VII. INCLUSION OF TIME-DEPENDENT SYSTEM HEALTH MONITORING
7.4. Use of crack inspection data within prognosis
likelihood function QM
i P r(Di|a0) is the conditional probability ofM crack inspection results given a particular EIFS valuea0, which can be written for each of the three possible outcomes as
Case 1: P r(Di|a0) = Z
f(aN)π(aN|a0)daN (7.16) Case 2: P r(Di|a0) =
Z
[1−f(aN)]π(aN|a0)daN (7.17) Case 3: P r(Di|a0) ∝π(am|a0) =
Z
π(am|aN)π(aN|a0)daN (7.18)
wheref(aN) is the probability of detection (POD) and can be obtained from Eq. 7.3; π(am|aN) is the conditional probability density function of measured crack size for a given actual crack size, which can be derived from Eq. 7.4. By assuming the regression coefficients β0 and β1 as constants, and assuming the residual term εm as a zero-mean normal random variable, this conditional probability can be derived as
π(am|aN) = 1
√2πσ2exp[−am−β0−β1aN)2
2σm2 ] (7.19)
where σm is the standard deviation of εm.
7.4.2 Strategy of prognosis for components in a fleet
For a fleet of aerospace mechanical components in service, an ideal case will be to conduct structural health monitoring for each component, followed by individual prognosis. However, due to budget or technical limitations, load monitoring data and crack inspection data may or may not be available for every component in the fleet. Four scenarios of prognosis can be classified based on the availability of monitoring data as shown below.
Scenario 1: Prognosis for components with OLM and CSM
This is the best situation where prognosis is tied to the individual component. The future
loading can be predicted using the Bayesian ARIMA model presented in Sections 6.2.3 and 6.3.2, and the initial values of the ARIMA model are the latest recorded loading amplitudes.
Before crack inspection, the prognosis starts from the EIFS with the prior probability distribution derived from Eq. 7.6. After the crack inspection, the prognosis of future crack growth starts from the current crack size, which can be obtained using the measured crack size by rewriting Eq. 7.4 as
a = 1 β1
(am−β0−εm) (7.20)
Scenario 2: Prognosis for components without OLM but with CSM
For components without loading monitored, future loading predictions are generated with random initial values using the ARIMA model estimated in Scenario 1. The extra uncertainty due to the random initial value will require more Monte Carlo simulations for prognosis than in Scenario 1.
The use of the prior EIFS distribution before inspection and the estimation of the current crack size based on measured crack size is the same as in Scenario 1.
Scenario 3: Prognosis for components with OLM but without CSM
In this case, the prediction for future loading is the same as in Scenario 1. The use of the prior EIFS distribution before inspections is the same as in Scenario 1. However, after inspections, since the crack size is not measured, crack growth simulation based on updated EIFS is used to infer the probability distribution of current crack size. Note that this procedure also applies to (i) components without any crack inspection and (ii) components with crack detection but without size measurement. The updated probability distribution of EIFS is obtained using crack inspection data via the Bayesian approach as presented in Section 7.4.1.
Scenario 4: Prognosis for components with no OLM or CSM
This is the worst case, where no load monitoring data or inspection data are available for an individual component. In such a situation, the prognosis for such a component has to be
based on data obtained on other components. In this case, the prediction for future loading is the same as in Scenario 2. The use of the prior EIFS distribution before inspections is the same as in Scenario 1, and the estimation of the current crack sizes is the same as in Scenario 3.
7.4.3 Validation of prognosis with new crack inspection data
It is important to assess the validity and performance of prognostic algorithms based on the comparison between predictions and observed data, and various graphical and quantitative methods have been developed for this purpose [Oberkampf and Barone, 2006; Hills and Leslie, 2003; Saxena et al., 2010; Rebba et al., 2006]. A detailed illustration and discussion of the existing quantitative model validation methods has been given in Section 2.3 and Chapter IV.
The Bayesian equality hypothesis testing-based approach illustrated in Section 4.3 is used here since it takes into account the entire probability distribution of model output, instead of only the distribution parameters, and a confidence metric for model prediction can be easily derived based on the calculated validation metric. Two hypotheses are compared, namely H0 - the null hypothesis that the proposed method gives correct predictions, and H1 - the alternative hypothesis that the proposed method gives incorrect predictions. The validation metric - Bayes factor - is equal to the ratio of the likelihood functions of these two hypotheses [O’Hagan, 1995]
B = P r(D|H0) P r(D|H1) =
Q
i
R P r(Di|aN)π0(aN)daN
Q
i
R P r(Di|aN)π1(aN)daN
(7.21)
where P r(Di|aN) is the conditional probability of obtaining the inspection data Di for a given actual crack size. Eqs. 7.2 and 7.19 are used to calculate this conditional probability for three different types of inspection results. In Eq. 7.21,π0(aN) is the PDF of the actual crack size under the null hypothesis, and π1(aN) is the PDF of the actual crack size under
the alternative hypothesis. π0(aN) is the same as the PDF of crack size predicted by the prognosis. In order to calculateπ1(aN), we assume that under the alternative hypothesis the crack size follows a uniform distribution, i.e., the crack sizes within a certain interval are equally probable. Note that the boundaries of this interval will affect values of the estimated Bayes factor.
In addition to crack inspection data, load monitoring data can also be used for validation.
An example of using loading data to validate the prediction of the ARMA model can be found in Section 6.3.4. If new SHM data on system health status or damage response are available, further validation can be performed by comparing the system response estimated based on crack size predictions against the SHM data.