考虑相关竞争故障过程及变动阈值的可靠性评估

Journal of Southeast University(English Edition) Vo1.29，No.1，PP.52-56 Mar.2013 ISSN 1003-7985Reliability assessment conside ringde；ndent Lpeting failure md shifting thresholdpe com [l process an ll - Su Chun Qu Zhongzhou Hao Huibing(School of MechanicEngineering，Southeast University，Nanjing 21 1 189，China)Abstract：The reliability assessment problem for productssubiect to degradation and random shocks is investigated。

Two kinds of probabilistic models are constructed，in whichthe dependent competing failure process is considered.First，based on the assumption of cumulative shock，the probabilisticmodels for hard failure and soft failure are built respectively。

On this basis， the dependent competing failure modelinvolving degradation and shock processes is established。

Furthermore，the situation of the shifting-threshold is alsoconsidered， in which the hard failure threshold valuedecreases to a lower level after the arival of a certain numberof shocks.A case study of fatigue crack growth is given toillustrate the proposed models.Numerical results show thatshock has a significant effect on the failure process；meanwhile，the efect wil be magnified when the value of thehard threshold shifts to a lower leve1。

Key words：degradation；hard failure；dependent competingfailure process；cumulative shock model；shifting-thresholddoi：10.3969/ .issn.1003-7985.2013.01.011F or mechanical parts，failures usualy result from the cptio betwen soft failre(degadain)andhard failure， and shock processes can speed up both ofthe failures. In recent years， studies on competing fail-ures have been extensively explored； but most of themassume that the two kinds of failure processes are inde-pendent of each other 。

In real circumstances，however，there exist correlationsbetween the degradation process and random shocks.Forexample，the degradation process makes the system morevulnerable to random shocks，and random shocks can ac-celerate the degradation process. Up to now， differentcategories of random shock assumptions have been con-structed、 including the cumulative shock model ， theextreme shock modell 6j the mixed model and the 6-shock model 081. M oreover， effects of correlations， corn-Received 20l2 9.16。

Biography：Su Chun(1 970- )，male，doctor，associate professor，sa-chun### seu.edu.cn。

Foundation items：The Nationa1 Natural Science Foundation of Chinaf No 50405021 .Graduate Training Innovative Projects Foundation ofJiangsu Province(No.CXLX120081)。

Citation：Su Chun，Qu Zhongzhou，Hao Huibing.Reliability asess-ment considering dependent competing failure process and shifting-threshold[J.Journal of Southeast University(English Edition)，2013，29(1)：52-56.fdoi：10.3969/j.issn.1003-7985.2013.01.011]peting processes between shocks and degradation have al-so been studied.Su et a1. proposed a reliability assess-ment method considering competing failure based on theW iener process.Peng et a1.[1Ol developed reliability mod-els for systems that undergo multiple dependent compe-ring failure processes， in which two kinds of failureprocesses are dependent upon each other due to the impactfrom the same shock processes. Investigating both thedegradation process and the shock process， W ang eta1. constructed a system reliability model on competi。

tive failure processes wi山 fuzzy degradation data. whichwas evaluated with a multi-state system reliability theory。

In this paper， the reliability analysis is conducted onthe basis of the dependent competing failure process.Inwhich，hard failure is caused by the shock process，whilesoft failure is the result of continuous degradation and ab-rupt damage from the same process. The cumulativeshock model iS applied for the sake of establishing twoprobabilistic models， and the correlation between hardfailure and soft failure is considered.Based on that，relia-bility is consequently estimated including the situation ofthe shifting-threshold value.The case study with sensitiv-ity an alysis implies that the proposed models are in linewith the actual situation，which also demonstrates that theproposed models can be applied to the components thatendure dependent failure processes。

1 Dependent Competing Failure Process(DCFP)As shown in Fig.1，let Xff)be the wear volume ofthecontinuous degradation by time f.and it is n1OnOtOnical1vincreased with time.Shock loads will cause additional ab-rupt damage (i1，2，)and speed up the degradationprocess.Soft failure wil occur when the overl degrada-tion，X (f)，is beyond the critical strength level H。

Fig.1 Soft failure processJiang et a1.1l 2pointed out that when sustaining shocks。

Reliability assessment considering dependent competing failure process and shifting-threshold 53components become more susceptible to hard failures。

功 Hs.the same random shock process can also result inhard failure.As seen in Fig.2(a)，let denote the mag-nitude ofthe i-th shock(f：1，2， )，hard failure occurswhen the cumulative shock load magnitude exceeds thethreshold value D..The system wil f 1 when either ofthe two failures occurs.For most materiais，the strengthwill gradually decrease with time.Fig.2(b)shows thatthe critical strength value decreases from D to D，afterthe arrival of a run of m shock loads。

(b)Fig.2 Hard failure process. (a)Fixed threshold；(b)Shiftingthreshold2 Reliability M odeling for DCFP2.1 Shock process analysisIt is assumed that random shocks arrive according to ahomogeneous Poisson process with rate A.Let N(t)de。

note the number of shocks until time t，thenPⅣ( e f-o，1，2... (1)As shown in Fig.2(a)，in the cumulative shock mod-eI，Iet r be the time that the system incurs hard failure。

an d the system will not fail until the cumulative shockdamage exceeds the threshold value D .Thus，the relia。

bility function of the hard failure process can take theform as， 州t)R P(c≤ P( <即< ( ) (3)where f·)is the cumulative distribution function of astandard norm aily distributed variable。

As shown in Fig.2(b)，the hard failure threshold val-ue decreases from D，to fight after the arrival of a runof m shocks.In such a shifting.threshold situation.theequivalent reliability function isR ：P(t< )：P(∑

Each random shock can cause abrupt damage.The ab-rupt damage in the overall degradation are measured bythe shock damage sizes asY1，y2，.The cumulativedamage size due to ran dom shocks until time t is givenas[10]Ⅳ(t)s( )：f ( )>0 (5)0 if N( )：0where N(t)is the total number of shocks to the systemuntil time t。

Then the overall degradation of the system includinggradual degradation and shock damage can be expressedas X (t)X(t)S(t).Soft failure will occur when theoverail degradation is beyond the threshold value H。

Thus.出e probability that the component survives isP( ( )< )∑P( ( )s( )I/v(t) )·iop(N(t)i) (6)In this paper，it is assumed that the magnitude of grad-ual degradation，X(t)，at time t follows a normal distri-bution as N(/z(t)，o- (t)).And the shock damage sizesare also i.i.d variables taking the form as Yi～N Y，2y).Considering the independence of the two randomV ab1es，X。(t)takes the distribution as N( (t)N(t)/zy， (t)Ⅳ(t) 2r).Reliability is equal to theprobability that the total dam age to the system has not ex-ceeded the failure threshold.Using Eqs.(1)and(6)，the reliability function for the soft failure process can be(2) obtained asSpecifcaly，when the magnitudes of the shocks are in-dependent identically distributed(i.i.d)random variablesfolowing a normal distribution as ～N(/zw， 2w)，thereliability function can be obtained as ( )e 烈 (7)∑-Dm。

Su Chun，Qu Zhongzhou，and Hao Huibing2.3 System reliability analysis2.3.1 Reliability analysis under a cum ulative shockm odelFig.2(a)shows a cumulative shock model；that is tosay.a system is considered to be failed only when the cu。

mulative magnitudes exceed the threshold value D，. A1though the two failure processes are dependent for beingaffected by the same shock process，it is still reasonableto assume that they are physically independent of eachother .Thus，the probability that the component sur-vives from failure can be expressed asN( )P(X

P(N(t)i) (8)It is also assumed that the shock process follows thehomogeneous Poisson process described in Section 2.1。

Based on the specific assumptions ofX(t)，We，Yi(i1，2，)，and using Eqs.(3)and(7)，the reliabilityfunction of the dependent competing failure process canbe obtained as ( )eIi i 4 or1/ ( t 、 Ⅳ 、 √ (厂 ； 1 (A ) / 12.3.2 Reliability analysis due to a shifting-thresholdIn Fig.2(b)，the hard failure threshold decreases fromJD to D right after the arrival ofm shocks.This kind ofproblem was considered by Jiang et al 12 J. In their re-search，a generalized run shock model was given.In thispaper，we propose a different approach for reliabilityanalysis based on the cumulative shock mode1.Similarly，by using Eqs.(4)and(7)，we can obtain the reliabilitythat the component survives from failure asR(t)P( (t)<日)JP(N(t)：0)P(毫 <。。)P( ( ) jl

tion about the shocking process is used to demonstrate theproposed mode1。

Fig.3 shows the fatigue crack growth path.W e select1 3 samples out of a total of 2 1 samples since the remai。

ning sam ples are not completed.The least squares methodis employed to evaluate the parameters，and the resultsar e肛(K)：4.58×10 K0.874 5o-(K)1.33 x10 K0.000 8where K is the number of cyclesCycle/t04Fig.3 Fatigue crack growth paths of samplesSizes of random shock loads， ( 1，2，)，whichare measured in units of component life .are assumed tofollow a norm al distribution， ～J7v(2，0.5)；the shockdam age sizes take the form as，.～Ⅳ(0.02，0.01)fori1.2. ，and the threshold value of soft failure H 2.0inches(1 inch25.4 mm)：the threshold value of hardfailure DI35 units；and the arrival rate A 0.5×1O～。

In addition.we assume that m 3.and the lower level ofhard failure threshold value D 25 units。

According to Eqs.(9)and(11)，the reliability curvesOf the two models constructed in Section 2.3 are plottedin Fig.4，respectively.For Case l，reliability almost re-mains at 1 when K <5 x 10 cycles.This is because theeffect of random shocks is not significant and the gradualwear degradation amount is not large enough to cause anyfailure.In the next time period.with the gradual degra。

、l， D< ∑㈦日2.5 x 10 cycles.Analysis ofCase 2 can be obtained similarly。

Fig.4 Reliability function of different modelsIn addition，the reliability curve based on degradationwithout shocks is also provided in comparison with thetwo shock-considering models in Fig.4.According to theresults，it can be concluded that shocks wil accelerate thefailure process，which validates the effectiveness of mod。

els constructed in Section 2.Moreover，when the hardfailure threshold value decreases from D to D，.failurewill OCCur sooner。

Fig.5 shows the reliabi1ity distributions with differentrates of random shocks in Case 2.A higher arrival ratewil make the reliability drop more quickly.It is reasona-ble because more intensively frequent random shocks，lar。

ger sizes of shock loads and shock damages will resultant。

Cycle/10Fig.5 Sensitivity analysis for A4 ConclusionThis Paper focuses on the reliability analysis of compo-nents with dependent competing failure processes due tohard failure an d soft failure. Generalized probabilisticmethods based on the cumulative shock model are pro-posed and two specific models with normal distributionare obtained. Compared with the degradation processwithout shocks，models established in this Paper indicatethat random shocks have significant effects on the failureprocess.When considering the shifting-threshold situa-tion.reliability decreases even faster。

The two proposed models only consider one degrada-tion path.In real situations.components may have multi-pie degradation measures，an d this will be the focus ofour future research。

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占I1l BI茸lII I1 H56 Su Chun，Qu Zhongzhou，and Hao Huibing考虑相关竞争故障过程及变动阈值的可靠性评估苏 春 瞿众洲 郝会兵(东南大学机械工程学院，南京211189)摘要：研究了受退化和随机冲击共同作用的产品可靠性评估问题.考虑相关竞争故障过程，建立了2类概率模型.首先，基于累积冲击假设，分别建立了硬故障概率模型和软故障概率模型.以此为基础，考虑退化和冲击过程建立了相关竞争故障模型.此外，研究了因多次冲击栽荷作用而导致的硬故障阈值下降的情形，分析故障阈值的变动对产品可靠性的影响.以-组裂纹增长数据为例，验证模型的有效性.研究结果表明：冲击对产品故障过程有着显著影响，同时硬故障阈值的下降会进-步加剧上述效应。

关键词：退化；硬故障；相关竞争故障过程；累积冲击模型；变动闽值中图分类号：TH17