INSPECTION PROGRAM DEVELOPMENT FOR FATIGUE CRACK GROWTH MODEL WITH TWO RANDOM PARAMETERS

To keep the fatigue failure probability of an aircraft fleet at or below a certain level, an inspection program is appointed to discover fatigue cracks before they decrease the residual strength of some structurally significant item of the airframe lower than the level allowed by regulations. In this article, the p-set function for random vector, which, in fact, is a generalization of p-bound for random variable, and minimax approach to the problem of inspection number choice are used. It is supposed that the exponential approximation of a fatigue curve with two random parameters can be used in the interval when the fatigue curve becomes detectable and then grows to critical size. For estimation of distribution parameters, results of an approval test are used. A numerical example is given.


Introduction
The development of an inspection program is necessary in order to provide reliability in a complex system. Examples of a solution to this problem and a lot of references can be found in books [1,2,3]. As a rule, a solution to this problem is provided under the condition that the cumulative distribution function (cdf) of time to failure is known. But really we should make some estimation of the cdf or at least the parameters of the cdf on the basis of processing the lifetime test result. A confidence interval is usually used for the estimation of the lifetime distribution parameter and then for the estimation of reliability. It is always very difficult to find a compromise between required reliability and confidence probability. But if we process some approval test data when we make some redesign of the tested system if some requirements are not met, then, as it will be shown later, it is possible to use the minimax approach, which provides required reliability independently of unknown parameters of lifetime distribution without using a confidence probability. For this purpose, the p-set function definition is used. Here we consider some example of p-set function application to the problem of development and control of an inspection program. We make the assumption that some structurally significant item (SSI), the failure of which is failure of the system being considered, are characterized by a random vector (r.v.) (T d , T c ), where T c is critical lifetime (up to failure) and T d 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 the required operational life of the system is limited by the so-called specified life (SL), SL t , when the system is discarded from service. In previous publications, we consider the case when in the equation of fatigue crack model there is only one random parameter [4][5][6][7]. This time we consider the case with two random parameters.

P-set function definition
P-set function for random vector is a special statistical decision function that, in fact, is a generalization of p-bound for a random variable, the definition of which was introduced much earlier. P-set function for random vector is defined in following way. Let Z and X be random vectors of m and n dimensions, and we suppose that the class {P θ , θ ∈ Ω} is known.
The probability distribution of the random vector W = (Z, X) is assumed to belong to this class. Of the parameter θ, which labels the distribution, it is assumed known only that it lies in a certain set Ω, the parameter space. If For the development of the inspection program, the p-set function defines the sequence of inspection moments, which defines some set S z (x) of values of r.v. ( , ) d c Z T T = .

Development of inspection program
By processing the results of some special approval test (full-scale fatigue test of airframe, for instance), we can get the estimate θˆ of parameter θ . The problem is to find (in a general case) a vector function a way that the failure probability of the SSI under consideration does not exceed some small value : random variables. We suppose that we begin commercial production and operation only if some specific requirements are met. For example, the following requirements have to be met: 1) ˆR n n ≤ , 2) ˆc θ ∉ Θ (estimate of required inspection number for some fixed ε exceeds some threshold R n or estimate of expectation value of c T , ˆc t is too small in comparison with R t ), and then we redesign the SSI in such a way that probability of failure after this redesign will be equal to zero.
Let us define ,0 is the strategy (decision function) for which the required reliability R is provided.
where a 0 is a(0), d a is a crack size when the probability to discover it is equal to unit, and a c is a crack size that corresponds to the maximum residual strength of an aircraft component allowed by special design regulation. We see that parameters C c and C d can be derived can each be derived from the: From the analysis of the fatigue test data, it can be assumed that the logarithm of time required for the crack to grow to its critical size (logarithm of durability) is distributed normally:   In this paper we suppose that parameters , and X Y r σ σ depend on technology that does not change (for a new aircraft) and that these parameters can be estimated using information from previous designs. We suppose that they are fixed and are known values.
Then unknown parameter, θ , have only two components: And for the considered decision-making procedure, the mean probability of fatigue failure , we made similar processing the observations of several cracks (see bottom part of figure  3), which, we assume, grow under the same stress level. In following calculation, the vector ( X σ , Y σ , r) was considered some constant.

Numerical example
Let us demonstrate the approach described in previous sections on the numerical example. Suppose that we have only one fatigue crack observation and make an estimation of ˆY µ (Fig 3). And the vector ( X σ , Y σ , r) is known. Around the point (ˆX µ , ˆY µ ) we choose some area  ).
For these examples we assume that only one full-scale test was performed and we have data on just one single crack growth (crack #75: ln 8.588527, ln 1.905525 . Let us say that we have to ensure the probability of failure not exceeding 0.0326 and that we will return for redesign all projects when required number of inspections exceeds R n = 5. If we perform modelling using various values of failure probability ε , we will get a set of "surfaces" The maximum values of the function ( , ) w θ ε , have to use the value ε = * ε = 0.003 (it is worth mentioning that * w is ten times higher than * ε !) (Fig 8). The required number of inspections in our example is * n =5 (the same data gives the required number of inspections n =4 for ε =0.0326).

Conclusion
This procedure for the development of an inspection program is offered for the case when the exponential model of fatigue crack growth has two random parameters. The p-set function and minimax approach are offered for the choice of inspection number using the results of a full-scale fatigue test on an airframe. It is shown that if instead of using unknown parameters of the exponential fatigue crack growth model, we can use the estimates of a parameter (processing only one fatigue crack observation), then the real probability can be 10 times more than one that was used for the inspection number calculation. For the case of approval fatigue test, when we redesign the tested airframe if some requirements are not met, the minimax statistical decision functions allow us to find a decision that provides the required reliability of airframe in operation.