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BINARY LAMBDA-SET FUNCTION AND RELIABILITY OF AIRLINE
Y. Paramonov, S. Tretyakov,
M. Hauka •
Riga Technical University, Aeronautical Institute, Riga, Latvia
e-mail: [email protected] [email protected] [email protected]
ABSTRACT
A definition of binary X - set function is introduced. It is used for the inspection interval planning in order to limit a probability of fatigue failure rate (FFR) of an airline (AL). A solution of this problem is based on a processing of the result of the acceptance full - scale fatigue test of a new type of an aircraft. Numerical example is given.
1 INTRODUCTION
This paper is really the addition to the previous author paper [1] devoted to the reliability of an aircraft (AC) and presentation [2]. Here we consider the reliability of the process of operation of an airline when after the specified life is reached (retirement time), fatigue failure discovery or fatigue failure takes place a new aircraft is acquired and the operation of airline is continued up to infinity. We consider again the case when for solution of the problem of limitation of fatigue failure rate (FFR) of airline we know the type of distribution function of fatigue life of AC but do know the parameter of this function, for estimation of which we have the result of the acceptance full - scale fatigue test of a new type of an aircraft.
Despite of all the simplicity, the equation a(t) = aexp(Qt) gives us rather comprehensible description of fatigue crack growth in the interval (td, tc), where (we recall) id is the time when a fatigue crack becomes detectable (a(td) = ad) and ic is the time when the crack reaches its critical size (a(tc) = a) and fatigue failure takes place. It can be assumed that corresponding random variables Td = (log ad - log a)/ Q = Cd / Q and Tc = (log ac - log a)/ Q = Cc / Q have the lognormal distribution because, as it is assumed usually, normal distribution of logT can take place only if either both log C and logQ are normally distributed or if one of these components is normally distributed while another one is constant. We suppose also, that vector (X,Y) = (log(Q),log(C)) has two dimensional normal distribution with vector-parameter 0 = (yx,ax,ar, r). It is worth to note, that for the case when a and a are constants then cdf of C is completely defined by the distribution of Cc because Cd = Cc -8 , where 8=log(ac /a)•
First, we consider solution of the problem for the known distribution parameter and then for the unknown one. Numerical example will be given.
2 PROBABILITY MODEL FOR THE KNOWN 0
Just as in paper [1] for the known 6, there are two decisions: 1) the aircraft is good enough and the operation of this aircraft type can be allowed, 2) the operation of the new type of AC is not allowed. A redesign of AC should be made. In the case of the first decision, the vector tVn = (ty,...,tn), where t;, i = l,...,n, is the time moment of i-th inspection, should be defined also. If 6 is known the different rules can be offered for the choice of structure of the vector tl n: 1) every interval between the inspections is equal to the constant tSL / (n +1) , where t5i is the aircraft specified life (SL) (the retirement time), 2) the probabilities of a failure in every interval are equal to the same value ... In this paper we consider the first approach, but really, our considerations can be applied in more general case when the vector t is defined by two parameters, the fixed t and
the the number of inspections, n, in such a way that probability of failure tends to zero when n tends to infinity.
For the substantiation of the choice of inspection number we should know FFR and the gain of AL as a functions of vector t . For this purpose the process of an operation of AL can be considered as an Markov chain (MCh) with (n + 4) states. The states E,E>■••>E+i correspond to an AC operation in the time intervals [t0,^),[t,t2),..,\tn,tn+1), t0 = 0, f„+1 = f5L. StatesE+2, £„+3 and E correspond to the following events : the specified life SL is reached , fatigue failure (FF) or fatigue crack detection (CD) take place. In all these three cases the acquisition of new AC takes place.
E1 E2 E3 En-1 En En+1 En+2 (SL) En+3 (FF) En+4 (CD)
E1 0 u1 0 0 0 0 0 q1 v1
E2 0 0 u2 0 0 0 0 q2 v2
E3 0 0 0 0 0 0 0 q3 v3
En-1 0 0 0 0 un-1 0 0 qn-1 vn-1
En 0 0 0 0 0 un 0 qn vn
En+1 0 0 0 0 0 0 un+1 qn+1 vn+1
En+2 (SL) 1 0 0 0 0 0 0 0 0
En+3 (FF) 1 0 0 0 0 0 0 0 0
En+4 (CD) 1 0 0 0 0 0 0 0 0
Figure 1. Matrix of transition probabilities p
In the corresponding transition probability matrix, PAL, let vi be the probability of a crack detection during the inspection number i , let q be the probability of the failure in service time interval (t,-_j, ti ], and let ui = 1 -vt- qt be the probability of successful transition to the next state. In our model we also assume that an aircraft is discarded from a service at t5i even if there are no any crack discovered by inspection at the time moment t5i . This inspection at the end of (n +1) -th interval (in state En+1) does not change the reliability but it is made in order to know the state of an aircraft (whether there is a fatigue crack or there is no fatigue crack). Here it is supposed that fatigue crack is discovered with probability equal to unit if inspection is made in interval (Td ,TC). It can be shown that
u = pt >tt T >tt-o, q = P(t- <T <Tc <MT >t—, v = 1-u -q, i=1,-,n+1. (i) In the three last lines of the matrix PAL there are three units in the first column, corresponding to renewal of an operation of the airline (the AL operation returns to the first interval). All the other entries of this matrix are equal to zero, see Fig.1.
Using the theory of semi-Markov process with rewards and definition of PAL we can get the vector of stationary probabilities, 7 = (7 ,..,n„+A) which is defined by the equation system
i=1
and the airline gain
where
n+4
g ( n)=& g' ( «) » (3)
i=1
\atut + biqi + cv J = l,..^ n +1,
gi (n)=l J ■ ^ A . (4)
[ ai ,i= n + 2,., n + 4,
at is the reward defined by the successful transition from one operation interval to the following one and the cost of one inspection; b, C and d correspond to transition to the states En+3 (FF), En+4 (CD) and then to the state El (the "cost" of FF of AC, fatigue crack detection, acquisition of new AC) . Let us note that if a = t - t-i, b = c = d = 0 then
g («) = Y?,g, (n) = gt (n) = & (f, - t'-i) (5)
n + 4
( n )=]
i=1 i=1
and Lj = gt(n,d)l 7tj defines the mean step number of MCh to return to the same state E}-, XF(n,e) = 1lLn+3At , where 4 = ^ l(n + 1), is the FFR.
If e is known we calculate the gain as a function of n, g(n,e), and choose the number ng corresponding to the maximum of the gain :
ng (e) = arg max g (n,e). (6)
Then we calculate FFR as function of n, \ (n,e), and choose n^ in such a way that for any n > nA the function Xp (n,e) will be equal or less than some value 2:
nA(A,e) = minjn (n,e) <2, for all n > nx(2,0)} . (7)
And finally
n = ng2(2,e) = max(ng,nx). (8)
3. SOLUTION FOR AN UNKNOWN 0
In [1] the problem of a limitation of fatigue failure probability in an operation of one AC (FFP1) was considered using the definition of binary p-set function
3.1. Binary p-set function
Now let us take into account that we consider the case when the for the estimate of unknown parameter 6 =§ixx,..., x„), the result of acceptance test is used and the operation of a new type of aircraft will not take place if the result of the fatigue test in a laboratory is "too bad" (previously, the redesign of the new type of AC should be made). We say that in this case the event , ©0 c ©
takes place (for example, $g©0 if the test fatigue life Tc is lower than some limit; or n(p,6p) is too large,...).
Let us define some binary set function
c I r+1£07) if £e©n,
0, if &<£©0
where S(n) = {(td,tc):<td,C -1}, t = ihh/ (n +1),i = 1,. .,n +1; 0 is an empty set. It can be shown that for very wide range of the definition of the set © and the requirements to limit FFP1 by the value p*, where (1 - p*) is a required reliability, there is a preliminary "designed allowed FFP1" , pFD, such that corresponding set function S(&,©,n(^FD,6pj) is binaryp -set function of the level p* for the vector Z = (Td ,TC) on the base of the estimate
n+1
*
suVeYJPiZ^S1{n{pPDMf\^®0) = P ■ (10)
7=1
This means that FFP1 will be limited by the value p* for any unknown 6 e©. 3.2. Binary 2 -set function
In similar way, it can be shown that for very wide range of the definition of the set © and the
requirements to limit FFR of AL by the value A*, where A* is a required fatigue failure intensity, there is a preliminary "designed allowed FFR" , Ad , such that corresponding set function
S ($ ©, n^ (Ad fy) is a binary A-set function of the level A* for the vector Z = (Td ,TC) on the base
of the estimate &:
sup6E((A(nA(AFD,#e©0)*P(#e©0)) = A* . (11)
This means that FFR will be limited by the value A* for any unknown 6 e©.
Let us note, that instead of the words a binary A -set function we should use the words binary Ag -set function if instead of n(A>,€ we use ngA(AFD,<£).
For the requirement of a high reliability the choice of an inspection number will be defined by the limitation of FFR. For very high "cost" of FF of AC it will be defined by the maximum of the gain.
4 NUMERICAL EXAMPLE
The example of the solution of the reliability problem of aircraft fleet is considered in [1, 2]. Here we consider only the problem of reliability of AL.
We use the following definitions of the components of an AL income: for all i = i,..., n+1 a = a0(n) + dinsptSL, where a0(n) = a01tSL / (n +1), - is the reward related to successful transition from one operation interval to the following one, a defines the reward of operation in one time unit (one hour or one flight); dinsptSL is the cost of one inspection (negative value) which is supposed to be proportional to t5i ; bt = b01tSL is related to FF (negative value), c = CiO)(n) is the reward related to transitions from any stateEx,...,En+l to the state Ek+4(it is supposed to be proportional to a0because it is a part of a0); dt = dmtSL is negative reward, the absolute value of which is the cost of new aircraft acquisition after events SL, FF or CD and transition to Ei takes place. In numerical example we have used the following values: tSL = 40000, b01 =-0.3; d = - 0.05; a01 =1; c01 =0.1; d01 =- 0.3.
Suppose we have the following estimate of parameter 0 = (ux,u,ax,ar,r): #=(-8.58688044, 1.9424608, 0.155, 0.0778895, 0.796437) (see Fig 2.2 and Table 2.1 in [3]). It was assumed that the set ©0 corresponds to the decision to make redesign if the estimate of critical time to failure tc = exp(UU -UU) is too small: tC < 0.3/SL.
Contour Plot of the lambda
Figure 2. The value of w(9,KD, 0O ) for five value of ¿uY (1.2415 </uY < 2.6435) in vicinity of maximum value of w(9,KD, ®o ) which is equal to 0.9041*10-6 for (ax ,ar, r) =(0.155128668,
0.0778895, 0.796437) .
Calculation of w(9,K>,®0) = EiK(&Kd,®o)} was made for (7.2029<9.9709), (1.3972 </liy < 2.4877) assuming that the vector(ax,aY, r) is the same for all different vectors (yx). It was found that for ÀPD =0.1*10-6 the maximum value of w(9,Kd, ®o ) is equal to 0.9041*10-6 .
Suppose that the value 0.9041*10-6 is required reliability. Then for the known estimate of the parameter the calculation of nx(KD,9) for ÀPD =0.1*10-6 gives us the required number of inspection. It is equal to 6. For the considered estimate of 9 tC ( realization of rc is equal to
37.4574e+003 so the redesign is not needed. After the necessary calculation of g(n,d) it is found n = 4. So finally, the required number of inspections n = max(ng, n ) is equal to 6.
8 7
J5 4
x 10
Figure 3. The value of nz(APD,§) for five value of¡uY (1.24152.6435) for XPD =0.1*10-6 as function of equivalent mean value of Tc which was calculated as exp(^ - )
CONCLUSIONS
6
5
3
2
0
The problem of inspection planning on the bases of the result of acceptance full-scale fatigue test of AC structure is the choice of the sequence {^, t, , tSL} providing the limitation of FFR of AL if
some requirements to the result of acceptance full-scale fatigue test are met. If these requirements are not met the redesign of the new type of an aircraft should be made.
The definition of binary 2 -set function is introduced for description of corresponding mathematical procedure, based on the observation of some fatigue crack during the acceptance full-scale fatigue test of aircraft structure. In general case the the desire to increase the gain of airline service can be taken into account but under condition that required reliability is already provided. The limitation of FFR is provided for any unknown parameter of the fatigue crack model. The method of necessary calculation is provided.
REFERENCES
1. Paramonov Yu, Tretjakov S, Hauka M. Fatigue-prone aircraft fleet reliability based on the use of a p-set function. RT&A, #01(36) (Vol. 10) 2015, pp.40-49.
2. Hauka M, Tretjakov S, Paramonov Yu (2014). Minimax inspection program for reliability of aircraft fleet and airline. In: Proceedings for 8th IMA International Conference on Modelling in Industrial Maintenance and Reliability (MIMAR), Oxford, Institut of Mathematics and its Applications, 10 - 12 July 2014, pp. 120-124
3. Paramonov Yu., Kuznetsov A., Kleinhofs M. (2011). Reliabilty offatigue-prone airframes and composite materials. Riga: RTU. (http://gnedenko-forum.org/library/Paramonov/Reliability Paramonov.pdf)
