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 哈尔滨工程大学学报  2019, Vol. 40 Issue (4): 689-695  DOI: 10.11990/jheu.201709063 0

### 引用本文

YU Pengyao, WANG Tianlin, ZHAO Yong, et al. Water entry of 2-D wedge by hydroelastic theory[J]. Journal of Harbin Engineering University, 2019, 40(4), 689-695. DOI: 10.11990/jheu.201709063.

### 文章历史

Water entry of 2-D wedge by hydroelastic theory
YU Pengyao , WANG Tianlin , ZHAO Yong , SU Shaojuan , ZHEN Chunbo
College of Naval Architecture and Ocean Engineering, Dalian Maritime University, Dalian 116026, China
Abstract: In this study, a hydroelastic analysis method for water entry of a 2-D wedge is established by combining modal information provided by the finite element method with the analytical hydrodynamic model.A feasible numerical method is used to solve the issues introduced by the finite element mode.By comparing the calculation results with different mesh densities and modal numbers, we can show that the proposed method has better convergence.The accuracy of the method is verified by comparison with results of published studies.Compared with the hydroelastic analysis method based on the analytical slamming model, the method presented in this paper does not require the analytic modal shape of structural models, which is applicable to complex structures.Finally, the superiority of the proposed method is demonstrated by an example of a wedge with complex boundaries.
Keywords: water entry    finite element method    normal mode    hydrodynamic    hydroelasticity    slamming    fluid-structure interaction    2-D wedge

1 水弹性分析 1.1 结构动力学方程

 $\mathit{\boldsymbol{M\ddot U}} + \mathit{\boldsymbol{KU}} = \mathit{\boldsymbol{P}}$ (1)

 $\mathit{\boldsymbol{U}} = \mathit{\boldsymbol{DP}} = \sum\limits_{r = 1}^m {{\mathit{\boldsymbol{D}}_r}{p_r}\left( t \right)}$ (2)

 $\mathit{\boldsymbol{a\ddot p}} + \mathit{\boldsymbol{cp}} = \mathit{\boldsymbol{Z}}$ (3)

 $\mathit{\boldsymbol{D}}_r^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{D}}_r} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 0,\\ 1, \end{array}&\begin{array}{l} r \ne s\\ r = s \end{array} \end{array}} \right.$
 $\mathit{\boldsymbol{D}}_r^{\rm{T}}\mathit{\boldsymbol{K}}{\mathit{\boldsymbol{D}}_r} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 0,\\ \omega _r^2, \end{array}&\begin{array}{l} r \ne s\\ r = s \end{array} \end{array}} \right.$

1.2 弹性结构的广义力

 Download: 图 1 入水砰击的坐标系与参数 Fig. 1 Coordinates and parameters of water entry
 $\varphi \left( {x,0,t} \right) = - V\sqrt {{c^2}\left( t \right) - {x^2}}$ (4)

 $c\left( t \right) = \frac{{{\rm{ \mathsf{ π} }}V}}{{2\tan \beta }}$ (5)

$c\left( t \right) = s\left( t \right)\cos \beta ;x = \xi \cos \beta$，可得：

 $\varphi \left( {\xi ,0,t} \right) = - V\cos \beta \sqrt {{s^2}\left( t \right) - {\xi ^2}}$ (6)
 $\dot s\left( t \right) = \frac{{{\rm{ \mathsf{ π} }}V}}{{2\sin \beta }}$ (7)

 $\varphi \left( {\xi ,0,t} \right) = \left( { - V \cdot \cos \beta + \bar {\dot w}\left( {s\left( t \right)} \right)} \right) \cdot \sqrt {{s^2}\left( t \right) - {\xi ^2}}$ (8)

 $\begin{array}{*{20}{c}} {\dot s\left( t \right) = \frac{{{\rm{ \mathsf{ π} }} \cdot \left( {V\cos \beta - \bar {\dot w}\left( {s\left( t \right)} \right)} \right)}}{{2\sin \beta cos\beta }} = }\\ {\frac{{{\rm{ \mathsf{ π} }} \cdot \left( {V\cos \beta - \bar {\dot w}\left( {s\left( t \right)} \right)} \right)}}{{\sin \left( {2\beta } \right)}}} \end{array}$ (9)

 $\begin{array}{*{20}{c}} {P\left( {\xi ,t} \right) = - \rho \frac{{\partial \varphi }}{{\partial t}} = - \rho \bar {\ddot w}\left( {s\left( t \right)} \right) \cdot \sqrt {{s^2}\left( t \right) - {\xi ^2}} = }\\ { - \rho \left[ { - V\cos \beta + \bar {\dot w}\left( {s\left( t \right)} \right)} \right] \cdot \frac{{s\left( t \right)}}{{\sqrt {{s^2}\left( t \right) - {\xi ^2}} }} \cdot \dot s\left( t \right)} \end{array}$ (10)

 $\begin{array}{*{20}{c}} {{P_r}\left( {\xi ,t} \right) = \rho V \cdot \cos \beta \cdot \frac{{s\left( t \right)}}{{\sqrt {{s^2}\left( t \right) - {\xi ^2}} }} \cdot \dot s\left( t \right)}\\ {{P_e}\left( {\xi ,t} \right) = - \rho \bar {\ddot w}\left( {s\left( t \right)} \right) \cdot \sqrt {{s^2}\left( t \right) - {\xi ^2}} - }\\ {\rho \bar {\dot w}\left( {s\left( t \right)} \right) \cdot \frac{{s\left( t \right)}}{{\sqrt {{s^2}\left( t \right) - {\xi ^2}} }} \cdot \dot s\left( t \right)} \end{array}$

 ${f_i}\left( {s\left( t \right)} \right) = \int\limits_0^{s\left( t \right)} {P\left( {\xi ,t} \right) \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi }$

${f_i}\left( {s\left( t \right)} \right) = {f_{{\rm{exc,}}\mathit{i}}}\left( {s\left( t \right)} \right) + {f_{{\rm{ela}},i}}\left( {s\left( t \right)} \right)$, 则有:

 $\begin{array}{l} {f_{{\rm{exc}},i}}\left( {s\left( t \right)} \right) = \int\limits_0^{s\left( t \right)} {{P_r}\left( {\xi ,t} \right) \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi } \\ {f_{{\rm{ela}},i}}\left( {s\left( t \right)} \right) = \int\limits_0^{s\left( t \right)} {{P_e}\left( {\xi ,t} \right) \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi } \end{array}$

 $\begin{array}{*{20}{c}} {{f_{{\rm{ela}},i}}\left( {s\left( t \right)} \right) = \int\limits_0^{s\left( t \right)} {\left[ { - \rho \bar {\ddot w}\left( {s\left( t \right)} \right) \cdot \sqrt {{{\left( {{s^k}\left( t \right)} \right)}^2} - {\xi ^2}} - } \right.} }\\ {\left. {\rho \bar {\dot w}\left( {s\left( t \right)} \right) \cdot \frac{{s\left( t \right)}}{{\sqrt {{{\left( {s\left( t \right)} \right)}^2} - {\xi ^2}} }} \cdot \dot s\left( t \right)} \right] \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi } \end{array}$ (11)

 $\left\{ \begin{array}{l} \bar w\left( {s\left( t \right)} \right) = \sum\limits_{j = 1}^m {{{\bar \psi }_j}\left( {s\left( t \right)} \right)} \cdot {p_j}\left( t \right)\\ \bar {\dot w}\left( {s\left( t \right)} \right) = \sum\limits_{j = 1}^m {{{\bar \psi }_j}\left( {s\left( t \right)} \right)} \cdot {{\dot p}_j}\left( t \right)\\ \bar {\ddot w}\left( {s\left( t \right)} \right) = \sum\limits_{j = 1}^m {{{\bar \psi }_j}\left( {s\left( t \right)} \right)} \cdot {{\ddot p}_j}\left( t \right) \end{array} \right.$ (12)

 ${{\bar \psi }_j}\left( {s\left( t \right)} \right) = \int_0^{s\left( t \right)} {\frac{{{\psi _j}\left( \xi \right)}}{{s\left( t \right)}}{\rm{d}}\xi }$

 $\begin{array}{*{20}{c}} {{f_{{\rm{ela}},i}}\left( {s\left( t \right)} \right) = - \sum\limits_{j = 1}^m {{A_{ij}}\left( {s\left( t \right)} \right)} \cdot {{\ddot p}_j}\left( t \right) - }\\ {\sum\limits_{j = 1}^m {{B_{ij}}\left( {s\left( t \right)} \right)} \cdot {{\dot p}_j}\left( t \right)} \end{array}$

 ${A_{ij}}\left( {s\left( t \right)} \right) = \int\limits_0^{s\left( t \right)} {\rho {{\bar \psi }_j}\left( {s\left( t \right)} \right)} \cdot \sqrt {{{\left( {s\left( t \right)} \right)}^2} - {\xi ^2}} \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi$
 $\begin{array}{*{20}{c}} {{B_{ij}}\left( {s\left( t \right)} \right) = \int\limits_0^{s\left( t \right)} {\rho {{\bar \psi }_j}\left( {s\left( t \right)} \right)} \cdot \frac{{s\left( t \right)}}{{\sqrt {{{\left( {s\left( t \right)} \right)}^2} - {\xi ^2}} }} \cdot }\\ {\dot s\left( t \right) \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi } \end{array}$

 $\begin{array}{*{20}{c}} {{f_i}\left( {s\left( t \right)} \right) = - \sum\limits_{j = 1}^m {{A_{ij}}\left( {s\left( t \right)} \right)} \cdot {{\ddot p}_j}\left( t \right) = }\\ { - \sum\limits_{j = 1}^m {{B_{ij}}\left( {s\left( t \right)} \right)} \cdot {{\dot p}_j}\left( t \right) + {f_{{\rm{exc}},i}}\left( {s\left( t \right)} \right)} \end{array}$ (13)
1.3 水弹性方程的建立与求解

 $\left( {\mathit{\boldsymbol{a}} + \mathit{\boldsymbol{A}}} \right)\mathit{\boldsymbol{\ddot p}} + \mathit{\boldsymbol{B\dot p}} + \mathit{\boldsymbol{cp}} = {\mathit{\boldsymbol{F}}_{{\rm{exc}}}}$ (14)

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {{A_{11}}\left( t \right)}& \cdots &{{A_{1m}}\left( t \right)}\\ \vdots &{}& \vdots \\ {{A_{m1}}\left( t \right)}& \cdots &{{A_{mm}}\left( t \right)} \end{array}} \right];$
 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {{B_{11}}\left( t \right)}& \cdots &{{B_{1m}}\left( t \right)}\\ \vdots &{}& \vdots \\ {{B_{m1}}\left( t \right)}& \cdots &{{B_{mm}}\left( t \right)} \end{array}} \right];$

$\mathit{\boldsymbol{\dot p = }}\left[ {\begin{array}{*{20}{c}} {{{\dot p}_1}\left( t \right)}\\ \vdots \\ {{{\dot p}_m}\left( t \right)} \end{array}} \right];\mathit{\boldsymbol{\ddot p = }}\left[ {\begin{array}{*{20}{c}} {{{\ddot p}_1}\left( t \right)}\\ \vdots \\ {{{\ddot p}_m}\left( t \right)} \end{array}} \right];{\mathit{\boldsymbol{F}}_{{\rm{exc}}}} = \left[ {\begin{array}{*{20}{c}} {{f_{exc, 1}}\left( t \right)}\\ \vdots \\ {{f_{exc, m}}\left( t \right)} \end{array}} \right];m$为计入的弹性模态数目。

 $\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot p}}}\\ {\mathit{\boldsymbol{\dot q}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{q}}\\ {\frac{{{\mathit{\boldsymbol{F}}_{{\rm{exc}}}} - \mathit{\boldsymbol{Bq}} - \mathit{\boldsymbol{cp}}}}{{\mathit{\boldsymbol{a}} + \mathit{\boldsymbol{A}}}}} \end{array}} \right]$ (15)

2 矩阵系数的数值处理

 Download: 图 2 线元网格与浸湿长度的关系 Fig. 2 Relationship between mesh element and wetted length

 ${T_{1,i}}\left( {{s^k}} \right) = \int\limits_0^{{s^k}} {\sqrt {{{\left( {{s^k}} \right)}^2} - {\xi ^2}} } \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi$
 ${T_{2,i}}\left( {{s^k}} \right) = \int\limits_0^{{s^k}} {\frac{{{s^k}}}{{\sqrt {{{\left( {{s^k}} \right)}^2} - {{\left( \xi \right)}^2}} }}} \cdot {\psi _i}\left( \xi \right){\rm{d}}\xi$
 ${T_{3,i}}\left( {{s^k}} \right) = {{\bar \psi }_j}\left( {{s^k}} \right) = \int_0^{{s^k}} {\frac{{{\psi _j}\left( \xi \right)}}{{{s^k}}}{\rm{d}}\xi }$

t时刻s(t)位于线元的中间时，采用线性插值的方法确定瞬时时刻的上述积分T1, i(s(t))、T2, i(s(t))、T3, i(s(t))，则t时刻Aij(s(t))、Bij(s(t))和fexc, i(s(t))可以表达为：

 ${A_{ij}}\left( s \right) = \rho {T_{3,i}}\left( s \right){T_{1,i}}\left( s \right)$
 ${B_{ij}}\left( s \right) = \rho {T_{3,i}}\left( s \right){T_{2,i}}\left( s \right)\dot s\left( t \right)$
 ${f_{{\rm{exc}},i}}\left( s \right) = \rho V\cos \beta \cdot {T_{2,i}}\left( s \right) \cdot \dot s\left( t \right)$

3 收敛性分析

4 算法验证

 Download: 图 5 本文方法与文献[6, 11]结果的对比 Fig. 5 Comparison of this method with literature [6, 11] results
5 复杂边界形式的水弹性分析