﻿ 圆柱壳水下振动的截断模态规律性研究
 舰船科学技术  2018, Vol. 40 Issue (12): 22-26, 32 PDF

Study on the law of truncation mode of cylindrical shells underwater vibration
TONG Bo, LI Yong-qing, ZHU Xi, ZHANG Yan-bing
Department of Naval Architecture Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: In order to improve the rapidity and accuracy of numerical calculation on underwater vibration of cylindrical shell, the selection rules of truncated mode are studied. At first, a sample of 35 cylindrical shells with different scales was calculated, the mean square velocity of the cylindrical shell was defined as the convergence target, and the Rayleigh-ritz method was applied to solve the vibration equation of cylindrical shell. The calculated results show that the axial truncation mode m is related to both the aspect ratio and the ring frequency. As the aspect ratio is consistent, with the increasing of ring frequency, the value of truncation mode m decreases gradually and the downward tend to flat. When the ring frequency is consistent, the larger the aspect ratio, the higher the value of truncation mode. The circumferential truncation mode n is only related to the ring frequency, and with the increasing of ring frequency, the value of truncation mode n decreases gradually. Under the condition of high precision calculation on the mean square velocity, the axial wavelength corresponding to the truncation mode is about two times that of the circumferential wavelength. The analytical results of natural frequency and mean square velocity are compared with that of finite element simulation, which verifies the accuracy of the theoretical calculation and the rationality of the truncation mode selection.
Key words: cylindrical shell     underwater vibration     truncation mode     aspect ratio     ring frequency
0 引　言

1 圆柱壳水下振动方程推导

 图 1 有限长圆柱壳模型 Fig. 1 Finite cylindrical shell model
 $\left( {\begin{array}{*{20}{c}}{{l_{11}}} & {{l_{12}}} & {{l_{13}}}\\{{l_{21}}} & {{l_{22}}} & {{l_{23}}}\\{{l_{31}}} & {{l_{32}}} & {{l_{33}}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}u\\v\\w\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0\\0\\{\displaystyle\frac{{{R^2}\left( {1 - {\mu ^2}} \right)}}{{Eh}}({F_r} - {p_r})}\end{array}} \right)\text{，}$ (1)

 $\begin{array}{l}u(x,\theta ) = \displaystyle\sum\limits_{mn} {{U_{mn}}} \cos n\theta \sin {k_m}x\text{，}\\v(x,\theta ) = \displaystyle\sum\limits_{mn} {{V_{mn}}} \sin n\theta \cos {k_m}x\text{，}\\w(x,\theta ) = \displaystyle\sum\limits_{mn} {{W_{mn}}} \cos n\theta \cos {k_m}x\text{。}\end{array}$ (2)

 $\begin{array}{l}{F_{{r}}}(x,\theta ) = \displaystyle\sum\limits_{mn} {{F_{mn}}} \cos n\theta \cos {k_m}x\text{，}\\{p_{{r}}}(x,\theta ) = \displaystyle\sum\limits_{mn} {{{\rm{p}}_{mn}}} \cos n\theta \cos {k_m}x\text{。}\end{array}$ (3)

 ${\dot w_{mn}} = \frac{{{F_{mn}}}}{{{Z^M}_{mn} + {Z^s}_{mn}}}\text{，}$ (4)

 ${Z^M}_{mn} = \frac{{ - iEh\left| A \right|}}{{\omega {R^2}(1 - {\mu ^2})\left| B \right|}}\text{，}$ (5)

 $\frac{{{\partial ^2}p}}{{\partial {r^{\rm{2}}}}} + \frac{1}{r}\frac{{\partial p}}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}p}}{{\partial {\theta ^2}}} + \frac{{{\partial ^2}p}}{{\partial {x^{\rm{2}}}}} + {k^2}p = 0\text{。}$ (6)

 ${Z^s}_{mn} = \left\{ {\begin{array}{*{20}{c}} {\displaystyle\frac{{ - i\rho \omega {H_n}^{(2)}\left(\sqrt {{k^2} - {k_m}^2} R\right)}}{{\sqrt {{k^2} - {k_m}^2} {H_n}{{^{(2)}}^\prime }\left(\sqrt {{k^2} - {k_m}^2} R\right)}},{k_m} \leqslant k}\text{，} \\ {\displaystyle\frac{{ - i\rho \omega {H_n}^{(2)}\left(\sqrt {{k_m}^2 - {k^2}} R\right)}}{{\sqrt {{k_m}^2 - {k^2}} {H_n}{{^{(2)}}^\prime }\left(\sqrt {{k_m}^2 - {k^2}} R\right)}},{k_m} > k} \text{。}\end{array}} \right.$ (7)

 $\left\langle {\dot w(x,\theta ) \times {{\dot w}^*}(x,\theta )} \right\rangle = \frac{1}{{2S}}\int\limits_S {\dot w(x,\theta ) \times {{\dot w}^*}(x,\theta ){\rm{d}}s} \text{，}$ (8)

 $\dot w(x,\theta ) = \sum\limits_{mn} {{{\dot w}_{mn}}} \cos n\theta \cos {k_m}x\text{。}$ (9)

 $\left\langle {\dot w(x,\theta ) \times {{\dot w}^*}(x,\theta )} \right\rangle = \frac{1}{4}\sum\limits_{mn} {\frac{1}{{{\varepsilon _n}}}} {\dot w_{mn}}{\dot w^*}_{mn}\text{，}$ (10)

 ${L_W} = 10\lg \frac{{\left\langle {\dot w(x,\theta ) \times {{\dot w}^*}(x,\theta )} \right\rangle }}{{{v_0}^2}}\text{。}$ (11)

2 圆柱壳的模态截断

2.1 圆柱壳模型

 ${{{f}}_{{r}}} = \frac{1}{{2{\text{π}} R}}\sqrt {\frac{E}{{{\rho _{\rm{s}}}}}} \text{。}$ (12)
2.2 截断模态研究

 ${{{f}}_{{r}}} = 1000 \times {10^{\frac{{3{{j}}}}{{30}}}}{\rm{Hz}},\;\;\left( {{{j}} = \cdots - 2, - 1,0,1,2 \cdots } \right)\text{。}$

 图 2 不同轴向截断模态数时均方振速曲线对比（L=R=1 m） Fig. 2 Comparison of mean square vibration velocity curves at different axial truncated modes (L=R=1 m)

 图 3 不同轴向截断模态数时均方振速曲线对比（L=R=1 m） Fig. 3 Comparison of mean square vibration velocity curves at different circumferential truncated modes (L=R=1 m)

 图 4 不同长径比时圆柱壳轴向截断模态数m随环频率变化趋势对比 Fig. 4 Change trend of the axial truncated modal number with the ring frequency at different ratios of length to diameter

 图 5 不同长径比圆柱壳环向截断模态数n随环频率变化趋势对比 Fig. 5 Change trend of the circumferential truncated modal number with the ring frequency at different ratios of length to diameter

 图 6 轴向和环向截断模态波长比值 Fig. 6 Wavelength ratios of axial and circumferential truncated modes
3 有限元仿真对比分析

 图 7 圆柱壳柱形计算域 Fig. 7 Columnar calculation domain of cylindrical shells

 图 8 圆柱壳表面网格及边界条件 Fig. 8 Surface meshes and boundary conditions of cylindrical shells

 图 9 有限元法和理论方法计算均方振速对比 Fig. 9 Comparison of the mean quadratic velocity calculated by the finite element method and the theoretical method
4 结　语

1）对于轴向截断模态m，其大小与长径比和环频率均相关。长径比相同，随着环频率的增大，截断模态m逐渐降低，且下降趋势逐渐平缓；环频率相同，长径比越大，截断模态m值越高。

2）对于环向截断模态n，其大小仅与环频率相关。随着环频率的增大，n值逐渐减小。

3）在保证圆柱壳环频率以下的表面均方振速级最大误差不超过1%的情况下，截断模态对应的轴向波长与环向波长之比大概满足2倍的关系。

4）采用理论方法和有限元法计算了圆柱壳的固有频率和均方振速，通过对比，验证了理论计算的正确性和截断模态选取的合理性。

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