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Comparison of different life distribution schemes for prediction of crack propagation in an aircraft wing
Engineering Failure Analysis 96 (2019) 241–254
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Comparison of different life distribution schemes for prediction of crack propagation in an aircraft wing
T
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Moez ul Hassan , Fabiha Danish, Waleed Bin Yousuf, Tariq Mairaj Rasool Khan Department of Electronics and Power Engineering, Pakistan Navy Engineering College (PNEC), National University of Sciences and Technology (NUST), Karachi 75350, Pakistan
A R T IC LE I N F O
ABS TRA CT
Keywords: Aircraft failure Fatigue crack growth Degradation Life prediction
Estimation of remaining useful life/prognostics of an aircraft structure permits aerospace industry to timely schedule maintenance activities. Accurate planning through prognostics ensures safer flight operation and lower downtimes. The core of any prognostic algorithm is the state transition/degradation model. In the reported research work, Particle Filter (PF) based prognostic algorithm is used to predict crack growth with three different state transition/life distribution models. PF (Bayesian Sequential Monte Carlo) allows using of non-linear state-transition and non-Gaussian/multimodal noise distributions. The typical candidate life distributions for modeling crack growth are Exponential, Weibull and Lognormal distributions. A framework is proposed where effectiveness of the candidate distribution for modeling degradation phenomenon can be adjudged. The algorithm is tested on actual historical NDT data of crack growth on countersunk (CSK) rivet holes on an Airbus A310 aircraft’s wing. Historical data is bifurcated into two periods i.e. training and validation periods. The most appropriate distribution based on the comparison of the above mentioned candidate distributions is proposed for prediction of the degradation/ flaw propagation.
1. Introduction Structural failure during flight can result into catastrophic consequences [1]. In general, failures occur when a crack on aircraft structure is no longer able to withstand the stresses imposed on it during operation [2]. Therefore, it is necessary to predict the crack growth before a crack reaches the critical value [3]. Accurate crack growth estimation has a potential to decrease aircrafts operating cost through improved maintenance/overhaul scheduling which, in turn, reduces the downtime. Crack damage is normally originated on structural components subject to fatigue loading. Fatigue loading initiate cracks on aircraft structure whose subsequent growth leads to sudden failure [4]. During the complete flight cycle, aircraft structure remains under the influence of cyclic stresses. NDT (non-destructive testing) data acquired from actual aircraft is used in the reported research work. The NDT data is acquired at rivet holes on aircraft’s wing. The holes are one of the most fatigue sensitive sub components of aircraft structures [5]. Accurate crack growth prediction is a challenging task due to the influence of different varying factors. Different physical fatigue models have been suggested in recent literature for crack initiation and propagation [6,7]. An application of such local unified approach has been presented in [8]. A notched rectangular plate, made of P355NL1 steel is modelled in order to generate S–N curves for distinct stress R-ratios. Enrique Castillo and Alfonso Fernández-Canteli proposed stochastic method to predict crack propagations
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Corresponding author. E-mail address: [email protected] (M.u. Hassan).
https://doi.org/10.1016/j.engfailanal.2018.10.010 Received 21 November 2017; Received in revised form 29 September 2018; Accepted 22 October 2018 Available online 23 October 2018 1350-6307/ © 2018 Elsevier Ltd. All rights reserved.
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by using three main fatigue approaches [9]. In [10] a procedure for estimating probabilistic S-N curve was proposed using strain life data. Crack growth prediction using constant amplitude loading (CAL) is commonly found in the literature [11]. Whereas fatigue crack initiation and crack propagation under variable amplitude loading (VAL) is less commonly found in the literature. In [12–14] crack growth prediction is investigated by artificially produced fatigue cracks in the laboratory specimen of aero structures. Simulation is then performed to predict the remaining useful life of damaged aircraft structures while applying VAL. Reference [15, 16] processed the entire crack growth simulation using finite element method. Physical modeling of three dimensional fatigue crack progression does not account comprehensive information due to inadequate model parameters. Similarly, loads applied on test specimen in laboratory do not exactly replicate the real stresses due to underlying uncertainty in the environmental conditions. In this paper, we used actual crack growth data which captures the actual environmental conditions the aircraft wing is subjected to. Historical data gathered over a period of time is used to determine the remaining useful life (RUL) of a specific component [1]. Statistical models have a potential to predict crack growth with improved accuracy even in the presence of uncertainties [17]. In this paper, particle filtering (PF) based statistical method is applied on historical data acquired from rivet holes of a civil aviation aircraft. PF is a recursive Bayesian technique using sequential Monte Carlo simulations [18]. Using PF technique, crack growth can be predicted using suitable degradation model. The research work reports a PF based framework for crack growth determination. Three candidate degradation models are used in the PF framework to find the most appropriate degradation model. The crack growth results are then compared with actual historical data to find the most appropriate degradation model. Use of actual historical data with crack growth models improves the accuracy of the proposed estimation. However the contemporary methods comprising of simulation methods [14–16] do not possess this degree of accuracy. Accurate crack growth prediction offers enhanced aircraft safety through implementation of timely maintenance strategies [19]. Next section describes the ultrasonic NDT data acquisition around countersunk (CSK) rivet holes. PF based crack growth prediction is elaborated in the research methodology section. Then the results and discussions are given. Finally, conclusion and future work are presented. 2. Case study Wings are the most critical structural unit of any aircraft. Their physical condition indicates the aircraft’s safety to a significant extent. Rivets are used to join different metallic sheet together in an aircraft wing structure. Load is transferred from one metallic sheet to another through riveted joints. These joints are subjected to stress during the bending of the aircraft wings. Presence of CSK rivets in the region of joints makes this stress state highly complicated as residual stress introduces around the holes. Under continuous cyclic loading, crack initiation and its subsequent growth in this high stress concentration region lead to structural failure. The loading cycle comprises of take-off, cruise and landing phases. Wings mainly bend during take-off and landing phases. Fatigue cracks on wings are encouraged by the dynamic load acting on an aircraft wing during the loading cycle [20]. Therefore, in this paper crack growth is computed with respect to the loading cycles. Flaw growth around a rivet-hole is a three dimensional entity as shown in Fig. 1. The ultrasonic thickness gauging sensors are used to measure the flaw dimension around the countersunk rivet-hole in an aircraft wing structure. The NDT measurements are acquired after 0, 16067, 18634, 21201, 23715 and 25816 loading cycles. The loading cycle’s frequency to schedule inspections are specified by OEM to avert fatal failures. Considering the rivet hole as center, the three dimensional damage around CSK rivet holes is expressed in terms of flaw length and flaw depth. The recommended procedure is to collect length-depth pair readings with the angular step size of 45° around the CSK rivet hole. The length–depth pair data is tabulated in Table 1 and graphically presented in Fig. 2. Cubic spline interpolation method was used to enhance data points. As displayed in Table 1, last readings were acquired after 16067 loading cycles. Whereas, the current readings were recorded after 18643 loading cycles with the difference of 2567 loading cycles. 3. Methodology 3.1. Statistical framework Flaw growth prediction from NDT measurements is a typical nondestructive evaluation (NDE) inverse problem. Since the crack is exposed to harsh environmental conditions, its subsequent growth is stochastic in nature [21]. A statistical framework is adopted to
Fig. 1. Illustration of Flaw growth around CSK rivets [5]. 242
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Table 1 Details of damage around Countersunk Rivet Hole (CSK) [5] ANGLE
0 45 90 135 180 225 270 315
Flaw length (mm)
Flaw depth (mm)
Last Reading
Current Reading
Last Reading
Current Reading
6 4 4 4 6 0 0 0
8 8 10 10 10 15 11 6
5.12 5.12 5.02 5.02 5.12 0.52 0.62 0.52
6.12 5.52 5.42 5.42 5.52 5.42 5.52 5.42
Fig. 2. Damage length around CSK rivet hole [5].
Fig. 3. Effect of β on Weibull distribution.
deal with the underlying uncertainty [5,22]. The Bayesian filtering approach is used to encounter the uncertainty in the final solution [22]. Bayesian filter is applied by treating state uc as the actual crack dimensions (length and depth) where c refers to loading cycle index (c=1, 2, 3, ...C) and zc is considered to be the corresponding noisy measurements from the ultrasonic gauging sensor. As all the preceding measurementsz1:c are known, the posterior density p(uc| z1:c) can be estimated using the prior density p(uc| uc−1) and the likelihood density p(zc| uc) [23] as given in (1).
p (uc | z1: c ) =
p (z c | uc ). p (uc | z1: c − 1 ) p (z c | z1: c − 1 )
(1)
Crack growth prediction from historical data is a challenging task. Now, predictive density p(uC+F| z1:c) can be computed as: 243
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Fig. 4. Different degradation models.
Fig. 5. Particle filter algorithm scheme.
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Fig. 6. Actual flaw data of length at 45 degree.
Fig. 7. Flaw length prediction at angle 90° of CSK rivet.
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Fig. 8. Flaw depth prediction at angle 45° of CSK rivet. C+F
p (uC + F | z1: c ) =
∫ p (uc | z c ). ∏
p (uF | uF − 1 ) duF
(2)
F=C+1
Where, the term uC+F indicates the future state at C+F cycles (‘F’ is the total number of predicted states). There are several prognostic algorithms used for flaw growth estimation [12,13]. For Prediction of future health states, a generalized method known as Particle Filter (PF) [5,15,22] is used in this work. However, PF works more effectively in the presence of uncertainty [24,25]. PF using Sequential Monte Carlo (SMC) methods is implemented in this paper to compute posterior PDFs of crack growth in future. The Posterior PDF for each loading cycle is represented in terms of Ns samples uci, in terms of samples and their associated Ns
weights, wci where i=1:Ns, (Ns=1000 samples). The weights are normalized such that ∑ wci = 1and the posterior density is estimated i=1
as: Ns
p (uc | z c ) ≈
∑ wci δ (uc − uci)
(3)
i=1
The samples are drawn fromq(u1:c| z1:c)and the weights are given by:
wci ∝
p (u1:i c | z1: c ) q (u1:i c | z1: c )
(4)
p(u1:c| z1:c) is represented with a new set of samples when the observation zc is known at the loading cycle c. The weights after loading cycle c can be calculated using wc−11:Ns recursively in the weight update equation as:
wc ∝ wc − 1
p (z c | uc ) p (uc | uc − 1 ) q (uc | uc − 1 , z c )
(5) 246
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Fig. 9. Comparison of Flaw Length at different angles of CSK A-03 using different degradation models
3.2. Prior density As mentioned above, commonly used standard prior distributions for modeling crack growth are Weibull, Log-Normal and Exponential distributions [26]. The shape of the distribution can be controlled by the distribution parameters. Each prior distribution p(uc| u1−c), offers different flaw growth rates. The exponential density based degradation model allows lesser change in the flaw dimensions as the age of aero structures evolve. However, Log-Normal distribution allows abrupt increase in flaw size as the age grows. The Weibull probability distribution function (PDF) is given in (6)
f (uk ; β, k ) =
⎧β k ⎨0 ⎩
u uk β − 1 − k β e k k
( )
( )
⎫ uk < 0 ⎬ ⎭
uk ≥ 0
(6)
Where β is shape parameter & k is scale parameter. In reliability engineering, Weibull is the most versatile lifetime distribution that takes the form of other distributions depending on the value of shape parameter β as shown in Fig. 3 where x axis represents the change in crack/flaw dimension. It is evident from Fig. 3 that when 0 < β < 1, there is a high probability that the crack size remains the same. Therefore, Weibull degradation model resist the change in crack dimension. When β=1 Weibull PDF (probability density function) models the life behavior of exponential distribution illustrating smooth flaw growth. When β > 1, there are high chances that the crack dimension increases in spite of the resisting change. For suitable value of β > 1, PDF of Weibull distribution may model the life behavior of Lognormal. When β=2, Weibull distribution reduces to Rayleigh distribution. In our research we have tested our algorithm for different values of shape parameter in order to check how much degree of variability our model can handle. The exponential density expression is given in (7). Previous state is taken as the mean of the exponential density. 247
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Fig. 10. Comparison of Flaw Depth at different angles of CSK A-03 using different degradation models
Table 2 Comparison table of prediction errors (mm) in flaw length at 23000 loading cycle (Fig. 10). Angles
Weibull β = 1.2 k=1
Log-normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
0 45 90 135 180 225 270 315 RMSE
−2.4921 −4.0760 0.8028 0.9177 −0.2671 −5.6017 −0.4012 −6.6294 3.5332
−2.5374 −4.6025 1.0681 −0.1056 −0.2712 −5.0846 −0.2130 −8.9094 4.0945
−2.0095 −4.5329 0.0604 1.4840 −0.5089 −3.5508 0.8791 −5.8553 2.5032
−2.7370 −4.3429 0.5558 −1.8757 −0.7796 −10.6937 −0.2082 −9.4235 5.4082
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values.
p (uc | uc − 1) = e−
‖uc − uc − 1 ‖a a
(7)
Where a is a scalar value that controls the variability in the predicted state value. The Log-Normal distribution expression is given in (8).
f (uc ) =
1 uc 2πσ 2
−
e
(ln uc − μ)2 2σ 2
(8)
Whereuc the current state, μ is the mean of the distribution and σ is the standard deviation. 248
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Table 3 Comparison table of flaw length RMSE (mm) for 24000 and 25000 loading cycles (Fig. 10). Loading cycles
Weibull β = 1.2 k=1
Log-Normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
24000 25000
3.9250 4.3137
4.5866 4.7510
3.0981 3.3526
5.8780 5.9453
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values. Table 4 Comparison table of prediction errors (mm) in flaw depth at 23000 loading cycle (Fig. 11). Angles
Weibull β = 1.2 k=1
Log-Normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
0 45 90 135 180 225 270 315 RMSE
−1.3881 −0.3849 −0.3204 −0.9455 −0.9765 −3.2049 −2.6425 −2.4051 1.8393
−1.4638 −0.3899 −0.3223 −0.9446 −0.9765 −3.2959 −2.9548 −3.0955 2.0439
−0.5519 −0.0884 −0.0975 −0.6451 −0.8265 −1.8627 −1.7279 −2.1568 1.2514
−1.8391 −0.5455 −0.3927 −1.0202 −1.5883 −3.5920 −3.5413 −3.3228 2.3421
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values. Table 5 Comparison table of flaw depth RMSE (mm) for 24000 and 25000 loading cycles (Fig. 11). Loading Cycles
Weibull β = 1.2 k=1
Log-normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
24000 25000
2.0340 2.1430
2.2129 2.2274
1.5115 1.6093
2.4490 2.5303
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values. Table 6 Comparison table of mean length and depth (mm). Leading cycles
Weibull β = 1.2 k=1
Log-Normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
23000 24000 25000
2.6862 2.9795 3.2283
3.0692 3.3998 3.4892
1.8773 2.3048 2.4809
3.8751 4.1635 4.2378
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values.
The prior distribution and their suitable parameters used in this study are displayed in Fig. 4. The shape parameters ‘β’ is selected for three different ranges i.e. β < 1, β=1, β > 1 while the scale parameter ‘k’ is kept near to 1 for Weibull, Exponential and Lognormal distributions respectively. The pattern of the state transition degradation models shown in Fig. 4 indicate that Lognormal is useful to predict abrupt flaw growth whereas Exponential and Weibull (β=0.8) is useful to predict constant flaw growth. On the other hand Weibull with β=1.2 offers smooth and slow changes in crack size. These state transition degradation models were chosen to evaluate our algorithm for different behavior of degradations and the degree of variability the models can tolerate. In the statistical inverse problem, the state transition function corresponds to the prior PDF. Likewise, the measurement function is analogous to the likelihood PDF. In particle filter; posterior PDF is expressed in terms of samples and associated weights at each location [21]. Weights are assigned and updated using likelihood PDF as expressed in weight update Eq. (5). The error is the degree of deviation of the actual data from the computed data using the measurement model. Depending on that error, weights are assigned. Weight assign to a sample is high if the error of that sample is small and vice versa. The measurement model relates the states to measurements. The measurement model is developed using the available states and subsequent measurements. Assuming that measurements are direct observation of the states, where the only source of uncertainty is 249
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Fig. 11. Comparison of actual & predicted states at 23000 loading cycle of CSK rivet.
the random noise nc, measurements are generated from the actual states as in [22].
z c = f (uc , nc )
(9) th
Now, the measurement model is represented by a H
order polynomial, which is expressed as follows:
H
zc =
∑ mh uh
(10)
h=1
The polynomial coefficients m1, . . ……mH are computed using available states and the measurements [21,22]. Due to the absence of future measurements, the update step can only be implemented till the current state. Therefore, only the prediction step of the particle filter is implemented for future states. The overall scheme of particle filter algorithm is shown in Fig. 5. The details of PF based inversion methods can also be found in references [5, 22]. In order to find the suitable degradation model for crack size prediction in future, the historical crack size database is divided into training and validation databases. The crack growth prediction capability of each degradation model is quantified using the computed errors in the prediction and validation data. Error is simply calculated as the difference between the predicted crack value and the actual crack dimensions in the validation database. 4. Results and discussion The proposed framework is tested on the actual historical flaw data of countersunk rivet-holes of an in-service Airbus A310 aircraft’s wing. The data was acquired at the airline NDT center at different time intervals of around 3000 cycles with an angular step of 45o for length and thickness of the sheet around countersunk rivet holes till 25000 flights/loading cycles. The data is developed for 1 degree resolution through cubic spline interpolation method. 250
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Fig. 12. Comparison of actual & predicted states at 24000 loading cycle of CSK rivet.
An interval of 1000 cycles is adopted between 25000 flight/loading cycles hence 25 data points are available for examination of proposed algorithm. The historical data was divided into Training and Validation Periods as displayed in Fig. 6. The data till 22000 loading cycles was used as training period (1 to 22 state) whereas the data from 23000 loading cycle till 25000 is used as validation period (23 to 25 state). This data is applied on PF framework using four state transition models i.e. Weibull (β= 0.8 and β= 1.2), Lognormal and Exponential for the prediction of crack growth. Weibull, Lognormal and Exponential distributions have been used extensively for analysis of life time data [5]. Each life time distribution depends on different parameters. Weibull depends on shape β and scale parameter k. Lognormal depends on mean μ and standard deviation σ. Exponential depends on scale parameter k. β has significant impact on behavior of Weibull distribution. For suitable value of β, Weibull may reduce to other forms of distributions. There are several methods to estimate the parameters for chosen distribution. To reduce the complexity of determining different parameters for each distribution. The shape parameter of 1parameter Weibull is hypothesized to approximate exponential and lognormal distributions. The behavior of Weibull is captured in all three regions β < 1, β=1, β > 1. Goodness of fit of chosen parameters is computed using Particle Filter based prognostic technique The results obtained using the three degradation models at a particular angle of CSK rivet are shown in Fig. 7 and Fig. 8. Both figures illustrate that Exponential distribution as degradation model offer improved crack growth prediction as compared to other candidate distributions. The similar scheme is adopted for all lengths and depths for CSK rivets. A comparison of actual states with crack growth prediction of Weibull (β=0.8 and β=1.2), Lognormal, and Exponential Distributions at different angles are displayed in Fig. 9 and Fig. 10. Root Mean Square Error metric is used to estimate the degree of deviation of predicted crack values from the actual crack values.
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Fig. 13. Comparison of actual & predicted states at 25000 loading cycle of CSK rivet. Table 7 Comparison table of mean length and depth (mm) for 20000 loading cycles training data. Leading cycles
Weibull β = 1.2 k=1
Log-normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
21000 22000 23000
2.7681 3.2721 3.4382
3.0490 3.4269 3.7178
2.6549 3.0195 3.2879
3.5208 3.6668 4.0059
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values.
RMSE =
1 n
n
∑ ei2
(11)
i=1
Where ei is the difference between predicted & actual crack sizes. In order to determine the prediction efficacy of the three candidate distributions as degradation distributions, we computed errors in crack dimensions’ prediction after 23000, 24000 & 25000 loading cycles once update step is not available after 22000 loading cycles. Distribution wise flaw length prediction errors for 23000 loading cycles at each angle are tabulated in Table 2. RMSE results are also displayed in the last row of this table. Similar distribution wise flaw length prediction errors were computed for 24000 and 25000 loading cycles. RMSE results for 24000 and 25000 loading cycles are then tabulated in Table 3. As update step is not being applied to the prediction at 23000 and 24000 loading cycles, it is evident from Table 2 and 3 that RMSE increases as more predictions are made without update step. The RMSE, calculated at 25000 loading cycles, is the largest for all distribution. Using the same 252
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calculation method flaw depth prediction errors and RMSE for 23000, 24000 and 25000 loading cycles are tabulated in Table 4 and Table 5 To conclude the quantitative results, Mean RMSE of both length and depth has been calculated from Tables 2-5. The final results are displayed in Table 6. Exponential Distribution offers best approximation as compared to Weibull and Lognormal. Comparative results are also displayed using 3D polar plots. The actual & estimated states after 23000, 24000, and 25000 loading cycles are displayed in Fig. 11, 12 & 13 respectively. In the figures, the distance from the CSK rivet hole is in the unit of 10-1mm whereas the color intensities are showing the crack depth values in mm [5]. We are predicting the crack growth for future time instant. There is a difference of 1000 loading cycles between each time instant. During each loading cycle wings is subjected to varying loads. The crack growth is stochastic in nature [21] [27] and is challenging to match exact crack growth after 1000, 2000 and 3000 loading cycles, without update step, as displayed in Fig. 11-13 respectively. However, the lesser RMSE of the prediction in comparison to actual growth still showcase the efficacy of the proposed technique Life distribution of any structural material follows constant failure rate. Prediction results calculated using exponential degradation model offers better performance. In order to further check the effectiveness of exponential degradation model for crack size prediction, the proposed framework is also tested using shorter training period. The data till 20000 loading cycles was used as training period (1 to 20 state) whereas the data from 21000 loading cycle till 23000 loading cycles is used as validation period (21 to 23 state). Mean Length and Mean Depth of RMSE for the prediction of 21000, 22000 and 23000 loading cycles is shown in Table 7. It is evident that Exponential distribution based model offers best estimation of actual crack size even in shorter training period. Note that the exponential degradation model offers less abrupt flaw growth which is appropriate to the reported case study. To further reduce the prediction errors, correction loops can be incorporated within the algorithm [5]. 5. Conclusion and future work To enhance reliability of aircraft cost effective maintainability schemes are required to be incorporated in the aircraft maintenance system. Early flaw detection can help in reducing the aircraft’s failure risk. This can be achieved by using suitable and effective life prediction algorithms. Particle Filter which is a statistical based prognostic algorithm also known as Sequential Monte Carlo is used to predict the future health mode of CSK rivet holes of an aircraft wings. A PF based framework is developed in which Weibull, Exponential and Lognormal distributions are used to model crack growth. A statistical tool Root Mean Square Error (RMSE) is used to narrate the forecasting errors of these models. The results reveals that Exponential distribution as the degradation model has the superior capability to predict crack growth of countersunk rivet hole of an in service airbus A310 of aircraft wing as compared to other distributions. This proposed work can be implemented in predictive maintenance of various components of aerospace & marine structures. In future work, maximum likelihood estimation (MLE) can be used to estimate parameters of standard distributions fitted on the historical degradation data. Acknowledgements The authors are grateful to Pakistan International Airlines NDT Center for provision of data/ knowledge necessary. References [1] C. Xiongzi, Y. Jinsong, T. Diyin, W. 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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Comparison of different life distribution schemes for prediction of crack propagation in an aircraft wing
T
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Moez ul Hassan , Fabiha Danish, Waleed Bin Yousuf, Tariq Mairaj Rasool Khan Department of Electronics and Power Engineering, Pakistan Navy Engineering College (PNEC), National University of Sciences and Technology (NUST), Karachi 75350, Pakistan
A R T IC LE I N F O
ABS TRA CT
Keywords: Aircraft failure Fatigue crack growth Degradation Life prediction
Estimation of remaining useful life/prognostics of an aircraft structure permits aerospace industry to timely schedule maintenance activities. Accurate planning through prognostics ensures safer flight operation and lower downtimes. The core of any prognostic algorithm is the state transition/degradation model. In the reported research work, Particle Filter (PF) based prognostic algorithm is used to predict crack growth with three different state transition/life distribution models. PF (Bayesian Sequential Monte Carlo) allows using of non-linear state-transition and non-Gaussian/multimodal noise distributions. The typical candidate life distributions for modeling crack growth are Exponential, Weibull and Lognormal distributions. A framework is proposed where effectiveness of the candidate distribution for modeling degradation phenomenon can be adjudged. The algorithm is tested on actual historical NDT data of crack growth on countersunk (CSK) rivet holes on an Airbus A310 aircraft’s wing. Historical data is bifurcated into two periods i.e. training and validation periods. The most appropriate distribution based on the comparison of the above mentioned candidate distributions is proposed for prediction of the degradation/ flaw propagation.
1. Introduction Structural failure during flight can result into catastrophic consequences [1]. In general, failures occur when a crack on aircraft structure is no longer able to withstand the stresses imposed on it during operation [2]. Therefore, it is necessary to predict the crack growth before a crack reaches the critical value [3]. Accurate crack growth estimation has a potential to decrease aircrafts operating cost through improved maintenance/overhaul scheduling which, in turn, reduces the downtime. Crack damage is normally originated on structural components subject to fatigue loading. Fatigue loading initiate cracks on aircraft structure whose subsequent growth leads to sudden failure [4]. During the complete flight cycle, aircraft structure remains under the influence of cyclic stresses. NDT (non-destructive testing) data acquired from actual aircraft is used in the reported research work. The NDT data is acquired at rivet holes on aircraft’s wing. The holes are one of the most fatigue sensitive sub components of aircraft structures [5]. Accurate crack growth prediction is a challenging task due to the influence of different varying factors. Different physical fatigue models have been suggested in recent literature for crack initiation and propagation [6,7]. An application of such local unified approach has been presented in [8]. A notched rectangular plate, made of P355NL1 steel is modelled in order to generate S–N curves for distinct stress R-ratios. Enrique Castillo and Alfonso Fernández-Canteli proposed stochastic method to predict crack propagations
⁎
Corresponding author. E-mail address: [email protected] (M.u. Hassan).
https://doi.org/10.1016/j.engfailanal.2018.10.010 Received 21 November 2017; Received in revised form 29 September 2018; Accepted 22 October 2018 Available online 23 October 2018 1350-6307/ © 2018 Elsevier Ltd. All rights reserved.
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by using three main fatigue approaches [9]. In [10] a procedure for estimating probabilistic S-N curve was proposed using strain life data. Crack growth prediction using constant amplitude loading (CAL) is commonly found in the literature [11]. Whereas fatigue crack initiation and crack propagation under variable amplitude loading (VAL) is less commonly found in the literature. In [12–14] crack growth prediction is investigated by artificially produced fatigue cracks in the laboratory specimen of aero structures. Simulation is then performed to predict the remaining useful life of damaged aircraft structures while applying VAL. Reference [15, 16] processed the entire crack growth simulation using finite element method. Physical modeling of three dimensional fatigue crack progression does not account comprehensive information due to inadequate model parameters. Similarly, loads applied on test specimen in laboratory do not exactly replicate the real stresses due to underlying uncertainty in the environmental conditions. In this paper, we used actual crack growth data which captures the actual environmental conditions the aircraft wing is subjected to. Historical data gathered over a period of time is used to determine the remaining useful life (RUL) of a specific component [1]. Statistical models have a potential to predict crack growth with improved accuracy even in the presence of uncertainties [17]. In this paper, particle filtering (PF) based statistical method is applied on historical data acquired from rivet holes of a civil aviation aircraft. PF is a recursive Bayesian technique using sequential Monte Carlo simulations [18]. Using PF technique, crack growth can be predicted using suitable degradation model. The research work reports a PF based framework for crack growth determination. Three candidate degradation models are used in the PF framework to find the most appropriate degradation model. The crack growth results are then compared with actual historical data to find the most appropriate degradation model. Use of actual historical data with crack growth models improves the accuracy of the proposed estimation. However the contemporary methods comprising of simulation methods [14–16] do not possess this degree of accuracy. Accurate crack growth prediction offers enhanced aircraft safety through implementation of timely maintenance strategies [19]. Next section describes the ultrasonic NDT data acquisition around countersunk (CSK) rivet holes. PF based crack growth prediction is elaborated in the research methodology section. Then the results and discussions are given. Finally, conclusion and future work are presented. 2. Case study Wings are the most critical structural unit of any aircraft. Their physical condition indicates the aircraft’s safety to a significant extent. Rivets are used to join different metallic sheet together in an aircraft wing structure. Load is transferred from one metallic sheet to another through riveted joints. These joints are subjected to stress during the bending of the aircraft wings. Presence of CSK rivets in the region of joints makes this stress state highly complicated as residual stress introduces around the holes. Under continuous cyclic loading, crack initiation and its subsequent growth in this high stress concentration region lead to structural failure. The loading cycle comprises of take-off, cruise and landing phases. Wings mainly bend during take-off and landing phases. Fatigue cracks on wings are encouraged by the dynamic load acting on an aircraft wing during the loading cycle [20]. Therefore, in this paper crack growth is computed with respect to the loading cycles. Flaw growth around a rivet-hole is a three dimensional entity as shown in Fig. 1. The ultrasonic thickness gauging sensors are used to measure the flaw dimension around the countersunk rivet-hole in an aircraft wing structure. The NDT measurements are acquired after 0, 16067, 18634, 21201, 23715 and 25816 loading cycles. The loading cycle’s frequency to schedule inspections are specified by OEM to avert fatal failures. Considering the rivet hole as center, the three dimensional damage around CSK rivet holes is expressed in terms of flaw length and flaw depth. The recommended procedure is to collect length-depth pair readings with the angular step size of 45° around the CSK rivet hole. The length–depth pair data is tabulated in Table 1 and graphically presented in Fig. 2. Cubic spline interpolation method was used to enhance data points. As displayed in Table 1, last readings were acquired after 16067 loading cycles. Whereas, the current readings were recorded after 18643 loading cycles with the difference of 2567 loading cycles. 3. Methodology 3.1. Statistical framework Flaw growth prediction from NDT measurements is a typical nondestructive evaluation (NDE) inverse problem. Since the crack is exposed to harsh environmental conditions, its subsequent growth is stochastic in nature [21]. A statistical framework is adopted to
Fig. 1. Illustration of Flaw growth around CSK rivets [5]. 242
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Table 1 Details of damage around Countersunk Rivet Hole (CSK) [5] ANGLE
0 45 90 135 180 225 270 315
Flaw length (mm)
Flaw depth (mm)
Last Reading
Current Reading
Last Reading
Current Reading
6 4 4 4 6 0 0 0
8 8 10 10 10 15 11 6
5.12 5.12 5.02 5.02 5.12 0.52 0.62 0.52
6.12 5.52 5.42 5.42 5.52 5.42 5.52 5.42
Fig. 2. Damage length around CSK rivet hole [5].
Fig. 3. Effect of β on Weibull distribution.
deal with the underlying uncertainty [5,22]. The Bayesian filtering approach is used to encounter the uncertainty in the final solution [22]. Bayesian filter is applied by treating state uc as the actual crack dimensions (length and depth) where c refers to loading cycle index (c=1, 2, 3, ...C) and zc is considered to be the corresponding noisy measurements from the ultrasonic gauging sensor. As all the preceding measurementsz1:c are known, the posterior density p(uc| z1:c) can be estimated using the prior density p(uc| uc−1) and the likelihood density p(zc| uc) [23] as given in (1).
p (uc | z1: c ) =
p (z c | uc ). p (uc | z1: c − 1 ) p (z c | z1: c − 1 )
(1)
Crack growth prediction from historical data is a challenging task. Now, predictive density p(uC+F| z1:c) can be computed as: 243
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Fig. 4. Different degradation models.
Fig. 5. Particle filter algorithm scheme.
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Fig. 6. Actual flaw data of length at 45 degree.
Fig. 7. Flaw length prediction at angle 90° of CSK rivet.
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Fig. 8. Flaw depth prediction at angle 45° of CSK rivet. C+F
p (uC + F | z1: c ) =
∫ p (uc | z c ). ∏
p (uF | uF − 1 ) duF
(2)
F=C+1
Where, the term uC+F indicates the future state at C+F cycles (‘F’ is the total number of predicted states). There are several prognostic algorithms used for flaw growth estimation [12,13]. For Prediction of future health states, a generalized method known as Particle Filter (PF) [5,15,22] is used in this work. However, PF works more effectively in the presence of uncertainty [24,25]. PF using Sequential Monte Carlo (SMC) methods is implemented in this paper to compute posterior PDFs of crack growth in future. The Posterior PDF for each loading cycle is represented in terms of Ns samples uci, in terms of samples and their associated Ns
weights, wci where i=1:Ns, (Ns=1000 samples). The weights are normalized such that ∑ wci = 1and the posterior density is estimated i=1
as: Ns
p (uc | z c ) ≈
∑ wci δ (uc − uci)
(3)
i=1
The samples are drawn fromq(u1:c| z1:c)and the weights are given by:
wci ∝
p (u1:i c | z1: c ) q (u1:i c | z1: c )
(4)
p(u1:c| z1:c) is represented with a new set of samples when the observation zc is known at the loading cycle c. The weights after loading cycle c can be calculated using wc−11:Ns recursively in the weight update equation as:
wc ∝ wc − 1
p (z c | uc ) p (uc | uc − 1 ) q (uc | uc − 1 , z c )
(5) 246
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Fig. 9. Comparison of Flaw Length at different angles of CSK A-03 using different degradation models
3.2. Prior density As mentioned above, commonly used standard prior distributions for modeling crack growth are Weibull, Log-Normal and Exponential distributions [26]. The shape of the distribution can be controlled by the distribution parameters. Each prior distribution p(uc| u1−c), offers different flaw growth rates. The exponential density based degradation model allows lesser change in the flaw dimensions as the age of aero structures evolve. However, Log-Normal distribution allows abrupt increase in flaw size as the age grows. The Weibull probability distribution function (PDF) is given in (6)
f (uk ; β, k ) =
⎧β k ⎨0 ⎩
u uk β − 1 − k β e k k
( )
( )
⎫ uk < 0 ⎬ ⎭
uk ≥ 0
(6)
Where β is shape parameter & k is scale parameter. In reliability engineering, Weibull is the most versatile lifetime distribution that takes the form of other distributions depending on the value of shape parameter β as shown in Fig. 3 where x axis represents the change in crack/flaw dimension. It is evident from Fig. 3 that when 0 < β < 1, there is a high probability that the crack size remains the same. Therefore, Weibull degradation model resist the change in crack dimension. When β=1 Weibull PDF (probability density function) models the life behavior of exponential distribution illustrating smooth flaw growth. When β > 1, there are high chances that the crack dimension increases in spite of the resisting change. For suitable value of β > 1, PDF of Weibull distribution may model the life behavior of Lognormal. When β=2, Weibull distribution reduces to Rayleigh distribution. In our research we have tested our algorithm for different values of shape parameter in order to check how much degree of variability our model can handle. The exponential density expression is given in (7). Previous state is taken as the mean of the exponential density. 247
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Fig. 10. Comparison of Flaw Depth at different angles of CSK A-03 using different degradation models
Table 2 Comparison table of prediction errors (mm) in flaw length at 23000 loading cycle (Fig. 10). Angles
Weibull β = 1.2 k=1
Log-normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
0 45 90 135 180 225 270 315 RMSE
−2.4921 −4.0760 0.8028 0.9177 −0.2671 −5.6017 −0.4012 −6.6294 3.5332
−2.5374 −4.6025 1.0681 −0.1056 −0.2712 −5.0846 −0.2130 −8.9094 4.0945
−2.0095 −4.5329 0.0604 1.4840 −0.5089 −3.5508 0.8791 −5.8553 2.5032
−2.7370 −4.3429 0.5558 −1.8757 −0.7796 −10.6937 −0.2082 −9.4235 5.4082
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values.
p (uc | uc − 1) = e−
‖uc − uc − 1 ‖a a
(7)
Where a is a scalar value that controls the variability in the predicted state value. The Log-Normal distribution expression is given in (8).
f (uc ) =
1 uc 2πσ 2
−
e
(ln uc − μ)2 2σ 2
(8)
Whereuc the current state, μ is the mean of the distribution and σ is the standard deviation. 248
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Table 3 Comparison table of flaw length RMSE (mm) for 24000 and 25000 loading cycles (Fig. 10). Loading cycles
Weibull β = 1.2 k=1
Log-Normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
24000 25000
3.9250 4.3137
4.5866 4.7510
3.0981 3.3526
5.8780 5.9453
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values. Table 4 Comparison table of prediction errors (mm) in flaw depth at 23000 loading cycle (Fig. 11). Angles
Weibull β = 1.2 k=1
Log-Normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
0 45 90 135 180 225 270 315 RMSE
−1.3881 −0.3849 −0.3204 −0.9455 −0.9765 −3.2049 −2.6425 −2.4051 1.8393
−1.4638 −0.3899 −0.3223 −0.9446 −0.9765 −3.2959 −2.9548 −3.0955 2.0439
−0.5519 −0.0884 −0.0975 −0.6451 −0.8265 −1.8627 −1.7279 −2.1568 1.2514
−1.8391 −0.5455 −0.3927 −1.0202 −1.5883 −3.5920 −3.5413 −3.3228 2.3421
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values. Table 5 Comparison table of flaw depth RMSE (mm) for 24000 and 25000 loading cycles (Fig. 11). Loading Cycles
Weibull β = 1.2 k=1
Log-normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
24000 25000
2.0340 2.1430
2.2129 2.2274
1.5115 1.6093
2.4490 2.5303
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values. Table 6 Comparison table of mean length and depth (mm). Leading cycles
Weibull β = 1.2 k=1
Log-Normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
23000 24000 25000
2.6862 2.9795 3.2283
3.0692 3.3998 3.4892
1.8773 2.3048 2.4809
3.8751 4.1635 4.2378
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values.
The prior distribution and their suitable parameters used in this study are displayed in Fig. 4. The shape parameters ‘β’ is selected for three different ranges i.e. β < 1, β=1, β > 1 while the scale parameter ‘k’ is kept near to 1 for Weibull, Exponential and Lognormal distributions respectively. The pattern of the state transition degradation models shown in Fig. 4 indicate that Lognormal is useful to predict abrupt flaw growth whereas Exponential and Weibull (β=0.8) is useful to predict constant flaw growth. On the other hand Weibull with β=1.2 offers smooth and slow changes in crack size. These state transition degradation models were chosen to evaluate our algorithm for different behavior of degradations and the degree of variability the models can tolerate. In the statistical inverse problem, the state transition function corresponds to the prior PDF. Likewise, the measurement function is analogous to the likelihood PDF. In particle filter; posterior PDF is expressed in terms of samples and associated weights at each location [21]. Weights are assigned and updated using likelihood PDF as expressed in weight update Eq. (5). The error is the degree of deviation of the actual data from the computed data using the measurement model. Depending on that error, weights are assigned. Weight assign to a sample is high if the error of that sample is small and vice versa. The measurement model relates the states to measurements. The measurement model is developed using the available states and subsequent measurements. Assuming that measurements are direct observation of the states, where the only source of uncertainty is 249
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Fig. 11. Comparison of actual & predicted states at 23000 loading cycle of CSK rivet.
the random noise nc, measurements are generated from the actual states as in [22].
z c = f (uc , nc )
(9) th
Now, the measurement model is represented by a H
order polynomial, which is expressed as follows:
H
zc =
∑ mh uh
(10)
h=1
The polynomial coefficients m1, . . ……mH are computed using available states and the measurements [21,22]. Due to the absence of future measurements, the update step can only be implemented till the current state. Therefore, only the prediction step of the particle filter is implemented for future states. The overall scheme of particle filter algorithm is shown in Fig. 5. The details of PF based inversion methods can also be found in references [5, 22]. In order to find the suitable degradation model for crack size prediction in future, the historical crack size database is divided into training and validation databases. The crack growth prediction capability of each degradation model is quantified using the computed errors in the prediction and validation data. Error is simply calculated as the difference between the predicted crack value and the actual crack dimensions in the validation database. 4. Results and discussion The proposed framework is tested on the actual historical flaw data of countersunk rivet-holes of an in-service Airbus A310 aircraft’s wing. The data was acquired at the airline NDT center at different time intervals of around 3000 cycles with an angular step of 45o for length and thickness of the sheet around countersunk rivet holes till 25000 flights/loading cycles. The data is developed for 1 degree resolution through cubic spline interpolation method. 250
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Fig. 12. Comparison of actual & predicted states at 24000 loading cycle of CSK rivet.
An interval of 1000 cycles is adopted between 25000 flight/loading cycles hence 25 data points are available for examination of proposed algorithm. The historical data was divided into Training and Validation Periods as displayed in Fig. 6. The data till 22000 loading cycles was used as training period (1 to 22 state) whereas the data from 23000 loading cycle till 25000 is used as validation period (23 to 25 state). This data is applied on PF framework using four state transition models i.e. Weibull (β= 0.8 and β= 1.2), Lognormal and Exponential for the prediction of crack growth. Weibull, Lognormal and Exponential distributions have been used extensively for analysis of life time data [5]. Each life time distribution depends on different parameters. Weibull depends on shape β and scale parameter k. Lognormal depends on mean μ and standard deviation σ. Exponential depends on scale parameter k. β has significant impact on behavior of Weibull distribution. For suitable value of β, Weibull may reduce to other forms of distributions. There are several methods to estimate the parameters for chosen distribution. To reduce the complexity of determining different parameters for each distribution. The shape parameter of 1parameter Weibull is hypothesized to approximate exponential and lognormal distributions. The behavior of Weibull is captured in all three regions β < 1, β=1, β > 1. Goodness of fit of chosen parameters is computed using Particle Filter based prognostic technique The results obtained using the three degradation models at a particular angle of CSK rivet are shown in Fig. 7 and Fig. 8. Both figures illustrate that Exponential distribution as degradation model offer improved crack growth prediction as compared to other candidate distributions. The similar scheme is adopted for all lengths and depths for CSK rivets. A comparison of actual states with crack growth prediction of Weibull (β=0.8 and β=1.2), Lognormal, and Exponential Distributions at different angles are displayed in Fig. 9 and Fig. 10. Root Mean Square Error metric is used to estimate the degree of deviation of predicted crack values from the actual crack values.
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Fig. 13. Comparison of actual & predicted states at 25000 loading cycle of CSK rivet. Table 7 Comparison table of mean length and depth (mm) for 20000 loading cycles training data. Leading cycles
Weibull β = 1.2 k=1
Log-normal σ = 1.5 μ = 0.8
Exponential λ = 1.1
Weibull β = 0.8 k=1
21000 22000 23000
2.7681 3.2721 3.4382
3.0490 3.4269 3.7178
2.6549 3.0195 3.2879
3.5208 3.6668 4.0059
Exponential Distribution RMSE values are bold because they are lowest as compared to Weibull and Lognormal values.
RMSE =
1 n
n
∑ ei2
(11)
i=1
Where ei is the difference between predicted & actual crack sizes. In order to determine the prediction efficacy of the three candidate distributions as degradation distributions, we computed errors in crack dimensions’ prediction after 23000, 24000 & 25000 loading cycles once update step is not available after 22000 loading cycles. Distribution wise flaw length prediction errors for 23000 loading cycles at each angle are tabulated in Table 2. RMSE results are also displayed in the last row of this table. Similar distribution wise flaw length prediction errors were computed for 24000 and 25000 loading cycles. RMSE results for 24000 and 25000 loading cycles are then tabulated in Table 3. As update step is not being applied to the prediction at 23000 and 24000 loading cycles, it is evident from Table 2 and 3 that RMSE increases as more predictions are made without update step. The RMSE, calculated at 25000 loading cycles, is the largest for all distribution. Using the same 252
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calculation method flaw depth prediction errors and RMSE for 23000, 24000 and 25000 loading cycles are tabulated in Table 4 and Table 5 To conclude the quantitative results, Mean RMSE of both length and depth has been calculated from Tables 2-5. The final results are displayed in Table 6. Exponential Distribution offers best approximation as compared to Weibull and Lognormal. Comparative results are also displayed using 3D polar plots. The actual & estimated states after 23000, 24000, and 25000 loading cycles are displayed in Fig. 11, 12 & 13 respectively. In the figures, the distance from the CSK rivet hole is in the unit of 10-1mm whereas the color intensities are showing the crack depth values in mm [5]. We are predicting the crack growth for future time instant. There is a difference of 1000 loading cycles between each time instant. During each loading cycle wings is subjected to varying loads. The crack growth is stochastic in nature [21] [27] and is challenging to match exact crack growth after 1000, 2000 and 3000 loading cycles, without update step, as displayed in Fig. 11-13 respectively. However, the lesser RMSE of the prediction in comparison to actual growth still showcase the efficacy of the proposed technique Life distribution of any structural material follows constant failure rate. Prediction results calculated using exponential degradation model offers better performance. In order to further check the effectiveness of exponential degradation model for crack size prediction, the proposed framework is also tested using shorter training period. The data till 20000 loading cycles was used as training period (1 to 20 state) whereas the data from 21000 loading cycle till 23000 loading cycles is used as validation period (21 to 23 state). Mean Length and Mean Depth of RMSE for the prediction of 21000, 22000 and 23000 loading cycles is shown in Table 7. It is evident that Exponential distribution based model offers best estimation of actual crack size even in shorter training period. Note that the exponential degradation model offers less abrupt flaw growth which is appropriate to the reported case study. To further reduce the prediction errors, correction loops can be incorporated within the algorithm [5]. 5. Conclusion and future work To enhance reliability of aircraft cost effective maintainability schemes are required to be incorporated in the aircraft maintenance system. Early flaw detection can help in reducing the aircraft’s failure risk. This can be achieved by using suitable and effective life prediction algorithms. Particle Filter which is a statistical based prognostic algorithm also known as Sequential Monte Carlo is used to predict the future health mode of CSK rivet holes of an aircraft wings. A PF based framework is developed in which Weibull, Exponential and Lognormal distributions are used to model crack growth. A statistical tool Root Mean Square Error (RMSE) is used to narrate the forecasting errors of these models. The results reveals that Exponential distribution as the degradation model has the superior capability to predict crack growth of countersunk rivet hole of an in service airbus A310 of aircraft wing as compared to other distributions. This proposed work can be implemented in predictive maintenance of various components of aerospace & marine structures. In future work, maximum likelihood estimation (MLE) can be used to estimate parameters of standard distributions fitted on the historical degradation data. Acknowledgements The authors are grateful to Pakistan International Airlines NDT Center for provision of data/ knowledge necessary. References [1] C. Xiongzi, Y. Jinsong, T. Diyin, W. 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