ANALYSIS OF THE AIRCRAFT OPERATION IN THE CONTEXT OF SAFETY AND EFFECTIVENESS

Transcription

1 Proceedings of he 6h Inernaional Conference on Mechanics and Maerials in Design, Ediors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-3 July 215 PAPER REF: 5466 ANALYSIS OF THE AIRCRAFT OPERATION IN THE CONTEXT OF SAFETY AND EFFECTIVENESS Jozef Zurek 1(*), Anoni Jankowski 1, Jan Rajchel 2 1 Air Force Insiue of Technology, Warszawa, Poland 2 Polish Air Force Academy, Deblin, Poland (*) jozef.zurek@iwl.pl ABSTRACT The paper presens a mehod for evaluaing he echnical objec submerged in a logisical sysem from he poin of view of susainabiliy, availabiliy and uilizaion. These facors deermine he effeciveness of he objec. The probabilisic models of reliabiliy heory and semi-markov process were used o he evaluaion. An example of a echnical objec in he paper is he airplane. The operaion of he aircraf consiss in usage of is funcioning resource accumulaed during he manufacuring process and periodic reproducion of his resource. The decisive facor of he effeciveness of he aircraf operaion process is he readiness o perform is airborne asks and he exen usage of aircraf during heir implemenaion. An imporan parameer is he damage sream parameer for serviceable componens. The basic informaion is he answer, wheher i falls wihin he olerance lane. To calculae he limi values of he damage sream parameer, he relaionship beween he upime probabiliy densiy funcion and damage sream parameer was used. The aricle gives an example of he damage sream parameer of aging elemens. In he readiness analysis, he semi-markov exploiaion process model was brough o hree saes incidence marix. Keywords: aircraf, aircraf operaion, reliabiliy heory, he semi-markov process, secure susainabiliy, readiness index. INTRODUCTION The aircraf, operaed in he logisic sysem is characerized by a specific funcion; a limied usage abiliy; mainenance, maerial and energy; a finie service life (Zurek, Ziolkowski, 25). The operaion of he aircraf is o use he funcioning resource accumulaed during he manufacuring process and he periodic reproducion of his resource, in order o preserve he abiliy of he vessel o coninue operaing. The safey of aircraf depends iner alia on he inensiy of he processes of desrucion of individual componens, funcional circuis and sysems generaing failures, including he incidence of heir damage(woropay, Zurek, 23). The aircraf, operaed in he logisic sysem is characerized, like every echnical objec, by cerain general characerisics, namely: has a specific funcion (a se of asks o be performed); has a limied usage abiliy (limied exploiaion poenial); has cerain mainenance, maerial and energy, informaion and oher needs; has a finie service life. The operaion of he aircraf is o use he funcioning resource accumulaed during he manufacuring process and he periodic reproducion of his resource, in order o preserve he abiliy of he vessel o coninue operaing. For simpliciy, following he airborne vessel will -193-

2 Track_B Tesing and Diagnosic be named as he aircraf (including in his concep he oher means of ranspor). An imporan feaure of he airplane life qualiy is operaing safey and efficiency. The complexiy of issues of safey and effeciveness, as well as he diversiy of definiions, require an individual approach o each ype of sysem and o he purposes for which effeciveness measures are deermined. The safey of aircraf depends iner alia on he inensiy of he processes of desrucion of individual componens, funcional circuis and sysems generaing failures, including he incidence of heir damage. Facors deermining he efficiency of aircraf operaion in he logisic sysem can be: readiness o perform he airborne asks; degree of an aircraf uilizaion during he realisaion of asks. SAFE SERVICE LIFE ANALYSIS METHOD For he purposes of he analysis, he elemens of machine srucure, funcional sysems and echnical sysems can be divided by he damage effecs. Due o he division, we spli as generaing failures and damaging wihou consequences in he considered ime horizon. Devices ha generae failures can be classified ino conrolled and unconrolled ones. During operaion, he unconrolled devices, causing failures should have adequae surplus in he reliabiliy srucure and he abiliy o regiser he number of failures and he number of funcioning, such as: he load cycles, he working ime, he number of sars, ec. and he opporuniy of evaluaing echnical condiion a seleced momens of ime. These daa, obained mosly in he laboraory, are he basis o deermine he degree of correlaion wih he amoun of operaion. Examples of mehods for he analysis of he process of desrucion averaged in he elemen or assembly se may be based on incidence of damages or on a se of realizaions of random variables of he assembly echnical parameers (Barbu, Bulla, Maruoi, 212). An imporan delineaed parameer in he exploiaion process operaion wih compuer sysem reliabiliy analysis, ha informs abou componens aging and wear, is he parameer damage sream o serviceable componens w (). The basic informaion is here a change in he w () parameer a he ime, and he answer o he quesion of wheher i falls wihin he olerance lane. For he calculaion of he parameer damage sream limis one uses a relaionship beween he upime probabiliy densiy funcion f (), and he damage sream parameer w (). w( ) = f ( ) + w( ) f ( ) d. (1) The approximae value of w () we calculae from he relaionship: n j( ) w( ) = w( j ) = wj = N, (2) where: f(), f() probabiliy densiy funcions of he objec faulless operaion; w(-) value he objec damage sream a he iniial ime; w() value described above a he ime of he final examinaion; n j ( ) he number of objecs ha have been damaged in he j-h ime inerval [j, (j+1) ]; -194-

3 Proceedings of he 6h Inernaional Conference on Mechanics and Maerials in Design, Ediors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-3 July 215 N number of objecs in exploiaion. Using he Laplace ransform for he funcion f() and W() and some simplificaions, (w = λ, where: (λ he inensiy of defecs in he exponenial model of reliabiliy)) he soluion can be obained for he exponenial disribuion of failures and one can adop a crierion (Janssen, Manca, 26): λ for < b, λ( ) = λ+ λ1( b) for b. An imporan issue is he search for he poin b, he increase in he value of parameer λ. Knowledge of b poins on he componens or assemblies exploiaion process ime axis, allows he raional disribuion of whole objec mainenance projecs and deermine he periods of safe exploiaion. (3) RESULTS The aircraf echnical readiness for operaion is measured by he probabiliy of he aircraf presence or hold of a any ime in a subse of saes, giving he abiliy o use and perform he inended ask. I can also be measured wih he expeced value of aircraf residence ime in seleced saes and oher merics, based on purpose analysis of he exploiaion process. Technical readiness is he condiional indicaor of an aircraf availabiliy o perform he ask a he ime under he condiion, ha he oher requiremens ha deermine he possibiliies of he aircraf operaion are me (eg. requiremens for he fuel supply, gases, saring equipmen, ec.). An example graph of he damage sream parameer of aging elemens is presened in Fig. 1. λ () λmax λ b T Fig. 1 - An example graph of he damage sream parameer of aging elemens As an example of he analysis mehod, one can deermine he probabiliy of aircraf saying in echnical readiness sae on he four sae model of exploiaion process: H 1 sae of usage (fligh), H 2 on call sae (o mainain a consan readiness), -195-

4 Track_B Tesing and Diagnosic H 3 sae of prevenive suppor and planned repairs, H 4 sae of random repairs. The residence imes in he various saes of aircraf mainenance are random variables wih known disribuions: { kh } F ( ) = P T < he disribuion funcion of he k-h fligh duraion, k = 1, 2, 3; kh1 1 { kh } F ( ) = P T < he disribuion funcion of he k-h on call ime, k = 1, 2, 3,...; kh2 2 { } F( ) = P TkH 3 < he disribuion funcion of suppor and scheduled repairs ime; { } F( ) = P TkH 4 < he disribuion funcion of random repair ime. For an objec implemening he muli-sae exploiaion process, willingness o underake and realize he asks can be deermined by analysis of semi-markov process wih a finie number of phase saes, exending in coninuous ime. To be able o use he semi-markov model of he objec exploiaion mapping process, he following assumpions mus be mee: All objec ransiions from he operaional saus \[i\in E\] o he operaional saus \[j\in E\] ake place in seps. The objec residence ime in he i sae, before moving on o he j sae is a T random variable, receiving values independenly of he calendar ime. T random variables are muually independen for each i, j = 1, 2, 3,..., r. Consider he aircraf exploiaion process of a finie se of operaing condiions, E = {1,2,...,r} represened by he random process {Z : } defined as follows: Z = i, if a he ime process is in he sae i. Z = j,, if he process a he ime is in he sae j. The residence ime of he process in he i sae and before moving on o he j sae is a random variable wih probabiliy disribuion funcion F (). The probabiliies of ransiion from he i sae o he subsequen sae j are given as a marix r r P = [P ], which defines he so-called Markov chain insered in he process. For any E he funcion Q () is deermined, represening he cumulaive probabiliy of even consising in ha he process residence ime in he i sae is smaller han and ha process from he i sae moves o a j sae. This funcion saisfies he condiions: Q () = ; i, j E, Q ( ) = P = 1, i E. (4) j E j E Se of Q () funcion creaes a Q() marix of disribuions of he {Z : } process ransiion. Uncondiional probabiliy disribuion of he exploiaion process residence ime in he sae i defines he funcion F i (): F ( ) = Q ( ). (5) i The iniial sae of he process deermines he r-dimensional vecor of iniial probabiliies: j E -196-

5 Proceedings of he 6h Inernaional Conference on Mechanics and Maerials in Design, Ediors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-3 July 215 where: P = [P 1, P 2,...,P r ], Pi = 1. (6) i E In his way, he specific random process of Z : is a semi-markov process, predeermined by rio (r, p, Q). For such formulaed exploiaion process one can creae algorihms for deermining a number of aircraf readiness indicaors relaive o he subse of asks or o aciviy in general. Since in our example operaing saes H 1 and H 2 form a subse of H 12 echnical preparedness o ac, hen semi-markov exploiaion process model can be reduced o hree saes H 12, H 3, H 4 incidence marix: and disribuions marix of he process ransiion: ( ) 1 1 I = 1 (7) 1 ( ) ( ) ( ) ( ) Q Q = 21 Q Q. (8) Q23 The funcion Q () depends on he disribuions GR ( ), F ( ), FH ( ), F ( ) 3 H 4. Afer deermining he funcion Q () and iniial disribuion P = {P i, i = 1,2,3}, he aircraf echnical readiness indicaors can be deermined, assuming ha he sae of H 12 is a subse of readiness saes. Change of he H 12 sae o he H 3 sae occurs, when he following evens occur: The proper operaion ime R TR is greaer han he TBO consumpion ime > ; T, R usage ime o he momen is no greaer han, ( S ). { R } The cumulaive even ( ) ( > ) has a probabiliy funcion: [ ] Q12 ( ) = 1 GR( y) df ( y). (9) Change in he H 12 sae o he H 4 sae occurs, when following evens occur: he proper operaion ime is no greaer han,( ); TBO consumpion ime is greaer han he proper operaion ime R ( > R). The cumulaive even {( < ) ( > )} has a probabiliy funcions: R R R Q13 ( ) = [1 F ( y)] dgr ( y). (1) -197-

6 Track_B Tesing and Diagnosic Change in he H 3 sae o he H 12 sae occurs when and he H 4 sae o he H 12 sae where ( T < ), so: H3 { } Q ( ) 21 = P TH 3 < (11) ( T < ), hen: H4 { } Q ( ) 23 = P TH 4 <. (12) Assuming ha he H 12 sae is he iniial sae of he process, specific iniial process vecor p = [1,,] was obained. The probabiliy of aircraf say in echnical sandby H 12 by he ime is 1 F 12 (). On he basis of he formula (13) he value was deermined: P ( ) = 1 F ( ) = 1 [ Q ( ) + Q ( )] = = 1 1 ( ) ( ) + 1 ( ) ( ) [ Gr y ] df y [ F y ] dgr y. (13) Asympoic probabiliy P 12 () of aircraf say in he echnical sandby by he ime = (he ime of fligh) is he produc of: P ( ) = P ( ) [ 1 12( )] [ 1 12( )] F d, (14) F d ( ) where P 12 is he limi probabiliy of he aircraf presence in he H 12 sae, when he process ime is very long,. The value of The value of ( ) P 12 characerizes he aircraf funcional readiness (insananeous readiness). ( ) P 12 is deermined on he basis of heorem. If he semi-markov process {Z, }abou finie se E is irreducible and aperiodic and random variables T Hj have finie posiive expeced values E[T Hj ], he boundary ransiion probabiliies exiss: p = lim p ( ) i, j E, (15) ( ) π ( ) ( ) ( ) je T Hj p = p j =, (16) π E( T ) where: π j is he boundary probabiliy of he Markov chain insered ino process. I is he incidence of he airplane presence in he j-h exploiaion sae (subse of saes). Probabiliy, π j, j = 1,2,3,..., can be deermined: r E r Hr -198-

7 Proceedings of he 6h Inernaional Conference on Mechanics and Maerials in Design, Ediors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-3 July 215 π D j j =, (17) Di i E where: D i principal minor of he marix P, obained by deleing he i-h column and j-h row, means a marix of ransiion probabiliies insered in he Markov chain process. p = lim Q ( ). (18) Individual values of p in he considered model are: The marix P has he form: The marix D is expressed as: wherein: p = lim Q ( ) = [1 G ( y)] df ( y), (19) p [ ] p13 = Q13 ( ) = 1 F ( y) dgr ( y), (2) ( ) 12 = p = lim Q ( ) = 1, (21) p = lim Q ( ) = 1. (22) p12 p13 P = 1. (23) 1 1 p12 p13 D = 1 1, (24) 1 1 π π 1 1 =, (25) 3 p p p = p12 p, (26) 13 π p = p12 p, (27) 13 E[ TH ] 12. (28) E[ T ] + p E[ T ] + p E[ T ] H12 12 H3 13 H4 Expeced values of random variables TH, T, 12 H T 3 H, which denoe aircraf mean residence imes 4 in differen exploiaion saes can be deermined from he relaionship: -199-

8 Track_B Tesing and Diagnosic, (29) E[ T ] = [1 G ( )] df ( ) + [1 F ( )] dg ( ) H12 R R E[ T ] = df ( ), (3) H 3 H3 The expeced value of a random variable E[ T ] = df ( ). (31) H 4 H4 TH 12 is also a echnical readiness indicaor, characerizing he aircraf mean residence ime in he required subse of saes. Deermined by formula (21) he probabiliy of aircraf presence in he echnical readiness sae is rue, when he condiion is saisfied, ha he iniial sae of he exploiaion process is he sae from he se {H 1, H 2 }. If he iniial sae of he exploiaion process is differen from he readiness sae, one should deermine he cumulaive disribuion funcion of he random variable T ia a he ime of he firs ransiion from he iniial sae i E being no a readiness sae, o a subse A of readiness saes A E. The cumulaive disribuion funcion of random variable T iż can be deermined on he basis of heorem. If A is a subse of saes reachable from every sae i E i A of he semi-markov process wih a finie se of saes E, and funcions Q () are coninuous, here are cumulaive disribuion funcions F ia () of he random variable T ia saisfying he sysem of equaions:. (32) F ( ) = Q ( ) + F ( x) dq ( x) ia ka ik j A k k E For he considered aircraf exploiaion model: The probabiliy P ( ) ( ) 3,1,2 F ( ) ( ) 3 A = F, (33) H3 F ( ) ( ) 4 A = F. (34) H4 of aircraf presence in echnical readiness sae (H 12 ) for a ime afer he exploiaion process ime, when he iniial sae was he (H 3 ) sae, can be deermined from he formula: ( ) 3,1,2 = 12 + H 3 P ( ) [1 F ( x)] df ( x), (35) where he value 1 F12 ( + x) is deermined from he equaion (1). The probabiliy of P ( ) ( ) 3,1,2 characerizes he aircraf echnical readiness o operae a ime afer ime from he beginning of he process, ha sars in he H 3 sae. For he process saring a ime from he H 4 sae, we have respecively: ( ) 4,1,2 ( ) = [1 12( + )] H ( ). (36) 4 P F x df x -2-

9 Proceedings of he 6h Inernaional Conference on Mechanics and Maerials in Design, Ediors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-3 July 215 A deeper analysis of he aircraf echnical readiness can be carried ou on a model wih a higher level of complexiy. The degree of aircraf usage can be measured by he real fligh-log, and in he case of passenger aircraf, also by he filling degree of he passenger adminisered seas. Toal aircraf operaional ime in a subse of H 12 saes o he ime is a S random variable wih a probabiliy formula: ( n) ( n 1) n P{ S < x} = F ( x) F ( x) F ( x), (37) H1 H2 H2 n= 1 where he n symbol is he n-fold convoluion of disribuion funcions. The incremen process of he aircraf operaion ime as a funcion of calendar ime has asympoically normal disribuion for. The expeced value of he disribuion is: and variance: E TH 1 [ ] [ ] E S = [ ] [ ] = m( ) (38) E TH + E T 1 H2 D ( T )( E[ T ]) + D [ T ]( E[ T ]) D S H1 H2 H2 H1 2 [ ] ~ = [ σ ( )] 3 ( E[ TH ] + E[ T ]) 1 H2 Aircraf uilizaion facor for aircraf saying on readiness sae is: where: E T H1, [ ] H 2 E T H1 ρ = E T + E T [ ] H1 H 2. (39), (4) E T he expeced values of fligh ime and on call-ime. The fill facor of passenger seas γ is: where: k i N u ki =, (41) u N i= 1 γ number of passenger seas filled in he i-h fligh (i = 1, 2,..., u); number of he adminisered passenger seas in he aircraf. The overall raing of exploiaion effec is: and i akes values from he inerval [, 1]. E = ρ γ CONCLUSION Wear and aging processes of he various elemens are very differenly correlaed wih he ime or he amoun of operaion or calendar ime. Presened above mehod of deermining he period of safe durabiliy concerns he elemens -21-

10 Track_B Tesing and Diagnosic having srongly correlaed parameers, deermining he airworhiness saus wih he amoun of usage or operaional ime, wha can be idenified wih he exisence of memory on he pas. Sored in he exploiaion daabase lifeime facions of individual objecs and assemblies and pars beween seleced evens or classes of evens, creae appropriae ses of ime random variables in he probabilisic model of he exploiaion process. The disribuions of hese random variables can be described by heoreical models, whose parameers characerize he paricular qualiy feaures of he exploiaion process and he properies of objecs. A measure of he aircraf readiness characerizes he objec and he logisics sysem, in which he given objec is immersed. Probabiliy value expressing readiness, ranslaes o he percenage of he aircraf ime, in which i is a he disposal of he head of he ranspor sysem. This value depends on he exploiaion program se by he manufacurer and comprising: schedule of mainenance and diagnosic checks during use, echnical suppor procedures and oher exploiaion underakings. Deermined in his manner objec readiness index in he logisic sysem expresses a poenial opporuniy for effecive performance. The degree of developmen of his poenial for he passenger aircraf represens he fill facor. The overall efficiency raio characerizing availabiliy and marke demand can be expressed by he produc of he above-menioned readiness and fill facors. REFERENCES [1]-Barbu, V. S., Bulla, J., Maruoi, A. Esimaion of he Saionary Disribuion of a Semi- Markov Chain, Journal of Reliabiliy and Saisical Sudies, 212, Vol. 5, p [2]-Janssen J., Manca, R. Applied Semi-Markov Processes. Springer, New York, 26. [3]-Woropay M., Zurek J. Model of he Readiness for Impac of Inervenional Vehicles in he Transpor Sysem, Bydgoszcz-Radom, 23 (in Polish). [4]-Zurek J., Ziolkowski J. Model of he Readiness for Impac of Inervenional Vehicles. X Congress of Mainenance of Technical Devices, Sare Jablonki, 25 (in Polish). -22-