Fatigue Lifetime Assessment of Aircraft Engine Disc via Multi-source Information Fusion Abstract: Keywords: PACS (2010).

Transcription

1 DOI /tjj Int. J. Turo Jet-ngines 2014; 31(2): Hong-Zong Huang* Ping-Liang Cui Weiwen Peng Hui-Ying Gao and Hai-Kun Wang Fatigue Lifetime Assessment of Aircraft ngine Disc via Multi-source Information Fusion Astract: Fatigue lifetime assessment for aircraft engine discs is an important issue for te operation and ealt management of aircraft engines. Due to te lack of field test data traditional metods can ardly meet te requirements of fatigue lifetime assessment of aircraft engine discs. By comining a multi-source information fusion metod wit a Bayesian inference tecnique tis paper develops a practical approac for fatigue lifetime assessment of aircraft engine discs. Sujective information and istorical data are comined coerently wit te sparse test data to generate a credile fatigue lifetime assessment of aircraft engine discs. Metods for quantifying sujective information cecking different experts information and fusing multiple prior distriutions are presented to facilitate te implementation of fatigue lifetime assessment. An illustrative example is presented to demonstrate te procedures and te implication of te proposed metod. Keywords: fatigue lifetime assessment aircraft engine disc Bayesian inference multi-source information fusion PACS (2010) Tt *Corresponding autor: Hong-Zong Huang: Scool of Mecanical lectronic and Industrial ngineering University of lectronic Science and Tecnology of Cina Cengdu Sicuan Cina. -mail: zuang@uestc.edu.cn Ping-Liang Cui Weiwen Peng Hui-Ying Gao Hai-Kun Wang: Scool of Mecanical lectronic and Industrial ngineering University of lectronic Science and Tecnology of Cina Cengdu Sicuan Cina 1 Introduction Due to te fast development of modern aircraft industry and te continual growt of market demand aircrafts are putting on te market in an ever-increasing pace. Manufacturers and companies are under great pressure to deliver aircrafts wit ig safety and reliaility under limited time. Moreover te evolving of customer demands as laid a great imperative for manufacturers and companies to develop new types of aircrafts efficiently. Increased expectation wit lower cost and ig reliaility as ecome critical issue in aircraft industry. Aircraft engine discs as een recognized as one of te critical components tat contriute to te failure of aircrafts. Fatigue lifetime assessment of aircraft engine discs as consequently ecome a key issue ot for te development of aircrafts wit ig reliaility in limited time and for te operation of aircrafts wit ig safety under limited cost. Commonly fatigue lifetime assessment of aircraft engine discs is carried out ased on te test data gatered from relevant reliaility tests. However due to te ig reliaility and cost of te aircraft engine discs it is ard to implement reliaility test of te aircraft engine discs wit large sample size and long test time. As a result common metods for fatigue lifetime assessment wic are mainly ased on te aundance of test data can ardly lead to a credile result. Fatigue lifetime assessment of aircraft engine discs is consequently encountered te difficulty introduced y small sample size and sparse failure data. In fact oter tan te data gatered from reliaility test various types of reliaility information can e gatered and integrated for te fatigue lifetime assessment of te aircraft engine discs. Tanks to te continue evolving and te gradual development of te aircraft engines tere are plenty of sujective information and istorical data for te reliaility analysis of aircraft engines. In detail for te aircraft engine discs sujective information can e gatered from experts wo design te discs and from te engineers wo repair te engines. Moreover istorical data can e otained from test data of similar discs and from te maintenance records of similar engines. Tese types of reliaility information can e incorporated and comined wit te reliaility test data to mitigate te difficulty discussed aove. Accordingly a coerent metod is needed to implement te sujective information quantification and multi-source information fusion. To andle te prolems introduced y small sample size and spare test data Bayesian metod as een frequently used in fatigue lifetime assessment. A coerent framework can e constructed for te incorporation of multiple sources information using Bayesian metod [1 2]. Sujective information and istorical data can e

2 168 H.Z. Huang et al. Fatigue Lifetime Assessment of Aircraft ngine Disc quantified and incorporated troug te proper derivation of prior distriution. Te information contained in te sparse test data is descried y te relevant likeliood function. Tese various types of information are ten e comined troug te Bayesian formulation. Te posterior distriution is otained ased on te prior distriutions and likeliood function. Various studies ave discussed te reliaility analysis wit different data types and reliaility information using Bayesian metod [3 5]. Peng et al. [6] introduced a framework for reliaility analysis of multicomponent system y incorporating multilevel eterogeneous data suc as pass/fail data lifetime data and degradation data. Reese et al. [7] proposed a Bayesian framework for lifetime assessment of complex system wit multiple level lifetime data. Wang et al. [8] introduced a metod for reliaility assessment and prediction wit evolving insufficient and sujective data set ased on te Bayesian teory. Huang et al. [9] presented a framework for reliaility assessment using Bayesian metod y considering te situation of fuzzy lifetime data. Liu [10] applied various qualitative and quantitative metods to syntesize multiple sources of information. Liu [11] discussed information pretreatment including collecting and converting prior information and metods of distriution type test of prior information. Based on te literature review presented aove one practical way to andle te issues introduce y te fatigue lifetime assessment of aircraft engine discs is to construct a coerent framework y incorporating te Bayesian teory. In tis paper a teoretically sound model and a practically usale metod are uild up ased on te Bayesian teory for te fatigue lifetime assessment of te aircraft engine discs. Tree critical expects are considered in te proposed metod i.e. (1) quantification of sujective information gatered from multiple experts were a proaility encoded metod is used (2) cecking of prior consistency for information otained from different experts were a common recognition extent is defined (3) fusion of multiple prior distriution otained from different sources were a linear weigted comining metod is used. Te proposed metod is demonstrated wit a step-y-step implication to te fatigue lifetime assessment of aircraft engine discs. Te remainder of tis paper is presented as follows. Section 2 riefly introduces te multi-source information fusion. Section 3 presents te Bayesian metod for te fatigue lifetime assessment wit specific detail on te construction of Bayesian model te discussion of sujective information quantification multiple experts information cecking and multiple prior distriutions fusion. Te proposed metod is ten demonstrated troug a fatigue lifetime reliaility assessment of an aircraft engine disc in Section 4. We ten conclude te paper in Section 5 wit some remarks on te future researc. 2 Multi-source information fusion Multi-source information fusion metod was originated from multi-sensor fusion metod. Te fundamental idea of multi-source information fusion is to comine te information gatered from various sources were te issues of information quantification coerency cecking and distriution fusion are andled properly [4]. Te implication of multi-source information fusion in fatigue lifetime assessment is aimed to mitigate te insufficient of spare reliaility test data y incorporating information otained from various sources as discussed in te introduction part. In order to otain more reliale and accurate information tan a single source of information it makes use of different sources of information troug a teoretically sound algoritm. Researcers ave done a significant amount of work to apply te information fusion metod to analyze te reliaility and lifetime of aircraft engine discs [12 14]. Zang [15] applied information fusion teory to perform accuracy evaluation and reliaility assessment of te weapon systems and proposed an effective solution to te reliaility information fusion for similar system in different environments and different conditions. Feng [16] made a compreensive analysis on reliaility information fusion metod applied to a complex system wit small samples and developed te weigted fusion metod y syntesizing multi-source prior information. To construct a coerent framework for te fatigue lifetime assessment of aircraft engine discs using te multisource information fusion te following aspects need to e andled properly. 1. A practical model is cosen for te modeling of fatigue lifetime of aircraft engine disc. It is cosen ased on te fitness of te reality of te disc y te suggestion of experts in tat domain or troug te goodnessof-fit cecking of test data. Moreover to facilitate te implementation of multi-source information fusion a parametric model is preferale tan a non-parametric model especially for te one wit practical meaning of model parameters suc as Weiull model [17]. 2. A usale metod is needed for te quantification of sujective information gatered from experts. Since te sujective information can e presented in various forms a metod is needed to unify te procedure for

3 H.Z. Huang et al. Fatigue Lifetime Assessment of Aircraft ngine Disc 169 sujective information derivation [18]. Moreover proaility distriution for te model parameters cosen aove can e properly derived using tis metod. 3. A proper metod is required for te consistency cecking of information gatered from multiple experts. Te information otained from different experts can e consistent or conflict wit eac oter. Moreover some information from particular experts is more valuale tan te oters. As a result a proper metod sould e introduced to andle te issue of consistency cecking. 4. An efficient metod is needed for te fusion of multiple proaility distriutions. Te ultimate target for te implication of multi-source information fusion for te fatigue lifetime assessment is to otain te joint proaility distriution for model parameters cosen aove. In detail different proaility distriutions need to e compressed into a joint proaility distriution. A criteria needs to e introduced for te evaluation of te importance of te distriutions for te joint distriution. Moreover te efficiency of generation samples from tis joint distriution needs to e guaranteed for te calculation of Bayesian inference. Based on te discussion aout multiple information fusion igligted aove te proposed metod for te fatigue lifetime assessment of aircraft engine discs is presented step y step in te following sections. 3 Reliaility assessment of aircraft engine discs 3.1 Bayesian model Bayesian metod is used to integrate te sujective information and istorical data wit te sparse lifetime data wic come from reliaility test wit limited sample size and test time. Te fundamental principle of Bayesian metod can e descried as P q T ( q T) q lt q( T ) lt q( T ) π q q π q q dq q q were q are parameters of te model cosen for te fatigue lifetime modeling of an aircraft engine disc. T denotes te sparse lifetime data otained from reliaility test of te aircraft engine disc. πq ( q ) is te prior distriution for model parameters. Te multi-source prior information are (1) quantified and formulated as tis prior distriution. Te information contained in te sparse lifetime T is delivered troug te likeliood function lt q ( T q ) wic is otained ased on te proaility density function of te cosen fatigue lifetime model. All of te information is integrated and presented in te joint posterior distriution of model parameters Pq T ( q T). Te fatigue lifetime assessment and posterior analysis are ased on tis joint posterior distriution. For te fatigue lifetime of an aircraft engine disc a two-parameter Weiull distriution is generally cosen ased on experts testimony and istorical experience. Te corresponding Bayesian model is given as: P α t ( α T) πα ( α ) Lt α ( T α ) α Lt α ( T ) α α π α α dαd 1 n t i πα( α) ti exp i 1 α α α 1 n ti πα( α) ti exp dαd i 1 α α α were P ( ) α t α T is joint posterior distriution of model parameters. Lt α ( T α ) is te likeliood for te collected lifetime data T. πα ( α ) is te prior distriution wic is generated from istorical data and sujective information. 3.2 Quantitative metod for sujective information Derivation of prior distriution is one of te key questions in a Bayesian analysis of lifetime data. In tis paper prior distriution is derived from information of relevant experts and istorical data of te same product lines. Primary prior information for an aircraft engine disc is expert experience and knowledge. Sujective information and opinions from expert can e used to increase te sample size wic will e very elpful for improving te accuracy of reliaility assessment in te small sample situation. Te prior distriution is a random variale wit a certain distriution in Bayesian metod wile actual sujective data is igly personal representation. It is our priority to quantify sujective information. Proaility encoding metod is simple and manipulale [19]. It is a process to otain statistical information in te form of proaility density functions discrete proaility values (2)

4 170 H.Z. Huang et al. Fatigue Lifetime Assessment of Aircraft ngine Disc or occurrence rates. Spetzler et al. [20] provide an overview of proaility encoding metods. Te inverse transformation of a parametric cumulative distriution function (CDF) can fully fit te process of canging te encoded survey data into parameter data of te relative prior distriution [19]. Te inverse transformation of normal cumulative proaility function can e descried as follows: Vα α α α P f αm ; σ dα (3) were P α and V α are otained from experts and σ α is known and te only unknown yper-parameter m α can e otained y te inverse transformation of te CDF. Tere are some asic steps to implement te proaility encoding metod [20]. Step 1: Specify uncertain quantities interested. Step 2: Coose proaility encoding metods. Step 3: Interview sujects and collect sujective data sets. Step 4: Coose suitale Bayesian updating models. Step 5: Transfer te sujective data to model parameter data y inverse CDF analysis. Step 6: Update model parameters y integrating model parameter data using te Bayesian updating tecnique. An illustration of te proaility encoding metod is presented in Section Multiple expert information cecking In te actual assessment process more tan one expertise knowledge can e collected [9]. xpert information is quantified as prior proaility distriution troug te proaility encoding metod. Te entire distriution will e gotten y fusing all prior distriutions. Since simple weigted metod cannot meet requirements a more suitale metod needs to e found. Assuming two prior proaility distriutions otained from two experts is f a (x) and f (x) we can define a common recognition extent (CR) as te fusion of two expert information: CR f ( a f) { min a } { max a } f x f x dx f x f x dx were te numerator represents te intersection of te area in wic two proaility distriutions are surrounded (4) y te X-axis. Te denominator represents te union of te area in wic two proaility distriutions are surrounded y te X-axis. Te ratio is te degree of common recognition extent. If fatigue life distriution function transformed from expert information i is assumed to e fi( q ) te matrix CR can e otained according to te degree of coincidence etween te expert information. 1 CR 1n CR CRn1 1 n ( i ) ij ( 12 ) CR CR i n i 1 i 1 j i ( i ) ( i 12 n ) i CR wi n CR were CR( i ) means te sum of coincidence degree tat expert I wit respect to te oter N-1 experts. After normalization we can calculate te weigt w i wic sows te importance of eac expertise. If w i 0.10 it means tat expert I as a large conflict wit oter experts and sould e removed. In particular case a standard is required according to te requirements. 3.4 Multiple prior distriution fusion In addition to te information gatered from te expert knowledge tere is also istorical information from te similar disc. We can use data from te similar disc to otain prior distriution troug Bayesian model wit non-informative priors. Different prior distriutions ave different goodness of fit to te experimental data [21]. How to effectively reflect te prior distriution tat fits field test is considered in te paper. Te reference [22] provides a metod to comine two groups of prior distriutions. It is one of te most widely used approaces for comining proaility distriution. It is given as: were f fc + ( 1 ) C (5) f q lf q l f q (6) q is prior distriution otained from expertise. q is prior distriution otained from similar disc. l and 1 l are te weigts assigned for te corresponding two priors and f ( q ) represents te comined proaility distriution.

5 H.Z. Huang et al. Fatigue Lifetime Assessment of Aircraft ngine Disc 171 Te idea of tis metod is a measure of prior distriution fitting to field data. Te prior distriution tat fits te field data more closely will e assigned a iger weigt in te linear pooling. Te comined distriution inerits major elief of two distriutions and maintains caracteristics of relevant field data. 4 Case study 4.1 Prolem definition In order to demonstrate te proposed metod te fatigue lifetime of a type of aircraft engine disc is estimated ased on multi-source information fusion. Te fatigue lifetime model is assumed to follow te two-parameter Weiull distriution. Te field data is listed in Tale 1. Tale 1: Field fatigue lifetime data of aircraft engine disc 4.2 Quantifying te sujective information Te interested parameters are sape parameter and scale parameter wic are assumed to e statistically independent and to follow te normal distriution: ( ) N ( ) N : N m σ σ 0.1 m : : N m σ σ 20 m : Field fatigue lifetime data (cycles) As sujective information is collected and translated tese distriutions can e updated. Te uncertain quantities are te mean values of te parameters and. Te standard deviations are fixed. Proaility encoding metod is used in tis example and two experts are interviewed regarding te values ( V V ) of te scale parameter and sape parameter and corresponding proailities ( P P ). Te survey results are listed in Tale 2. Te corresponding mean value of eac survey point can e otained y using te inverse CDF transformation metod as depicted in q. (3). By taking te first data set of scale parameter [ ] from xpert 1 as example te computational process is sown as follows: (7) Tale 2: Results of te survey and inverse CDF analysis V ( m ; 20) P f d ( m ) exp d π were m can e acquired y solving te q. (8) ere it equals to Te oter results are given in Tale 2. To expert I if prior distriution of m is assumed to follow te normal distriution N( ). Treating encoded values in Tale 2 as field data togeter wit te assumed prior posterior distriution for m is updated to normal distriution N( ) using Bayesian metod. Similarly we can otain all posterior distriutions of te sujective information as sown in q. (9): ( ) N ( ) N ( ) N ( ) N : N m σ σ 0.1 m : : N m σ σ 20 m : : N m σ σ 0.1 m : : N m σ σ 20 m : Terefore te proaility density function of prior distriutions derived from expert s sujective knowledge is: f 1 f 2 V P m V P m xpert xpert ( t) ( t) 5.27 t t exp t t exp (8) (9) (10)

6 172 H.Z. Huang et al. Fatigue Lifetime Assessment of Aircraft ngine Disc Fig. 1: Proaility distriution function of priori distriution Clearly te proaility distriution function of prior distriution is derived and sown in Fig. 1. It sould e noted tat parameters of prior distriution deprived from different expert knowledge are different wic means different experts ave different perspectives to fatigue lifetime distriution. Teir common recognition extent (CR) needs to e found wic can e calculated y q. (4) as follows: CR f ( f ) { min 1 2 } { max 1 2 } f t f t dt f t f t dt 1 2 (11) By using computational formula in q. (5) te weigts w1 w2 0.5 can e calculated wic sows te importance of eac expertise. Terefore te fusion of multisource expert information priori proaility density parameter and is : N : N (12) Tale 3: Historical fatigue lifetime data from similar types of aircraft engine discs To incorporate distriution sown in q. (12) and q. (13) te multiple prior distriution fusion metod is used. Te Bayesian goodness-of-fit for tese two types of prior distriutions are otained using te metod mentioned in [18]. Te information fusion factor in tis stage is ten otained as B Historical fatigue lifetime data (cycles) B p B C l B + B p p C p p Finally te prior distriution is otained as follows: ( ) ( ) ( C ) (14) π π π (15) 4.3 Multiple prior distriution fusion Historical information from te similar disc is sown in Tale 3. Te data of te similar disc is used to otain te prior distriution troug te non-informative prior Bayesian metod. C C : N : N (13) 4.4 Reliaility assessment using te Bayesian metod Te field data is given in Tale 1 and te prior distriution is sown in q. (15). Using Bayesian metod te estimated results ased on te prior distriution and te field data are listed in Tale 4. Terefore fatigue lifetime model of aircraft engine disc is:

7 H.Z. Huang et al. Fatigue Lifetime Assessment of Aircraft ngine Disc 173 Tale 4: stimation results of te Bayesian metod Parameter Mean SD 2.5% 97.5% t F( t ) 1 exp (16) Te fatigue lifetime can e gotten using aove model wit certain reliaility degree. 5 Conclusions Tis paper proposed a fatigue lifetime assessment metod ased on te multi-source information fusion. Our study extends te application of Bayesian inference teory to fatigue lifetime assessment for aircraft engine discs. Firstly to make te fatigue lifetime assessment more reliale te proaility encoding metod is used to quantify te sujective information. A common recognition extent is used to get a multiple prior distriution. Secondly y fusing te actual field data and oter availale information te difficulty introduced y small sample size for lifetime analysis is mitigated to a certain extent. Finally te Bayesian goodness-of-fit metod is used to confirm te consistence degree of field data and expert knowledge wic effectively mitigate te sujectivity of fatigue lifetime estimation. However tere are some works need to e andled properly in our future work suc as a prior distriution considering more tan two expertise te uncertainty quantifying to improve te degree of accuracy. Tis researc was supported y te National Natural Science Foundation of Cina under te contract numer and Specialized Researc Fund for te Doctoral Program of Higer ducation of Cina under te contract numer Received: Novemer Accepted: Decemer References [1] M.S. Hamada H.F. Martz C.H. Reese et al. Bayesian Reliaility. Springer New York [2] J.O. Berger. Statistical Decision Teory and Bayesian Analysis. Springer New York [3] N.Z. Geraeel M.A. Lawley R. Li et al. Residual-life distriutions from component degradation signals: a Bayesian approac. II Transactions (6): [4] A.G. Wilson L.A. McNamara G.D. Wilson et al. Information integration for complex systems. Reliaility ngineering and System Safety (1): [5] N. Cen K.L. Tsui. Condition monitoring and remaining useful life prediction using degradation signals: revisited. II Transactions (9): [6] W.W. Peng H.Z. Huang M. Xie et al. A Bayesian approac for system reliaility analysis wit multilevel pass-fail lifetime and degradation data sets. I Transactions on Reliaility (3): [7] R.C. Sane A.G. Wilson J.Q. Guo et al. A Bayesian model for integrating multiple sources of lifetime information in system-reliaility assessments. Journal of quality tecnology (2): [8] P. Wang B.D. Youn Z. Xi et al. Bayesian reliaility analysis wit evolving insufficient and sujective data sets. Journal of Mecanical Design (11): [9] H.Z. Huang M.J. Zuo Z.Q. Sun. Bayesian reliaility analysis for fuzzy lifetime data. Fuzzy Sets and Systems (12): [10] X.R. Liu. Researc on metod of aero-engine disc reliaility estimation. Master Dissertation Nanjing University of Aeronautics and Astronautics [11] H. Liu. Researc on metods of reliaility assessment wit small sample test on te asis of Bayes teory. Master Dissertation National University of Defense Tecnology [12] S.P. Zu H.Z. Huang Z.L. Wang. Fatigue life estimation considering damaging and strengtening of low amplitude loads under different load sequences using fuzzy sets approac. International Journal of Damage Mecanics (6): [13] S.P. Zu H.Z. Huang V. Ontiveros et al. Proailistic low cycle fatigue life prediction using an energy-ased damage parameter and accounting for model uncertainty. International Journal of Damage Mecanics (8): [14] S.P. Zu H.Z. Huang L.P. He et al. A generalized energyased fatigue-creep damage parameter for life prediction of turine disk alloys. ngineering Fracture Mecanics : [15] X.P. Zang. Researces on te small sample statistical inference and fusion teory and its application to te assessment of weapon system. P.D. Dissertation National University of Defense Tecnology [16] J. Feng. Researc on metods and applications of reliaility information fusion of complex system wit small sample test. P.D. Dissertation National University of Defense Tecnology [17] D.N.P. Murty M. Xie R. Jiang. Weiull Models. Jon Wiley & Sons [18] A. O Hagan C.. Buck A. Daneska et al. Uncertain Judgements: liciting xperts Proailities. Jon Wiley & Sons New York [19] H.Q. Li R. Yuan W.W. Peng et al. Bayesian inference of Weiull distriution ased on proaility encoding metod. Quality Reliaility Risk Maintenance and Safety ngineering Conference 2011:

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