An Analytical Investigation on the Nonlinear Vibration Behavior of a New Hybrid Laminated Nanocomposite Cylindrical Shell Resting on the Three-Parameter Nonlinear Substrate

Abstract

Purpose

The current analytical study is devoted to examining the nonlinear free vibration behavior of the laminated nanocomposite cylindrical shells containing multi-scale hybrid reinforcements once a nonlinear three-parameter substrate surrounds the structures. The multi-phase material of the structures includes polymeric matrix, nano-scale GOPs, which are uniformly distributed through the thickness of each layer and macro-scale carbon fibers with various orientation angles.

Methods

The effective material properties of each multi-phase nanocomposite layer are calculated by implementing the modified Halpin–Tsai micromechanical scheme together with the extended rule of the mixture in a hierarchy. After that, with an incorporation of the improved Donnell’s shell theory and Hamilton’s principle, the governing equations are derived. Then by adopting a two-step solution technique, these nonlinear equations are transferred to one ordinary differential equation via Galerkins’ method, and in the next step, the nonlinear frequencies of the structure are obtained by employing the multiple scale technique.

Results

In the framework of various graphical results, a parametric analysis is conducted in detail to reveal the influences of the different parameters such as nonlinear foundation parameters, carbon fibers’ orientation angles, GOPs’ weight, and carbon fibers’ volume fractions and length-to-radius ratios on the nonlinear vibration characteristics of the multi-phase laminated nanocomposite cylindrical shell.

Conclusions

The results declare that the addition of the GO nanofillers along with the CFs and also embedding the structure on the nonlinear substrate can significantly enhance the vibrational behavior of the multi-phase laminated nanocomposite cylindrical shell.

Appendix

$$\begin{gathered} \tilde{L}_{{11}} = - \frac{{A_{{11}} m^{2} R^{2} \pi ^{2} + A_{{66}} n^{2} L^{2} }}{{\rho _{{eff}}^{T} L^{2} R^{2} }},{\text{ }} \hfill \\ \tilde{L}_{{12}} = \frac{{\left( {RA_{{12}} + RA_{{66}} + B_{{12}} + 2B_{{66}} } \right)mn\pi }}{{\rho _{{eff}}^{T} LR^{2} }}, \hfill \\ \tilde{L}_{{13}} = \frac{{m\pi \left( {L^{2} RA_{{12}} + m^{2} R^{2} \pi ^{2} B_{{11}} + n^{2} L^{2} (B_{{12}} + 2B_{{66}} )} \right)}}{{\rho _{{eff}}^{T} L^{3} R^{2} }},{\text{ }} \hfill \\ \tilde{L}_{{14}} = 0,{\text{ }} \hfill \\ \tilde{L}_{{21}} = \tilde{L}_{{12}} , \hfill \\ \tilde{L}_{{22}} = - \frac{{n^{2} L^{2} A_{{11}} + m^{2} R^{2} \pi ^{2} A_{{66}} + (2{{n^{2} L^{2} } \mathord{\left/ {\vphantom {{n^{2} L^{2} } {R)B_{{22}} + 4m^{2} R\pi ^{2} B_{{66}} + n^{2} L^{2} D_{{22}} + 4m^{2} R^{2} \pi ^{2} D_{{66}} }}} \right. \kern-\nulldelimiterspace} {R)B_{{22}} + 4m^{2} R\pi ^{2} B_{{66}} + n^{2} L^{2} D_{{22}} + 4m^{2} R^{2} \pi ^{2} D_{{66}} }}}}{{\rho _{{eff}}^{T} L^{2} R^{2} }},{\text{ }} \hfill \\ \tilde{L}_{{23}} = - \frac{{n\left( {m^{2} R^{3} \pi ^{2} \left( {B_{{12}} + 2B_{{66}} } \right) + L^{2} R\left( {B_{{22}} + RA_{{22}} } \right) + L^{2} n^{2} \left( {RB_{{22}} + D_{{22}} } \right) + m^{2} \pi ^{2} R^{2} \left( {D_{{12}} + 4D_{{66}} } \right)} \right)}}{{\rho _{{eff}}^{T} L^{2} R^{4} }},{\text{ }} \hfill \\ \tilde{L}_{{24}} = 0,{\text{ }} \hfill \\ \tilde{L}_{{31}} = \tilde{L}_{{13}} ,{\text{ }} \hfill \\ \tilde{L}_{{32}} = \tilde{L}_{{23}} ,{\text{ }} \hfill \\ \tilde{L}_{{33}} = \left[ {L^{2} R^{2} \left( { - A_{{22}} L^{2} + 2m^{2} \pi ^{2} RB_{{12}} + 2(L^{2} n^{2} /R)B_{{22}} + n^{2} \left( {L^{2} } \right.\left( - \right.\left. {k_{s} } \right) + m^{2} \left( { - 2D_{{12}} - 4D_{{66}} } \right)\left. {\pi ^{2} } \right)} \right) - } \right. \hfill \\ {\text{ }}{{\left( {L^{2} R^{4} } \right)\left( {k_{L} L^{2} + m^{2} k_{s} \pi ^{2} } \right) - \left. {\left( {D_{{22}} L^{4} n^{4} + D_{{11}} m^{4} R^{4} \pi ^{4} } \right)} \right]} \mathord{\left/ {\vphantom {{\left( {L^{2} R^{4} } \right)\left( {k_{L} L^{2} + m^{2} k_{s} \pi ^{2} } \right) - \left. {\left( {D_{{22}} L^{4} n^{4} + D_{{11}} m^{4} R^{4} \pi ^{4} } \right)} \right]} {\rho _{{eff}}^{T} L^{4} R^{4} }}} \right. \kern-\nulldelimiterspace} {\rho _{{eff}}^{T} L^{4} R^{4} }} \hfill \\ \tilde{L}_{{34}} = 0,{\text{ }} \hfill \\ \tilde{L}_{{35}} = 0,{\text{ }} \hfill \\ \tilde{L}_{{36}} = 0, \hfill \\ \tilde{L}_{{37}} = - \frac{{9A_{{11}} \left( {n^{4} L^{4} + m^{4} R^{4} \pi ^{4} } \right) + 2\left( {A_{{12}} + 2A_{{66}} } \right)m^{2} R^{2} \pi ^{2} n^{2} L^{2} }}{{32\rho _{{eff}}^{T} L^{4} R^{4} }} \hfill \\ \end{gathered}$$