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6
PLANNING OF INSPECTION PROGRAM OF FATIGUE-PRONE
AIRFRAME
Yu. Paramonov, A. Kuznetsov •
Aviation Institute, Riga Technical University, Riga, Latvia 1 Lomonosova Str., Riga, Latvia e-mail: rauprm@junik.lv, andreyk@hotbox.ru
Abstract. To keep the fatigue ageing failure probability of an aircraft fleet on or below the certain level an inspection program is appointed to discover fatigue cracks before they decrease the residual strength of the airframe lower the level allowed by regulations. In this article the Minimax approach with the use one- and two-parametric Monte Carlo modelling for calculating failure probability in the interval between inspections is offered. Keywords: failure probability, Minimax approach, inspection program, approval test
INTRODUCTION
Inspection program development should be made on the base of processing of approval lifetime test result, when we should make some redesign of the tested system if any requirement is not met. Here we consider some example of p-set function application to the problem of development and control of inspection program. We make assumption that some Structural Significant Item (SSI), the failure of which is the failure of the whole system, is characterized by a random vector (r.v.) (Td, Tc), where Tc is critical lifetime (up to
failure), Td is service time, when some damage (fatigue crack) can be detected. So we have some time
interval, such that if in this interval some inspection will be fulfilled, then we can eliminate the failure of the SSI. We suppose also that a required operational life of the system is limited by so-called Specified Life (SL), tSL, when system is discarded from service. P-set function for random vector is a special statistical decision function, which is defined in following way. Let Z and X are random vectors of m and n dimensions and we suppose that it is known the class (Pe, 0e Q} to which the probability distribution of the random vector W=(Z,X) is assumed to belong. The only thing we assume to be known about the parameter e is that it
lies in a certain set Q, the parameter space. If SZ (x) = U SZi (x) is such set of disjoint sets of z values as
i
function of x that sup ^ P(Z e SZ i (x)) < p then statistical decision function Sz(x) is p-set function for r.v.
e i
Z on the base of a sample x=(xi,...,xn).
Later on the value x, observation of the vector X, would be interpreted as estimate 0 of parameter e, Z would be interpreted as some random vector-characteristic of some SSI in service: Z = (Td,Tc) .
FATIGUE CRACK GROWTH MODEL
The fatigue crack growth process flows in accordance with quite complicated rules, which depend on a big number of factors. An analytical approach describing that process could be considered as almost impossible. Nevertheless, it can be shown, that in general case crack growth process could be well enough approximated with the formula:
a(t) = a-eQt, 1.
where a(t) is a fatigue crack size at time t (the number of flight blocks); a is so called equivalent initial crack size (as if the airframe has been initially produced with the crack of such small size; a corresponds to the best fit of test data); and parameter Q defines the speed of growth of fatigue crack and depends on the loading mode (on the stress range in case of cycling loading).
For further needs, let us take a logarithm of both left and right sides of equation 1:
ln a(t) = lna + Qt. 2.
Thus, ln a(t) - lna
t _ Q ' 3'
so the time when crack becomes detectable and the time when crack reaches its critical size can be calculated as
T _ ln adet - lna _ Cd/ T _ ln acnt - lna _ C/
Td _ Q " /Q, T _ Q " /Q, 4.
where adet is a crack size, at which chances to discover it tends to unit, acrit is a crack size, which corresponds to the minimum residual strength of an aircraft component allowed by regulations, Td is a time for crack to grow to its detectable size and Tc is a time for crack to grow to its critical size, Cc and Cd are appropriate constants.
Let us define failure as the situation, when we were unable to discover cracks with the size adet < a < acrit, or, in other words, if there are no inspections performed in [Td; Tc] time interval.
It is clear, that varying the number of inspections ni ctions in the service interval [0, tSL ] we will
discover a different number of cracks; therefore, the estimate of failure probability will vary as well. Unfortunately, we don't know the real values of parameters, so we are using theirs estimates from a small number (one, seldom two) of available observations (fatigue cracks during fatigue test) instead.
USING MONTE CARLO MODELLING TO ESTIMATE FAILURE PROBABILITY
We use the Monte Carlo method to generate a set of cracks to be processed in accordance with procedure, described in Section 0. The parameters for modelling can be derived from the full-scale fatigue tests or from other real crack observations.
We can never know how the certain fatigue crack curve will look like. Thus, performing approximation of that fatigue crack curve with a certain model, the fatigue crack growth model parameters (FCGMP) - we have two FCGM parameters X _ ln Q and Y _ ln CC - will vary as well, so they are
random values, and these random values have theirs own parameters of distribution. To perform Monte Carlo modelling of the fatigue crack growth process we have to know FCGMPs' distribution types and parameters, i.e. c.d.f. of each FCGMP. From the analysis of the fatigue test data it can be assumed, that the logarithm of time required the crack to grow to its critical size is distributed normally:
lnTc ~ N(VlTc ,< ). 5.
From formulas 2 and 4 follows: ln Tc _ ln Cc - ln Q. 6.
From additive property of normal distribution comes that lnTc could be normally distributed either if
both ln Cc and ln Q are normally distributed:
X _ ln Q ~ N(pX ,a2x), Y _ ln Cc ~ N (pY ,cty2), 7.
or if one of them is normally distributed while another one is a constant. Thus, the value of logarithm of our FCGM parameters is distributed normally or, on other words, FCGM parameters have a log-normal distribution.
To get estimates of FCGMP distribution parameters (/} and a) we consider statistics of several crack observations. For each of those cracks we calculate estimates of distribution parameters lnQ and lnCc, and then gather all data together into the table with two columns: lnQ and lnCc. From that table we then derive estimates of mean value and standard deviation for each column, as well as estimate of correlation between lnQ and lnCc.
The Monte Carlo modelling in fact means the process of getting a big number of pairs [Td; Tc] with upper mentioned specific distribution parameters. Having the array of [Td; Tc] pairs we apply an inspection
program looking for failures - situations, when both Td and Tc are located between two consequent
inspections. For each interval between inspections [t^A] failure probability will be
Pfi = P(tt- < Td < Tc < tt ), 8.
and for the entire inspection program
P =Z P. 9.
i
MINIMAX DECISION MAKING APPROACH
As it was stated above, the goal is to develop an inspection program, defined by the vector of irnspection time moments
t = t2,L tn„p K 10.
i.e. to find a vector function t (6) (6 is the estimate of FCGMP distribution parameters, nTIP is the total number of inspections per inspection program, so t = tSL ) that limits aircraft failure probability at the required level Pf d with the minimum inspections nTIP undertaken in service interval [0, tSL ] ( tn = tSL ). In mathematical terms that can be presented as:
s?(Pf (0 r))< , 11.
6
where
nTip
Pf (6,t)=EP((-1 < Td < Tc < T). 12.
i=i
In expression 12 T1, T2,..., Tn are time moments of inspections: random value T = ((,-,Tnnr ) = ?(6), 13.
where T0 = 0, T = tSL, and nTIP = 0, 1, 2, ____The expression nTIP = œ symbolically means that
the aircraft must be returned for redesign to the design office.
The inspection program definition vector 10) is a function, where both number of inspections during service interval nTIP and disposition of inspection time moments Tx,T2,...,T during [0,tSL ] are to be
chosen as a function of 6 and some limitations. It is clear, that there might be many ways how to position inspection time moments on [0, tSL ] for a particular nTIP . Let us apply the following inspection time
moment disposition rule RD : the time of the first inspection Tx ( Tx is a random value because it is a
function of 6 ) will be defined by procedure similar to the safe life approach (probability of failure without inspections is less than some small value Pf 1 ), while all remaining inspections are distributed evenly in the
interval [[, tSL ]. Of course, this rule RD in general case does not minimise the total required number of inspections nTIP ; there are other rules that are more optimal, but our choice of rule RD is caused by its simplicity for further applications; inspection programs created by this rule are currently used in practice for commercial jet aircrafts.
To apply a particular rule RD we have to find the total required number of inspections nTIP , which depends on the limiting value of the failure probability Pf . d = 1 - Rrequired, where required reliability Rrequired is mandated, for example, by JAR regulations.
As it was shown above, the failure probability is a function of the number of inspections n and parameter 6; let us denote it as Pf (6, n ). We also suppose that Pf (6, n ) monotonically decreases when the
number of inspections n increases (at least when n is large enough) and lim Pf (ô, n) = 0 for all 0. Let nTIP is a solution of the equation
P (0, n) = P^d. 14.
Then let us denote
nT1p = p- (pf_d )= n(0,pf_d ) 15.
as the minimal inspection number at which failure probability Pf (ô, nTIP ) < Pf ^. But the true value of
the 0 in unknown, so HTIP = n(0, Pf and Pf = Pf (0, riTIP ) are random values. We suppose that we
begin the commercial production and operation of aircrafts only if some specific requirements to reliability are met. For the simplest case there is a limitation for the maximum allowed number of inspections nmax : we will return airframe project for redesign as unprofitable in case, when the required number of inspections in the inspection program nTIP exceeds nmax (we need to inspect aircraft too often to ensure required reliability). It can be assumed, that the probability of failure for the returned projects is equal to zero, i.e.
P =\Pf(0, nTlP) , nTlP < nmax 16
f— I 0 , nTIP > nmax. .
In the more complex case there is a set of limitations. For example, in addition to limitation on the expected number of inspections ncalculated = HTIP we will return airframe project for redesign if estimate of
expectation value of Tc (T ) is too small in comparison with tSL (breaking minimum threshold Tc );
if estimate of time between two consequent inspections Atcalculated is smaller than a threshold Atmin ; if
estimate of initial equivalent crack size acalculated exceeds crack detectable size adet and so on. Let us denote
the vector of calculated values of limiting values dL = dL (ô) as
n„
dL =
calculated
Atcalculated 17
and the set of its allowed values DL as
Dl =
(0, «max ] [At . ,to)
L mm' / 18
To <»)' '
L '■'min /
[0, «dot ]
Actually, the number of elements in dL and, therefore, the number of dimensions in the set of the allowed values DL may vary depending on modelling situation and specific requirements. For example, for inspection programs with the equal time between inspections in the whole service interval [0, tSL ] the time between two consequent inspections At = tSL / n, so it can be excluded from the set of limitations, but it is important in programs when the time between inspections may vary.
If vector of limiting values dL does not match the set of its allowed values DL, then the project is considered as unprofitably and is returned back for redesign in the design office. As we stated above, the probability of failure for returned projects is equal to zero, thus
P Jpf (dl ) , d L e DL 19
fcorrected I 0 , dL £ Dl
ry
calculated
The parameter 0, which defines the c.d.f. of vector (Td, Tc), is a vector parameter. For considered
case in this work, if both crack model parameters are random and have normal distribution, it consists of five components:
0 =
0 ,0, , 00 , o,
'in Cc 1 In Cc 0ln Q 1 In
20.
where 00 stands for a location and 0, stands for a scale parameter of the appropriate crack growth model parameter ln Cc or ln Q; r is a coefficient of correlation between ln Cc and ln Q, and
0 e 0 = j (-to, to); [0, to); (- to, to); [0, to); [0,l] j . As it was shown before, we don't know the real
value of 0, thus we use its estimate 0. A part of elements of 0) may be assumed as known. For example, 01ic , 0 and correlation coefficient r can be considered as constants, so processing fatigue crack growth
data we should estimate only two remaining parameters 00i c and 00i .
It can be shown that for considered decision making procedure random variable Pf has
J corrected
expectation value, which is a function of 0, and this function has a maximum value for 0e0. To prove that let us fix one of two crack model parameters and look how the probability of failure depends on another one. Let us consider that the equivalent initial crack size is a constant: a = const, i.e. 00 = /ua = const,
01a = ^ a =
In accordance with upper defined rules the probability of failure tends to zero when the crack growth speed representing parameter 00 = E{lnQ tends to zero: this is a case when the item is extremely reliable
and cracks are growing so slowly, that have no chance to grow up to acrit in interval [0, tSL ], thus there are no inspections required. The failure probability without inspections is defined by formula:
^ln tSL — MlnTc 1 21
J = P(T * tSL ) = O
or, in terms of reliability
CTln Tc
Kt = P(Tc > tSL ) = 1 - P(Tc * t sl ) = 1 -O
ln tSL -An
CTln Tc
22.
where ln Tc is distributed normally as ln Tc ~ N(^ln T T ) .
From other side, if the 00i is high, then the probability of failure tends to zero as well: with high probability we return for redesign all items due to the break of limiting rules, i.e. dL £ DL (see formulas 17, 18 and 19). Between these zero values of E{Pf } there can be non-zero values somewhere in between,
^ J corrected '
when the fatigue cracks maybe can reach theirs critical size during the time between inspections, maybe not, but there are no sufficient reasons to return project for redesign so far. Let us call a value of failure probabilitry used for calculations (at the choice of the number of inspections required, or choosing vector-function t ) as Pf . The following conclusion can be made from the upper mentioned: the dependence of
the probability of failure as a function of 0 is a function which has a maximum, the value of that maximal value is unknown, but somehow depends on the value of failure probability Pf used for calculations.
Let us call the value of expectation of failure probability for all allowable 0 as EPf . We have
fcorrected
named it as "corrected" to distinguish it from Pf , because we take into consideration some limitations. The
*
goal is to find such a maximum value of failure probability for calculations Pf that the corrected value of
failure probability pf does not exceed the required limiting value of failure probability
P = 1 - R •
frequired required ■
P * • P (P )< P ,
fcaic V fcaic ' frequired '
where
PPJ
f
= max
E < P
fcorrected
23.
24.
Graphically this approach for a two-dimensional case (when either a = const or Q = const) is presented in Figure 3:
f required
f required
Figure 3. Minimax approach example (a = const or ln Q = const) For the more complex case we get a three-dimensional picture like in Figure 4:
Figure 4. Minimax approach example (general case)
Depending on parameters the shapes of these two- or three-dimensional failure probability curves may vary, but this does not affect our conclusions.
NUMERICAL EXAMPLE AND CONCLUSION
The upper mentioned approach lets us to ensure reliability of the airframe on or above the required level by developing appropriate inspection program for the case of lack of the initial fatigue test data. There are examples of numerical modelling for one- and two-parametric models shown in Figure 5 and Figure 6below (please note: in pictures LQ=LN(Q)= 00 , LC=LN(Cc)= 00 ).
1.6
1.4
1.2
¿0.8
.2
0.6
0.4
0.2
ln Q
„„-3 P, ., as a function of Ln(Q) for T =1.10 and N „, =999
X 10 failure cm inspMax
flT-TITTI
f 100%
-9.4 -9.2
■7.8
0%
LQ
Figure 5. One-parametric numerical example (a = const)
Figure 6. Two-parametric numerical example (3D and projection)
REFERENCES
1. MSG-3. Operator/Manufacturer Scheduled Maintenance Development. Revision 2003.1. // Air Transport Association of America, Inc. Washington, D.C., 2003.
2. Y.Paramonov, A.Kuznetsov. Fatigue crack growth parameter estimation by processing inspection results // In: Third International conference on Mathematical Methods in Reliability MMR2002 -Trondheim, Norway, 2002.
3. Y.Paramonov, A.Kuznetsov. Inspection data use for airframe inspection interval correction // In: Aviation - Issue #6 - Vilnius, Technika, 2002.
4. Y.Paramonov, A.Kuznetsov. Planning of inspections of fatigue-prone airframe // In: Proceedings of International conference "Diagnosis of technical systems, numerical and physical non-destructive quality testing - 2004" - Vilnius, April 23, 2004.
5. Y.Paramonov, A.Kuznetsov. Switching to doubled aircraft inspection frequency strategy analysis for exponential fatigue crack growth model // In: Proceedings of International conference on Longevity, Ageing and Degradation models in Reliability, Medicine and Biology (LAD2004), Volume 1, pp.143-154 - St. Petersburg, Russia, 2004.
