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Radial nonlinear vibrations of thin-walled hyperelastic cylindrical shell composed of Mooney–Rivlin materials under radial harmonic excitation

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Abstract

In this paper, the radial nonlinear vibrations are investigated for a thin-walled hyperelastic cylindrical shell composed of the classical incompressible Mooney–Rivlin materials subjected to a radial harmonic excitation. Using Lagrange equation, Donnell’s nonlinear shallow-shell theory and small strain assumption, the nonlinear differential governing equation of motion is obtained for the incompressible Mooney–Rivlin material thin-walled hyperelastic cylindrical shell. The differential governing equation of motion is simplified to a generalized Duffing equation with the quadratic term. The second-order approximate analytical solutions are obtained by using the modified Lindstedt–Poincaré (MLP) method. The impacts of the parameters on the amplitude–frequency response curves and number of the equilibrium points are analyzed. According to Runge–Kutta method, the bifurcation diagrams, Lyapunov exponents and Poincaré maps are obtained. The chaotic behaviors are found in the radial nonlinear vibrations of the incompressible Mooney–Rivlin material thin-walled hyperelastic cylindrical shell. The results demonstrate that the nonlinear dynamic responses of the incompressible Mooney–Rivlin material thin-walled hyperelastic cylindrical shell are highly sensitive to the structural parameters and external excitation.

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Acknowledgements

The authors gratefully acknowledge the support of National Natural Science Foundation of China (NNSFC) through Grant Nos. 11832002 and 12072201.

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Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    F. Zheng & W. Zhang

  2. Department of Mechanics, GuangXi University, GuangXi, 530004, People’s Republic of China

    W. Zhang & Y. F. Zhang

  3. School of Science, Dalian Minzu University, Dalian, 116600, People’s Republic of China

    X. G. Yuan

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Appendix A

Appendix A

The elements of Eqs. (24)–(31) are given as follows

$$ \begin{aligned} & M_{11} = \pi Rhl\rho ,\;\;M_{22} = \pi Rhl\rho \left( {\frac{{\pi^{2} h^{2} m^{2} }}{{12l^{2} }} + 1} \right), \\ & & M_{12} = M_{21} = 0,\;\;k_{11} = \frac{{4Rhm^{2} \pi^{3} \left( {\mu_{1} + \mu_{2} } \right)}}{l} \\ & k_{12} = k_{21} = - 2hm\pi^{2} \left( {\mu_{1} + \mu_{2} } \right),\;\;k_{22} = \pi hl\left( {\mu_{1} + \mu_{2} } \right)\left( {\frac{4}{R} + \frac{{m^{4} h^{2} \pi^{2} R}}{{3l^{4} }}} \right) \\ & k_{2}^{11} = A_{1} + B,\;\;k_{2}^{12} = k_{2}^{21} = B + S, \\ & k_{2}^{22} = S + U,\;\;k_{3}^{11} = C + D + E,\;\;k_{3}^{12} = k_{3}^{21} = G + H + I, \\ & k_{3}^{22} = E + K + Y \\ \end{aligned} $$
(A1)

where

$$ \begin{aligned} A_{1} & = \frac{{16Rhm^{2} \pi^{3} \left( {\mu_{1} + \mu_{2} } \right)}}{{l^{2} }},\;B = \frac{{8mh\pi^{2} \mu_{2} }}{3l} - \frac{{8mh\pi^{2} \mu_{1} }}{l}, \\ C & = \frac{{45m^{4} \pi^{5} Rh\left( {\mu_{1} + \mu_{2} } \right)}}{{2l^{3} }},\;D = - \frac{{18hm^{3} \mu_{1} \pi^{4} }}{{l^{2} }}, \\ E & = \frac{{15h^{3} m^{6} \pi^{7} R\left( {\mu_{1} + \mu_{2} } \right)}}{{8l^{5} }} - \frac{{5hRm^{4} \pi^{5} \left( {\mu_{1} + \mu_{2} } \right)}}{{2l^{3} }} \\ & \quad + \frac{{27hm^{2} \pi^{3} \mu_{1} }}{4Rl} + \frac{{3hm^{2} \pi^{3} \mu_{2} }}{4Rl}, \\ S & = \frac{{4h^{3} m^{4} \pi^{5} R\left( {\mu_{1} + \mu_{2} } \right)}}{{3l^{4} }} - \frac{{8\pi h\mu_{2} }}{3R} \\ & \quad + \frac{{8\pi h\mu_{1} }}{R} - \frac{{8hRm^{2} \pi^{3} \left( {\mu_{1} + \mu_{2} } \right)}}{{3l^{2} }}, \\ G & = - \frac{{9hm^{3} \pi^{4} \mu_{1} }}{{l^{2} }}, \\ H & = \frac{{15Rh^{3} m^{6} \pi^{7} \left( {\mu_{1} + \mu_{2} } \right)}}{{4l^{5} }} - \frac{{5Rhm^{4} \pi^{5} \left( {\mu_{1} + \mu_{2} } \right)}}{{l^{3} }} \\ & \quad + \frac{{27hm^{2} \pi^{3} \mu_{1} }}{2Rl} + \frac{{3hm^{2} \pi^{3} \mu_{2} }}{2Rl} \\ I & = \frac{{3hm^{3} \mu_{1} \pi^{4} }}{{l^{2} }} - \frac{{9hm\pi^{2} \mu_{1} }}{{R^{2} }} - \frac{{9h^{3} m^{5} \mu_{1} \pi^{6} }}{{4l^{4} }}, \\ U & = \frac{{4hm\pi^{2} \left( {\mu_{1} + \mu_{2} } \right)}}{l} - \frac{{2h^{3} m^{3} \pi^{4} \mu_{1} }}{{l^{3} }} \\ & \quad + \frac{{2h^{3} m^{3} \pi^{4} \mu_{2} }}{{3l^{3} }} - \frac{{16hl\left( {\mu_{1} + \mu_{2} } \right)}}{{R^{2} m}} \\ K & = \frac{{6hm^{3} \mu_{1} \pi^{4} }}{{l^{2} }} - \frac{{18hm\mu_{1} \pi^{2} }}{{R^{2} }} - \frac{{9h^{3} m^{5} \mu_{1} \pi^{6} }}{{2l^{4} }}, \\ Y & = - \frac{{5Rh^{3} m^{6} \pi^{7} \left( {\mu_{1} + \mu_{2} } \right)}}{{4l^{5} }} + \frac{{27h^{3} m^{4} \pi^{5} \mu_{1} }}{{8Rl^{3} }} \\ & \quad + \frac{{3h^{3} m^{4} \pi^{5} \mu_{2} }}{{8Rl^{3} }} + \frac{{9Rhm^{4} \pi^{5} \left( {\mu_{1} + \mu_{2} } \right)}}{{2l^{3} }} \\ & \quad + \frac{{45\pi hl\left( {\mu_{1} + \mu_{2} } \right)}}{{2R^{3} }} - \frac{{9hm^{2} \mu_{1} \pi^{3} }}{2Rl} \\ & \quad + \frac{{3hm^{2} \mu_{2} \pi^{3} }}{2Rl} + \frac{{9Rh^{5} m^{8} \pi^{9} \left( {\mu_{1} + \mu_{2} } \right)}}{{32l^{7} }}. \\ \end{aligned} $$
(A2)

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Zheng, F., Zhang, W., Yuan, X.G. et al. Radial nonlinear vibrations of thin-walled hyperelastic cylindrical shell composed of Mooney–Rivlin materials under radial harmonic excitation. Nonlinear Dyn 111, 19791–19815 (2023). https://doi.org/10.1007/s11071-023-08878-w

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