﻿ 基于冲击的舰炮设备可靠性建模与评估
 舰船科学技术  2022, Vol. 44 Issue (14): 165-169    DOI: 10.3404/j.issn.1672-7649.2022.13.035 PDF

Research on reliability modeling and assessment of naval gun equipment based on impact theory
ZHU Han
The 713 Research Institute of CSSC, Zhengzhou 450015, China
Abstract: Under certain assumptions, the reliability model of naval gun equipment based on impact factors is established. The features of reliability functions and failure rates are analyzed and the reliability evaluation method is put forward. In the reliability evaluation method, the parameters in the equipment reliability model are estimated by using the performance degradation information and the fault information. Finally, an example is given to illustrate the applicability of the model and the effectiveness of the evaluation method in characterizing the reliability of naval gun equipment.
Key words: naval gun equipment     reliability modeling     reliability assessment     degradation data     impact theory
0 引　言

1 舰炮设备可靠性影响因素分析

2 舰炮设备可靠性模型 2.1 模型假设

 $P\left( {{N_1}(t) = n} \right) = \frac{{{{\left( {\lambda t} \right)}^n}}}{{n!}}\exp \left( { - \lambda t} \right)\text{，}n = 1,2, \cdots，$ (1)

 $P\left( {{N_2}(t) = n} \right) = \frac{{{{\left( {at} \right)}^n}}}{{n!}}\exp \left( { - at} \right)\text{，}n = 1,2, \cdots 。$ (2)

2.2 舰炮设备寿命及其分布

1）考虑致命冲击流导致的设备硬故障。

 $\begin{split} {F_1}(t) = P\left( {{T_1} \leqslant t} \right) & = 1 - P\left( {{T_1} > t} \right) = 1 - P\left( {{N_1}(t) = 0} \right) =\\ & 1 - \exp ( - \lambda t) 。\\[-10pt] \end{split}$ (3)

2）考虑非致命冲击流对设备性能指标的影响

 $X(t) = \sum\limits_{i = 1}^{{N_2}(t)} {{Y_i}}。$ (4)

 $X(t) \sim N\left( {{\mu _t},\sigma _t^2} \right)。$ (5)

 ${T_2} = \min \left( {t:\inf \left\{ {t:X(t) = {X_{cu}},t \geqslant 0} \right\}} \right) ，$

 $\begin{split} {F_2}\left( t \right) = P\left\{ {{T_2} \leqslant t} \right\} & = 1 - P\left\{ {{T_2} > t} \right\} = 1 - P\left\{ {\;X(t) \leqslant {X_{cu}}} \right\} = \\ & 1 - \Phi \left( {\frac{{{X_{cu}} - {\mu _t}}}{{{\sigma _t}}}} \right) 。\\[-20pt] \end{split}$ (6)

 $\begin{split} F\left( t \right) = P\left\{ {T \leqslant t} \right\} &= P\left\{ {\min ({T_1},{T_2}) \leqslant t} \right\} = 1 - P\left\{ {{T_1} > t,{T_2} > t} \right\} =\\ & 1 - \left[ {1 - {F_1}(t)} \right]\left[ {1 - {F_2}(t)} \right] =\\ & 1 - \exp \left( { - \lambda t} \right)\varPhi \left( {\frac{{{X_{cu}} - {\mu _t}}}{{{\sigma _t}}}} \right) 。\\[-20pt] \end{split}$ (7)
2.3 舰炮设备可靠性模型

 $\begin{split} R\left( t \right) = P\left\{ {T > t} \right\} &= P\left\{ {\min ({T_1},{T_1}) > t} \right\}= \\ & \left[ {1 - {F_1}(t)} \right]\left[ {1 - {F_2}(t)} \right] =\\ & \varPhi \left( {\frac{{{X_{cu}} - {\mu _t}}}{{{\sigma _t}}}} \right)\exp \left( { - \lambda t} \right) 。\\[-10pt] \end{split}$ (8)

 $\begin{split} {\lambda _s}\left( t \right)& = - \frac{{R'(t)}}{{R(t)}} = \lambda - \frac{{{\rm{d}}\varPhi \left( {({X_{cu}} - {\mu _t})/{\sigma _t}} \right)}}{{{\rm{dt}}}}{\varPhi ^{ - 1}}\left( {\frac{{{X_{cu}} - {\mu _t}}}{{{\sigma _t}}}} \right) =\\ & \lambda + \left( {\frac{{{X_{cu}}}}{{2t}} + \frac{{a\mu }}{2}} \right){e^{ - \frac{{{{\left( {{X_{cu}} - a\mu t} \right)}^2}}}{{2a({\mu ^2} + {\sigma ^2})t}}}}{\left[ {\int_{ - \infty }^{{X_{cu}}} {{e^{ - \frac{{{{\left( {{X_{cu}} - a\mu t} \right)}^2}}}{{2a({\mu ^2} + {\sigma ^2})t}}}}} {\rm{d}}x} \right]^{ - 1}} 。\\[-20pt] \end{split}$ (9)

 $\begin{split} & \mathop {\lim }\limits_{t \to + \infty } {\lambda _s}\left( t \right)=\\ &\;\; \lambda + \mathop {\lim }\limits_{t \to + \infty } \left( {\frac{{{X_{cu}}}}{{2t}} + \frac{{a\mu }}{2}} \right){e^{ - \frac{{{{\left( {{X_{cu}} - a\mu t} \right)}^2}}}{{2a({\mu ^2} + {\sigma ^2})t}}}}{\left[ {\int_{ - \infty }^{{X_{cu}}} {{e^{ - \frac{{{{\left( {{X_{cu}} - a\mu t} \right)}^2}}}{{2a({\mu ^2} + {\sigma ^2})t}}}}} {\rm{d}}x} \right]^{ - 1}} =\\ &\;\; \lambda + \frac{{a\mu }}{{2\sqrt {a({\mu ^2} + {\sigma ^2})} }}{e^{\frac{{a\mu {X_{cu}}}}{{({\mu ^2} + {\sigma ^2})}}}}\mathop {\lim }\limits_{t \to + \infty } \frac{{{e^{ - \frac{{a{X_{cu}}}}{{({\mu ^2} + {\sigma ^2})}}}}}}{{\sqrt t \int_{ - \infty }^{\frac{{{X_{cu}} - a\mu t}}{{\sqrt {a({\mu ^2} + {\sigma ^2})t} }}} {{e^{ - \frac{{{x^2}}}{2}}}} {\rm{d}}x}} ，\end{split}$

 ${\lambda _s} = \mathop {\lim }\limits_{t \to + \infty } {\lambda _s}\left( t \right) = \lambda + \frac{a}{2}\left( {\frac{{{\mu ^2}}}{{{\mu ^2} + {\sigma ^2}}}} \right)。$ (10)

3 舰炮设备可靠性评估

3.1 可靠性信息结构

 $\left( {{t_i},{X_{ij}}} \right),i = 1,2, \cdots ,m,j = 1,2, \cdots ,n 。$ (11)

3.2 模型参数估计

 $\hat \lambda = \left\{ \begin{gathered} \frac{T}{r},\quad \quad r > 0 ，\\ T,\quad \quad r = 0 。\\ \end{gathered} \right.$ (12)

 $X\left( t \right) \sim N\left( {ut,vt} \right)，$ (13)

 ${L_j}\left( {b,u,v} \right) = \prod\limits_{i = 1}^m {\frac{1}{{\sqrt {2\text{π} vt_j^{}} }}\exp \left( { - \frac{{{{\left( {{X_{ij}} - ut_j^{}} \right)}^2}}}{{2vt_j^{}}}} \right)}。$

 $\left\{ \begin{gathered} \hat u = {{\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^m {{X_{ij}}} } } \mathord{\left/ {\vphantom {{\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^m {{X_{ij}}} } } {\sum\limits_{j = 1}^n {mt_j^{}} }}} \right. } {\sum\limits_{j = 1}^n {mt_j^{}} }}，\\ \hat v = \frac{1}{{mn}}\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^m {\frac{{{{\left( {{X_{ij}} - t_j^{}u} \right)}^2}}}{{t_j^{}}}} } 。\\ \end{gathered} \right.$ (14)
3.3 可靠性指标估计

 ${\hat R^*}(t) = \varPhi \left( {\frac{{{X_{cu}} - \hat ut}}{{\sqrt {\hat vt} }}} \right)\exp \left( { - \hat \lambda t} \right)，$ (15)

 ${\hat \lambda _s} = \hat \lambda + \frac{{{{\hat u}^2}}}{{2\hat v}}。$ (16)
4 实例分析

1）致命冲击参数 $\lambda$ 的估计。

2）退化数据的预处理

 图 1 传感器零位电压随时间变化趋势 Fig. 1 The zero voltage of the sensor changes with time

3）退化模型参数估计

4）可靠度与故障率的估计

 图 2 传感器的可靠度曲线图 Fig. 2 Reliability curve of the sensor

 图 3 传感器的故障率曲线图 Fig. 3 Failure rate curve of the sensor
 $\begin{split} {\hat \lambda _s} =& \hat \lambda + \frac{{{{\hat u}^2}}}{{2\hat v}} = 4.6512 \times {10^{ - 4}}+ \\ &{(3.7917 \times {10^{ - 4}})^2}/(2 \times 7.0833 \times {10^{ - 5}})=\\ & 1.4798 \times {10^{ - 3}} 。\end{split}$

5 结　语

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