﻿ 基于可靠度的高铁制动机周期预防维修优化研究
 上海理工大学学报  2020, Vol. 42 Issue (4): 384-389 PDF

Periodic preventive maintenance optimization and a model for high-speed rail brake based on the limitation of reliability
DONG Hangyu, LIU Qinming, YE Chunming, LIU Wenyi
Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: Aiming at the problems of high-speed rail brake, such as difficult maintenance and high maintenance cost, an optimization model for periodic preventive maintenance was established. Based on the reliability theory, an improvement factor was introduced to describe the fault rate evolution rule with the lowest cost as an objective function. Taking the failure rate obeying Weibull distribution as an example, the optimal preventive maintenance times of equipments and the relationships between reliability and cost, reliability and improvement factors were obtained by solving the problem with the Matlab instrument. The validity of the model was verified by an example.
Key words: preventive maintenance     reliability     Weibull distribution     improvement factor     failure rate
1 问题的提出

 图 1 现阶段高铁维修模式 Fig. 1 Current high-speed rail maintenance mode

2 研究方法 2.1 问题描述

2.2 模型假设

a. 设备的初始状态为全新状态。

b. 设备采用修复非新的政策，修补后不改变故障率函数。

c. 系统后备维修资源充足，发现故障可立即维修。

d. 假设车辆的保养工作放到夜间进行，不占用工作时间。

e. 设备在运行工作中的可靠度需高于最低可靠度阈值。

2.3 维修策略

 图 2 预防性维修模式 Fig. 2 Preventive maintenance mode

 图 3 修补性维修模式 Fig. 3 Patch repair mode

 图 4 更换性维修模式 Fig. 4 Replacement maintenance mode
3 模型建立 3.1 可靠度

 $\begin{split}&\exp \left[ { - \int_0^{{T_1}} {{\lambda _{1j}}} \left( t \right){\rm{d}}t} \right] = {\rm{exp}}\left[ { - \int_0^{{T_2}} {{\lambda _{2j}}} \left( t \right){\rm{d}}t} \right] = \cdots = \\ &\quad\quad\quad\quad\quad \exp \left[ { - \int_0^{{T_{ij}}} {{\lambda _{ij}}} \left( t \right){\rm{d}}t} \right] = R_j\\[-14pt]\end{split}$ (1)

 $\begin{split}\int_0^{{T_{1j}}} &{{\lambda _{1j}}\left( t \right){\rm{d}}t} = \int_0^{{T_{2j}}} {{\lambda _{2j}}\left( t \right){\rm{d}}t} = \cdots = \\ &\int_0^{{T_{ij}}} {{\lambda _{ij}}\left( t \right){\rm{d}}t} = - \ln \;R_j\end{split}$ (2)

3.2 改善因子

 $\quad\quad\quad\quad\quad{\lambda }_{i+1}\left(t\right)={b}_{i}{\lambda }_{i}\left(t+{a}_{i}{T}_{i}\right) ,\; t\in \left(0,{T}_{i+1}\right)$ (3)

3.3 故障率函数

 ${\lambda }_{\left(i+1\right)}\left(t\right)={b}_{i}\frac{\beta }{\eta }{\left(\frac{t+{a}_{i}{T}_{i}}{\eta }\right)}^{\beta -1}$ (4)

 ${\lambda }_{\left(i+1\right)}\left(t\right)=\frac{\beta }{\eta }{\left(\frac{t+{T}_{i}}{\eta }\right)}^{\beta -1}$ (5)

 $F\left( t \right) = 1{\rm{0}}{{\rm{e}}^{\lambda t}} = 1 - {\rm{exp}}\left[ { - {b_i}\frac{\beta }{\eta }{{\left( {\frac{{t + {a_i}{t_i}}}{\eta }} \right)}^\beta }} \right]$ (6)

${a}_{i}=\dfrac{i}{3i+8}$ ${b}_{i}=\dfrac{13i+2}{12i+1}$

3.4 数学模型

 $\begin{split} {C_{zk}} =& N{C_{pk}} + {C_{ck}}\sum _{i = 1}^N\left( {\int\nolimits _0^{{T_{ik}}}{\lambda _{ik}}\left( t \right){\rm{d}}t} \right) +\\ &{C_{sk}}\left[ {N{T_2}{\rm{ + }}{{{T}}_3}\sum _{i = 1}^N\left( {\int\nolimits _0^{{T_{ik}}}{\lambda _{ik}}\left( t \right){\rm{d}}t} \right)} \right] + {C_{rk}} \end{split}$ (7)

 $\begin{split} {C_{{{zk}}}} =& N{C_{pk}}{{ + N}}{C_{ck}}( - {\rm{ln}}\;{R_{km}}) + {C_{rk}} + \\ &{C_{sk}}\left[ {N{T_2} + N{T_3}( - {\rm{ln}}\;{R_{{{km}}}})} \right]\end{split}$ (8)

 $\begin{split}\min \;&{C_k} = \frac{{{C_{zk}}}}{{{T_{zk}}}} = \\ & \frac{{{N_k}{C_{pk}} + {N_k}{C_{ck}} + {N_k}{C_{sk}}[{T_2} + {T_3}] + {C_{rk}}}}{{\displaystyle\sum\limits_{i = 1}^N {{T_{ik}} + N{T_2} + {T_4}} }}\end{split}$ (9)

 $\quad\quad\quad {T_{{{zk}}}}{\rm{ = }}\sum\limits_{i = 1}^N {{T_{ik}} + N{T_3}( - \ln\; {R_{km}}) + N{T_2} + {T_4}}$ (10)

 ${{}}\min \;{C_k} = \frac{{{C_{zk}}}}{{{T_{zk}}}} = \frac{{{N_k}{C_{pk}} + {N_k}{C_{ck}}( - \ln \;{R_{km}}) + {N_k}{C_{sk}}[{T_2} + {T_3}( - \ln \;{R_{km}})] + {C_{rk}}}}{{\displaystyle\sum\limits_{i = 1}^N {{T_{ik}} + N{T_2} + {T_4}} }}\\[-14pt]$ (11)

 $\min\; {C_k} = \frac{{{C_{zk}}}}{{{T_{zk}}}} = \frac{{{N_k}{C_{pk}} + {N_k}{C_{ck}} + {N_k}{C_{sk}}[{T_2} + {T_3}] + {C_{rk}}}}{{\displaystyle\sum\limits_{i = 1}^N {{T_{ik}} + N{T_2} + {T_4}} }}\\[-14pt]$ (12)
4 算例分析

4.1 设备故障率分析

Weibull分布的失效率[15]引用式（6）。

$F\left(t\right)=1{-\exp}\left[-{\left(\dfrac{t}{\alpha }\right)}^{\beta }\right]$

 $f\left(t\right)=\frac{\beta {t}\;^{\beta -1}}{{\eta }^{\beta }}{\exp}\left[-{\left(\frac{t}{\eta }\right)}^{\beta }\right]$ (13)

 $R\left(t\right)=1-F\left(t\right){=\exp}\left[-{\left(\frac{t}{\eta }\right)}^{\beta }\right]$ (14)

 $\ln \left[ {\ln \frac{1}{{1{\rm{ - }}F(t)}}} \right]{\rm{ = }}\beta \ln \; t{\rm{ - }}\beta \ln \; \eta$ (15)

y= $\ln \left[ {\ln \dfrac{1}{{1{\rm{ - }}F(t)}}} \right]$ , x=ln t, A=ln η, B=β，可得

 $y=Bx+A$ (16)
 $F({N}_{i})=\frac{{{i}}-0.3}{n+0.4}$ (17)

 图 5 拟合数据 Fig. 5 Fit data
 $\lambda (t){\rm{ = }}\frac{{1.26}}{{30\;242}}{\left( {\frac{t}{{30\;242}}} \right)^{0.26}}$ (19)
4.2 结果分析

 图 6 R=0.79时预防性维修次数与成本的关系 Fig. 6 Relationship between the times of preventive maintenance and cost when R=0.79

 图 7 考虑可靠度与不考虑可靠度时预防性维修次数与成本的关系 Fig. 7 Relationship between the times of preventive maintenance and cost when considering or not considering reliability

 图 8 考虑或不考虑改善因子时预防性维修次数与可靠度之间的关系 Fig. 8 Relationship between the times of preventive maintenance and reliability when considering or not considering improvement factors

5 结　论

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