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Nonlinear vibration response of a functionally graded carbon nanotube-reinforced composite conical shell using a stress function method

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Abstract

The nonlinear vibration response of a functionally graded carbon nanotube (CNT)-reinforced composite conical shell subjected to external excitation is investigated by a stress function method, and the motion state of the conical shell is revealed. Firstly, the motion equations of the conical shell are derived in the frame of the Hamilton principle and the von Kármán nonlinearity. Thereafter, the coupled nonlinear ordinary differential equations are obtained by the Galerkin method and are simplified by a stress function method. In the end, the nonlinear vibration response of the functionally graded carbon nanotube-reinforced composite conical shell is analyzed using the multi-scale method and the Runge-Kutta method. The effects of various parameters such as the volume fraction of carbon nanotube and the distribution patterns are investigated. Results indicate that the functionally graded carbon nanotube-reinforced composite conical shell with X type distribution having higher CNT volume fraction has the lowest vibration amplitude; considering the excitation amplitude as a parameter, the bifurcation behavior is observed under different semi-vertexes; the conical shell has three types of motion states, periodic motion, multi-periodic motion, and chaos motion.

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Acknowledgements

We would like to express our appreciation to the National Natural Science Foundation of China (Grant No. 51965042) for supporting this research.

Funding

National Natural Science Foundation of China, 51965042, Wenguang Liu.

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Authors and Affiliations

  1. School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang, 330063, Jiangxi, People’s Republic of China

    Yuhang Zhang & Wenguang Liu

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  1. Yuhang Zhang
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  2. Wenguang Liu
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Correspondence to Wenguang Liu.

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Appendix

Appendix

Some coefficients can be given, such as A1, A2, B1, B2, C1, D1, D2, E1, F1, F2.

$$\begin{gathered} \overline{A} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left( \begin{gathered} a_{1} M^{2} + 3a_{2} M^{2} - 2a_{1} N^{2} - \hfill \\ 5a_{3} M^{2} - 2a_{2} N^{2} - 3a_{4} M^{2} + a_{1} N^{4} \hfill \\ + a_{3} M^{4} - c_{1} N^{2} + a_{2} M^{2} N^{2} \hfill \\ + a_{4} M^{2} N^{2} + c_{1} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } & {\left( \begin{gathered} - 4a_{3} M^{3} + a_{2} M^{3} - a_{4} M^{3} \hfill \\ - 2a_{1} M - 2a_{2} M + \hfill \\ 2a_{3} M + 2a_{4} M - 3a_{2} MN^{2} \hfill \\ + a_{4} MN^{2} + 2c_{1} MN^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ { - \left( \begin{gathered} - 4a_{3} M^{3} + a_{2} M^{3} - a_{4} M^{3} \hfill \\ - 2a_{1} M - 2a_{2} M + \hfill \\ 2a_{3} M + 2a_{4} M - 3a_{2} MN^{2} \hfill \\ + a_{4} MN^{2} + 2c_{1} MN^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } & {\left( \begin{gathered} a_{1} M^{2} + 3a_{2} M^{2} - 2a_{1} N^{2} \hfill \\ - 5a_{3} M^{2} - 2a_{2} N^{2} - 3a_{4} M^{2} + a_{1} N^{4} \hfill \\ + a_{3} M^{4} - c_{1} N^{2} + a_{2} M^{2} N^{2} \hfill \\ + a_{4} M^{2} N^{2} + c_{1} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ \end{array} } \\ \end{array} } \right]^{ - 1}, \hfill \\ \left( {\begin{array}{*{20}c} {\overline{A}_{1} } \\ {\overline{A}_{2} } \\ \end{array} } \right){ = }\overline{A}\left[ {\begin{array}{*{20}c} { - \frac{{M^{2} \cos \left( \alpha \right)}}{{s_{1}^{3} \sin \left( \alpha \right)}}} \\ {\frac{M\cos \left( \alpha \right)}{{s_{1}^{3} \sin \left( \alpha \right)}}}, \\ \end{array} } \right] \hfill \\ \end{gathered}$$
$$\begin{gathered} \overline{B} = \left[ {\begin{array}{*{20}c} {\left( \begin{gathered} a_{1} + a_{1} M^{2} - 2a_{1} N^{2} + a_{3} M^{2} \hfill \\ + a_{1} N^{4} + a_{3} M^{4} + a_{2} M^{2} N^{2} \hfill \\ + a_{4} M^{2} N^{2} + c_{1} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } & {\left( \begin{gathered} a_{4} MN^{2} - a_{2} MN^{2} \hfill \\ + a_{2} M^{3} - a_{4} M^{3} \hfill \\ + a_{2} M - a_{4} M \hfill \\ \end{gathered} \right)} \\ { - \left( \begin{gathered} a_{4} MN^{2} - a_{2} MN^{2} \hfill \\ + a_{2} M^{3} - a_{4} M^{3} \hfill \\ + a_{2} M - a_{4} M \hfill \\ \end{gathered} \right)} & {\left( \begin{gathered} a_{1} + a_{1} M^{2} - 2a_{1} N^{2} + a_{3} M^{2} \hfill \\ + a_{1} N^{4} + a_{3} M^{4} + a_{2} M^{2} N^{2} \hfill \\ + a_{4} M^{2} N^{2} + c_{1} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ \end{array} } \right]^{ - 1}, \hfill \\ \left( {\begin{array}{*{20}c} {\overline{B}_{1} } \\ {\overline{B}_{2} } \\ \end{array} } \right) = \overline{B}\left[ {\begin{array}{*{20}c} {\left( \begin{gathered} b_{2} + b_{2} M^{2} + b_{3} M^{2} - 2b_{2} N^{2} \hfill \\ + b_{3} M^{4} + b_{2} N^{4} + b_{1} M^{2} N^{2} \hfill \\ + b_{4} M^{2} N^{2} - c_{2} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ {\left( \begin{gathered} b_{1} M_{3} - b_{4} M^{3} \hfill \\ + b_{1} M - b_{4} M \hfill \\ - b_{1} MN^{2} + b_{4} MN^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} }, \\ \end{array} } \right] \hfill \\ \end{gathered}$$
$$\overline{C}_{1} = \left( {M^{2} \left( {N^{2} - 1} \right)/\left( {4s_{1}^{4} } \right)} \right)/\left( {\left( { - 2a_{1} - 2a_{2} + 2a_{3} + 2a_{4} } \right)/s_{1}^{4} }, \right)$$
$$\begin{gathered} \overline{D} = \left[ {\begin{array}{*{20}c} {\left( {4M^{2} \left( \begin{gathered} 4a_{3} M^{2} + a_{1} + \hfill \\ 3a_{2} - 5a_{3} - 3a_{4} \hfill \\ \end{gathered} \right)} \right)/s_{1}^{4} } & {\left( \begin{gathered} - 4a_{1} M - 4a_{2} M + 4a_{3} M \hfill \\ + 4a_{4} M + 8a_{2} M^{3} \hfill \\ - 32a_{3} M^{3} - 8a_{4} M^{3} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ { - \left( \begin{gathered} - 4a_{1} M - 4a_{2} M + 4a_{3} M \hfill \\ + 4a_{4} M + 8a_{2} M^{3} \hfill \\ - 32a_{3} M^{3} - 8a_{4} M^{3} \hfill \\ \end{gathered} \right)/s_{1}^{4} } & {\left( {4M^{2} \left( \begin{gathered} 4a_{3} M^{2} + a_{1} + \hfill \\ 3a_{2} - 5a_{3} - 3a_{4} \hfill \\ \end{gathered} \right)} \right)/s_{1}^{4} } \\ \end{array} } \right]^{ - 1}, \hfill \\ \left( {\begin{array}{*{20}c} {\overline{D}_{1} } \\ {\overline{D}_{2} } \\ \end{array} } \right) = \overline{D}\left[ {\begin{array}{*{20}c} {M\left( {1 - M^{2} - N^{2} } \right)/\left( {4s_{1}^{4} } \right)} \\ {3M^{2} \left( {1 - N^{2} } \right)/\left( {4s_{1}^{4} } \right)} ,\\ \end{array} } \right] \hfill \\ \end{gathered}$$
$$\overline{E}_{1} = \left( {M^{2} \left( {N^{2} - 1} \right)/\left( {4s_{1}^{4} } \right)} \right)/\left( {\left( {4N^{2} \left( {4a_{1} N^{2} - 2a_{1} - 2a_{2} - c_{1} } \right)} \right)/s_{1}^{4} }, \right)$$
$$\begin{gathered} \overline{F} = \left[ {\begin{array}{*{20}c} {\left( \begin{gathered} 4a_{1} M^{2} + 12a_{2} M^{2} - 8a_{1} N^{2} \hfill \\ - 20a_{3} M^{2} - 8a_{2} N^{2} - 12a_{4} M^{2} \hfill \\ + 16a_{1} N^{4} + 16a_{3} M^{4} - 4c_{1} N^{2} \hfill \\ + 16a_{2} M^{2} N^{2} + 16a_{4} M^{2} N^{2} + 16c_{1} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } & {\left( \begin{gathered} - 4a_{1} M - 4a_{2} M + 4a_{3} M \hfill \\ + 4a_{4} M + 8a_{2} M^{2} - 32a_{3} M^{2} \hfill \\ - 24a_{2} N^{2} - 8a_{4} M^{2} - 8a_{4} N^{2} - 16c_{1} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ { - \left( \begin{gathered} - 4a_{1} M - 4a_{2} M + 4a_{3} M \hfill \\ + 4a_{4} M + 8a_{2} M^{2} - 32a_{3} M^{2} \hfill \\ - 24a_{2} N^{2} - 8a_{4} M^{2} - 8a_{4} N^{2} - 16c_{1} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } & {\left( \begin{gathered} 4a_{1} M^{2} + 12a_{2} M^{2} - 8a_{1} N^{2} \hfill \\ - 20a_{3} M^{2} - 8a_{2} N^{2} - 12a_{4} M^{2} \hfill \\ + 16a_{1} N^{4} + 16a_{3} M^{4} - 4c_{1} N^{2} \hfill \\ + 16a_{2} M^{2} N^{2} + 16a_{4} M^{2} N^{2} + 16c_{1} M^{2} N^{2} \hfill \\ \end{gathered} \right)/s_{1}^{4} } \\ \end{array} } \right]^{ - 1}, \hfill \\ \left( {\begin{array}{*{20}c} {\overline{F}_{1} } \\ {\overline{F}_{2} } \\ \end{array} } \right) = \overline{F}\left[ {\begin{array}{*{20}c} {M\left( {M^{2} + N^{2} - 1} \right)/\left( {4s_{1}^{2} } \right)} \\ {M^{2} \left( {N^{2} - 1} \right)/\left( {4s_{1}^{2} } \right)}. \\ \end{array} } \right] \hfill \\ \end{gathered}$$

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Zhang, Y., Liu, W. Nonlinear vibration response of a functionally graded carbon nanotube-reinforced composite conical shell using a stress function method. Acta Mech 233, 3157–3174 (2022). https://doi.org/10.1007/s00707-022-03273-9

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  • DOI: https://doi.org/10.1007/s00707-022-03273-9

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