DELAY-TIME MODEL BASED ON IMPERFECT INSPECTION OF AIRCRAFT STRUCTURE WITHIN FINITE TIME SPAN

Cai Jing,Zuo Hongf u,Zhu Lei

(College of Civil Aviation,Nanjing University of Aeronautics and Astronautics,Nanjing,210016,P.R.China)

INTRODUCTION

Maintenance review board(MRB)report is one of the vital documents for the evaluation on the continuing airworthiness of civil aircrafts.A civil aircraft includes two main parts:the systems and the structure. The systems are usually imported together with a detailed maintenance analysis,while the structure is designed and manufactured by domestic researchers and the structure analysis thus is very important.Structural significant items (SSIs) in the structure are most important and need to be analyzed by MSG-3.Since most of the SSIs have the failure characteristics of delay-time,a delay time model can be adopted to implement the analysis.

Most of the existing inspection optimization models[1-4]assumed the inspection with the same inspection intervals.The assumption facilitated the modeling but failed to fit in the practical situation. Thresholds and repeated intervals should be considered together.

The structure consists of many components with multiple failure modes.Each of the existing delay-time models was designed for a single failure mode.And the existing delay-time models assumed that the number of previous inspections was very large and the inspections were perfect,which did not match the reality that the life time of the structure is limited and the inspections are imperfect over a finite time span.

Therefore it is necessary to develop a new delay-time model with imperfect inspection within a a finite time span for aircraft structure.

1 NONHOMOGENOUS POISSON PROCESS

Structure inspection has been widely developed as a fault counting process,which has been proved as a no nhomogen ous Poisson process(NHPP)[5-7].The no nhomogene ous process{N(t),t≥0}is with an intensity function r(t),and satisfies:

(1)P(N0=0)=1,

(2)? t≥ 0 and h＞ 0,if h→ 0,then

P(Nt,t+h≥2)=o(h),

(3)Independent increment in the process,

(4)P{N(t+h)-N(t)=1}=r(t)h+o(h).

If N(t)follows a Poisson distribution with mean value function m(t)

The mean value function m(t),which is the expected number of faults emerging until a certain time point t,can be expressed in terms of the fault rate of the program

2 DELAY-TIME MODEL FOR AIRCRAFT STRUCTURE

2.1 Assumption

A large number of failure modes emerge in aircraft structure. And the correction of one defect or failure has nominal impact on the steady state of the overall failure characteristics.To describe the characteristic of aircraft structure,the following assumptions are given:

(1) Because of fatigue damage and environmental degradation,the failure rate of aircraft structure exhibits aging effect(increasing failure rate).

(2)According to the use of thresholds and repeated intervals in aircraft maintenance program,the first inspection(threshold)takes place at k T time(k is a positive integer),then a repeated inspection takes place at each T time unit,and the time of inspection is negligible.

(3)According to NHPP,defects arise with the rate of defect occurrence r(t),per unit time.

(4)Since the flight cycles(FC)have effect on the aircraft structure,defects and failures only arise while the aircraft is operating.

(5)The minimal repair will be carried out when the defects or failures are found.After a minimal repair,the structure is as old as before.

Because many factors(such as the level of inspections, lighting conditions, surface conditions,material thickness and edge effect,access to view,human error)have an important influence on inspections,it is assumed that inspections are imperfect. For example,if a defect is present at an inspection,there is a probability p that the defect can be identified,which implies that there is a probability 1-p that the defect can be unnoticed.

2.2 Analysis of NHPP

It has been proved that the failure process over each inspection interval is an NHPP(Fig.1),and not identical over all the inspection intervals of the system,because all inspections are within a finite time span Ta.

Because the total life of the aircraft structure is Ta,the maximum number of the inspections within[0,Ta]is

Fig.1 Failure process in three imperfect inspections at points(i-1)T,i T,and(i+ 1)T

Fig.1 shows that the expected number of the defects occurs within[t,t+W t],(i-1)T≤t＜iT,is r(t)˙W t.If the inspection is perfect,the expected value of the failures caused by these defects is r(t)? W t? F(iT-t).Therefore the expected value of the failures within[(i-1)T,iT]can be obtained by integrating t into[(i-1)T,iT].

But not all the defects can be found because of the limitation of imperfect inspections. The following appropriate analysis is given to develop another way to obtain the expected value of the failures.

(1)It is assumed that the defects emerge within[0,k T].

If the defects is found at k T,then

If the defects are found at(k+j-1)T,j=2,…,N-k+ 1,then

If the failures happen within[0,k T],then

If the failure is found within[(k+j-2)T,(k+j-1)T],j=2,…,N-k+1,therefore˙[F((k+ j- 1)T- t)-

(2)It is assumed that the defects emerge within[(k+i-2)T,(k+i-1)T],i=2,…,N-k+1.

If the defects are found at(k+i-1)T,then

然而當(dāng)n>1時(shí)，入侵時(shí)間窗口T也是一個(gè)隨機(jī)變量，進(jìn)而無法確定T內(nèi)最大入侵次數(shù)k的準(zhǔn)確取值.但是因?yàn)樗耘c式(6)同理，可求得k的期望為

If the defects are found at(k+ j-1)T,2≤i＜ j≤N-k+1,then

If the failures happen within[(k+i-2)T,(k+i-1)T],then

If the failure is found within[(k+j-2)T,(k+j-1)T],2≤i＜ j≤ N-k+ 1,then

The probabilities of all the repairs for the defects are obtained as the following matrix

The probabilities of all minimal repairs for the failures are obtained as the following matrix

Therefore the total expected repair cost for the defects is given as

where c d is the repair cost of a defect.

The total expected repair cost for the failures is given as

where cfis the repair cost of a failure.

The total expected cost within a finite time span Ta is given as

where c i is the inspection cost and c p the preventive maintenance cost.

The expected cost rateis given as

2.3 Solution

To solve the optimal inspection interval T,the coefficient k of threshold inspection is involved.A step-by-step algorithm proposed by Li and Pham[4]based on the Nelder-Mead downhill simplex method is summarized in Fig.2,whereis an aver agevalue.

Fig.2 Algorithm flow

3 NUMERICAL EXAMPLE

The validation takes a piece of fuselage skin as an example[8-9].Because the statistical data of cracks from airlines have not been subdivided into three categories:fatigue crack,stress corrosion crack and corrosion fatigue crack,the causes of cracks are not been considered. Through parameter estimation and hypothesis testing,we can get

The values of k and T are now determined,therefore the average total cost perun it time EC(k,T)is minimized.

Since there are two decision variables k and T,(n+1)=3 in itial distinct vertices are needed,which areZ(1)=(2,4000),Z(2)=(4,2000),Z(3)=(9,1000).Set m=0.Based on Fig.2,the optimal values(Table1)are obtained.

Table 1illustrates the process of the Nelder-Mead algorithm. It shows thata set of the optimal values is k*=5.24,T*=2056and the corresponding cost value is(5.24,2056)=0.2567.If the proposed delay-time model is not used,namely,the total maintenance cost rate is calculated without considering the imperfect inspection and finite time,as described in general aircraft structure maintenance program, the threshold6000 FC is twice as much as the repeated inter-val 3000 FCis EC(2,3 000)=0.314 2 which is moreexpensive than(5.24,205 6)=0.256 7.Thus,it is proved that the proposed delay-time model is effective.

Table1 Optimal valuesk,T

Table 1 shows that the bigger the optimized variable inspection interval T or threshold coefficient k is,the higher the probability of the defects or failure will be,which results in a higher maintenance cost.When the optimized variable inspection interval T or threshold coefficient k is smaller,which means there are more inspections,the frequent inspections will reduce the probability of failure,while incurring additional cost.So the choice of T or k must be appropriate to minimize the maintenance cost rate.

4 CONCLUSIONS

(1)The delay-time model within a finitetime span is first presented and applied.

(2)The presented model is consistent with the practical situation of the aircraft structural maintenance.

(3)The example suggests that the threshold interval should be longer than the repeated inspection interval for the degraded system with increasing failure rate.

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