Bayesan Approach for Fatgue Lfe Predcton from Feld Inspecton Dawn An, and Jooho Cho School of Aerospace & Mechancal Engneerng, Korea Aerospace Unversty skal@nate.com, jhcho@kau.ac.kr Nam H. Km, and Srram Pattabhraman Dept. of Mechancal & Aerospace Engneerng, Unversty of Florda nkm@ufl.edu, psrram@ufl.edu Abstract: In the desgn consderng fatgue lfe of mechancal components, uncertantes arsng from the materals and manufacturng processes should be taken nto account for ensurng relablty. Common practce n the desgn s to apply safety factor n conjuncton wth the numercal codes for evaluatng the lfetme. Ths approach, however, most lkely reles on the desgner's experence. Besdes, the predctons often are not n agreement wth the real observatons durng the actual use. In ths paper, a more dependable approach based on the Bayesan technque s proposed, whch ncorporates the feld falure data wth the pror knowledge to obtan the dstrbuton of the unknown parameters of the fatgue lfe. A matter of pror knowledge s also consdered snce the dstrbuton s nfluenced by t. Posteror predctve dstrbutons and assocated values are estmated afterwards, whch represents the degree of our belef of the lfe condtonal on the observed data. As more data are provded, the values wll be updated to more confdent nformaton. The results can be used n varous needs such as a rsk analyss, relablty based desgn optmzaton, mantenance schedulng, or valdaton of relablty analyss codes. In order to obtan the dstrbuton, Markov Chan Monte Carlo (MCMC) technque s employed, whch s a modern statstcal computatonal method whch draws effectvely the samples of the gven dstrbuton. Feld data of turbne components are exploted to llustrate our approach, whch counts as a regular nspecton of the number of faled blades n a turbne dsk. Keywords: Fatgue lfe, Pror dstrbuton, Posteror dstrbuton, Bayesan approach, Markov Chan Monte Carlo Technque, Feld Inspecton, Turbne blade. Introducton Performance of mechancal components undergoes a change by uncertantes such as envronmental effects, dmensonal tolerances, loadng condtons, materal propertes and mantenance processes. Fatgue lves of the components n partcular are sgnfcantly nfluenced even by small changes. In the desgn for fatgue lfe, t s not feasble to consder all the uncertantes of the relevant varables, snce most of them are not characterzed n the desgn phase. Analytcal predcton of fatgue lfe s therefore, often not n agreement wth the feld data. Common practce n the desgn s then to apply proper safety factor when evaluatng fatgue lfe. Ths approach, however, causes overdesgn or rsk of desgn, snce t reles on the desgner s experence. Recently, for more relable lfe predcton, the study usng feld data have been undertaken (Marahleh et al., ). Feld data can be helpful n predctng fatgue lfe that has uncertantes due to the by authors. Prnted n SINGAPORE. REC - Author
Dawn An, Jooho Cho, Nam H. Km, and Srram Pattabhraman unknown potental nputs. Ths approach can be dealt wth Bayesan technque whch ncorporates the feld falure data wth the pror knowledge to obtan the dstrbuton of the unknown parameters of the fatgue lfe (Km et al., 9). As more data are provded, the values wll be updated to more confdent nformaton. In ths paper, Markov Chan Monte Carlo (MCMC) technque s employed as an effcent means to draw samples of gven dstrbuton (Andreu et al., ). Consequently, the dstrbuton of the unknown parameters of the fatgue lfe s obtaned n lght of the feld data collected from the nspecton of turbne blades. Subsequently, fatgue lfe of turbne blades s predcted a based on the drawn samples.. Bayesan Technque for Lfetme Predcton Bayesan technque s employed to update lfetme predcton usng analytcal model and feld data, whch s based on Bayes rule and defned as (Gelman et al., ) : f D L D f () L D s the lkelhood of observed data D t, y, n where condtonal on the gven model parameters (n the case of normal dstrbuton, are the mean and standard devaton ), f s the pror dstrbuton of, and f D s dstrbuton of condtonal on the D. The procedure to obtan dstrbuton f D s outlned as follows. In Eq.(), lkelhood s just a multplcaton of each bnomal PDF, gven as N D Bn, f t where L y n p () f p f t dt () lfe and N, t, y, n are gven n the Table I whch s a feld data for nspected turbne blades. In the case of the frst nspecton, we found falures out of blades at the tme of hours and N s whch s set of f t denotes PDF of lfetme of turbne blade, whch can be assumed to a nspected data. In Eq.(), lfe certan predetermned model, whch s n ths paper, a normal dstrbuton or webull dstrbuton. Consequently, the PDF n Eq. () s obtaned by multplcatyng Eq. () and the pror PDF f. Table I. Feld data for nspected turbne blades Engne Hours( t ) Faled( y )/Total( n ) Engne Hours( t ) Faled( y )/Total( n ) / / / 9 / / / 9 / / / / / ( N ) 7 / 7 / REC - Author
Bayesan Approach for Fatgue Lfe Predcton from Feld Inspecton In ths paper, the PDF of lfetme f lfe t and the pror PDF f are assumed as the followng cases: case. flfe t : normal dst. and f : non-nformatve pror case. flfe t : normal dst. and f, N 9,9 / N 79,79 / () case. flfe t : normal dst. and f, N, / () case. flfe t, m : Webull dst. and f, m LogN.,. LogN.7,. (7) () In terms of the lkelhood, normal and Webull dstrbuton are consdered. In the model, the assocated model parameters are n the case of normal and,m n the case of Webull dstrbuton, respectvely. These are taken to be unknown and are estmated usng the nspected data. In the Webull dstrbuton,,m denote scale and shape parameter, respectvely. In terms of pror, four cases are consdered to examne the effects of pror knowledge on the PDF of model parameters and predcted lfetme of turbne blades. If there s specfc pror nformaton, models lke the cases,, can be provded. Otherwse, nonnformatve pror lke the case s used. The results of case where the lkelhood s normal dstrbuton and non-nformatve pror are shown n Fgure. In Fgure (a), the contours of jont PDF of the unknown parameters are plotted. In these fgures, the updated pror and the lkelhood are obtaned from the dstrbuton prevously obtaned and the nspecton data, respectvely. The dstrbuton s obtaned by multplyng the pror and lkelhood, and s used n the next updatng step as the pror dstrbuton. In Fgure (b) and (c), PDF are plotted agan n the form of contour and D shape respectvely. It s shown that as more data are added, the locaton and range of, moves and narrows down to converge to a certan pont. The results ndcate our knowledge on the unknown parameters based on the feld nspecton. usng only thest data usng st~th data usng st~7th data usng st~th data updated pror updated pror updated pror usng st~th data updated pror (a) lkelhood functon and updated pror and dstrbuton usng st~th data usng st~7th data usng st~th data usng st~th data usng only thest data (b) contour of jont PDF REC - Author
Dawn An, Jooho Cho, Nam H. Km, and Srram Pattabhraman Fgure. Updated PDF of case. (c) -D plot of jont PDF The results of case ~ are shown n Fgure. The results of case where the lkelhood s stll normal dstrbuton but wth normally dstrbuted prors are shown n Fgure (a) and (b). The fgure (b) shows the convergent behavor on the two parameters as we collect more and more data. The results of case where the lkelhood s normal and the pror for the sgma s changed to ch-square dstrbuton, whch s more reasonable assumpton due to the non-negatvty, are shown n Fgure (c) and (d). The results of case where the lkelhood s Webull and the prors are lognormal are shown n Fgure (e) and (f). The Fgure (f) shows agan the convergent behavor. Ths case s the most reasonable because the Webull dstrbuton s the best model for the lfetme....... pror usng only thest data...... dst. contour st~th data st~7th data st~th data only st data st~th data.... (a) the results usng only the st data of case (b) updated PDF contours of case.......... usng only thest data dst. contour pror...... (c) the results usng only the st data of case (d) updated PDF contours of case.......... st~th data st~7th data st~th data st~th data only st data REC - Author
(scale parameter) (scale parameter) (scale parameter) Bayesan Approach for Fatgue Lfe Predcton from Feld Inspecton usng only thest data dst. contour........ pror 7 9 7 9 m(shape parameter) m(shape parameter) (e) the results usng only the st data of case (f) updated PDF contours of case Fgure. Updated PDF of case ~......... st~th data st~th data st~7th data st~th data only st data The fnal updated results of all cases are shown n Fgure. The Fgure (a) and (b) are the results of and of the normal dstrbuton and m and of the webull dstrbuton, respectvely. Case n Fgure (a) and the jont PDF on the rght n Fgure (b) are the results of the non-nformatve prors. As was expected, the results are much wder than the others due to the non-nformaton of the pror. If there s specfc pror nformaton, the precson of the dstrbuton s ncreased. dst. contour (lkel-n)..... case case case 7 (a) lkelhood s normal dstrbuton Fgure. Fnal updated results of all cases. dst. contour (lkel-w)........ case. usng pror non-nformatve (n case ) 7 9 m(shape parameter) (b) lkelhood s webull dstrbuton. Posteror Dstrbuton usng MCMC.. MCMC SIMULATION Once the expresson for PDF s avalable, one can proceed to sample from the PDF. Prmtve way s to compute the values at a grd of ponts after dentfyng the effectve range, and sample by nverse CDF method. The method, however, has several drawbacks such as the dffculty fndng correct locaton and scale of the grd ponts, spacng of the grd, and so on. MCMC smulaton s an effectve soluton n ths case (Andreu et al., ). The Metropols-Hastngs (M-H) algorthm s typcal method of MCMC, whch s gven n the case of a sngle parameter by the followng procedure: REC - Author
Probablty Densty Functon(PDF) Cumulatve Dstrbuton Functon(CDF) Dawn An, Jooho Cho, Nam H. Km, and Srram Pattabhraman (). Intalse x.. For to N Sample u ~ U. else, * * Sample x ~ q x x. * * * * p x q x x * f u A x, x mn, p x q x x x x x x () where x s the ntal value of an unknown parameter to estmate, N s the number of teratons or samples, U s the unform dstrbuton, p x s the PDF (target PDF), and q x s an arbtrary chosen proposal dstrbuton. A unform dstrbuton s used n ths study for the sake of smplcty. Then, * q x x and * q x x become constant, and s shown the samplng result of fcttous PDF gven as q x can be gnored. As an example of MCMC, n Fgure.exp..7exp. p x x x (9) Wth only, teratons, the samplng result follows the dstrbuton qute well........ Target MCMC.9..7.... Target MCMC.... - - random varable, X (a) PDF Fgure. Results of MCMC smulaton. - - random varable, X (b) CDF.. POSTERIOR DISTRIBUTION The jont PDF of the unknown parameters of the fatgue lfe usng only the frst data s shown n Fgure, whch represents the degree of belef on the concerned parameters n the form of PDF. The jont PDF usng grd method as well as MCMC samplng are shown n Fgure (a) and (b), respectvely. REC - Author
Bayesan Approach for Fatgue Lfe Predcton from Feld Inspecton In Table II, statstcal moments by the two methods are compared. As shown n the table, the two methods agree qute closely but MCMC used () samples, whereas grd used () samples. (a) usng grd method ( grd) Fgure. Jont PDF of case. (b) usng MCMC ( teratons) Table II. Statstcal moments by the two methods E E E E E Grd.7... -. MCMC.9.7.7... Posteror Predctve Dstrbuton The drawn samples of the parameters obtaned n secton are used for predctng the falure probablty. The ntally gven pror and fnal updated predctve dstrbutons of fatgue lfe are shown n Fgure. Due to the uncertantes of the model parameters, we have a CDF n a confdence bands. In the fgure, dotted and sold curve denote ntal and fnal CDF respectvely. Black, red and magenta colors denote medan, % lower bound and 9% upper bound of the CDF at each stage respectvely. In order to accommodate safety, t s advsed to take % lower bound of the CDF whch s colored as red. In Fgure (a), whch s the result of case, CDF of fatgue lfe after the fnal update s located to the left of the prevous one, whch reflects the feld data depcted as blue star n more sutable manner. In Fgure (b) 와 (c), however, the results are found dfferently,.e., the CDF after the fnal update s located to the rght of the ntal one n spte of added feld data. The reason mght be attrbuted to the wrong nformaton of the prors. In Fgure (d), the CDF of fatgue lfe after the fnal update s located to the left of the ntal one below the B lfe, whch s the part of our nterests. In Fgure (e), the CDFs of the four cases at % lower bound are plotted together. From the fgure, t s found that the CDF of the case s the most reasonable one because the CDF s overall located to the left of the feld data. From the results, confdence nterval of the % lfe and % lfe, whch are also called B lfe and B lfe, are gven n Table III. In the cases and, whch employed normal model, we have negatve values for the lfe, whch was due to the wrong assumpton of the model. On the other hand, the case s the reasonable REC - Author
CDF CDF CDF CDF CDF Dawn An, Jooho Cho, Nam H. Km, and Srram Pattabhraman model as mentoned before, whch shows the postve values. In case that we are gnorant of the type of the dstrbuton, t s advsed to choose the most conservatve one, whch s the case n ths study..9 fnal.9 ntal.9 ntal....7.7.7... st~9th data... fnal... fnal. medan. % lower bound. 9% upper bound feld data 7 Operaton Hours x. medan. % lower bound. 9% upper bound feld data 7 Operaton Hours x. medan. % lower bound. 9% upper bound feld data 7 Operaton Hours x (a) case (b) case (c) case.9. ntal.9..7. fnal.7..... medan. % lower bound. 9% upper bound feld data 7 Operaton Hours x (d) case Fgure. Fnal updated dstrbuton of fatgue lfe.. case. case. case. case feld data 7 Operaton Hours x (e) merged results of all cases Table III. Confdence nterval of fatgue lfe case case case case % lower 9% upper % lower 9% upper % lower 9% upper % lower 9% upper % P f -, -,99,9, -,77-7 % P f,7, 9,9,7,9 7,,7,7. Conclusons In ths paper, a Bayesan updatng technque s presented, whch ncorporates the statstcal predcton wth feld data. By usng MCMC smulaton, samples of model parameters (, or m, ) are drawn effectvely, whch are parameters of the fatgue lfe dstrbuton. After gettng samples for jont PDF of, the fatgue lfe predcton results are obtaned, whch have a CDF n a confdence bands due to the uncertantes of the model parameters. If there s specfc pror nformaton of model parameters, the REC - Author
Bayesan Approach for Fatgue Lfe Predcton from Feld Inspecton precson of the dstrbuton s ncreased. Cauton should be pad, however, that f the pror nformaton s wrong, the result s worse than the one wth non-nformaton as was evdenced n the case study. By usng the adequate pror wth proved accuracy, relablty s mproved as the number of feld data ncrease. In case that the type of the dstrbuton of the lkelhood s not known a pror, t s advsed to choose the most conservatve one after examnng several canddates as was found n ths study. Acknowledgements Ths research was supported by Basc Scence Research Program Through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Educaton, Scence and Technology (-- and 9- ) References Andreu, C., N. de Fretas, A. Doucet, and M. Jordan. An ntroducton to MCMC for Machne Learnng. Machne Learnng, :-,. Gelman, A., J. B. Carlm, H. S. Stern, and D. B. Rubn. Bayesan Data Analyss (second ed.). New York, Chapman & Hall/CRC,. Km, N. H., S. Pattabhraman, and L.A. Houck III. Bayesan Technque for Incorporatng Feld Experence nto Analytcal Model for Lfe Predctons. ASME Internatonal Mechancal Engneerng Congress and Exposton, Lake Buena Vsta, Florda, November, 9. Marahleh, G., A. R. I. Kheder, and H. F. Hamad. Creep-Lfe Predcton of Servce-Exposed Turbne Blades. Materals Scence, (),. REC - Author