- Email: [email protected]
Automated system for inspection planning
oon'qm.al;mr,D ~k I~iml onginooring PERGAMON
Computers & Industrial Engineering 37 (1999) 461-464
AUTOMATED SYSTEM FOR INSPECTION PLANNING Yu. M. Paramonov, Riga Aviation University, Riga, LATVIA V. 1. Abramov, A. A. Glagovsky, Aviation Complex named S. K Ilyushin, Moscow, Russia ABSTRACT The main functions of special automated system for airframe inspection problem development and a numerical example are described. Parameters of fatigue cracks growth model should be estimated by processing of airframe full-scale fatigue test results. For the choice of periodicity a minimax solution are offered. The calculations are complex enough so by the use of the Delphi's integrated development environment an automated system for inspection planning is developed. © 1999 Elsevier Science Ltd. All rights reserved. KEYWORDS Automated system, fatigue, inspection planning, reliability. INTRODUCTION The guidelines for developing structural inspection programs, that meet the new damage tolerance regulations, are given in the ATA document, MSG-3. A key element of the development is a probability calculation of fatigue failure. The calculations are complex enough so it will be useful to use a special automated system for the purpose. The first phase of system "Airframe reliability" is developed in Aircraft Structure & Strength Analysis Department of RAU. Main functions of it and statistical decision strategy for an inspection interval choice are described in the paper. The system is based on a complex of probabilistic models of fatigue crack growth process, probability characteristics of fatigue crack detection method and decision making procedure aiter fatigue crack discovery. For estimations of distribution function parameters of the models the results of full-scale fatigue test are used.
THE FUNCTIONS OF THE SYSTEM The functions of the system are : I ) the input of initial information ( full-scale fatigue test data of fatigue crack growth, test loads, service loads, etc.), 2) the fatigue test data processing and determination of fatigue crack growth parameters, 3) the calculation of a fatigue failure probability, 4) the visualization of the calculation results by the use of both a PC screen and a printer, 5) the visualization of Monte-Carlo realisations of inspection process, 6) the storage of both an initial information and calculation results in special data base, and 0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00118-7
462
Proceedings of the 24th International Conference on Computers and Industrial Engineering
7) the processing of data base information. The system is in a process of development and not all of these functions have adequate program code. The functions can be changed and in a near future some new functions can be added. The System comprises four main bloks : I) Data Base, 2) Fatigue Crack Growth Parameters Estimation, 3) Calculation of Fatigue Failure Probability, 4) Minimax Solution of Periodicity Choice Problem. DATA BASE. FATIGUE CRACK GROWTH PARAMETERS In Data Base there are initial information and some results of calculations. We suppose to know the trajectory of fatigue crack propagation during full-scale fatigue test, for instance, in the form of table {( t,, a, ), i= 1..... k } where a~ is a fatigue crack size at t, "service" time, k is number of experimental points. Data of full-scale fatigue test of Aviation Complex named S.V. Ilyushin was used for a numerical example. This information of fatigue crack growth in some part of passenger "aircraft wing can be seen in Fig.1 (table "TA1 ...TA9" and circles of central curve).
Fig. 1 It was supposed that the fatigue crack growth curve can be approximated by the well known formula which is developed from Paris's formula for the case of regular harmonic loading (see e.g. Paramonov 1995): a ( t ) = a exp( - ( I n ( 1 - / 2 w Q t ) ) / / 2 ) , i f /2 ~: 0, a(t) = a exp(Qt),
if /2 = 0,
where w = a ~ ; a , /2, Q are some parameters of fatigue crack; a ( so called equivalent initial flow size) and /2 depend on material characteristics, technology and structure, Q - on loading mode. By the use of least squares method we can get estimations of the parameters. These estimations are random variables. If the distribution function of the estimations and it's parameter 0 = ( 0 0, ~ ~. . . . ) are known then we can calculate the probability of fatigue failure for any inspection program. But it is very difficult problem to get the distribution function of the estimations. Really we'll never know it. But we can develop some approximation of it. We should remember that in fact we'll use the model only for comparison of fatigue characteristics of new aircraft and of some old one, fatigue characteristics of which in some measure are already
Proceedings of the 24th International Conference on Computers and Industrial Engineering
463
known. In the circumstances in the first version of the System as the preferable decision the most simple model should be used. Enough simple model is a model in which the following assumption is made: the scatter of fatigue crack trajectories in the fleet of different aircrafts of the same type is defined mainly by the scatter of stress concentration and operational loads. We suppose that parameters a and g axe under rigorous acceptance inspection , so they have relatively small scatter and in first approximation can be considered as some constants. But as consequence of operational conditions significant scatter we should consider Q as a random variable. More precisely, an assumption was made that lnQ has normal distribution with unknown mathematical expectation 00 and
known standard deviation
01=0.15 ( the value
taken from (Yang, 1980)). For estimation of parameters 0 o , o~, ~t a some modification of least square method is used. In the considered numerical example the System provides following estimations: 0 o = - 8.783, ct= 1.251, fl = 0.251. In the Fig.1 we can see the initial data (on the left side) and three curves (on the fight side) : the middle one corresponds to the estimations, the other two - to upper and lower confidence limits of 00 . INSPECTION PROGRAM DEVELOPMENT For adopted inspection technology the fatigue crack become visible at t d = 38224 flights (when
a(t) reaches 20 nun) and reaches critical value(when residual stress becomes less then maximum (in one flight) operational stress 69 N/ram 2 ) at t c = 43 601 flights. If we consider the mentioned above estimations of fatigue crack parameters as true values then the probability of failure p j is equal to 0.01 for inspection interval d = 4444 and p l = 0.003 for d =4000. Let D = {d~,d 2.... } be a some sequence such, that d, > d,÷~, limd, = 0; d(e,O) is maximal interval d ~ D for which p j (d,0) < c , where 6 is design allowable failure probability of first approximation. But true value of 00 is not known. So d = d ( 6 , 0 ) and bl = pr(cl,O)are random variables. Here we have to take into account that we must retrofit the tested airframe if results of full scale tests are not good enough. For example, we can take the decision to begin of the aircraft operation only if estimation of t, (time, when fatigue crack will grow up to critical value a,. ) will be more than some boundary t b . Let us denote the event as 0 E 60 . In the opposite case the tested airframe must be retrofitted and we assume that after this the corresponding failure probability will be equal to zero. For this type of strategy the mean probability of fatigue failure
Me~,i. = Mo(~. t IO¢ 6 o) * p(o ~ 6 o) is a function of
0
and for limited specified life tsL' and fixed tb it has a maximum,
depending o n e . Now we can choose 6 = ¢ '
in such a way that max Mo~ ¢ < _ l - R ,
where R is a required reliability. Then we should to return to the System block Fatigue Failure Probability Calculation and repeat the calculation and make the choice of d with E = e ' . For instance (for considered example tsL' =40000 and
t~ = 40000), if we decide that reliability
R = 1 - 0.0103 = 0.9897 is satisfactory, then for 6' interval tt = d ( 6 , 0 ) = 4000
-- 0.003 we should choose inspection
464
Proceedings of the 24th International Conference on Computers and Industrial Engineering
Fig. 2 instead of initial value 4444 for design allowable failure probability of first approximation c = 0.01 . It is worth to mention that max Mofo,j =0.0228 if e' = 0.01. Fig.2 shows Moil,1 as function of (00 -00o)/0 I, where 000 is estimation of 00(in these figures (00 -000)/0 ~ is denoted by (Th0-Yh00)/Thl), max of Me/3,j is denoted by MxMPf ). Left part of the figure corresponds to e" = 0.003, right part corresponds to 6" = 0.01. Note, that if we use the offered minimax decision function d = d(e" ,t)) for the inspection interval choice then expected value of fatigue failure probability is limited uniformly for all unknown parameters of fatigue crack growth model and we do not need to use a dubious compromise combination of required reliability and confidence probability.
REFERENCES Paramonov Yu.M.( 1992 ). Mathematical statistics application to the problem of aircraft fatigue life estimation and insurance (Russian).RAU: Riga. Paramonov Yu.M.(1995). Probability of fatigue crack detection in the case of Paris model. Transactions of Riga Aviation University Mechanical Department. (Russian).RAU: Riga. Vol. 1, pp.35-37. Yang J.N.(1980). Statistical Crack Growth in Durability and Damage Tolerant Analysis. In Proc.Int Conf. AIAA/ASME/ASCE/AHS 22-th Structures, Structural Dynamics and Materials. Vol. 1, pp.38-44.
Computers & Industrial Engineering 37 (1999) 461-464
AUTOMATED SYSTEM FOR INSPECTION PLANNING Yu. M. Paramonov, Riga Aviation University, Riga, LATVIA V. 1. Abramov, A. A. Glagovsky, Aviation Complex named S. K Ilyushin, Moscow, Russia ABSTRACT The main functions of special automated system for airframe inspection problem development and a numerical example are described. Parameters of fatigue cracks growth model should be estimated by processing of airframe full-scale fatigue test results. For the choice of periodicity a minimax solution are offered. The calculations are complex enough so by the use of the Delphi's integrated development environment an automated system for inspection planning is developed. © 1999 Elsevier Science Ltd. All rights reserved. KEYWORDS Automated system, fatigue, inspection planning, reliability. INTRODUCTION The guidelines for developing structural inspection programs, that meet the new damage tolerance regulations, are given in the ATA document, MSG-3. A key element of the development is a probability calculation of fatigue failure. The calculations are complex enough so it will be useful to use a special automated system for the purpose. The first phase of system "Airframe reliability" is developed in Aircraft Structure & Strength Analysis Department of RAU. Main functions of it and statistical decision strategy for an inspection interval choice are described in the paper. The system is based on a complex of probabilistic models of fatigue crack growth process, probability characteristics of fatigue crack detection method and decision making procedure aiter fatigue crack discovery. For estimations of distribution function parameters of the models the results of full-scale fatigue test are used.
THE FUNCTIONS OF THE SYSTEM The functions of the system are : I ) the input of initial information ( full-scale fatigue test data of fatigue crack growth, test loads, service loads, etc.), 2) the fatigue test data processing and determination of fatigue crack growth parameters, 3) the calculation of a fatigue failure probability, 4) the visualization of the calculation results by the use of both a PC screen and a printer, 5) the visualization of Monte-Carlo realisations of inspection process, 6) the storage of both an initial information and calculation results in special data base, and 0360-8352/99 - see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0360-8352(99)00118-7
462
Proceedings of the 24th International Conference on Computers and Industrial Engineering
7) the processing of data base information. The system is in a process of development and not all of these functions have adequate program code. The functions can be changed and in a near future some new functions can be added. The System comprises four main bloks : I) Data Base, 2) Fatigue Crack Growth Parameters Estimation, 3) Calculation of Fatigue Failure Probability, 4) Minimax Solution of Periodicity Choice Problem. DATA BASE. FATIGUE CRACK GROWTH PARAMETERS In Data Base there are initial information and some results of calculations. We suppose to know the trajectory of fatigue crack propagation during full-scale fatigue test, for instance, in the form of table {( t,, a, ), i= 1..... k } where a~ is a fatigue crack size at t, "service" time, k is number of experimental points. Data of full-scale fatigue test of Aviation Complex named S.V. Ilyushin was used for a numerical example. This information of fatigue crack growth in some part of passenger "aircraft wing can be seen in Fig.1 (table "TA1 ...TA9" and circles of central curve).
Fig. 1 It was supposed that the fatigue crack growth curve can be approximated by the well known formula which is developed from Paris's formula for the case of regular harmonic loading (see e.g. Paramonov 1995): a ( t ) = a exp( - ( I n ( 1 - / 2 w Q t ) ) / / 2 ) , i f /2 ~: 0, a(t) = a exp(Qt),
if /2 = 0,
where w = a ~ ; a , /2, Q are some parameters of fatigue crack; a ( so called equivalent initial flow size) and /2 depend on material characteristics, technology and structure, Q - on loading mode. By the use of least squares method we can get estimations of the parameters. These estimations are random variables. If the distribution function of the estimations and it's parameter 0 = ( 0 0, ~ ~. . . . ) are known then we can calculate the probability of fatigue failure for any inspection program. But it is very difficult problem to get the distribution function of the estimations. Really we'll never know it. But we can develop some approximation of it. We should remember that in fact we'll use the model only for comparison of fatigue characteristics of new aircraft and of some old one, fatigue characteristics of which in some measure are already
Proceedings of the 24th International Conference on Computers and Industrial Engineering
463
known. In the circumstances in the first version of the System as the preferable decision the most simple model should be used. Enough simple model is a model in which the following assumption is made: the scatter of fatigue crack trajectories in the fleet of different aircrafts of the same type is defined mainly by the scatter of stress concentration and operational loads. We suppose that parameters a and g axe under rigorous acceptance inspection , so they have relatively small scatter and in first approximation can be considered as some constants. But as consequence of operational conditions significant scatter we should consider Q as a random variable. More precisely, an assumption was made that lnQ has normal distribution with unknown mathematical expectation 00 and
known standard deviation
01=0.15 ( the value
taken from (Yang, 1980)). For estimation of parameters 0 o , o~, ~t a some modification of least square method is used. In the considered numerical example the System provides following estimations: 0 o = - 8.783, ct= 1.251, fl = 0.251. In the Fig.1 we can see the initial data (on the left side) and three curves (on the fight side) : the middle one corresponds to the estimations, the other two - to upper and lower confidence limits of 00 . INSPECTION PROGRAM DEVELOPMENT For adopted inspection technology the fatigue crack become visible at t d = 38224 flights (when
a(t) reaches 20 nun) and reaches critical value(when residual stress becomes less then maximum (in one flight) operational stress 69 N/ram 2 ) at t c = 43 601 flights. If we consider the mentioned above estimations of fatigue crack parameters as true values then the probability of failure p j is equal to 0.01 for inspection interval d = 4444 and p l = 0.003 for d =4000. Let D = {d~,d 2.... } be a some sequence such, that d, > d,÷~, limd, = 0; d(e,O) is maximal interval d ~ D for which p j (d,0) < c , where 6 is design allowable failure probability of first approximation. But true value of 00 is not known. So d = d ( 6 , 0 ) and bl = pr(cl,O)are random variables. Here we have to take into account that we must retrofit the tested airframe if results of full scale tests are not good enough. For example, we can take the decision to begin of the aircraft operation only if estimation of t, (time, when fatigue crack will grow up to critical value a,. ) will be more than some boundary t b . Let us denote the event as 0 E 60 . In the opposite case the tested airframe must be retrofitted and we assume that after this the corresponding failure probability will be equal to zero. For this type of strategy the mean probability of fatigue failure
Me~,i. = Mo(~. t IO¢ 6 o) * p(o ~ 6 o) is a function of
0
and for limited specified life tsL' and fixed tb it has a maximum,
depending o n e . Now we can choose 6 = ¢ '
in such a way that max Mo~ ¢ < _ l - R ,
where R is a required reliability. Then we should to return to the System block Fatigue Failure Probability Calculation and repeat the calculation and make the choice of d with E = e ' . For instance (for considered example tsL' =40000 and
t~ = 40000), if we decide that reliability
R = 1 - 0.0103 = 0.9897 is satisfactory, then for 6' interval tt = d ( 6 , 0 ) = 4000
-- 0.003 we should choose inspection
464
Proceedings of the 24th International Conference on Computers and Industrial Engineering
Fig. 2 instead of initial value 4444 for design allowable failure probability of first approximation c = 0.01 . It is worth to mention that max Mofo,j =0.0228 if e' = 0.01. Fig.2 shows Moil,1 as function of (00 -00o)/0 I, where 000 is estimation of 00(in these figures (00 -000)/0 ~ is denoted by (Th0-Yh00)/Thl), max of Me/3,j is denoted by MxMPf ). Left part of the figure corresponds to e" = 0.003, right part corresponds to 6" = 0.01. Note, that if we use the offered minimax decision function d = d(e" ,t)) for the inspection interval choice then expected value of fatigue failure probability is limited uniformly for all unknown parameters of fatigue crack growth model and we do not need to use a dubious compromise combination of required reliability and confidence probability.
REFERENCES Paramonov Yu.M.( 1992 ). Mathematical statistics application to the problem of aircraft fatigue life estimation and insurance (Russian).RAU: Riga. Paramonov Yu.M.(1995). Probability of fatigue crack detection in the case of Paris model. Transactions of Riga Aviation University Mechanical Department. (Russian).RAU: Riga. Vol. 1, pp.35-37. Yang J.N.(1980). Statistical Crack Growth in Durability and Damage Tolerant Analysis. In Proc.Int Conf. AIAA/ASME/ASCE/AHS 22-th Structures, Structural Dynamics and Materials. Vol. 1, pp.38-44.