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Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis

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Abstract

A study of free vibration of orthotropic circular cylindrical shells is presented. The vibration control equations of shells are based on Flügge classical thin shell theory. Wave approach is used in the analysis, in which the boundary conditions of shells can be simplified according to the associated beam. The free vibration frequencies of shells can be obtained from a frequency polynomial equation of order 6. The parametric analysis of the free vibration of orthotropic cylindrical shells is investigated using a statistical method. The effects of geometrical parameters and material characteristics upon frequencies are investigated here. Multivariate analysis (MVA) can be a useful tool for this parametric study. Some statistical characteristics, including correlation analysis and ANOVA are applied. ANOVA has been conducted to predict the statistical significance of the various factors. Calculations are performed in the Minitab statistical software. The results show that the L/R, h/R and m have larger effects on the lowest frequency. The importance of input parameters is ranked according to their contributions to the total variance. A knowledge and data visualization approach, Self-organizing mapping (SOM) is also adopted here for mining some intrinsic characteristics of shells.

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Authors and Affiliations

  1. Wuhan 2nd Ship Design and Research Institute, Wuhan, 430064, Hubei Province, People’s Republic of China

    Xuebin Li, Zhiqiang Wang & Lihua Huang

Authors
  1. Xuebin Li
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  2. Zhiqiang Wang
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  3. Lihua Huang
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Corresponding author

Correspondence to Xuebin Li.

Appendix

Appendix

$${\text{g}}_{6} = \frac{{h^{3} R^{6} \rho^{3} }}{{D_{x}^{3} }}$$
$$\begin{aligned} {\text{g}}_{4} = & - \frac{1}{{D_{x}^{3} }}h^{2} R^{2} \rho^{2} \left( {\left( {1 + n^{2} } \right)R^{2} D_{\theta } - 3PR^{4} k_{\text{ns}}^{2} + R^{4} D_{x} k_{\text{ns}}^{2} + R^{2} D_{{{\text{x}}\uptheta}} \left( {n^{2} + R^{2} k_{\text{ns}}^{2} } \right) + R^{4} k_{\text{ns}}^{4} K_{x} + n^{2} K_{{{\text{x}}\uptheta}} } \right. \\ & \;\left. { +\,3R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}\uptheta}} + 4n^{2} R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}\uptheta}} + K_{\theta } - 2n^{2} K_{\theta } + n^{4} K_{\theta } + n^{2} R^{2} k_{\text{ns}}^{2} K_{\theta } \nu_{x} + n^{2} R^{2} k_{\text{ns}}^{2} K_{x} \nu_{\theta } } \right) \\ \end{aligned}$$
$$g_{2} = \frac{1}{{R^{2} D_{x}^{3}}}h\rho ({3P^{2} R^{8} k_{\text{ns}}^{4} - 2PR^{8} D_{x} k_{\text{ns}}^{4} - 2PR^{8} k_{\text{ns}}^{6} K_{x} + R^{8} D_{x} k_{\text{ns}}^{6} K_{x} - R^{6} k_{\text{ns}}^{6} K_{x}^{2} - 2n^{2} PR^{4} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}}} - 6PR^{6} k_{\text{ns}}^{4} K_{{{\text{x}}{\uptheta}}} - 8n^{2} PR^{6} k_{\text{ns}}^{4} K_{{{\text{x}}{\uptheta}}} + 3R^{6} D_{x} k_{\text{ns}}^{4} K_{{{\text{x}}{\uptheta}}} + 4n^{2} R^{6} D_{x} k_{\text{ns}}^{4} K_{{{\text{x}}{\uptheta}}} + 3n^{2} R^{4} k_{\text{ns}}^{4} K_{x} K_{{{\text{x}}{\uptheta}}} + 3R^{6} k_{\text{ns}}^{6} K_{x} K_{{{\text{x}}{\uptheta}}} + 3n^{2} R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}}^{2} + 3n^{4} R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}}^{2} + 3n^{2} R^{4} k_{\text{ns}}^{4} K_{{{\text{x}}{\uptheta}}}^{2} - 2PR^{4} k_{\text{ns}}^{2} K_{\theta} + 4n^{2} PR^{4} k_{\text{ns}}^{2} K_{\theta} - 2n^{4} PR^{4} k_{\text{ns}}^{2} K_{\theta} + R^{4} D_{x} k_{\text{ns}}^{2} K_{\theta} - 2n^{2} R^{4} D_{x} k_{\text{ns}}^{2} K_{\theta} + n^{4} R^{4} D_{x} k_{\text{ns}}^{2} K_{\theta} + n^{2} K_{{{\text{x}}{\uptheta}}} K_{\theta} - 2n^{4} K_{{{\text{x}}{\uptheta}}} K_{\theta} + n^{6} K_{{{\text{x}}{\uptheta}}} K_{\theta} + 3R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}} K_{\theta} - 6n^{2} R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}} K_{\theta} + 3n^{4} R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}} K_{\theta} - 2n^{2} PR^{6} k_{\text{ns}}^{4} K_{\theta} \nu_{x} + n^{2} R^{6} D_{x} k_{\text{ns}}^{4} K_{\theta} \nu_{x} + n^{4} R^{2} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}} K_{\theta} \nu_{x} - 2n^{2} PR^{6} k_{\text{ns}}^{4} K_{x} \nu_{\theta} - R^{6} D_{x} k_{\text{ns}}^{4} K_{x} \nu_{\theta} + n^{2} R^{6} D_{x} k_{\text{ns}}^{4} K_{x} \nu_{\theta} + n^{2} R^{4} D_{x} k_{\text{ns}}^{2} K_{{{\text{x}}{\uptheta}}} \nu_{\theta} + n^{4} R^{2} k_{\text{ns}}^{2} K_{x} K_{{{\text{x}}{\uptheta}}} \nu_{\theta} - n^{2} R^{4} k_{\text{ns}}^{4} K_{x} K_{\theta} \nu_{x} \nu_{\theta} + R^{2} D_{{{\text{x}}{\uptheta}}} ({R^{6} k_{\text{ns}}^{6} K_{x} + n^{2} ({- 1 + n^{2}})^{2} K_{\theta} + R^{2} D_{\theta} ({n^{2} + n^{4} + R^{2} k_{\text{ns}}^{2} ({1 - n^{2} \nu_{x}})})} + R^{4} k_{\text{ns}}^{4} ({- 2PR^{2} + R^{2} D_{x} + 4n^{2} K_{{{\text{x}}{\uptheta}}} + n^{2} K_{\theta} \nu_{x} + n^{2} K_{x} ({1 + \nu_{\theta}})}) + {R^{2} k_{\text{ns}}^{2} ( {4({n^{2} + n^{4}})K_{{{\text{x}}{\uptheta}}} + K_{\theta} ({({- 1 + n^{2}})^{2} + n^{4} \nu_{x}}) + n^{2} ({- 2PR^{2} - R^{2} D_{x} \nu_{\theta} + n^{2} K_{x} \nu_{\theta}})})}) + R^{2} D_{\theta} ({n^{2} ({( {1 + n^{2}})K_{{{\text{x}}{\uptheta}}} + ({- 1 + n^{2}})^{2} K_{\theta}})} + R^{4} k_{\text{ns}}^{4} K_{x} ({n^{2} - \nu_{x}}) + R^{2} k_{\text{ns}}^{2} ({- 2PR^{2} - 2n^{2} PR^{2} - n^{2} K_{\theta} \nu_{x}} { {{+ n^{4} K_{\theta} \nu_{x} + K_{{{\text{x}}{\uptheta}}} ({3 - 6n^{2} + 4n^{4} + n^{2} \nu_{x}}) - n^{2} K_{x} \nu_{\theta} + n^{4} K_{x} \nu_{\theta} - ({1 + n^{2}})R^{2} D_{x} ({- 1 + \nu_{x} \nu_{\theta}})})})})$$
$$g_{0} = \frac{1}{{R^{4} D_{x}^{3}}}(D_{\theta} (- n^{4} ({- 1 + n^{2}})^{2} K_{{{\text{x}}{\uptheta}}}K_{\theta} + R^{6} k_{\text{ns}}^{6} K_{x} (n^{2} PR^{2} - PR^{2} \nu_{x} + 3K_{{{\text{x}}{\uptheta}}} \nu_{x} - 3n^{2} K_{{{\text{x}}{\uptheta}}} \nu_{x} +n^{2} K_{x} ({1 - \nu_{x} \nu_{\theta}})+ n^{2} R^{2} D_{x} ({- 1 + \nu_{x} \nu_{\theta}})) + R^{4} k_{\text{ns}}^{4} (3n^{2} ({- 1 + n^{2}})K_{{{\text{x}}{\uptheta}}}^{2} \nu_{x} - PR^{2} (({1 + n^{2}})PR^{2} - n^{2} ({- 1 + n^{2}})K_{\theta} \nu_{x} - n^{2} ({- 1 + n^{2}})K_{x} \nu_{\theta}) - R^{2} D_{x} (PR^{2} + n^{2} PR^{2} + ({- 3 +6n^{2} - 4n^{4}})K_{{{\text{x}}{\uptheta}}} - n^{2} ({- 1 +n^{2}})K_{\theta} \nu_{x} + n^{2} K_{x} \nu_{\theta} - n^{4} K_{x} \nu_{\theta})({- 1 + \nu_{x}\nu_{\theta}}) + K_{{{\text{x}}{\uptheta}}} ({PR^{2} ({3 -6n^{2} + 4n^{4} + n^{2} \nu_{x}}) + n^{4} K_{x} ({- 3 + \nu_{x} \nu_{\theta}})})) +n^{2} R^{2} k_{\text{ns}}^{2} (- 3({- 1 + n^{2}})^{2}K_{{{\text{x}}{\uptheta}}}^{2} + K_{{{\text{x}}{\uptheta}}} ({({1 + n^{2}})PR^{2} - n^{2} ({- 1 + n^{2}})K_{\theta} \nu_{x} - n^{2} ({- 1 + n^{2}})K_{x} \nu_{\theta}}) + ({- 1 + n^{2}})^{2} R^{2}K_{\theta} ({P + D_{x} ({- 1 + \nu_{x} \nu_{\theta}})}))) + k_{\text{ns}}^{2} (- R^{6} k_{\text{ns}}^{6} K_{x}({PR^{2} - R^{2} D_{x} + K_{x}})({PR^{2} -3K_{{{\text{x}}{\uptheta}}}}) + n^{2} ({- 1 + n^{2}})^{2} ({PR^{2} - 3K_{{{\text{x}}{\uptheta}}}})K_{{{\text{x}}{\uptheta}}} K_{\theta} + R^{4} k_{\text{ns}}^{4} (PR^{2} ({- ({3 +4n^{2}})PR^{2}+ 3n^{2} K_{x}})K_{{{\text{x}}{\uptheta}}} + 3n^{2} (PR^{2}- 3K_{x})K_{{{\text{x}}{\uptheta}}}^{2} + PR^{2} ({- n^{2} K_{\theta}\nu_{x} ({PR^{2} + K_{x} \nu_{\theta}}) + PR^{2}({PR^{2} - n^{2} K_{x} \nu_{\theta}})}) +R^{2} D_{x} (- 3n^{2} K_{{{\text{x}}{\uptheta}}}^{2} + K_{{{\text{x}}{\uptheta}}} ((3 + 4n^{2})PR^{2} - 3({- 1 + n^{2}})K_{x} \nu_{\theta}) + PR^{2} ({- PR^{2} + n^{2}K_{\theta} \nu_{x} + ({- 1 + n^{2}})K_{x} \nu_{\theta}}))) + k_{\text{ns}}^{2} (- ({- 1+ n^{2}})^{2} PR^{6} ({P - D_{x}})K_{\theta}+ 3n^{2} R^{4} K_{{{\text{x}}{\uptheta}}}^{2} ({({1 + n^{2}})P + ({- 1 + n^{2}})D_{x} \nu_{\theta}}) + K_{{{\text{x}}{\uptheta}}} (n^{2} PR^{4} (- PR^{2} +R^{2} D_{x} \nu_{\theta} + n^{2} K_{x} \nu_{\theta}) +K_{\theta} (3({- 1 + n^{2}})^{2} PR^{4} + n^{4}R^{2} \nu_{x} ({PR^{2} + K_{x} \nu_{\theta}}) +R^{4} D_{x} (- 3({- 1 + n^{2}})^{2} +n^{4} \nu_{x} \nu_{\theta}))))) + R^{2} D_{{{\text{x}}{\uptheta}}}(D_{\theta} (- n^{4} ({- 1 + n^{2}})^{2} K_{\theta} + ({1 + n^{2}})R^{6} k_{\text{ns}}^{6} K_{x} \nu_{x}+ n^{2} R^{2} k_{\text{ns}}^{2} (PR^{2} + n^{2} PR^{2} -4({- 1 + n^{2}})^{2} K_{{{\text{x}}{\uptheta}}} - ({- 1+ n^{2}})K_{\theta} \nu_{x} + n^{2} K_{x} \nu_{\theta}- n^{4} K_{x} \nu_{\theta}) + R^{4} k_{\text{ns}}^{4} (PR^{2} -n^{2} PR^{2} \nu_{x} - 4n^{2} K_{{{\text{x}}{\uptheta}}} \nu_{x}+ 4n^{4} K_{{{\text{x}}{\uptheta}}} \nu_{x} - n^{2} K_{\theta} \nu_{x}^{2} +n^{4} K_{\theta} \nu_{x}^{2} + R^{2} D_{x} ({- 1 + \nu_{x} \nu_{\theta}}) + n^{2} K_{x} ({- 2 - n^{2} +n^{2} \nu_{x} \nu_{\theta}}))) +k_{\text{ns}}^{2} (R^{6} k_{\text{ns}}^{6} K_{x} ({PR^{2} - R^{2} D_{x}+ K_{x}}) + n^{2} ({- 1 + n^{2}})^{2}K_{\theta} ({- 4K_{{{\text{x}}{\uptheta}}} + R^{2} ({P + D_{x}\nu_{\theta}})}) + R^{4} k_{\text{ns}}^{4} (PR^{2} (-PR^{2} + 4n^{2} K_{{{\text{x}}{\uptheta}}} + n^{2} K_{\theta}\nu_{x}) - n^{2} K_{x}^{2} \nu_{\theta} + R^{2} D_{x} ({PR^{2} - 4n^{2} K_{{{\text{x}}{\uptheta}}} - n^{2} K_{\theta} \nu_{x} +K_{x} \nu_{\theta}}) + n^{2} K_{x} (- 12K_{{{\text{x}}{\uptheta}}} -K_{\theta} \nu_{x} + PR^{2} ({1 + \nu_{\theta}}))) + k_{\text{ns}}^{2} (4n^{2} R^{4} K_{{{\text{x}}{\uptheta}}} ({({1 + n^{2}})P + ({- 1 + n^{2}})D_{x}\nu_{\theta}}) + n^{2} R^{4} (P({- PR^{2} + n^{2}K_{x} \nu_{\theta}}) + D_{x} \nu_{\theta}({- PR^{2} + ({- 1 + n^{2}})K_{x} \nu_{\theta}})) + K_{\theta} (({- 1 + n^{2}})^{2} PR^{4} + n^{4} R^{2} \nu_{x} ({PR^{2} + K_{x}\nu_{\theta}}) + R^{4} D_{x} ({-({- 1 + n^{2}})^{2} + n^{4} \nu_{x} \nu_{\theta}}))))))$$

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Li, X., Wang, Z. & Huang, L. Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis. Meccanica 52, 2349–2361 (2017). https://doi.org/10.1007/s11012-016-0587-8

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