{B} ^{{\text{T}}} \left. {{\varvec{\Upphi }}_{S}^{\prime } ,} \right|_{{x_{{\text{1}}} = d}} {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } },$$
$$\begin{aligned} {\mathbf{K}}_{{{\text{33}}}} & = \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z_{{\text{2}}} ^{2} E_{B} }}{{1 - \mu _{B} ^{2} }}{\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{{Bxx}}^{{{\prime\prime} {\text{T}}}} {\varvec{\Upphi}}_{{Bxx}}^{{\prime\prime} } {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } + \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z_{{\text{2}}} ^{2} E_{B} }}{{1 - \mu _{B} ^{2} }}{\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{{Byy}}^{{{\prime\prime} {\text{T}}}} {\varvec{\Upphi}}_{{Byy}}^{{\prime\prime} } {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } \\ & \quad + \int_{{ - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z_{{\text{2}}} ^{2} E_{B} \mu _{B} }}{{1 - \mu _{B} ^{2} }}{\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{{Byy}}^{{{\prime\prime} {\text{T}}}} {\varvec{\Upphi}}_{{Bxx}}^{{\prime\prime} } {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } + \int_{{ - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z_{{\text{2}}} ^{2} E_{B} \mu _{B} }}{{1 - \mu _{B} ^{2} }}{\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{{Bxx}}^{{{\prime\prime} {\text{T}}}} {\varvec{\Upphi}}_{{Byy}}^{{\prime\prime} } {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } \\ & \quad + 2\int_{{ - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\frac{{z_{{\text{2}}} ^{2} E_{B} }}{{1 + \mu _{B} }}{\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {{\varvec{\Upphi}}_{{Bxy}}^{{{\prime\prime} {\text{T}}}} {\varvec{\Upphi}}_{{Bxy}}^{{\prime\prime} } {\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } + \frac{{\Omega ^{2} }}{2}\int_{{ - {h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho _{B} {\text{d}}z_{{\text{2}}} } \int_{0}^{a} {\int_{{ - {b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{b \mathord{\left/ {\vphantom {b 2}} \right. \kern-\nulldelimiterspace} 2}}} {\left[ {\left( {a^{2} - y^{2} } \right){\varvec{\Upphi}}_{{Byy}}^{{{\prime\prime} {\text{T}}}} {\varvec{\Upphi}}_{{Byy}}^{{{\prime\prime} {\text{T}}}} } \right]{\text{d}}x_{{\text{2}}} {\text{d}}y_{{\text{2}}} } } . \\ \end{aligned}$$

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Zhao, T.Y., Wang, Y.X., Yu, Y.X. et al. Modeling and vibration analysis of a spinning assembled beam–plate structure reinforced by graphene nanoplatelets. Acta Mech 232, 3863–3879 (2021). https://doi.org/10.1007/s00707-021-03039-9

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