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Forced vibration of an axially moving laminated composite cylindrical shallow shell

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Abstract

An axially moving composite cylindrical shallow shell is presented in a wide class of engineering problem, such as the space sailors drag sail mechanism and the deployable antennas. It is very important to have a better understanding of the vibrational characteristics of an axially moving composite cylindrical shallow shell. Nonlinear forced vibrations of an axially moving thin laminated composite cylindrical shallow shell are investigated in the present work. The governing equation and the compatibility equation are derived based on the von Kármán plate theory and discretized by the Galerkin method in a system of ordinary differential equations. The ordinary differential equations are solved by employing the harmonic balance method to obtain the approximate analytical response. Moreover, the stability of the analytical response is also determined. The model is validated by comparing with the natural frequencies available in the literature and the dynamic response obtained through the Runge–Kutta method. Natural frequencies and frequency–response characteristics are discussed in detail. The effects of the movement speed, curvature radius on the natural frequencies are examined. The system has a strong hardening nonlinear behavior and complex dynamic behavior. The frequency–response curve possesses three peaks, due to the strong nonlinear modal coupling. It is further found that the effects of the movement speed, curvature radius, transverse excitation amplitude, damping coefficient and initial tension on the nonlinear dynamic response of an axially moving thin laminated composite cylindrical shallow shell are examined.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant No. 11772107].

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Authors and Affiliations

  1. National Key Laboratory of Science and Technology on Advanced Composites in Special Environment, Harbin Institute of Technology, Harbin, 150001, China

    Ming Li, Yanqi Li, Xiaohui Liu & Fuhong Dai

  2. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, China

    Dong Yu

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  1. Ming Li
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Appendix

Appendix

The coefficients in Eq. (12) and Eq. (13) are given by

$$\begin{gathered} p_{1} = \frac{{2\xi^{2} \left( {D_{12} + 2D_{66} + 4B_{16} B_{26} P_{12} - B_{16} B_{26} P_{33} - 2B_{16}^{2} P_{11} - 2B_{26}^{2} P_{22} } \right)}}{{D_{11} - B_{16}^{2} P_{33} }},p_{2} = \frac{{\xi^{4} \left( {D_{22} - B_{26}^{2} P_{33} } \right)}}{{D_{11} - B_{16}^{2} P_{33} }}, \hfill \\ p_{3} = \frac{{h^{2} \xi^{4} }}{{P_{22} \left( {D_{11} - B_{16}^{2} P_{33} } \right)}},p_{4} = \frac{\xi }{h}\left( {2B_{16} P_{11} - 2B_{26} P_{12} - B_{26} P_{33} } \right),p_{5} = \frac{1}{\xi h}\left( {2B_{26} P_{22} - 2B_{16} P_{12} - B_{16} P_{33} } \right), \hfill \\ p_{6} = \frac{{\xi^{2} \left( {P_{33} - 2P_{12} } \right)}}{{P_{22} }},p_{7} = \frac{{\xi^{4} P_{11} }}{{P_{22} }} \hfill \\ \end{gathered}$$

The coefficients in Eq. (19) are given by

$$\begin{gathered} K_{1} { = }1,K_{2} { = }c,K_{3} { = } - \frac{16}{3}\gamma ,K_{4} { = } - \pi^{2} \gamma^{2} + \frac{{\kappa^{2} p_{3} }}{{1 + p_{6} + p_{7} }}{ + }\pi^{2} N_{0}^{{}} + \pi^{4} \left( {1 + p_{1} + p_{2} } \right),K_{5} { = } - \frac{8}{3}c\gamma , \hfill \\ K_{6} { = }\frac{{16\kappa p_{3} }}{{1 + p_{6} + p_{7} }},K_{7} { = }\frac{{64\kappa p_{3} }}{{5\left( {1 + p_{6} + p_{7} } \right)}} + \frac{{512\kappa p_{3} }}{{5\left( {16 + 4p_{6} + p_{7} } \right)}},K_{8} { = }\frac{{512p_{3} }}{{9\left( {1 + p_{6} + p_{7} } \right)}},K_{9} { = }\frac{{2048p_{3} }}{{15\left( {1 + p_{6} + p_{7} } \right)}} + \frac{{16384p_{3} }}{{25\left( {16 + 4p_{6} + p_{7} } \right)}}, \hfill \\ K_{10} { = } - 4F_{0} \sin \left( {\pi x_{0} } \right)\sin \left( {\pi \theta_{0} } \right) \hfill \\ \end{gathered}$$

The coefficients in Eq. (20) are given by

$$\begin{gathered} T_{1} { = }1,T_{2} { = }\frac{16}{3}\gamma ,T_{3} { = }c,T_{4} { = }\frac{8}{3}c\gamma ,T_{5} { = } - 4\pi^{2} \gamma^{2} + \frac{{16\kappa^{2} p_{3} }}{{16 + 4p_{6} + p_{7} }}{ + 4}\pi^{2} N_{0}^{{}} + \pi^{4} \left( {16 + 4p_{1} + p_{2} } \right), \hfill \\ T_{6} { = }\frac{{128\kappa p_{3} }}{{5\left( {1 + p_{6} + p_{7} } \right)}} + \frac{{1024\kappa p_{3} }}{{5\left( {16 + 4p_{6} + p_{7} } \right)}},T_{7} { = }\frac{{2048p_{3} }}{{15\left( {1 + p_{6} + p_{7} } \right)}} + \frac{{16384p_{3} }}{{25\left( {16 + 4p_{6} + p_{7} } \right)}},T_{8} { = }\frac{{8192p_{3} }}{{25\left( {1 + p_{6} + p_{7} } \right)}}, \hfill \\ T_{9} { = } - 4F_{0} \sin \left( {2\pi x_{0} } \right)\sin \left( {\pi \theta_{0} } \right) \hfill \\ \end{gathered}$$

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Li, M., Li, Y., Liu, X. et al. Forced vibration of an axially moving laminated composite cylindrical shallow shell. Meccanica 58, 1581–1598 (2023). https://doi.org/10.1007/s11012-023-01693-y

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