Analytical approach for nonlinear vibration response of the thin cylindrical shell with a straight crack

Abstract

Thin cylindrical shells are susceptible to cracking under long-term load and external impact, and it is of considerable scientific and technical value to investigate the nonlinear vibration response characteristics and monitor the health condition of the shell structure. Based on the Flügge shell theory, the nonlinear dynamic model for the thin cylindrical shell is established. By the partial Fourier transform combined with the residue theorem, the forced vibration generation and propagation mechanism of the thin cylindrical shell are investigated, and the analytical solution of forced vibration displacement in the space domain is obtained. Then, the local flexibility matrix is derived from the perspective of fracture mechanics, and the continuous coordination condition on both sides of the straight crack is constructed using the linear spring model. Combined with the wave superposition principle, the analytical approach for nonlinear vibration response is proposed to reveal the evolution law of vibration characteristics of the thin cylindrical shell with a straight crack, and then, a straight crack identification method based on natural frequency isolines and amplitude maximization methods is presented. Finally, the effect of various morphological information of the straight crack on the nonlinear vibration response characteristics of the thin cylindrical shell is studied in detail, and a numerical case is conducted to verify the effectiveness of the proposed straight crack identification method.

Appendices

Appendix A

The detailed expressions of coefficient matrix $\left[ {N_{3 \times 3} } \right]$ are described in the following forms.

$$ N_{11} = \lambda^{2} + n^{{2}} \left( {{1} + \beta^{{2}} } \right)\frac{{{1} - \mu }}{{2}} - \Omega^{{2}} $$

$$ N_{12} = - in\frac{1 + \mu }{2}\lambda $$

$$ N_{13} = - i\beta^{2} \lambda^{3} - \left( {i\mu - in^{2} \beta^{2} \frac{1 - \mu }{2}} \right)\lambda $$

$$ N_{21} = in\frac{1 + \mu }{2}\lambda $$

$$ N_{22} = (1 + 3\beta^{2} )\frac{1 - \mu }{2}\lambda^{2} + n^{2} - \Omega^{2} $$

$$ N_{23} = n\beta^{2} \frac{3 - \mu }{2}\lambda^{2} + n $$

$$ N_{31} = - i\beta^{2} \lambda^{3} - \left( {i\mu - in^{2} \beta^{2} \frac{1 - \mu }{2}} \right)\lambda $$

$$ N_{23} = - n\beta^{2} \frac{3 - \mu }{2}\lambda^{2} - n $$

$$ N_{33} = - 1 - \beta^{2} (n^{2} - 1)^{2} + \Omega^{2} - 2n^{2} \beta^{2} \lambda^{2} - \beta^{2} \lambda^{4} $$

Appendix B

The coefficient $a_{i} \left( {i = 0,2,4,6,8} \right)$ of the algebraic equation in Sect. 2.1 are expressed as

$$ \begin{aligned} a_{0} & = \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)\left( {n^{2} - \Omega^{2} } \right) \\ & \quad \left( { - 1 - \beta^{2} (n^{2} - 1)^{2} + \Omega^{2} } \right) \\ & \quad + \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)n^{2} \\ \end{aligned} $$

$$ \begin{aligned} a_{2} & = \left( {n^{2} - \Omega^{2} + (1 + 3\beta^{2} )\frac{1 - \mu }{2}\left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)} \right) \\ & \quad \left( { - 1 - \beta^{2} (n^{2} - 1)^{2} + \Omega^{2} } \right) \\ & {\kern 1pt} \quad - 2n^{2} \beta^{2} \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)\left( {n^{2} - \Omega^{2} } \right) \\ & \quad + 2in^{2} \frac{1 + \mu }{2}\left( {i\mu - in^{2} \beta^{2} \frac{1 - \mu }{2}} \right) \\ {\kern 1pt} & \quad - \left( {i\mu - in^{2} \beta^{2} \frac{1 - \mu }{2}} \right)^{2} \left( {n^{2} - \Omega^{2} } \right) + n^{2} \\ & \quad + 2n^{2} \beta^{2} \frac{3 - \mu }{2}\left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right) \\ {\kern 1pt} & \quad -^{2} \left( { - 1 - \beta^{2} (n^{2} - 1)^{2} + \Omega^{2} } \right)\left( {n\frac{1 + \mu }{2}} \right) \\ \end{aligned} $$

$$ \begin{aligned} a_{4} & = (1 + 3\beta^{2} )\frac{1 - \mu }{2}\left( { - 1 - \beta^{2} (n^{2} - 1)^{2} + \Omega^{2} } \right) - 2n^{2} \beta^{2} (n^{2} - \Omega^{2} ) \\ & {\kern 1pt} \quad - 2n^{2} \beta^{2} (1 + 3\beta^{2} )\frac{1 - \mu }{2}\left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right) \\ & \quad - \beta^{2} \left( {n^{2} - \Omega^{2} } \right)\left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right) \\ & {\kern 1pt} \quad + 2in^{2} \beta^{2} \frac{3 - \mu }{2}\frac{1 + \mu }{2}\left( {i\mu - in^{2} \beta^{2} \frac{1 - \mu }{2}} \right) + 2in^{2} \frac{1 + \mu }{2}\left( {i\beta^{2} } \right) \\ & \quad + 2\beta^{2} \left( {\mu - n^{2} \beta^{2} \frac{1 - \mu }{2}} \right)\left( {n^{2} - \Omega^{2} } \right) \\ & \quad + \left( {\mu - n^{2} \beta^{2} \frac{1 - \mu }{2}} \right)^{2} \left( {(1 + 3\beta^{2} )\frac{1 - \mu }{2}} \right){\kern 1pt} + 2n^{2} \beta^{2} \frac{3 - \mu }{2} \\ & \quad + \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)\left( {n\beta^{2} \frac{3 - \mu }{2}} \right)^{2} + 2n^{2} \beta^{2} \left( {n\frac{1 + \mu }{2}} \right)^{2} \\ \end{aligned} $$

$$ \begin{aligned} a_{6} & = - 2n^{2} \beta^{2} (1 + 3\beta^{2} )\frac{1 - \mu }{2} - \beta^{2} \left( {n^{2} - \Omega^{2} } \right) \\ & \quad + \beta^{2} (1 + 3\beta^{2} )\frac{1 - \mu }{2}\left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right) - \beta^{4} n^{2} \frac{{\left( {1 + \mu } \right)\left( {3 - \mu } \right)}}{2} \\ & {\kern 1pt} \quad + \beta^{4} \left( {n^{2} - \Omega^{2} } \right) + 2\beta^{2} (1 + 3\beta^{2} )\frac{1 - \mu }{2}\left( {\mu - n^{2} \beta^{2} \frac{1 - \mu }{2}} \right) \\ & {\kern 1pt} \quad + \left( {n\beta^{2} \frac{3 - \mu }{2}} \right)^{2} {\kern 1pt} + \left( {n\beta \frac{1 + \mu }{2}} \right)^{2} \\ \end{aligned} $$

$$ a_{8} = \left( {\beta^{4} - \beta^{2} } \right)\left( {(1 + 3\beta^{2} )\frac{1 - \mu }{2}} \right) $$

Appendix C

The analytical solutions of axial $u(x)$, circumferential $v(x)$, and radial $w(x)$ displacement responses are obtained using the residue theorem.

$$ \begin{aligned} u(x) & = \frac{{\Omega^{2} F_{x} }}{{2\pi \rho_{s} hR\omega^{2} }} \cdot \left[ {2\pi i\sum\limits_{k = 1}^{m} {{\text{Res}}} [M_{13} (\lambda_{k} )] + \pi i\;\sum\limits_{k = 1}^{8 - 2m} {{\text{Res}}} [M_{13} (\lambda_{k} )]} \right] \\ & = \frac{{\Omega^{2} F_{x} }}{{2\pi \rho_{s} hR\omega^{2} }} \cdot {\text{Re}} \left[ {\left( \begin{gathered} 2\pi i \cdot \sum\limits_{k = 1}^{m} {\frac{{\alpha^{\prime}_{1} \lambda_{k}^{4} + \beta^{\prime}_{1} \lambda_{k}^{2} + \gamma^{\prime}_{1} \lambda_{k}^{0} }}{{8a^{\prime}_{8} \lambda_{k}^{7} + 6a^{\prime}_{6} \lambda_{k}^{5} + 4a^{\prime}_{4} \lambda_{k}^{3} + 2a^{\prime}_{2} \lambda_{k} }}} \hfill \\ + \pi i \cdot \sum\limits_{k = 1}^{8 - 2m} {\frac{{\alpha^{\prime}_{1} \lambda_{k}^{4} + \beta^{\prime}_{1} \lambda_{k}^{2} + \gamma^{\prime}_{1} \lambda_{k}^{0} }}{{8a^{\prime}_{8} \lambda_{k}^{7} + 6a^{\prime}_{6} \lambda_{k}^{5} + 4a^{\prime}_{4} \lambda_{k}^{3} + 2a^{\prime}_{2} \lambda_{k} }}} \hfill \\ \end{gathered} \right) \cdot \exp \left( {i\frac{{\lambda_{k} }}{R}x} \right)} \right] \\ \end{aligned} $$

$$ \begin{aligned} v(x) & = \frac{{\Omega^{2} F_{x} }}{{2\pi \rho_{s} hR\omega^{2} }} \cdot \left[ {2\pi i \cdot \sum\limits_{k = 1}^{m} {{\text{Res}}} [M_{23} (\lambda_{k} )] + \pi i \cdot \sum\limits_{k = 1}^{8 - 2m} {{\text{Res}}} [M_{23} (\lambda_{k} )]} \right] \\ & = \frac{{\Omega^{2} F_{x} }}{{2\pi \rho_{s} hR\omega^{2} }} \cdot {\text{Re}} \left[ {\left( \begin{gathered} 2\pi i \cdot \sum\limits_{k = 1}^{m} {\frac{{a^{\prime}_{2} \lambda_{k}^{4} + \beta^{\prime}_{2} \lambda_{k}^{2} + \gamma^{\prime}_{2} \lambda_{k}^{0} }}{{8a^{\prime}_{8} \lambda_{k}^{7} + 6a^{\prime}_{6} \lambda_{k}^{5} + 4a^{\prime}_{4} \lambda_{k}^{3} + 2a^{\prime}_{2} \lambda_{k} }}} \hfill \\ { + }\pi i \cdot \sum\limits_{k = 1}^{8 - 2m} {\frac{{a^{\prime}_{2} \lambda_{k}^{4} + \beta^{\prime}_{2} \lambda_{k}^{2} + \gamma^{\prime}_{2} \lambda_{k}^{0} }}{{8a^{\prime}_{8} \lambda_{k}^{7} + 6a^{\prime}_{6} \lambda_{k}^{5} + 4a^{\prime}_{4} \lambda_{k}^{3} + 2a^{\prime}_{2} \lambda_{k} }}} \hfill \\ \end{gathered} \right) \cdot \exp \left( {i\frac{{\lambda_{k} }}{R}x} \right)} \right] \\ \end{aligned} $$

$$ \begin{aligned} w(x) & = \frac{{\Omega^{2} F_{x} }}{{2\pi \rho_{s} hR\omega^{2} }} \cdot \left[ {2\pi i \cdot \sum\limits_{k = 1}^{m} {Res} [M_{33} (\lambda_{k} )] + \pi i \cdot \sum\limits_{k = 1}^{8 - 2m} {{\text{Res}}} [M_{33} (\lambda_{k} )]} \right] \\ & = \frac{{\Omega^{2} F_{x} }}{{2\pi \rho_{s} hR\omega^{2} }} \cdot {\text{Re}} \left[ {\left( \begin{gathered} 2\pi i \cdot \sum\limits_{k = 1}^{m} {\frac{{a^{\prime}_{3} \lambda_{k}^{4} + \beta^{\prime}_{3} \lambda_{k}^{2} + \gamma^{\prime}_{3} \lambda_{k}^{0} }}{{8a^{\prime}_{8} \lambda_{k}^{7} + 6a^{\prime}_{6} \lambda_{k}^{5} + 4a^{\prime}_{4} \lambda_{k}^{3} + 2a^{\prime}_{2} \lambda_{k} }}} \hfill \\ { + }\pi i \cdot \sum\limits_{k = 1}^{8 - 2m} {\frac{{a^{\prime}_{3} \lambda_{k}^{4} + \beta^{\prime}_{3} \lambda_{k}^{2} + \gamma^{\prime}_{3} \lambda_{k}^{0} }}{{8a^{\prime}_{8} \lambda_{k}^{7} + 6a^{\prime}_{6} \lambda_{k}^{5} + 4a^{\prime}_{4} \lambda_{k}^{3} + 2a^{\prime}_{2} \lambda_{k} }}} \hfill \\ \end{gathered} \right) \cdot \exp \left( {i\frac{{\lambda_{k} }}{R}x} \right)} \right] \\ \end{aligned} $$

where $\alpha^{\prime}_{1}$, $\beta^{\prime}_{1}$ $\gamma^{\prime}_{1}$, $\alpha^{\prime}_{2}$, $\beta^{\prime}_{2}$, $\gamma^{\prime}_{2}$, $\alpha^{\prime}_{3}$, $\beta^{\prime}_{3}$, $\gamma^{\prime}_{3}$, $a^{\prime}_{8}$, $a^{\prime}_{6}$, $a^{\prime}_{4}$, and $a^{\prime}_{2}$ are coefficient, which are given as:

$$ \alpha^{\prime}_{1} = \beta^{2} (1 + 3\beta^{2} )\frac{1 - \mu }{2} $$

$$ \begin{aligned} \beta^{\prime}_{1} & = \beta^{2} \left( {n^{2} - \Omega^{2} } \right) + \left( {\mu - n^{2} \beta^{2} \frac{1 - \mu }{2}} \right) \\ & \quad \left( {(1 + 3\beta^{2} )\frac{1 - \mu }{2}} \right) - \frac{1 + \mu }{2}\frac{3 - \mu }{2}n^{2} \beta^{2} \\ \end{aligned} $$

$$ \gamma^{\prime}_{1} = \left( {\mu - n^{2} \beta^{2} \frac{1 - \mu }{2} - \frac{1 + \mu }{2}} \right)n^{2} + \left( {\mu - n^{2} \beta^{2} \frac{1 - \mu }{2}} \right)\Omega^{2} $$

$$ \alpha^{\prime}_{2} = - n\frac{1 + \mu }{2}\beta^{2} - n\beta^{2} \frac{3 - \mu }{2} $$

$$ \begin{aligned} \beta^{\prime}_{2} & = n\frac{1 + \mu }{2}\left( {n^{2} \beta^{2} \frac{1 - \mu }{2} - \mu } \right) \\ & \quad - \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)\left( {n\beta^{2} \frac{3 - \mu }{2}} \right) \\ \end{aligned} $$

$$ \gamma^{\prime}_{2} = - \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)n $$

$$ \alpha^{\prime}_{3} = (1 + 3\beta^{2} )\frac{1 - \mu }{2} $$

$$ \begin{aligned} \beta^{\prime}_{3} & = n^{2} - \Omega^{2} + \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right) \\ & \quad (1 + 3\beta^{2} )\frac{1 - \mu }{2} + in\frac{1 + \mu }{2} \\ \end{aligned} $$

$$ \gamma^{\prime}_{3} = \alpha^{\prime}f^{\prime} = \left( {n^{2} (1 + \beta^{2} )\frac{1 - \mu }{2} - \Omega^{2} } \right)\left( {n^{2} - \Omega^{2} } \right) $$

$$ a^{\prime}_{8} = - a_{8} ,a^{\prime}_{6} = a_{6} ,a^{\prime}_{4} = - a_{4} ,a^{\prime}_{2} = - a_{2} $$