可修復(fù)的表決系統(tǒng)的可靠性分析

方永鋒,陳建軍,曹鴻鈞

(1.畢節(jié)學(xué)院機(jī)械工程學(xué)院,貴州畢節(jié) 551700; 2.西安電子科技大學(xué)電子裝備結(jié)構(gòu)設(shè)計(jì)教育部重點(diǎn)實(shí)驗(yàn)室,陜西西安 710071)

方永鋒1,2,陳建軍2,曹鴻鈞2

(1.畢節(jié)學(xué)院機(jī)械工程學(xué)院,貴州畢節(jié) 551700; 2.西安電子科技大學(xué)電子裝備結(jié)構(gòu)設(shè)計(jì)教育部重點(diǎn)實(shí)驗(yàn)室,陜西西安 710071)

根據(jù)強(qiáng)度應(yīng)力干涉理論,給出了kn表決系統(tǒng)的單元在多次隨機(jī)作用下且單元抗力退化情況下的動(dòng)態(tài)可靠性指標(biāo)計(jì)算模型,由可靠性指標(biāo)求得單元的動(dòng)態(tài)失效概率.根據(jù)單元的動(dòng)態(tài)失效概率,給出了在多次隨機(jī)外部作用下,單元失效數(shù)目變化的概率,再由可修復(fù)kn系統(tǒng)所具有的馬爾可夫性質(zhì),給出了在多次隨機(jī)外部作用下的可修復(fù)kn系統(tǒng)的轉(zhuǎn)移概率,由轉(zhuǎn)移概率獲得概率密度矩陣,通過求解微分方程組計(jì)算出系統(tǒng)的動(dòng)態(tài)可靠度.最后通過算例說明該方法方便易行,計(jì)算結(jié)果符合工程實(shí)際.

隨機(jī)作用;可修復(fù);kn系統(tǒng);動(dòng)態(tài);可靠性

筆者在相關(guān)文獻(xiàn)的基礎(chǔ)上,研究在多次隨機(jī)作用下,可修復(fù)的kn表決系統(tǒng)配有一個(gè)維修工情況下的動(dòng)態(tài)可靠性問題,建立該系統(tǒng)的動(dòng)態(tài)可靠性預(yù)測模型,預(yù)測系統(tǒng)的可靠度隨時(shí)間的變化規(guī)律.

1 多次隨機(jī)作用下可修復(fù)的表決系統(tǒng)

在n個(gè)單元組成的系統(tǒng)中,若至少有k個(gè)單元同時(shí)正常的系統(tǒng)才能正常工作,此種系統(tǒng)即為kn表決系統(tǒng).顯然,當(dāng)k=1時(shí),該系統(tǒng)成為并聯(lián)系統(tǒng),而當(dāng)k=n時(shí)成為串聯(lián)系統(tǒng).

隨機(jī)外部作用m次時(shí)的等效作用smax的概率分布函數(shù)為

這里,G(smax)為smax的概率分布函數(shù),其對應(yīng)的概率密度函數(shù)為g(smax),smax的均值和均方差分別為μ(smax)和σ(smax).

通常隨機(jī)外部作用的發(fā)生是服從參數(shù)為λLt的泊松(Possion)分布[9],由此可得在t時(shí)刻應(yīng)力的概率為

式中,N(t)為時(shí)段[0,t]內(nèi)隨機(jī)外部作用出現(xiàn)的總次數(shù),N(0)為0初始時(shí)刻隨機(jī)外部作用出現(xiàn)的次數(shù),m為從0到t時(shí)刻隨機(jī)外部作用發(fā)生的次數(shù).

經(jīng)式(2)的處理,隨機(jī)作用次數(shù)m已被轉(zhuǎn)化為時(shí)間.現(xiàn)將式(2)中的smax換為s,則式(2)可表示為

由代數(shù)綜合法求得P(s)的均值μ(s)和均方差σ(s)分別為

另外,考慮到系統(tǒng)中各工作單元的抗力r將是隨作用次數(shù)增長而退化的隨機(jī)變量,當(dāng)作用t時(shí)刻后,系統(tǒng)中工作單元的剩余抗力r(t)為[14]

式中,r(0)是工作單元的初始抗力,T是單元的生命周期,s同式(3),c是材料指數(shù).由式(6),可求得剩余抗力隨機(jī)變量r(t)的均值和均方差分別為

式中,μ(r(0))、σ(r(0))為系統(tǒng)單元的抗力的初始的均值與均方差;μ(r(1))、σ(r(1))為系統(tǒng)單元的抗力被破壞時(shí)的均值與均方差,μ(r(1))=μ(s),σ(r(1))=σ(s),r(1)為工作單元的服役期滿時(shí)的抗力.

式中,Φ為標(biāo)準(zhǔn)正態(tài)分布函數(shù).

記pij(Δτ)表示在Δτ時(shí)刻內(nèi)系統(tǒng)由狀態(tài)i轉(zhuǎn)移到狀態(tài)j的概率,即

P(N(τ)=j|N(τ+Δτ)=i)表示由狀態(tài)i轉(zhuǎn)移到狀態(tài)j的概率.

記aij表示t時(shí)刻一次外部作用之后系統(tǒng)中單元故障數(shù)量由i增加到j(luò)的概率,即

這里Pf=1-Rr(t).

利用以上關(guān)系,由轉(zhuǎn)移狀態(tài)可得系統(tǒng)的轉(zhuǎn)移概率函數(shù)為

式中,o(Δτ)表示Δτ的高階小量.

系統(tǒng)的概率密度矩陣中的元素為

由馬爾可夫鏈所具有的穩(wěn)態(tài)分布性質(zhì),得到系統(tǒng)的穩(wěn)態(tài)概率為

再由概率的完備性P0+P1+…+Pn=1進(jìn)行回代,利用追趕法即可解出全部的穩(wěn)態(tài)概率.

2 多次隨機(jī)外部作用下可修復(fù)的表決系統(tǒng)的可靠性分析

其初始條件為

由式(16)和式(17)求得Q0(τ),Q1(τ)分別為

式中,s1,s2的表達(dá)式為

式(20)即為在多次隨機(jī)外部作用下配有一個(gè)維修工的(n-1)/n表決系統(tǒng)動(dòng)態(tài)可靠度預(yù)測的計(jì)算表達(dá)式.對于配有一個(gè)維修工的kn表決系統(tǒng)的可靠性預(yù)測分析過程亦基本類似.

3 算 例

這里以配有一個(gè)維修工的3/4表決系統(tǒng)為例,系統(tǒng)中各單元的初始抗力為r(0)～N(600,60)MPa,單元的工作壽命周期T=10 000 h,外部隨機(jī)作用量的幅值為L～N(400,40)MPa,作用強(qiáng)度λL=1.5/h,修理時(shí)間參數(shù)λt=1,材料系數(shù)c=4.092.求該系統(tǒng)的動(dòng)態(tài)可靠度.

對此系統(tǒng),當(dāng)1個(gè)單元失效時(shí)則就判定該系統(tǒng)失效.根據(jù)式(9)計(jì)算得到系統(tǒng)中工作單元的動(dòng)態(tài)可靠性指標(biāo)為

對式(22)的數(shù)值計(jì)算結(jié)果見圖1.當(dāng)外部作用于系統(tǒng)至t=6 528 h時(shí),系統(tǒng)中工作單元的可靠性指標(biāo)開始急速下降,表明系統(tǒng)工作單元失效的可能性將快速增加.因此,就用這一時(shí)刻的系統(tǒng)單元失效概率計(jì)算系統(tǒng)的aij.現(xiàn)將t=6 528 h代入式(22),得

由式(12)計(jì)算得到

由式(23)求得系統(tǒng)的概率密度矩陣為

再由式(20)和式(24)解得系統(tǒng)的動(dòng)態(tài)可靠度為

圖1 3/4可修復(fù)系統(tǒng)單元?jiǎng)討B(tài)可靠性指標(biāo)

圖2 3/4可修復(fù)系統(tǒng)的動(dòng)態(tài)可靠度

對式(25)的數(shù)值計(jì)算結(jié)果見圖2.分析式(25)和圖2可以看出,當(dāng)外部作用時(shí)間多于6 500 h之后,即系統(tǒng)服役到4 333.3 h之后,系統(tǒng)的可靠度下降至0.917 9.由于對該系統(tǒng)配有一個(gè)維修工,使從此時(shí)刻開始系統(tǒng)的可靠度相比前面的服役時(shí)刻下降有所減慢.當(dāng)外部作用至10 000 h時(shí),即系統(tǒng)服役到6 667 h時(shí),系統(tǒng)的可靠度下降到0.876 0,已到了系統(tǒng)的服役期限.實(shí)際上,在系統(tǒng)的服役期限的時(shí)間節(jié)點(diǎn)之后,隨著外部作用次數(shù)的增加,維修時(shí)間的延長,該系統(tǒng)的可靠度將還會再進(jìn)一步以更快的速度下降,這些變化趨勢與工程實(shí)際情況是完全相符的.

4 結(jié)束語

(1)筆者給出了系統(tǒng)單元的動(dòng)態(tài)可靠性指標(biāo)計(jì)算公式,通過該公式得到了多次外部隨機(jī)作用下的配有一個(gè)維修工的kn表決系統(tǒng)的概率密度矩陣以及它的穩(wěn)態(tài)分布計(jì)算模型,然后計(jì)算了多次外部隨機(jī)作用下的配有一個(gè)維修工的(n-1)/n系統(tǒng)的動(dòng)態(tài)可靠度模型.

(2)算例表明,文中方法建立的多次隨機(jī)作用下配有一個(gè)維修工的kn表決系統(tǒng)動(dòng)態(tài)可靠性預(yù)測模型在工程應(yīng)用中方便可行,計(jì)算結(jié)果符合工程實(shí)際.

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(編輯:王 瑞)

Analysis of dynamic reliability of the repairable k-out-of-n system under several times random shocks

FANG Yongfeng1,2,CHEN Jianjun2,CAO Hongjun2
(1.School of Mechanical Engineering,Bijie University,Bijie 551700,China;2.Ministry of Education Key Lab.of Electronic Equipment Structure,Xidian Univ.,Xi’an 710071,China)

The model of unit dynamic reliability of the reparable k-out-of-n system is given under several times random shocks with the strength of unit degradation according to the strength-stress theory and the unit failure probability can be observed by using the dynamic reliability.The unit failure number can be obtained under several times random shocks according to the unit failure probability.The conversion probability of the reparable k-out-ofn system is given by its Markov property under several random shocks.The probability density matrix is observed by the conversion probability,and then the dynamic reliability of the reparable k-out-of-n system is obtained by solving the differential equation systems.Finally,it is illustrated that the method is prachcable and convenient. The calculated result conforms to the engineering practice.

random shock;reparable;k-out-of-n system;dynamic;reliability

TH122

A

1001-2400(2014)05-0180-05

2013-05-16< class="emphasis_bold">網(wǎng)絡(luò)出版時(shí)間:

時(shí)間:2014-01-12

國家自然科學(xué)基金資助項(xiàng)目(50905134);中央高校基本科研基金資助項(xiàng)目(JY10000904012)

方永鋒(1975-),男,副教授,博士,E-mail:fangyf_9707@126.com.

http://www.cnki.net/kcms/doi/10.3969/j.issn.1001-2400.2014.05.030.html

10.3969/j.issn.1001-2400.2014.05.030