An outline of the method for determining the density function of changes in diagnostic parameter deviations with the use of the Weibull distribution

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1 An oline of he mehod for deermining he densiy fncion of changes... 5 SCIENTIFIC PROBLEMS OF MACHINES OPERATION AND MAINTENANCE (66) MARIUSZ WAŻNY * An oline of he mehod for deermining he densiy fncion of changes in diagnosic parameer deviaions wih he se of he Weibll disribion K e y w o r d s A diagnosic parameer he Weibll disribion densiy fncion reliabiliy. S ł o w a k l c o w e Paramer diagnosycny rokład Weiblla fnkcja gęsości nieawodność. S m m a r y De o he nare of asks performed by an aircraf one of he mos essenial crieria deermining he qaliy of is mainenance process is he reliabiliy of devices and sysems insalled on an aircraf. The assrance of he reliabiliy of aircraf devices a an adeqaely high level minimises he cases of failres. Unfornaely he inflence of desrcive facors conneced among oher hings wih he impac of changing ambien condiions overload effecs and he inflence of ageing processes cases he echnical parameers of devices deeriorae. Mehods for describing diagnosic parameer changes de o he effecs of desrcive facors have been presened in he lierare [ ]. This aricle consies a new aemp of an analyical descripion of he changes in diagnosic parameer vales describing he echnical sae of devices based on he mehod * Miliary Universiy of Technology General Sylweser Kaliski Sree -98 Warsaw 49 Poland.

2 6 M. Ważny for deermining he densiy fncion of changes in diagnosic parameer deviaions wih he se of he Weibll disribion. Inrodcion One of he echnical objecs whose mainenance process is sbjec o a special spervision is an aircraf which is conneced wih nmeros facors relaed mainly o he environmen in which i is operaed i.e. airspace. The changeabiliy of environmenal parameers in which an aircraf moves and he inflence of varios exernal and inernal desrcive forces case ha he mainenance process of an aircraf shold be as complee as possible i.e. he process shold provide condiions enabling he performance of a schedled ask. Depending on he prpose of an aircraf asks can be divided ino differen elemens. Noneheless considering a general model of is operaion we can assme ha he basic fncion involves he execion of he flying fncion i.e. ake-off fligh o a arge desinaion and landing. This fncion can be execed in a safe way only when here is an appropriae and complex model for he performance of he mainenance process. The framework of he scope and characer of he mainenance process for a newly consrced aircraf is developed along wih he lanch of works on he concep of an objec. The main deerminan oriening he hogh process involves defining wo grops of erms i.e. [3]: Aircraf properies and Aircraf qaliies. Aircraf properies consie a se of fncions deermined a he sage of design and consrcion. They concern sch parameers as aircraf measremens and mass he drive ni srcre resisance o faige processes fel weigh consrcion-relaed condiions of operaion mainenance and repair operaional poenial and payload. On he oher hand aircraf qaliies are deermined by sch parameers as fncionaliy reliabiliy readiness siabiliy drabiliy service life and sscepibiliy. Each propery and qaliy defined in accordance wih he crierion assmed can be examined in hree dimensions conneced wih heory pracice and nderakings aimed a he modificaion of parameer vales describing he properies and qaliies [3]. One of he mos imporan parameers menioned above is reliabiliy referring boh o devices and sysems insalled on an aircraf and also o he aircraf iself. The reliabiliy parameer is inexricably linked wih he safey parameer.

3 An oline of he mehod for deermining he densiy fncion of changes... 7 Tesing he reliabiliy and safey of aircraf in he mainenance process is relaed o he predicion of he echnical sae -- devices sysems and aircraf. Desrcive processes manifesing hemselves in he form of overload fricion vibraions wear ec. have a crcial effec on echnical sae changes in aircraf devices. Today he limis deermining mainenance ime inervals are no direcly conneced wih he sage of eiher ping a device ino operaion or wihdrawing a device from operaion. The p-o-dae knowledge acqired hrogh he mainenance and operaion of a echnical device enables s o develop he mainenance process a he concepal sage while analysing he feasibiliy of a given projec. I is clearly depiced in broadly defined aeronaical engineering where an ieraive process is sed o develop a prodc which in rn is sed o obain cerain benefis/profis. Fig. shows he independence graph of he life cycle of a echnical device [4]. Fig.. A graph of he scopes of life of a device and he changes of he reliabiliy characerisics in he mainenance process Rys.. Wykres akresów życia rądenia ora mian charakerysyki nieawodnościowej w procesie eksploaacyjnym The echnical sae of an aircraf device is mainly evalaed hrogh a se of diagnosic parameers. The effec of desrcive processes manifess iself in he change of diagnosic parameer vales casing a rise in he deviaion from he nominal vales of hese parameers. The vales of deviaions from he nominal vales are sed o esimae he reliabiliy of a device.

4 8 M. Ważny The effec of desrcive processes on he change of diagnosic parameer vales can be divided ino hree grops: Elemens of a device ha have srongly correlaed vales of diagnosic parameer deviaions wih ime or he workload of a device; Elemens of a device ha have weakly correlaed vales of diagnosic parameer deviaions wih ime or he workload of a device; and Elemens of a device ha lack correlaion wih ime or he workload of a device. De o he diversiy of componens real devices sally comprise all he above-menioned grops. However someimes i can be shown ha some of he grops are dominan and hey can be sed o predic he reliabiliy and drabiliy of elemens and devices. The developmen of probabilisic models based on he above-menioned processes and he se of heir resls for deermining he reliabiliy and drabiliy of elemens and devices are imporan isses. This aricle is an aemp o bild a model enabling he deerminaion of he reliabiliy characerisics for elemens sbjeced o he effec of desrcive processes. The analyical descripion of device reliabiliy sars wih making iniial assmpions concerning he developed mehod [ 6 7]. The following findings and assmpions were made for he prpose of developing he model: The echnical sae of a device is deermined by one dominan diagnosic parameer. Is crren vale is denoed by x. The change of a diagnosic parameer vale de o he desrcive effec of ageing processes occrs wih he passing of calendar ime. The deviaion of a diagnosic parameer from he nominal vale is = x p x n x he measred vale of a diagnosic parameer p x n he nominal vale of a diagnosic parameer. If [ d ] hen an elemen of a device is regarded as operable; oherwise an elemen of a device is regarded as inoperable. An increase of a diagnosic parameer deviaion in he fncion of calendar ime saisfies he relaionship c d = c () d he mean vale a variable velociy depending on ageing processes he calendar ime.

5 An oline of he mehod for deermining he densiy fncion of changes... 9 Deermining he densiy fncion of changes in vales of diagnosic parameer deviaions One of he elemens describing device reliabiliy is he densiy fncion of changes in diagnosic parameer deviaions. For he prpose of deermining he fncion i was assmed ha he inensiy of he growh in deviaion has he form () λ( ) = () i he consans in he Weibll disribion wih he following denoaions: he shape facor he scale facor. The random dynamics of changes of diagnosic parameer vales inclding he deviaion is described by he difference eqaion. Le U denoe he probabiliy ha a he ime he vale of a diagnosic parameer deviaion adops he vale. The differeniaed eqaion has he following form: U = U U (3) he increase in deviaion of a diagnosic parameer over he ime inerval. By convering Eqaion (3) ino he fncion noaion we obain he Eqaion (4) as follows: ( ) = ( ) ( ) (4) ( ) he densiy fncion of a diagnosic parameer deviaion; he probabiliy ha over he ime inerval here is no parameer deviaion;

6 M. Ważny he probabiliy ha over he ime inerval here is he increase in he parameer deviaion ; and he following condiion is me. In order o deermine he densiy fncion of changes in diagnosic parameer deviaions we shall conver he fncion noaion of Eqaion (4) ino a parial differenial eqaion. We assme he following approximaion:. = = (5) By sbsiing he relaions presened in (5) ino Eqaion (4) we obain he following relaions: = = =. Hence = (6)

7 An oline of he mehod for deermining he densiy fncion of changes... For he prpose of frher ransformaions i was assmed ha = c = c c = which specifies ha we examine he increase of parameer deviaion per ime ni (when = ) where c denoes he deviaion increase per a ni of ime. By sbsiing he above assmpion ino Eqaion (6) we obain is final form ( ) c = 3 γ ( ) ( ) β ( ) ( ) c 443 (7) As i can be seen in Eqaion (7) he form of he coefficiens depends on he parameer vales. For = he coefficiens has he following form: c γ ( ) = ; c β =. For = he coefficiens has he following form: γ c c =. = ; β The solion of Eqaion (7) has he following form: ( ) = πa e ( B( )) A( ) (8) B() he average vale of a parameer deviaion for he ime of he service life B = γ d (9)

8 M. Ważny A() he vale of he variance of a diagnosic parameer deviaion for he ime of he service life. = A β d () Calclaing he Inegrals (9) and () we obain he following: B c c = c d = d = = = c c c c = = A = c () () Hence Relaionship (8) has he following form: ( ) c c = e (3) c π Relaionship (3) presens he densiy fncion of a diagnosic parameer deviaion from he nominal vale. Le c = b and c = a. Hence relaionship (3) has he following form: ( ) ( b ) a = e (4) πa By sing he densiy fncion (4) we can deermine he relaionship for he reliabiliy of a device in erms of an examined diagnosic parameer. This relaionship has he form of Eqaion (5)

9 An oline of he mehod for deermining he densiy fncion of changes... 3 d = ( ) R d (5) d he permissible deviaion vale of he diagnosic parameer a is deermined by relaionship (4). Smmary The operaion of echnical devices insalled on an aircraf depends on he inflence of changeable condiions boh amospheric condiions and mechanical ones ha are conneced inclding in-fligh overload effecs. These facors case he accmlaion of desrcive elemens ha in rn case cerain elemens of echnical sysems o lose nominal operaing parameers. The noaion presened in his paper enables one o deermine he reliabiliy of devices and consies he basis for frher analyses aimed a deermining he densiy fncion of he ime of exceeding he limi sae by he diagnosic parameer being discssed de o he inflence of desrcive facors. By sing he densiy fncion of he ime of exceeding he limi sae by he diagnosic parameer we are able o deermine he residal drabiliy of he device being considered in his paper which will consie he direcion of frher sdies. The finding obained in his way can be sed for he modificaion of he mainenance process of a given echnical objec. Scienific work fnded by he Naional Cenre for Research and Developmen in 3 as a research projec. References [] Abega G.: Rachnek probabilisycny. Poradnik. Wydawnicwo Minisra Obrony Narodowej 973. [] Gniedenko B. Bielajew J. Sołowiew A.: Meody maemaycne w eorii nieawodności. Wydawnicwo Nakowo-Technicne 968. [3] Lewiowic J. Ksroń K.: Podsawy eksploaacji saków powiernych. Własności i właściwości eksploaacyjne sak powiernego. Wydawnicwo ITWL 3. [4] Mrhy D.N.P. Xie Min Jiang R.: Weibll Models John Wiley & Sons Inc. Hoboken New Jersey 4. [5] Tomasek H. Wróblewski M.: Podsawy oceny efekywności eksploaacji sysemów brojenia lonicego. Dom Wydawnicy Bellona. [6] Tomasek H. Żrek J. Loroch L.: Zarys meody oceny nieawodności i rwałości elemenów konsrkcji lonicych na podsawie opis procesów desrkcyjnych. Zagadnienia Eksploaacji Masyn 4 Zesy 3 (39):

10 4 M. Ważny [7] Tomasek H. Scepanik R.: Meoda określania rokład cas do prekrocenia san granicnego. Zagadnienia Eksploaacji Masyn 5 Zesy 4 (44): [8] Ważny M. Jasal M.: Zarys meody określania prawdopodobieńswa danej pracy rądenia w aspekcie mian warości paramer diagnosycnego. Eksploaacja i Nieawodność Mainenance and Reliabiliy 7 : [9] Ważny M.: Meoda wynacania casookres prebywania wybranego rądenia sysem nawigacyjno-celownicego w sysemie żykowania. Eksploaacja i Nieawodność Mainenance and Reliabiliy 8 : 4. [] Ważny M. Wojowic K.: Analia sysem eksploaacji wojskowego sak powiernego w aspekcie jego moderniacji. Eksploaacja i Nieawodność Mainenance and Reliabiliy 8 3: 4. [] Ważny M.: Meoda oceny rwałości reskowej wybranych rądeń sysem awionicnego. Eksploaacja i Nieawodność Mainenance and Reliabiliy 9 3: Zarys meody określenia fnkcji gęsości mian odchyłek paramerów diagnosycnych wykorysaniem rokład Weiblla S r e s c e n i e Z wagi na charaker adań realiowanych pre saek powierny jednym podsawowych kryerim określającym jakość jego proces eksploaacyjnego jes nieawodność rądeń i sysemów ainsalowanych na jego pokładie. Zapewnienie warości nieawodności rądeń sak powiernego na odpowiednio wysokim poiomie minimalije prycyny wysępowania skodeń. Niesey oddiaływanie cynników desrkcyjnych wiąanych m.in. wpływem miennych warnków oocenia w kórym nasępje rch sak powiernego oddiaływania preciążeń cy eż wpływ procesów sareniowych powodje że paramery echnicne charakeryjące pracę rądeń legają pogorseni. Meody opis mian warości paramerów diagnosycnych w wynik oddiaływania cynników desrkcyjnych predsawiane były w poycjach lierarowych [ ]. Niniejsy arykł jes próbą analiycnego opis mian warości paramerów diagnosycnych opisjących san echnicny rądenia w oparci o meodę określania fnkcji gęsości mian odchyłek paramerów diagnosycnych rądeń echnicnych wykorysaniem rokład Weiblla.