﻿ 一种含内部结构的水下圆柱壳振动声辐射计算方法
 舰船科学技术  2021, Vol. 43 Issue (12): 122-127    DOI: 10.3404/j.issn.1672-7649.2021.12.022 PDF

A method for vibration and sound radiation analysis of an immersed cylindrical shell with internal structures
ZHANG Lei, MA Xun-jun, LU Min-yue, WANG Li
Wuhan Second Ship Design and Research, Wuhan 430064, China
Abstract: In present paper, a hybrid analytical-numerical approach is adopted to investigate vibration and acoustic radiation characteristics of a submerged cylindrical shell coupled with interior structures. The entire structure is divided into a fluid-loaded cylindrical shell and interior structures. Flügge shell equations and Helmholtz equation in the cylindrical coordinate system are adopted to describe the shell and surrounding fluid, respectively. Interior structures are modelled through finite element method (FEM). By utilizing the continuity conditions between the shell and internal structures, the final governing equations are established. Once vibration responses of the structure are determined, acoustic radiation from the cylindrical shell can be computed through the boundary element method (BEM). The validity of present method is demonstrated by comparing the results with those obtained from the coupled finite element/boundary element method. Present method is highly efficient and has significant advantages in the comparative analysis of multiple schemes.
Key words: cylindrical shells     hybrid analytical-numerical approach     acoustic radiation
0 引　言

Pan等[5]采用波动法研究了加筋圆柱壳的自由振动特性，环肋采用平摊法处理，该方法局限于均匀分布的小尺寸环向加筋。庞福振等[6]采用精细传递矩阵法评估水下加筋圆柱壳的辐射声压，应用叠加原理、增量存储以及改进非齐次项求解方法，提出了改进的精细传递矩阵法，并进一步分析了边界条件、流体介质以及圆柱壳厚度对辐射声压的影响。肖邵予[7]基于Hamilton变分原理研究了两端用端板封口的加筋圆柱壳声辐射特性，以及圆柱壳内声场对其声振特性的影响。陈美霞等[8]采用波动法研究了多舱段加筋圆柱壳的振动特性，加筋采用圆环板模型，舱壁采用圆板模型，给出了舱段截断时边界条件的选取原则。Wei等[9]采用波动法研究了非均匀加筋圆柱壳的自由振动特性。

1 数理模型

 图 1 含内部结构圆柱壳示意图 Fig. 1 Schematic diagram of the cylindrical shell with internal structures
1.1 水下圆柱壳建模分析

 $\begin{split}\dfrac{{Eh}}{{1 - {\mu ^2}}}\left[ {\boldsymbol{L}} \right]\left\{ {\begin{array}{*{20}{c}} u \\ v \\ w \end{array}} \right\} =& \left\{ {\begin{array}{*{20}{c}} {{F_x} + \dfrac{1}{{2R}}\dfrac{{\partial {M_r}}}{{\partial \theta }}} \\ {{F_\theta } + \dfrac{1}{R}{M_x} - \dfrac{1}{2}\dfrac{{\partial {M_r}}}{{\partial x}}} \\ {{F_r} - \dfrac{1}{R}\dfrac{{\partial {M_x}}}{{\partial \theta }} + \dfrac{{\partial {M_\theta }}}{{\partial x}}} \end{array}} \right\}\times\\ &\delta (x - {x_0})\delta (\theta - {\theta _0}) - \left\{ {\begin{array}{*{20}{c}} 0 \\ 0 \\ p \end{array}} \right\} \text{。}\end{split}$ (1)

 $\left\{ \begin{gathered} u = \sum\limits_{m = 1}^M {\sum\limits_{n = - N}^N {{U_{mn}}\cos ({k_m}x)\exp (jn\theta )\exp (j\omega t)} }\text{，} \hfill \\ v = \sum\limits_{m = 1}^M {\sum\limits_{n = - N}^N {{V_{mn}}\sin ({k_m}x)\exp (jn\theta )\exp (j\omega t)} }\text{，} \hfill \\ w = \sum\limits_{m = 1}^M {\sum\limits_{n = - N}^N {{W_{mn}}\sin ({k_m}x)\exp (jn\theta )\exp (j\omega t)} }\text{。} \hfill \\ \end{gathered} \right.$ (2)

 $c(P)p(P) = \int\limits_S \left[ {p(Q)\frac{{\partial G(P,Q)}}{{\partial {n_q}}} + j{\rho _0}\omega \bar v(Q)G(P,Q)} \right]{\text{d}}{S_q}\text{。}$ (3)

 $c(P) = 1 + \int\limits_S {\frac{\partial }{{\partial {n_q}}}\left[ {\frac{1}{{4\text{π} \bar R(P,Q)}}} \right]{\text{d}}{S_q}}\text{。}$ (4)

 图 2 轴对称结构示意图 Fig. 2 Schematic diagram of a axisymmetric structure
 $\begin{split} & p(P) = \sum\limits_{n = 0}^\infty {\left[ {p_n^{c * }\cos (n{\theta _p}) + p_n^{s * }\sin (n{\theta _p})} \right]} \text{，} \\ & p(Q) = \sum\limits_{n = 0}^\infty {\left[ {p_n^c\cos (n{\theta _q}) + p_n^s\sin (n{\theta _q})} \right]} \text{，} \\ & \bar v = \sum\limits_{n = 0}^\infty {\left[ {\bar v_n^c\cos (n{\theta _q}) + \bar v_n^s\sin (n{\theta _q})} \right]} \text{，} \end{split}$ (5)

 $\begin{split} & c(P)p_n^{c * }(P) = \int_\Gamma {\left[ {p_n^cH_n^\prime + j{\rho _0}\omega \bar v_n^c{H_n}} \right]{r_q}{\text{d}}{\varGamma _q}} \text{，} \\ & c(P)p_n^{s * }(P) = \int_\Gamma {\left[ {p_n^sH_n^\prime + j{\rho _0}\omega \bar v_n^s{H_n}} \right]{r_q}{\text{d}}{\varGamma _q}} \text{。} \end{split}$ (6)

 ${H_n} = \int_0^{2\text{π} } {\frac{{{e^{ - jk\bar R}}}}{{4\text{π} \bar R}}\cos (n\theta ){\text{d}}\theta }\text{。}$ (7)

${H_n}$ 可通过高斯积分计算。

 $r = \sum\limits_{i = 1}^3 {{r_i}{N_i}(\xi )} ,\;\;z = \sum\limits_{i = 1}^3 {{z_i}{N_i}(\xi )} \text{。}$ (8)

 $\begin{split} {\text{d}}\Gamma =& \sqrt {{{{\text{(d}}r)}^2} + {{({\text{d}}z)}^2}} = \sqrt {{{\left(\frac{{{\text{d}}r}}{{{\text{d}}\xi }}\right)}^2} + {{\left(\frac{{{\text{d}}z}}{{{\text{d}}\xi }}\right)}^2}} {\text{d}}\xi = \\ &\sqrt {{{\left(\sum\limits_{i = 1}^3 {{r_i}\frac{{{\text{d}}{N_i}}}{{{\text{d}}\xi }}} \right)}^2} + {{\left(\sum\limits_{i = 1}^3 {{z_i}\frac{{{\text{d}}{N_i}}}{{{\text{d}}\xi }}} \right)}^2}} {\text{d}}\xi = J{\text{d}}\xi\text{，}\end{split}$ (9)

 $\begin{split} & p_n^c = \sum\limits_{i = 1}^3 {p_{ni}^c{N_i}(\xi )} ,\;\;\;\bar v_n^c = \sum\limits_{i = 1}^3 {\bar v_{ni}^c{N_i}(\xi )} \text{，} \\ & p_n^s = \sum\limits_{i = 1}^3 {p_{ni}^s{N_i}(\xi )} ,\;\;\;\bar v_n^s = \sum\limits_{i = 1}^3 {\bar v_{ni}^s{N_i}(\xi )} \text{。} \end{split}$ (10)

 $\begin{split} c(P)p_n^{c * }(P) =& \sum\limits_{e = 1}^{{N_e}} \left[ \int_{ - 1}^1 {p_{en}^cH_{en}^\prime {r_{eq}}{J_e}{\text{d}}\xi } + \right.\\ &\left.j{\rho _0}\omega \int_{ - 1}^1 {\bar v_{en}^c{H_{en}}{r_{eq}}{J_e}{\text{d}}\xi } \right]\text{，} \\ c(P)p_n^{s * }(P) = &\sum\limits_{e = 1}^{{N_e}} \left[ \int_{ - 1}^1 {p_{en}^sH_{en}^\prime {r_{eq}}{J_e}{\text{d}}\xi } +\right.\\ &\left.j{\rho _0}\omega \int_{ - 1}^1 {\bar v_{en}^s{H_{en}}{r_{eq}}{J_e}{\text{d}}\xi } \right]\text{，} \end{split}$ (11)

 $\begin{split} c(P)p_n^{c * }(P) =& \sum\limits_{e = 1}^{{N_e}} \left[ \sum\limits_{g = 1}^{{N_g}} {{\kappa _g}p_{en}^c({\zeta _g})H_{en}^\prime ({\zeta _g}){r_{eq}}({\zeta _g}){J_e}{\text{(}}{\zeta _g}{\text{)}}} + \right.\\ &\left.j{\rho _0}\omega \sum\limits_{g = 1}^{{N_g}} {{\kappa _g}\bar v_{en}^c({\zeta _g}){H_{en}}({\zeta _g}){r_{eq}}({\zeta _g}){J_e}{\text{(}}{\zeta _g}{\text{)}}} \right] \text{，} \\ c(P)p_n^{s * }(P) =& \sum\limits_{e = 1}^{{N_e}} \left[ \sum\limits_{g = 1}^{{N_g}} {{\kappa _g}p_{en}^s({\zeta _g})H_{en}^\prime ({\zeta _g}){r_{eq}}({\zeta _g}){J_e}{\text{(}}{\zeta _g}{\text{)}}} +\right.\\ &\left.j{\rho _0}\omega \sum\limits_{g = 1}^{{N_g}} {{\kappa _g}\bar v_{en}^s({\zeta _g}){H_{en}}({\zeta _g}){r_{eq}}({\zeta _g}){J_e}{\text{(}}{\zeta _g}{\text{)}}} \right] \text{。} \end{split}$ (12)

 $\begin{split} &{\boldsymbol{{\rm A}}}_n^c \cdot {\boldsymbol{p}}_n^c = {\boldsymbol{B}}_n^c \cdot {\boldsymbol{\bar v}}_n^c \text{，} \\ &{\boldsymbol{{\rm A}}}_n^s \cdot {\boldsymbol{p}}_n^s = {\boldsymbol{B}}_n^s \cdot {\boldsymbol{\bar v}}_n^s \text{。} \end{split}$ (13)

 $\begin{split}p =& C\bar v = j\omega Cw = \\ &j\omega C\sum\limits_{m = 1}^M {\sum\limits_{n = - N}^N {{W_{mn}}\sin ({k_m}x)\exp (jn\theta )\exp (j\omega t)} }\text{，} \end{split}$ (14)

 $\left[ {{\boldsymbol{K}} + j\omega {\boldsymbol{C}}} \right]{\left\{ {{U_{mn}}\;\;{V_{mn}}\;\;{W_{mn}}} \right\}^{\rm T}} = {\boldsymbol{F}} \text{。}$ (15)

 $\begin{split}{{\boldsymbol{U}}_0} = &\sum\limits_{m = 1}^M \sum\limits_{n = - N}^N {\left[ {{{\boldsymbol{R}}_{mn}}} \right]{{\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{U}}_{mn}}}&{{{\boldsymbol{V}}_{mn}}}&{{{\boldsymbol{W}}_{mn}}} \end{array}} \right\}}^{\rm T}}} = \\ &\sum\limits_{m = 1}^M {\sum\limits_{n = - N}^N {\left[ {{{\boldsymbol{R}}_{mn}}} \right]{{\left[ {{\boldsymbol{K}} + j\omega {\boldsymbol{C}}} \right]}^{ - 1}}} {\boldsymbol{F}}} =\\ &\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{D}}_1}}&{{{\boldsymbol{D}}_2}} \end{array}} \right]{\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{F}}_1}}&{{{\boldsymbol{F}}_2}} \end{array}} \right\}^{\rm T}}\text{。}\end{split}$ (16)

1.2 内部结构建模分析

 $\left( {\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{{ K}}}_{mm}}}&{{{\boldsymbol{{ K}}}_{ms}}} \\ {{{\boldsymbol{{ K}}}_{sm}}}&{{{\boldsymbol{{ K}}}_{ss}}} \end{array}} \right] - {\omega ^2}\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{{ M}}}_{mm}}}&{{{\boldsymbol{{ M}}}_{ms}}} \\ {{{\boldsymbol{{ M}}}_{sm}}}&{{{\boldsymbol{{ M}}}_{ss}}} \end{array}} \right]} \right)\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{U}}_m}} \\ {{{\boldsymbol{U}}_s}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{F}}_m}} \\ {\mathbf{0}} \end{array}} \right\} \text{。}$ (17)

 ${{\boldsymbol{U}}_s} = - {\left( {{{\boldsymbol{{ K}}}_{ss}} - {\omega ^2}{{\boldsymbol{{\rm M}}}_{ss}}} \right)^{ - 1}}\left( {{{\boldsymbol{{ K}}}_{sm}} - {\omega ^2}{{\boldsymbol{{\rm M}}}_{sm}}} \right){{\boldsymbol{U}}_m}\text{，}$ (18)

 $\left[ {{{\boldsymbol{Z}}_b}} \right]\left\{ {{{\boldsymbol{U}}_m}} \right\} = \left\{ {{{\boldsymbol{F}}_m}} \right\}\text{。}$ (19)

 $\begin{split} {{\mathbf{Z}}_b} =& \left( {{{\boldsymbol{K}}_{mm}} - {\omega ^2}{{\boldsymbol{M}}_{mm}}} \right) - \left( {{{\boldsymbol{K}}_{ms}} - {\omega ^2}{{\boldsymbol{M}}_{ms}}} \right)\times\\ &{\left( {{{\boldsymbol{K}}_{ss}} - {\omega ^2}{{\boldsymbol{M}}_{ss}}} \right)^{ - 1}}\left( {{{\boldsymbol{K}}_{sm}} - {\omega ^2}{{\boldsymbol{M}}_{sm}}} \right)\text{，}\end{split}$ (20)

 ${{\boldsymbol{U}}_m} = {\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{U}}_{b1}}}&{{{\boldsymbol{U}}_{b2}}} \end{array}} \right\}^{\rm T}}\text{，}$ (21)

 $\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{Z}}_{b11}}}&{{{\boldsymbol{Z}}_{b12}}} \\ {{{\boldsymbol{Z}}_{b21}}}&{{{\boldsymbol{Z}}_{b22}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{U}}_{b1}}} \\ {{{\boldsymbol{U}}_{b2}}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{F}}_{b1}}} \\ {{{\boldsymbol{F}}_{b2}}} \end{array}} \right\} \text{。}$ (22)

1.3 圆柱壳与内部结构的耦合

 ${{\boldsymbol{U}}_{b2}} = {{\boldsymbol{U}}_0} \text{，}$ (23)
 ${{\boldsymbol{F}}_{b2}} = - {{\boldsymbol{F}}_2}\text{。}$ (24)

 $\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{Z}}_{b11}}}&{{{\boldsymbol{Z}}_{b12}}{{\boldsymbol{D}}_2}} \\ {{{\boldsymbol{Z}}_{b21}}}&{{{\boldsymbol{Z}}_{b22}}{{\boldsymbol{D}}_2} + {\boldsymbol{I}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{U}}_{b1}}} \\ {{{\boldsymbol{F}}_2}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{F}}_{b1}} - {{\boldsymbol{Z}}_{b12}}{{\boldsymbol{D}}_1}{{\boldsymbol{F}}_1}} \\ { - {{\boldsymbol{Z}}_{b22}}{{\boldsymbol{D}}_1}{{\boldsymbol{F}}_1}} \end{array}} \right\}\text{。}$ (25)

2 结果讨论 2.1 方法验证

 图 3 水中含内部基座圆柱壳有限元模型示意图 Fig. 3 Schematic diagram of the finite element model of an immersed cylindrical shell with a base

 图 4 本文方法与FEM计算水中含内部基座圆柱壳振动响应对比 Fig. 4 Comparisons of vibration responses of present method and FEM

 图 5 本文方法与FEM/BEM计算水中含内部基座圆柱壳辐射声压对比 Fig. 5 Comparisons of the sound radiation of present method and FEM/BEM

 图 6 本文方法与FEM/BEM计算水中含内部基座圆柱壳辐射声压指向性对比 Fig. 6 Comparisons of the sound directivity of present method and FEM/BEM

2.2 周向模态对辐射声压贡献量分析

 图 7 周向模态辐射声压贡献量 Fig. 7 Contribution of circumferential mode to sound pressure
3 结　语

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