﻿ 水下爆炸载荷的统计特性
 舰船科学技术  2017, Vol. 39 Issue (9): 12-16 PDF

Statistic characteristic of underwater explosion load
ZHANG Jing, YUAN Hai, WANG Chun-yu, YAN Yan
School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: In order to study probability density of underwater explosion load, density of explosive and density of water are considered as basic random variables. 100 samples of random variables are obtained using the Fortran program by which random numbers could be generated. The simulation of underwater explosion load is carried out by LS-DYNA, peak pressure of shock wave is obtained and verify whether it obey the normal distribution. Verify the correctness of the maximum entropy method, maximum entropy method is used to fit the probability density function of the peak pressure in different blast distance and different time. The method can be used to solve the random statistics model of explosion load, and it can provide the theoretical basis for the reliability analysis of the structure.
Key words: underwater explosion load     peak pressure     maximum entropy method     probability density
0 引 言

1 最大熵法

 $S = - \int_R {f(X)\ln [f(X)]{\rm d}X}\text{，}$ (1)

 $S = - \sum\limits_{i = 1}^n {f({x_i})\ln [f({x_i})]} \text{，}$ (2)

 ${\rm Max}.\;\;S = - \int_R^{} {f(X)\ln \left[ {f(X)} \right]} {\rm d}X\text{，}$ (3)
 ${\rm S.t.}\;\;\;\int_R^{} {f(X){\rm d}X = 1} \text{，}$ (4)
 $\int_R {{X^i}f(X){\rm d}X = {m_i}} \text{。}$ (5)

 $\bar S = S + ({\lambda _0} + 1)[\int_R {f(X){\rm d}X} - 1] + \sum\limits_{i = 1}^N {{\lambda _i}} [\int_R {{X^i}f({X_i}){\rm d}X - {m_i}} ]\text{，}$ (6)

 $\frac{{\partial \bar S}}{{\partial f(X)}} = 0\text{，}$ (7)

 $\int_R^{} {\left[ { - \ln f\left( X \right) - 1 + {\lambda _0} + 1 + \sum\limits_{I = 1}^N {{\lambda _i}{X^i}} } \right]} {\rm d}X = 0\text{。}$ (8)

 $\ln f(X) = {\lambda _0} + \sum\limits_{i = 1}^N {{\lambda _i}} {X^i}\text{，}$ (9)

 $f(X) = \exp ({\lambda _0} + \sum\limits_{i = 1}^N {{\lambda _i}} {X^i})\text{。}$ (10)

 $\int_R {\exp ({\lambda _0}} + \sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X = 1}$ (11)

 ${e^{ - {\lambda _0}}} = \int_R {\exp (} \sum\limits_{i = 1}^N {{\lambda _i}{X^i}){\rm d}X} \text{，}$ (12)
 ${\lambda _0} = - \ln [\int_R {\exp (} \sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} ]\text{。}$ (13)

 $\frac{{\partial {\lambda _0}}}{{\partial {\lambda _i}}} = - \int_R {{X^i}\exp ({\lambda _0} + \sum\limits_{i = 1}^N {{\lambda _i}{X^i}){\rm d}X} } = - {m_i}\text{，}$ (14)

 $\frac{{\partial {\lambda _0}}}{{\partial {\lambda _i}}} = - \frac{{\int_R {{X^i}\exp (\sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} } }}{{\int_R {\exp (} \sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} }}\text{，}$ (15)

 ${m_i} = \frac{{\int_R {{X^i}\exp (\sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} } }}{{\int_R {\exp (} \sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} }}\text{。}$ (16)

 ${Q_i} = 1 - \frac{{\int_R {{X^i}\exp (\sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} } }}{{{m_i}\int_R {\exp (} \sum\limits_{i - 1}^N {{\lambda _i}{X^i}){\rm d}X} }}\text{。}$ (17)

 ${\rm{Min }}{Q^2} = \sum\limits_{i = 1}^N {Q_i^2}\text{。}$ (18)

 $f\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\exp \left( { - \frac{{{x^2}}}{2}} \right)\text{。}$ (19)

 $f\left( x \right) = \frac{1}{{{\sigma _Y}\sqrt {2\pi } }}\frac{1}{x}\exp \left[ { - \frac{1}{2}{{\left( {\frac{{\ln x - {\mu _Y}}}{{{\sigma _Y}}}} \right)}^2}} \right]\text{。}$ (20)

Weibull分布密度函数为

 $f\left( x \right) = \frac{\beta }{k}{\left( {\frac{x}{k}} \right)^{\beta - 1}}\exp \left[ { - {{\left( {\frac{x}{k}} \right)}^\beta }} \right]\text{。}$ (21)

 $f\left( x \right) = \exp \left[ { - \exp \left( { - \alpha \left( {p - u} \right)} \right) - \alpha \left( {p - u} \right)} \right]\alpha \text{。}$ (22)

 图 1 概率密度函数比较图 Fig. 1 Comparison of probability density function
2 水下爆炸载荷的统计特性 2.1 有限元模型的建立

 图 2 有限元模型 Fig. 2 Finite element model
2.2 随机变量的确定 2.2.1 确定随机变量的均值和变异系数

2.2.2 随机数的生成

 ${y_{i + 1}} = \alpha {y_i}\left( {od M} \right)\text{，}\left( {i = 1,2,3, \cdots } \right)\text{，}$ (23)
 ${r_i} = \frac{{{y_i}}}{M}\text{。}$ (24)

2.2.3 数值仿真计算结果

 图 3 冲击波传播过程 Fig. 3 Shock wave propagation

 图 4 有限元仿真与经验公式[6] Fig. 4 Comparison of finite element simulation and empirical formula

1）随着爆距的增加，水下爆炸冲击波的峰值压力衰减较快，随着冲击波的传播，当爆距大于0.5 m时，冲击波的峰值压力的衰减速率相对减小；

2）水下爆炸冲击波的峰值压力与装药量呈线性相关，随着装药量的增加，峰值压力相应增大，同时传播介质（水）也影响着冲击波峰值压力大小，装药量相同的情况下随着水的密度增加冲击波峰值压力也相应增大，这表明了传播介质也影响了冲击波的压力峰值。

2.2.4 结果统计分析 2.2.4.1 W检验

W检验，又称Shapiro-Wilk检验[9]，是一种基于相关性的算法。通过计算可得到一个相关系数，如果数值越接近1，就表明数据和正态分布拟合得越好。

 $W = \frac{{{{(\sum\limits_{i = 1}^n {{a_i}{x_i}} )}^2}}}{{\sum\limits_{i = 1}^n {{{({x_i}{\rm{ - }}\bar x)}^2}} }}\text{，}$ (25)

①将数据按数值大小重新排列，使 ${x_1} \leqslant {x_2} \leqslant \cdots \leqslant {x_n}$

②计算上式分母；

③计算α值；

④计算检验统计量W

⑤若W值小于判断界限值Wα，按显著性水平α舍弃正态性假设；若WWα，接受正态性假设。

2.2.4.2 峰值压力概率密度统计

 $\mu = 450,\;\;\sigma = 10.15\text{，}$

 $f = \frac{1}{{\sqrt {2\pi } \times 450}}{e^{ - \frac{{{{(x - 450)}^2}}}{{2 \times {{(10.15)}^2}}}}}\text{。}$

 图 5 频数直方图 Fig. 5 The statistic plot on columns of peak pressure

 图 6 冲击波峰值压力的概率密度曲线 Fig. 6 Probability density curve of peak pressure
2.2.4.3 不同时刻概率密度统计

 $\mu = 421,\;\;\sigma = 12.41$

 $f = \frac{1}{{\sqrt {2\pi } \times 421}}{e^{ - \frac{{{{(x - 421)}^2}}}{{2 \times {{(12.41)}^2}}}}}\text{，}$

 图 7 0.002 48 ms冲击波峰值压力频数直方图 Fig. 7 Statistic plot on columns of peak pressure at 0.002 48 ms

 图 8 0.002 48 ms冲击波峰值压力的概率密度曲线 Fig. 8 Probability density curve of peak pressure at 0.002 48 ms

 图 9 0.007 47 ms冲击波峰值压力频数直方图 Fig. 9 Statistic plot on columns of peak pressure at 0.007 47 ms

 图 10 0.007 47 ms冲击波峰值压力的概率密度曲线 Fig. 10 Probability density curve of peak pressure at 0.007 47 ms

 图 11 0.125 ms冲击波峰值压力频数直方图 Fig. 11 Statistic plot on columns of peak pressure at 0.125 ms

 图 12 0.125 ms冲击波峰值压力的概率密度曲线 Fig. 12 Probability density curve of peak pressure at 0.125 ms
3 结 语

1）水（传播介质）的密度影响了冲击波压力峰值，水（传播介质）的密度越大冲击波峰值压力也相应增大。

2）建立基于较少数量的样本数据得到水下爆炸载荷的随机信息的样本拟合法。给出了随机变量正态分布类型的检验方法以及利用最大熵法拟合非正态分布随机变量概率密度函数的方法。最大熵法可较好地拟合爆炸载荷的随机数理统计模型。

3）本文方法能够对较少数量的样本进行拟合的基础上，充分利用随机变量的具体信息，采用最大熵法拟合水下爆炸载荷的概率密度，方法简便，避免了成本较高的水下爆炸试验。

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