SPIS TREŚCI - CONTENTS. Spis treści

SPIS TREŚCI - CONTENTS Sps treśc Nauka Technka Jan SZYBKA, Zdzsław BRONIEC, Robert PILCH Forecastng the falure of a thermal ppelne on the bass of rsk assessment and explotaton analyss Prognozowane uszkodzena cepłocągu na podstawe oceny ryzyka eksploatacj... 5 Awram LEWI, Darn PEEW On the evaluaton of explotaton qualtes of Prsta Super 25W40 motor ol wth DIA method Ocena cech eksploatacyjnych oleju slnkowego Prsta Super 25W40 metodą DIA... Potr CZECH, Henryk MADEJ Applcaton of cepstrum and spectrum hstograms of vbraton engne body for settng up the clearance model of the pston-cylnder assembly for RBF neural classfer Wykorzystane hstogramów wdma cepstrum drgań korpusu slnka do budowy wzorców luzu w układze tłok-cylnder dla klasyfkatora neuronowego RBF... 5 Longnas LIUDVINAVIČIUS, Leonas Povlas LINGAITIS, Gntautas BUREIKA Investgaton on wheel-sets slp and slde control problems of locomotves wth ac tracton motors Badane problemów buksowana kołowego sterowana procesu ślzgana w lokomotywach z slnkam trakcj asynchroncznej... 2 Henryk TOMASZEK, Mchał JASZTAL, Marusz ZIEJA A smplfed method to assess fatgue lfe of selected structural components of an arcraft for a varable load spectrum Uproszczona metoda oceny trwałośc zmęczenowej wybranych elementów konstrukcj statku powetrznego dla zmennego wdma obcążena... 29 Leszek DZICZKOWSKI, Andrzej BUCHACZ Modfcaton of a calbraton method for conductometers Modyfkacja metody skalowana konduktometru... 35 Anton ŚWIĆ, Lech MAZUREK Modelng the relablty and effcency of flexble synchronous producton lne Modelowane nezawodnośc wydajnośc synchroncznej elastycznej ln produkcyjnej... 4 Zhong Hua CHENG, Yong Sheng BAI, L Yng CAI, Lu Chao WANG, Peng Ju LI, Lng CHEN Research on Warranty Interval of Mult-component System wth Falure Interacton Badana okresu gwarancyjnego dla systemu weloskładnkowego, w którym zachodzą nterakcje uszkodzenowe... 49 Krzysztof BRZOZOWSKI, Jacek NOWAKOWSKI Toxcty of exhaust gases of compresson gnton engne under condtons of varable load for dfferent values of engne control parameters Toksyczność spaln slnka o zapłone samoczynnym w warunkach zmennego obcążena dla różnych wartośc parametrów regulacyjnych... 56 S. SHAKUNTLA, A. K. LAL, S. S. BHATIA Comparatve study of the subsystems subjected to ndependent and smultaneous falure Badana porównawcze podsystemów ulegających uszkodzenom nezależnym jednoczesnym... 63 Jarosław LATALSKI Modellng of macro fber composte pezoelectrc actve elements n ABAQUS system Modelowane w systeme ABAQUS pezoelektrycznych elementów aktywnych typu MFC... 72 Tomasz NOWAKOWSKI Problems of relablty modellng of multple-phased systems Problemy modelowana nezawodnośc systemów welofazowych... 79 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

W SKRÓCIE - ABSTRACTS SZYBKA J, BRONIEC Z, PILCH R. Forecastng the falure of a thermal ppelne on the bass of rsk assessment and explotaton analyss. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 5-0. In ths paper, a procedure for determnng the utlsaton perod of a thermal ppelne, exploted n mnng plants was presented. On the bass of analyses conducted n the real lfe stuatons, the ppelne falure, whch could be caused by ts burstng due to exceedng ppelne strength, was forecasted. The explotaton condtons of the thermal ppelne and factors contrbutng to ts degradaton were also analysed. Usng the probablty samplng method, the explotaton rsk was assessed and the lfespan of the ppelnes, after whch the ppelnes shall be replaced by new structures, were determned. LEWI A, PEEW D. Ocena cech eksploatacyjnych oleju slnkowego Prsta Super 25W40 metodą DIA. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: -4. The paper consders one of the few studes of motor ol usng Dfferental Impedance Analyss (DIA) method. The mpedance data of Prsta Super 25W40 motor ol are analyzed through ths method, whch s more advanced technque comparng to Electrochemcal Impedance Spectroscopy. Structural changes n samples of long exploted ol are found. The analyss of samples of less used ol regsters one mportant phenomenon, whch s known from the practce. The possbltes for vscosty ndces determnaton are dscussed. General mpedance data analyss s also conducted. CZECH P, MADEJ H. Applcaton of cepstrum and spectrum hstograms of vbraton engne body for settng up the clearance model of the pston-cylnder assembly for RBF neural classfer. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 5-20. The paper presents an attempt to evaluate the wear of pston-cylnder assembly wth the ad of vbraton sgnal recorded on spark gnton (SI) engne body. The subject of the study was a four-cylnder combuston engne. dm 3. Dagnosng combuston engnes wth vbraton methods s specfcally dffcult due to the presence of multple sources of vbraton nterferng wth the symptoms of damages. Dagnosng engnes wth vbro-accoustc methods s dffcult also due to the necessty to analyse non-statonary and transent sgnals. Varous methods for selecton of usable sgnal are utlsed n the dagnosng process. Changes of the engne techncal condton resultng from early stages of wear are dffcult to detect for the effect of mechancal defect maskng by adaptve engne control systems. Accordng to the studes carred out, t s possble to utlse artfcal neural networks for the evaluaton of the clearance n pstoncylnder assembly. LIUDVINAVIČIUS L, LINGAITIS L P, BUREIKA G. Investgaton on wheel-sets slp and slde control problems of locomotves wth ac tracton motors. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 2-28. The artcle s dedcated to the soluton of drve parameter adjustment and correcton problems of AC tracton motors. Havng assessed dfferent ral adheson ratos and AC electrcal machne operaton peculartes, there have been proposed new dynamc slp and slde process control methods to DC/ AC, AC/AC current system locomotves. There s descrbed the nfluence of mechancal characterstcs of AC tracton motors to the formaton of wheel slp process and there are provded structural control schemes of the dynamc slp process. The slp process formaton and control condtons for wheel pars are provded n mathematcal and graphcal forms. There are provded automatc control parameters of dynamc slp and slde process for AC/AC current locomotves. TOMASZEK H, JASZTAL M, ZIEJA M. A smplfed method to assess fatgue lfe of selected structural components of an arcraft for a varable load spectrum. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 29-34. The assessment of fatgue lfe of an arcraft s structural component operatng under varable load spectrum causes many and varous problems, hence the need for smplfed methods that facltate t. The presented study covers the queston of rearrangng an actual spectrum wth varable values of cycles nto a homogeneous spectrum wth weghted cycles. A method for the evaluaton of fatgue lfe of some selected arcraft s structural component wth an ntal crack has been presented usng a rearranged spectrum. To model an ncrement n the SZYBKA J, BRONIEC Z, PILCH R. Prognozowane uszkodzena cepłocągu na podstawe oceny ryzyka eksploatacj. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 5-0. W artykule przedstawono procedurę postępowana w wyznaczena okresu użytkowana cepłocągu eksploatowanego w zakładach górnczych. Na podstawe badań przeprowadzonych w warunkach rzeczywstej eksploatacj prognozowano awarę rurocągu, która może być spowodowana pęknęcem na skutek przekroczena jego wytrzymałośc. Opsano warunk eksploatacj cepłocągu czynnk wpływające na jego degradację. W ujęcu probablstycznym oszacowano ryzyko eksploatacj wyznaczono okres użytkowana, po którym pownna nastąpć wymana rurocągów. LEWI A, PEEW D. On the evaluaton of explotaton qualtes of Prsta Super 25W40 motor ol wth DIA method. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: -4. Artykuł pośwęcony jest badanom oleju slnkowego przy pomocy Analzy Dyferencjalnej Impedancj (DIA). Zbadano tą metodę, która jest doskonalsza od Elektrochemcznej Spektroskop Impedancyjnej, zmany mpedancj nowego oleju Prsta Super 25W40 w okrese eksploatacj. Wynk wykazują, że po pewnym krótkm okrese, używany olej ma lepsze właścwośc smarne względem próbk oleju neużywanego. Omówono także możlwośc oszacowana olestośc przeprowadzono analzę wynków. CZECH P, MADEJ H. Wykorzystane hstogramów wdma cepstrum drgań korpusu slnka do budowy wzorców luzu w układze tłok-cylnder dla klasyfkatora neuronowego RBF. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 5-20. W artykule przedstawono próbę oceny zużyca złożena tłok-cylnder za pomocą sygnału drgań rejestrowanego na kadłube slnka ZI. Obektem badań był czterocylndrowy slnk spalnowy o pojemnośc, dm 3. Dagnozowane slnka spalnowego metodam drganowym jest szczególne utrudnona ze względu na występowane welu źródeł drgań, co jest przyczyną wzajemnego zakłócana symptomów uszkodzeń. Dagnozowane uszkodzeń slnków metodam wbroakustycznym jest trudne także ze względu na koneczność analzy sygnałów nestacjonarnych mpulsowych. W procese dagnozowana stosuje sę różne sposoby selekcj sygnału użytecznego. Zmany stanu techncznego slnka wywołane wczesnym fazam jego zużyca są trudne do wykryca ze względu na maskowana usterek mechancznych przez adaptacyjne układy sterowana slnka. Z przeprowadzonych badań wynka, że stneje możlwość wykorzystana sztucznych sec neuronowych do oceny luzu w układze tłokcylnder. LIUDVINAVIČIUS L, LINGAITIS L P, BUREIKA G. Badane problemów buksowana kołowego sterowana procesu ślzgana w lokomotywach z slnkam trakcj asynchroncznej. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 2-28. Artykuł jest przeznaczony dla rozwązana problemów regulowana korekcj parametrów pracy przekładn lokomotywy z slnkem trakcyjnym AC. Ocenając różne czynnk przyczepnośc koła lokomotywy z torem osoblwośc maszyn elektrycznych typu AC, w artykule przedstawono nowe metody buksowana dynamcznego sterowana procesu ślzgana przekładn lokomotywy dla systemów prądu AC/DC, AC/AC. Ocenając wpływ rodzaju charakterystyk mechancznych slnków trakcyjnych AC na formowane procesu buksowana kół, w artykule przedstawono strukturalne schematy sterowana procesów buksowana dynamcznego. Formowane procesu buksowana par kół warunk jego sterowana przedstawono w wyrażenu matematycznym forme grafcznej. W artykule zaprezentowano parametry buksowana dynamcznego sterowana automatycznego procesu ślzgana dla lokomotywy systemów prądu AC/AC.. TOMASZEK H, JASZTAL M, ZIEJA M. Uproszczona metoda oceny trwałośc zmęczenowej wybranych elementów konstrukcj statku powetrznego dla zmennego wdma obcążena. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 29-34. Ocena trwałośc zmęczenowej elementu konstrukcj pracującego pod wpływem zmennego wdma obcążena przysparza welu trudnośc. Stąd potrzeba poszukwana uproszczonych metod umożlwających tą ocenę. Przedstawona praca obejmuje przekształcene wdma rzeczywstego o zmennych wartoścach cykl w wdmo jednorodne o cyklach ważonych. Wykorzystując wdmo przekształcone przedstawono metodę oceny trwałośc zmęczenowej wybranego elementu konstrukcj statku powetrznego z początkowym pęknęcem. Do 2 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

W SKRÓCIE - ABSTRACTS crack length a dfference equaton has been appled whch, after rearrangement, resulted n a partal dfferental equaton of the Fokker-Planck type. A densty functon of the crack length s a partcular soluton to ths equaton. Usng the densty functon of a crack length, fatgue lfe of the structural component has been determned for the crack that keeps growng up to the permssble value l d lower than the crtcal value l kr. What has been gven consderaton n ths study s the case when the exponent of the Pars equaton m 2. DZICZKOWSKI L, BUCHACZ A. Modfcaton of a calbraton method for conductometers. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 35-40. The paper presents an nnovatve and mproved method for doubled calbraton of an eddy-current conductometer. Implementaton of the proposed method makes t possble to acheve ndependence of the nstrument ndcatons on surface condton of the examned parts. The calbratng functon s determned on-lne when the measurements are n progress on the bass of parameters that are assgned to the contact probe col and calculated wth use of general mathematc equatons. The calculated correcton parameter can be consdered as a measure of the surface roughness. Applcaton of the modfcaton as descrbed n ths paper can contrbute to effcency mprovement of operatonal examnatons for components made of non-ferrous materals durng ther lfetme. ŚWIĆ A, MAZUREK L. Modelng the relablty and effcency of flexble synchronous producton lne. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 4-48. The paper ntroduces a mathematcal model of operaton of a flexble synchronous producton lne (FSPL) of multfunctonal CNC machnes that ncludes one redundant multfunctonal CNC machne whch can take over the functons of every FSPL machne. The graph of FSPL state, relatons and equatons used to calculate relablty and productvty are shown. Maple, the software used for relablty and productvty calculatons and modellng, as well as the mathematcal results are presented. CHENG Z, BAI Y, CAI L, WANG L, LI P, CHEN L. Research on Warranty Interval of Mult-component System wth Falure Interacton. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 49-55. Based on the analyss of falure nteracton, mperfect preventve warranty polcy s adopted for the mult-component system. Average falure rate of each warranty nterval s studed and warranty cost model and avalablty model are bult as vewed from nteractve falure rate. Then Warranty perod project s brought forward as an example, whch can valdate the feasblty of model and show the advantage of the project. The research can provde technque and methods for determnng Warranty Perod of mult-component system, whch further enrches and perfects the warranty theory. BRZOZOWSKI K, NOWAKOWSKI J. Toxcty of exhaust gases of compresson gnton engne under condtons of varable load for dfferent values of engne control parameters. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 56-62. Ths work presents a procedure leadng to emprcal dentfcaton of the dependency of smoke level and emsson of harmful compounds on selected engne control parameters n an compresson gnton engne. In the course of experments dentfcaton of emsson of ntrogen oxdes, carbohydrates, carbon oxde and smoke level of exhaust gases was carred out, dependng on: rotatonal speed, fuel amount, exhaust gas recrculaton factor and njecton tmng. Usng artfcal neural networks has been proposed to generalze the results of experments. The computed values of coeffcents used to evaluate approxmaton errors and predcton of smoke level and emsson of harmful compounds confrm effectveness of the proposed method of generalzng the dentfcaton procedure. SHAKUNTLA S, LAL A K, BHATIA S S. Comparatve study of the subsystems subjected to ndependent and smultaneous falure. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 63-7. The paper dscusses the comparson between avalablty of a ppe manufacturng ndustry when the sub systems are subjected to smultaneously and ndependent falure. The falure rates of the sub-systems are constant and the repar rates are varable. The governng dfferental equatons of the system are solved usng Lagrange s Method. The performance evaluaton of system s done by means of long run avalablty makng use of software package Matlab 7.0.4. The tables modelowana przyrostu długośc pęknęca wykorzystano równane różncowe z którego po przekształcenu otrzymano równane różnczkowe cząstkowe typu Fokkera-Plancka. Rozwązanem szczególnym tego równana jest funkcja gęstośc długośc pęknęca elementu. Wykorzystując następne funkcję gęstośc długośc pęknęca określono trwałość zmęczenową elementu konstrukcj dla pęknęca narastającego do wartośc dopuszczalnej l d mnejszej od wartośc krytycznej l kr. W pracy rozpatruje sę przypadek, gdy wykładnk równana Parsa m 2. DZICZKOWSKI L, BUCHACZ A. Modyfkacja metody skalowana konduktometru. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 35-40. Zaproponowano orygnalną, ulepszoną metodę podwójnego skalowana konduktometru wroprądowego. Realzując tę metodę można uzyskać nezależność wskazań przyrządu od stanu powerzchn badanych elementów. Funkcja skalująca konduktometr jest wyznaczana dynamczne w czase trwana pomaru na podstawe parametrów przypsanych eksperymentalne sondze-cewce stykowej oraz unwersalnych równań matematycznych. Oblczona poprawka może być traktowana jako mara chropowatośc powerzchn. Stosując opsaną modyfkację można znaczne usprawnć badana eksploatacyjne elementów wykonanych z metal neżelaznych. ŚWIĆ A, MAZUREK L. Modelowane nezawodnośc wydajnośc synchroncznej elastycznej ln produkcyjnej. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 4-48. Przedstawono model matematyczny funkcjonowana synchroncznej elastycznej ln produkcyjnej (SELP) z obrabarek welozadanowych CNC, w skład której wchodz maszyna technologczna rezerwowa. Maszyna technologczna rezerwowa może przejmować funkcje każdej z obrabarek SELP. Przedstawono graf stanu SELP, zależnośc, równana do oblczana nezawodnośc wydajnośc SELP. Opracowano program oblczeń nezawodnośc wydajnośc (Maple) oraz zaprezentowano wynk modelowana optymalzacj lośc obrabarek. CHENG Z, BAI Y, CAI L, WANG L, LI P, CHEN L. Badana okresu gwarancyjnego dla systemu weloskładnkowego, w którym zachodzą nterakcje uszkodzenowe. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 49-55. W oparcu o analzę nterakcj uszkodzenowych, przyjęto dla systemu weloskładnkowego poltykę gwarancyjną obejmującą nepełną odnowę proflaktyczną.zbadano średną ntensywność uszkodzeń dla każdego okresu gwarancyjnego oraz skonstruowano modele kosztów obsług gwarancyjnej oraz dostępnośc borąc pod uwagę ntensywność uszkodzeń nterakcyjnych. Jako przykład podano projekt okresu gwarancyjnego, który może potwerdzć poprawność przyjętego modelu oraz przedstawono zalety takego projektu. W badanach opracowano technkę metody ustalana okresu gwarancyjnego dla systemów weloskładnkowych, które stanową stotny wkład do teor gwarancj. BRZOZOWSKI K, NOWAKOWSKI J. Toksyczność spaln slnka o zapłone samoczynnym w warunkach zmennego obcążena dla różnych wartośc parametrów regulacyjnych. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 56-62. W pracy przedstawono postępowane prowadzące do eksperymentalnej dentyfkacj zależnośc zadymena emsj zwązków szkodlwych od wybranych parametrów regulacyjnych slnka o zapłone samoczynnym. W ramach badań przeprowadzono dentyfkację emsj tlenków azotu, węglowodorów, tlenku węgla oraz zadymena spaln w zależnośc od: prędkośc obrotowej, dawk palwa, stopna recyrkulacj spaln kąta wyprzedzena wtrysku. Do uogólnena wynków badań eksperymentalnych zaproponowano wykorzystane sztucznych sec neuronowych. Oblczone wartośc wskaźnków wykorzystanych do oceny błędów aproksymacj oraz predykcj zadymena emsj zwązków szkodlwych spaln potwerdzają skuteczność zaproponowanej metody uogólnena badań dentyfkacyjnych. SHAKUNTLA S, LAL A K, BHATIA S S. Badana porównawcze podsystemów ulegających uszkodzenom nezależnym jednoczesnym. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 63-7. W artykule porównano dostępność zakładu przemysłowego produkującego rury w przypadkach występowana jednoczesnych nezależnych uszkodzeń podsystemów. Badana prowadzono przy stałych ntensywnoścach uszkodzeń podsystemów zmennych ntensywnoścach napraw. Konstytutywne równana różnczkowe systemu rozwązano przy użycu metody Lagrange'a. Oceny wydajnośc systemu dokonano na podstawe długotrwałej dostępnośc z wykorzystanem paketu oprogramowana Matlab 7.0.4. Przedstawono tabele dla Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 3

W SKRÓCIE - ABSTRACTS for varous parameters are gven whch can be useful to the plant management for mprovng and plannng the mantenance schedule. LATALSKI J. Modellng of macro fber composte pezoelectrc actve elements n ABAQUS system. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 72-78. The paper presents an approach to effectve modellng of pezoelectrc transducers n fnte element method based software. A macroscopc model of an actve element made of macro fber composte (MFC type) exhbtng d 33 effect s developed n ABAQUS system. Next, a multlayer composte beam wth the dscussed pezoelectrc actuator s analysed. Both a drect and a converse pezo effects are analyzed numercally, calculatng respectvely charge on transducer s poles subject to forced dsplacements and beam statc deflectons wth respect to assumed supply voltage. The outcomes of the numercal smulatons are compared to the laboratory test results. Next, the worked-out FEM model of MFC actuator/sensor s used to detect the smulated defect n composte materal. NOWAKOWSKI T. Problems of relablty modellng of multplephased systems. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 79-84. Ths artcle dscusses the basc problems connected wth modellng the relablty of multple-phased systems. Operaton and mantenance of such systems s assocated wth executon of varous tasks whch together lead to the achevement of the fnal goal. Such systems nclude logstc and transport systems. Two types of models are dscussed: synthetc models, whch cover a system s entre operatng lfe and models n whch the partcular phases are consdered separately. As an example, a k out of n system s used wth dfferent k parameters for each phase. The advantages and dsadvantages of three models are dscussed: the conservatve model whch s based on an analyss of the block relablty model, a Markov model wth fxed duraton of each phase and a Markov model wth random phase duraton. różnych parametrów, które mogą być wykorzystywane przez osoby zarządzające produkcją przy poprawanu planowanu harmonogramów przeglądów. LATALSKI J. Modelowane w systeme ABAQUS pezoelektrycznych elementów aktywnych typu MFC. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 72-78. W pracy przedstawono sposób efektywnego modelowana kompozytowych elementów pezoelektrycznych metodą elementów skończonych. W systeme ABAQUS przygotowano makroskopowy model elementu aktywnego typu MFC wykorzystującego efekt pezoelektryczny d 33. W dalszej kolejnośc przeanalzowano welowarstwową belkę kompozytową z naklejonym badanym elementem aktywnym. Numeryczne zbadano prosty odwrotny efekt pezoelektryczny, wyznaczając odpowedno wartośc napęć na zacskach elementu aktywnego przy wymuszonym odkształcenu układu oraz ugęca statyczne przy różnych wartoścach napęca zaslającego. Rezultaty tych analz porównano z wynkam rzeczywstych pomarów przeprowadzonych na stanowsku laboratoryjnym. Opracowany model numeryczny wykorzystano do próby wykryca symulowanego uszkodzena materału kompozytowego. NOWAKOWSKI T. Problemy modelowana nezawodnośc systemów welofazowych. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 79-84. W artykule omówono podstawowe problemy zwązane z modelowanem systemów welofazowych. Eksploatacja takch systemów zwązana jest z realzacją różnych zadań, które składają sę na osągnęce celu końcowego. Do takch systemów można zalczyć systemy logstyczne systemy transportowe. Omówono dwa rodzaje model: modele syntetyczne ujmujące cały okres eksploatacj systemu modele, w których poszczególne fazy są rozpatrywane oddzelne. Wykorzystano przykładowy system o strukturze progowej zmennej w kolejnych trzech fazach eksploatacj. Przedstawono zalety wady korzystana z modelu konserwatywnego bazującego na analze modelu blokowego model Markowa z ustaloną losowo zmennych czasem trwana poszczególnych faz. 4 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

nauka technka Artcle ctaton nfo: SZYBKA J., BRONIEC Z., PILCH R.: Forecastng the falure of a thermal ppelne on the bass of rsk assessment and explotaton analyss. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 5-0. Jan SZYBKA Zdzsław BRONIEC Robert PILCH Forecastng the falure of a thermal ppelne on the bass of rsk assessment and explotaton analyss Prognozowane uszkodzena cepłocągu na podstawe oceny ryzyka eksploatacj* In ths paper, a procedure for determnng the utlsaton perod of a thermal ppelne, exploted n mnng plants was presented. On the bass of analyses conducted n the real lfe stuatons, the ppelne falure, whch could be caused by ts burstng due to exceedng ppelne strength, was forecasted. The explotaton condtons of the thermal ppelne and factors contrbutng to ts degradaton were also analysed. Usng the probablty samplng method, the explotaton rsk was assessed and the lfespan of the ppelnes, after whch the ppelnes shall be replaced by new structures, were determned. Keywords: thermal ppelne, falure, ppelne resstance, explotaton rsk, forecastng the falure. W artykule przedstawono procedurę postępowana w wyznaczena okresu użytkowana cepłocągu eksploatowanego w zakładach górnczych. Na podstawe badań przeprowadzonych w warunkach rzeczywstej eksploatacj prognozowano awarę rurocągu, która może być spowodowana pęknęcem na skutek przekroczena jego wytrzymałośc. Opsano warunk eksploatacj cepłocągu czynnk wpływające na jego degradację. W ujęcu probablstycznym oszacowano ryzyko eksploatacj wyznaczono okres użytkowana, po którym pownna nastąpć wymana rurocągów. Słowa kluczowe: cepłocąg, uszkodzene, wytrzymałość rurocągu, ryzyko eksploatacj, prognozowane uszkodzena.. Introducton Ensurng the falure free work of the analysed thermal ppelne s a prorty for the enterprse, on ste of whch the ppelne s exploted. Even a short break n hot water supply can cause serous complcatons for the normal work of the plant and can cause sgnfcant fnancal losses. The assessment of the falure hazard s an mportant nformaton, as t s possble to plan preventatve actons n the explotaton systems, whch wll prevent the damage, whch could otherwse occur n the near future. Problems of relablty and mantenance rsk of ppelnes operatons are presented n artcles [5, 8, 9]. The exstng lterature manly focuses on the nfluences of corroson and ageng elements of the ppelne [3, 4, 6]. The mportant problem, assocated wth forecastng falures of the thermal ppelne s a preventve mantenance of the ppelne [3], and n the analysed case, t s an ssue of buldng an addtonal, spare ppelne, whch wll be used nstead of the exstng one. The so far exploted thermal ppelne, after renovaton, wll be used as an addtonal ppelne n case of any falures and durng the planned repar works. Two thermal ppelnes, mutually supplementng each other durng the explotaton, guarantee hgh relablty of hot water supply. In ths paper, the authors presented the methodology for forecastng the ppelne falure and or determnng the lfespan of a thermal ppelne n a real lfe stuaton. (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 5

NAUKA I TECHNIKA 2. Structural explotaton characterstcs and dentfcaton of damage to the thermal ppelne The analysed object the ppelne s used for transportng hot water, of the average temperature around 463 K. The values of work pressures of the transported water vary from,33 to 2,44 MPa wthn a 24 hour perod and a maxmum flow output reaches 700 t/h. The ppelne was bult of seamless steel ppes, manufactured of a low carbon, R35 steel (P235GH) PN- 80/H-7429. The requred mnmum value of the yeld pont of the ppe materal, at 473 K equals R et 85 MPa. The ppelne s protected wth a thermal nsulaton of the thckness of 25 cm, comprsng of a layer of glass wool mat, covered wth a protectve layer of tar paper and alumnum sheetng. The most mportant factor n assessng the techncal condton of the ppelne and determnng ts damage due to explotaton s the change of the ppe wall thckness, beng a result of pttng [3, 4, 0]. The analysed ppelne s also subjected to adverse nfluence of the chemcal envronment of the transported medum and, occurrng smultaneously, mpulse dynamc loads. The progressng degradaton of the steel mcrostructure occurs as a result of the nfluence of the chemcally aggressve compounds, stresses, hgh temperature and tme. The basc cause of the changes n the thckness of the ppelne walls s the electrochemcal corroson, beng a result of hgh mneralsaton of used water. Addtonal cause of a quck oxdaton of the ppe surfaces s hgh temperature of water, what furthers the corroson. The presence of numerous chemcal compounds, partculate suspended n the water and hgh temperature also contrbute to an accelerated oxdaton of the surface of the metal and, to a certan degree, to the abrason of the surface of the ppelne [0]. The results of the conducted measurements of the thckness of the ppelne walls show, that corroson pttng n some places s several mllmetres deep. Ths leads to lowerng the area of dangerous cross-sectons of the ppelne and to ncreasng the summarsed compound stresses. Another type of destructve effect of chemcal corroson are deep pttng on the surface of the metal. The local change n the thckness of the ppe wall result n ppelne deformaton proportonal to the temperature dfference. When applyng the superposton prncple for deformatons caused by the explotaton stresses (pressure of the medum and ts changes n tme), t s also mportant to nclude deformatons caused by thermal stresses. The changng load of the ppelne s also vsble n hgh values of ppelne shft on erected supportng structures and towers. On the bass of observaton, conducted over many years, the change of the ppelne shft on the supportng structures n all drectons, x, y, z, was also proven. Ths shft, reachng up to several dozen centmetres, lead to changes n the ppelne route, and n one case, t even led to the ppelne slppng of the supportng structure. The ppelne deformatons, caused by ts shft on the supportng structures are partcularly hazardous for weakened cross-sectons. 3. Assessng the stresses n the selected sectons of the ppelne In thn wall ppes, subjected to an nternal pressure p (the thn wall condton: h/r 0,2 s fulflled) crcumferental stresses σ occur. Those stresses assume the same values for the entre thckness of the ppe (for the specfc cross-secton) [7]: σ = r p h ( D + D ) () Z W r = (2) 4 where: h thckness of the ppe wall (mm), r average radus of the ppe (mm), D z external radus of the ppe (mm), D w nternal radus of the ppe (mm), R average radus f the ppe bend (mm). For pressure changes, determned on the bass of analyses conducted over many years, the followng values were determned: p śr =,75795 MPa, p mn =,33457 MPa, p max = 2,44297 MPa and for ppe thckness (φ508x) n selected measurng ponts, the run of the changes of the crcumferental stress σ were determned. Crcumferental stresses (caused by the nternal ppe pressure) along the crcumference of the ppe change n places, where the ppelne course changes from a straght secton to a bent secton. Average crcumferental stress n the wall, on the nternal sde of the bend of a curved ppelne equals: σ wew ( ) ( ) σ R 05, r = R r and on the external sde of the bend equals: σ zew ( ) ( ) σ R+ 05, r = R+ r For the φ508x ppe and for the average bendng radus R = 750 mm the followng stress values were obtaned: σ σ wew = zew = (3) (4), 247 σ (5) 0, 876 σ (6) Crcumferental stress along the straght secton of the ppe (φ508x) for the cted pressure changes equals: σ = 248, 5 p Takng nto consderaton the regstered pressure values, the followng was obtaned: σ mn = 30,493 MPa, σ śr = 39,739 MPa, σ max = 55,89 MPa Takng nto consderaton the formula for calculatng the average crcumferental stress on the nternal sde of the bend of the curved ppelne, the followng was obtaned: σ wew, mn = 37,5962 MPa, σ wew, śr = 49,5225 MPa, σ wew, max = 68,8208 MPa Impulse (dynamc) effect of the pressure can double the ppelne shft on the supportng structures [2] as compared to the effects of the statc pressure. As a result, the ppelne shfts on the supportng structures and ts fxng (supports) become unstablsed. A justfed, engneerng soluton to the problem s, n ths case, the change of the constructon and the methods of suspendng and supportng the compensator. Also a result of an mpulse load [2] along the length of the ppelne or crosswse to the course of the ppelne, s a double ncrease of crcumferental stress, as compared to to the effects of the statc pressure. (7) 6 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology The propertes of the materals show a sgnfcant dsperson of expermental data, obtaned whle analysng ts mechancal propertes. Ths dsperson s also present when analysng the fatgue strength and statc strength, e.g. the yeld pont R e or mmedate strength R m, thus the resstance of the materal can be consdered to be a random varable. The coeffcents of varaton υ R charactersng the dsperson of the mechancal propertes can be obtaned from statstcal analyses, accordng to Warszyńsk [2], for steel υ R = 0,05 0, and t s an ndcator of the materal qualty, whch depends on the manufacturng and further treatment condtons. The average value of R e, śr and the mnmum value of R e, mn of the yeld pont can be estmated wth the use of the followng formula: ( ) R = R u v (8) e,mn e, sr R where the value of u s assumed: u= 2 3, that s the value, ensurng, wth the probablty exceedng 0,98, that the resstance wll be greater than R e, mn. Utlsng the afore recommendatons and formulas, the dstrbuton parameters of the yeld pont R e were determned n the followng chapters. 4. Assessment of the probablty of damage the ppelne The analysed thermal ppelne s exploted n dffcult and varable external condtons. Both, the pressure and the temperature of the transported medum change n tme. In addton, the transported medum can be charactersed by ts chemcal composton, whch, n conjuncton wth hgh temperature, causes a fast progressng corroson of the thermal ppelne walls. Recedng thckness of the ppelne walls and the dynamc character of the load are a factor contrbutng to a constant rsk of exceedng the ultmate lmt state and the damage occurrng. Probablty of damage of the thermal ppelne can be determned wth the use of the results of the measurements of the wall thckness of the thermal ppelne and the results of the regstered changes of the thermal load (pressure and temperature). Fg.. Strength and stress dstrbuton [] The approxmated estmaton of the probablty of the damage of the structure can be determned usng the Streleck method. The probablty of damage P s determned on the bass of load dstrbuton and the structure strength (Fg. ) and t s expressed wth the dependency []: ω ω < P < ω + ω ω ω (9) 2 2 2 where: ω and ω 2 areas under the strength densty dstrbuton curve and the load nduced, stress dstrbuton curve were presented n Fg.. On the bass of the strength data of the materal, of whch the ppelne was made (data ncluded n paragraphs 2 and 3), the dstrbuton of ts yeld pont was descrbed by a normal dstrbuton N (264; 39,5). The stress dstrbuton was determned on the bass of the measured wall thckness and the regstered pressure changes. The probablty of ppelne damage, determned wth the use of the Streleck method, for the average wall thckness and for the worst measured pont, n varous years of ppelne explotaton, were presented n table. Due to the small values of the probablty of ppelne damage, those values were expressed as a percentage. The determned above probablty of damage, for both: the average wall thckness and the worst measured pont, only gves the pont assessment. However, much more hazardous ponts, where the wall thcknesses can be lower, can also occur n the analysed secton of the ppelne. Thus, another method of makng a rsk assessment was proposed. Accordng to ths method, the wall thckness dstrbuton, of thcknesses measured at a certan date, shall be determned and referred to the determned mnmum wall thckness of the thermal ppelne. On the bass of the strength condton: Thus t can be noted as: h r σ = p R e,mn (0) h mn DZ p = () 2 R + p e,mn Takng nto consderaton the maxmum measured pressure n the ppelne, and ts dynamc propertes, the h mn =6,64 mm was determned. Next, the dstrbuton of the wall thcknesses, measured at certan dates, was determned and the probablty, that n the system, the value equal to or lower than h mn wll occur, was establshed (Fg. 2). Ths probablty determnes the possblty of wall thckness equal to or lower than the mnmum requred thckness occurrng n the analysed secton of the ppelne. It was assumed, that ths s also a rsk of a damage to the ppelne occurrng. The determned values of probablty of damage, determned wth the use of the descrbed method were presented n table 2. Tab.. Probablty of damage of the thermal ppelne, determned wth the use of the Streleck method [] Average from the measurements The worst pont Year, the measurements were taken 999 200 200 999 200 200 Consecutve year the ppelne was exploted 6 8 7 6 8 7 ω 0,002476 0,00327 0,00703 0,0095 0,0895 0,04033 ω 2 0,00656 0,00234 0,005385 0,009 0,0743 0,04704 Probablty of ppelne damage P [%] > 0,0004 0,000723 0,00379 0,00023 0,033 0,9 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 7

NAUKA I TECHNIKA Fg. 2. Dstrbuton of wall thckness of the thermal ppelne, measured at dfferent dates Tab. 2. Probablty of damage of the thermal ppelne [] Year, the measurements were taken 999 200 200 Consecutve year the ppelne was exploted 6 8 7 Wall thckness dstrbuton N(2,8; 0,9) N(8,85;,46) N(7,5;,3) Probablty of ppelne damage P P 0 P=0,065 P=0,223 The probabltes of the ppelne falure, calculated wth the use of the proposed method are much greater than those, calculated wth the use of the Streleck method. Ths s due to the fact that n accordance to the specfed wall thckness dstrbuton, t s also possble, that the walls are thnner than t was determned by conductng measurements n selected ponts. Ths approach better represents the analysed, real lfe segment of the thermal ppelne, on whch, only a mnmal percent of ts actual surface was analysed when takng the measurements. 5. Forecastng the falure of the thermal ppelne The probablty of falure of the thermal ppelne, as explaned n Secton 4, was determned for ths pont on the ppelne, n whch the wall thnnng acheved maxmum values. Ths pont was marked wth number 2 (Fg. 3) and was regarded as the most hazardous. The measurements of the ppelne wall thckness were taken n 6 dfferent ponts, marked n Fg. 3 wth numbers from 2 to 46. In four cases, the thnnng of the ppelne wall could be regarded as hazardous, due to the lowerng of the ppelne strength and n those places, the ppelne falure can occur n a near future. Those partcularly hazardous spots add to 25% of all measurements taken on the ppelne. A precse assessment of the techncal condton of the ppelne s not possble due to the fact, that the ppelne s thermally nsulated. Consderng, that the taken measurements are representatve, t can be assumed that f the research was conducted, more ponts, beng a potental centre of ppelne falures would be found. A precse assessment of the techncal condton of the ppelne s not possble due to the fact, that the ppelne s thermally nsulated. It can be assumed that by contnung the measurements, the rato of the partcularly hazardous areas to the general number of measured ponts wll reman unchanged. Damage of a ppelne s a serous threat to the normal functonng of the enterprse, thus all feasble techncal ntatves, preventng the ppelne falure, are undertaken. Takng nto consderaton dmnshng wth tme strength of the ppelne, the erecton of a second thermal ppelne, supplantng the exstng one, s consdered. Ths can lead to erectng two ppelnes, mutually supplementng each other and guaranteeng hgh relablty of hot water supply. The problem of determnng, when the further explotaton of the ppelne can be so rsky, that the constructon of the second lne wll become necessary, becomes more and more mportant. Takng nto consderaton the dvagatons presented n Secton 4, t was assessed, that for the weakest pont on the ppelne (pont no: 2) the damage wll occur after 20 years of explotaton []. Ths concluson was confrmed by the dstrbuton of the falure frequency after 7 years of ppelne explotaton, as presented n fgure 2, where around 22% of the values of the densty functon can exceed the thckness border value h mn = 6,64 mm. An addtonal crteron, determnng the necessty of replacng the thermal ppelne are the economc reasons. The expected value of the falure occurrng can be calculated on the bass of the renewal functon: ( ( )) ( ) N()= t + N t x df x (2) where: N(t) renewal functon, t-x remanng explotaton tme after the frst falure, t explotaton tme, F(x) dstrbuton of a random varable, beng the ppelne explotaton tme. Ppelne falure generates the necessty of conductng repar works, the costs of whch can be estmated. The frst ppelne falure, beng a result of corrosve processes shall be a sgnal that the subsequent falures can occur n a near future and n varous zones of the thermal ppelne. Ths s also a sgnal that the ppelne strength was also lowered and the tme has come to consder replacng the ppelne. By contnung the use of the ppelne, threatened by falures, one can generate proft, whch can be calculated. For the 8 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology analysed case, the rato of the proft from ppelne explotaton to fnancal losses, consderng the repar costs was 90 to. Takng nto consderaton ths rato, the probablty of generatng proft and the rsk of ncurrng losses were presented n fgure 4. The pont, n whch the curves crossed determned the tme, after whch further ppelne explotaton became unproftable. For the analysed case, ths tme perod was 20 years. The probablty of generatng proft and ncurrng losses after 6 years of ppelne explotaton was presented n fgure 4. Ths secton shows the explotaton perod before and after the frst ppelne falure occurred. The probablty of ths falure was calculated on the bass of the value of the renewal functon. Ths s a senstve perod n the ppelne explotaton perod; sgnallng the necessty of ppelne replacement. The determnaton of the explotaton perod of the thermal ppelne was conducted wth the use of the probablty samplng method. The worst possble cases were consdered and the presented verson s the pessmstc assessment. The appled, sgnfcant safety margn was determned by the fact that buldng a new thermal ppelne s a complcated and a costly techncal economcal venture, whch cannot be completed n a short perod of tme. Fg. 3. Locaton of measurement ponts spots n whch the thckness of the ppelne wall was measured and locaton of spots n whch thermo vson measurements were taken on the hot water (2,, 46 exact ponts n whch the measurements were taken) Fg. 4. Determnng the explotaton perod of the thermal ppelne Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 9

NAUKA I TECHNIKA 6. Conclusons In explotaton processes, determnaton of the devces and nstallaton explotaton perod s an mportant manageral ssue. Unjustfed lengthenng of the explotaton perod wll most often lead to sgnfcant losses and ecologcal hazard. Makng such decson s never easy. The proposal of employng the methods of assessng relablty of devces and nstallatons presented n ths artcle s a unversal one and t can be used for solvng ths types of problems n varous branches of the machne ndustry. 7. References. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. 3. Bołotn W. Metody statystyczne w mechance budowl. Wydawnctwo Arkady, Warszawa, 968. Brodny J. Wyznaczane współczynnka dynamcznego przy mpulsowym dzałanu sły. Prace naukowe GIG. Górnctwo Środowsko. Wyd. spec. nr 7, Katowce 2008. Caleyo F, Gonzalez J L, Hallen J M. A study on the relablty assessment methodology for ppelnes wth actve corroson defects. Internatonal Journal of Pressure Vessels and Ppng 2002; 79: 77-86. Qan G, Nffenegger M, L S. Probablstc analyss of ppelnes wth corroson defects by usng FITNET FSS procedure. Corroson Scence 20; 53: 855-86. Kahaleel M A, Smonen F A. Effect of through-wall stress gradents on ppng falure probabltes. Nuclear Engneerng and Desgn 2000; 97: 89-06. Kołowrock K, Kwatuszewska-Sarnecka B. Relablty and rsk analyss of large systems wth ageng components. Relablty Engneerng and Systems Safety 2008; 93: 82-829. Nezgodzńsk M E, Nezgodzńsk T. Wzory, wykresy tablce wytrzymałoścowe. WNT, Warszawa, 2006. Ten S-W, Hwang W-T, Tsa C-H. Study of rsk-based ppng nspecton gudelne system. ISA Transactons 2007; 46: 9-26. Soszyńska J. Relablty and rsk evaluaton of a port ol ppelne transportaton system n varable operatons condtons. Internatonal Journal of Pressure Vessels and Ppng 200; 87: 8-87. Stachowak A, Zwerzyck W. Verfcaton of computatonal model for corrosve and mechancal wear. Eksploatacja Nezawodnosc Mantenance and Relablty 2009; 4(44): 28-36. Szybka J, Plch R, Bronec Z, Tarnowsk J. Ocena ryzyka eksploatacj cepłocągu. Problemy Eksploatacj Zeszyty Naukowe Instytutu Technolog Eksploatacj - PIB nr. Radom, 20; 80: 75-83. Warszyńsk M. Nezawodność w oblczenach konstrukcyjnych. PWN, Warszawa, 988. Song Y, Ma L, Morrs J. A practcal approach for relablty predcton of ppelne systems. European Journal of Operatonal Research 2009; 98: 20-24. Prof. dr hab. nż. Jan SZYBKA Dr nż. Zdzsław BRONIEC Dr nż. Robert PILCH AGH Unversty of Scence and Technology Faculty of Mechancal Engneerng and Robotcs Department of Machne Desgn and Technology Al. A. Mckewcza 30, 30-059 Kraków E-mal: szybja@agh.edu.pl 0 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: LEWI A, PEEW D. On the evaluaton of explotaton qualtes of Prsta Super 25W40 motor ol wth DIA method. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: -4. Awram LEWI Darn PEEW On the evaluaton of explotaton qualtes of Prsta Super 25W40 motor ol wth DIA method Ocena cech eksploatacyjnych oleju slnkowego Prsta Super 25W40 metodą DIA The paper consders one of the few studes of motor ol usng Dfferental Impedance Analyss (DIA) method. The mpedance data of Prsta Super 25W40 motor ol are analyzed through ths method, whch s more advanced technque comparng to Electrochemcal Impedance Spectroscopy. Structural changes n samples of long exploted ol are found. The analyss of samples of less used ol regsters one mportant phenomenon, whch s known from the practce. The possbltes for vscosty ndces determnaton are dscussed. General mpedance data analyss s also conducted. Keywords: Dfferental Impedance Analyss, Motor Ol, Olness, Qualty Control. Artykuł pośwęcony jest badanom oleju slnkowego przy pomocy Analzy Dyferencjalnej Impedancj (DIA). Zbadano tą metodę, która jest doskonalsza od Elektrochemcznej Spektroskop Impedancyjnej, zmany mpedancj nowego oleju Prsta Super 25W40 w okrese eksploatacj. Wynk wykazują, że po pewnym krótkm okrese, używany olej ma lepsze właścwośc smarne względem próbk oleju neużywanego. Omówono także możlwośc oszacowana olestośc przeprowadzono analzę wynków. Słowa kluczowe: Analza Dyferencjalnej Impedancj, olej slnkowy, olestość, charakterystyk lepkośc, analza jakoścowa oleju.. Introducton Recently a number of real-tme motor ol condton systems have been developed. Usually the purpose of such systems s to determne the approprate moment for the ol change. Ths s due to the fact that economcal losses arse n both cases - f the ol has been changed before or after the useful lfe of the lubrcant has been exhausted. The second case leads to a fast machne wear. The economcal effect n the large commercal fleets s substantal. The most commonly used crteron for the ol change nterval determnaton s the elapsed mleage. Ths doesn t ensure the optmal use of the lubrcant, because of the varous factors that affect ts useful lfe. All the laboratory methods requre a consderable amount of tme and resources. For ths reason the systems for real-tme analyss fnd a wde applcaton. Frequently as ntal parameters for the analyss are used the specfc resstvty and the delectrc permttvty. The hgh precson of the methods whch use the Electrochemcal Impedance Spectroscopy (EIS) s well known [2, 9]. Durng the past years an advanced technque for mpedance data processng named Dfferental Impedance Analyss (DIA) was developed [9]. It does not requre an ntal workng hypothess and thus overcomes some prncpal dsadvantages of the known systems. The new capabltes of the DIA method permt to evaluate of vscosty ndces of the motor ol. The present study s dedcated to the applcaton of the DIA method to the analyss of the explotaton qualtes of Prsta Super 25W40 motor ol. 2. Basc equatons and problem formulaton The mpedance measurements are performed usng hgh precson Frequency Response Analyzer - Solartron 260 FRA at room temperature wthn a frequency range MHz - 0.Hz wth a densty of 5 ponts per decade. The graphcal representaton of the results for dfferent ol samples dependng on the explotaton cycle are shown on fgure. The mpedance of the cell Z( jω ) typcally has a resstancecapactve character (ω - angular frequency). There are two mechansms whch cause the total decreasng of the mpedance by the explotaton perod. The frst s the ol oxdaton whch result n ncreasng the concentraton of polar components, such as carboxylc acds, ketones and aldehydes et al. Ths leads to an ncreased dpolar and onc nature of the soluton [8]. The second s the presence of dfferent contamnants - soot, antfreeze, water and metal partcles from the machne wear causng the reducton of the specfc resstvty. The spectral tale at the lowest frequency regon s one of the reasons that obstruct modelng wth smple equvalent crcut. Ths s due to the sgnfcant growth of the delectrc permttvty between the frequences 0.Hz - 0 Hz, possbly caused by the formaton of mcelles. General mpedance data analyss Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

NAUKA I TECHNIKA Fg.. Nyqust plot the fresh and the exploted ols cannot nvestgate the non-lnear character of the result and the dentfcaton of the motor ol explotaton qualtes. 3. Soluton of the problem For the purposes of Dfferental Impedance Analyss n the tme doman (Temporal Analyss) t s used the effectve tme constant dstrbuton lgt = F(lgf - ), where f = ω/2π s the frequency of the stmulus sgnal; T effectve tme-constant of the local scannng model accordng to DIA []. The calculaton of the effectve tme-constant for specfc frequency s based on: T(ω) = dl eff (ω)/dr eff (ω) () where: dl eff (ω) - the effectve nductance as a functon of the angular frequency, Reff ( ω)= Re Z ( jω) - the real part of the mpedance. The basc property of the ol s the lubrcatng ablty. It depends on the vscosty. The vscosty defne the lubrcatng effcency when the ol flm s thn enough and then the vscosty doesn t effect the lubrcaton [3, 7]. If the vscosty s nsuffcent the frcton and the machne wear s consderable, specally n the pston-cylnder group [4]. The vscosty s hgher n case of an easer molecule polarzaton processes [0]. The tme-constant dstrbuton for the partcular ol samples s shown on the fgure 2. The dagrams dsplay a frequency dependent and an approxmately frequency nvarant regons [0]. The decreasng of the T wth elapsed mleage when the shape of the dstrbuton remans the same means an mproved polarzaton ablty. The specfc negatve peak (area A on fgure 2.) s hgher at the fresh ol comparng to the exploted ol. At the sample wth longest explotaton cycle the representatve tme-constant has the shortest frequency Fg. 2. Temporal plot for the ol samples 2 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology A = 0 dp k db where: B = lg(f - ), P = lgt, k 0 - coeffcent of a proportonalty (const). The result s shown on the fgure 3. Ths way of calculaton s sutable only n case of a low nose and a hgh measurement precson. In general case the estmaton of the ampltude of the spectral lnes could be done by means of an accumulaton of the calculated results wth smlar values. A hgh nose mmunty s thus acheved [, 5]. The fgure 3 shows the tme-constant spectrum for a three ol samples. The representatve tme-constant of fresh ol dstnctly predomnates and s concentrated n a narrow scope of values. For the longest exploted ol of the excerpt that parameter feebly prevals over the spectrum and s more dstrbuted. Ths ndcates an ntal destructon of the polymer macromolecules,. e. certan structural changes [0]. The Spectral Analyss confrms the result from the Temporal Analyss that the ol at 000-2000 km has a better lubrcatng ablty comparng to the fresh ol. Its representatve tme-constant s smaller and has a hgher ampltude A than the fresh ol sample. Such a result can hardly be recognzed wth other known methods of analyss. The relaxaton tme wthn 0Hz-MHz s approxmately ms. Accordng to some authors the relaxaton processes at the frequency of several klohertz s related to the nverse mcelles, formed by hydrophlc core and hydrophobc hydrocarbon tals [8]. (2) Fg. 3. Spectral plot for the effectve tme-constant length (area B). Ths ndcates structural changes whch correspond to the destructon of the polymer macromolecules. By means of a Spectral Analyss some addtonal nformaton can be extracted. The spectral lnes formaton s conducted through a spectral transform procedure [9]. The ampltude A of the lne can be calculated through : 4. Conclusons The mpedance of the analyzed motor ols generally decreases wth the explotaton cycle. The Dfferental Impedance Analyss can detect changes n the molecular structure of the lubrcant. It regsters an mproved lubrcatng ablty of the ol at nearly 2000 km,.e. there s some actvaton perod of the ol. The DIA method dsplays a nose-mmunty and a hgh potental for dentfcaton. It produces better results comparng to the Electrochemcal Impedance Spectroscopy Technque [6], because of the absence of lmtatons from equvalent model. ********** The study was supported by contract BG05PO00-3.3.04/28, "Support for the Scentfc Staff Development n the Feld of Engneerng Research and Innovaton". The project s funded wth support from the Operatonal Programme "Human Resources Development" 2007-203, fnanced by the European Unon. ********** 5. References. Atkns P. Physcal Chemstry. Oxford: Oxford Unversty Press, 990. 2. Bard A, Faulkner L. Electrochemcal Methods: Fundamentals and Applcatons, New Jersey: J. Wley Int. Inc., 2000. 3. Hamrock B, Schmd S, Jacobsen B. Fundamentals of flud flm lubrcaton. New York: Marcel Dekker Publ., 2004. 4. Lščák S, Revaj V, Šulgan M. Vehcle s techncal condton and emsson. Eksploatacja Nezawodnosc - Mantenance and Relablty 2008; : 6-63. 5. Macdonald J R. Impedance spectroscopy. Annals of Bomedcal Eng. 992; 20: 289-305. 6. Manfeld F. Electrochemcal Impedance Spectroscopy (EIS) as a new tool for nvestgaton methods of corroson protecton. Electrochmca Acta 990; 35: 533-544. Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 3

NAUKA I TECHNIKA 7. Pnkus O, Sternlcht B. Theory of hydrodynamc lubrcaton. New York: Mc Graw-Hll Publ., 96. 8. Smechowsk M. Electrochemcal characterzaton of lubrcants for mcrofabrcated sensor applcatons. Case Western Reserve Unversty: PhD Thess. Department of Chemcal Eng., 2005. 9. Stoynov Z, Vladkova D. Dfferental Impedance Analyss [n Bulgaran]. Sofa: M. Drnov Acad. Publ. House, 2005. 0. Stoynov Z, Zhaohu X, Vladkova D. Dfferental mpedance analyss of motor ols. Bulgaran Chemcal Communcatons 2003; 35: 62 66. Prof. Awram LEWI, Ph.D. Darn PEEW, M.Sc. Department of Electroncs A.Kynczew Unversty, Ul. Studentska nr 8, 707 Ruse, Bulgara E-mal: alev@ecs.un-ruse.bg 4 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: CZECH P, MADEJ H. Applcaton of cepstrum and spectrum hstograms of vbraton engne body for settng up the clearance model of the pston-cylnder assembly for RBF neural classfer. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 5-20. Potr CZECH Henryk MADEJ Applcaton of cepstrum and spectrum hstograms of vbraton engne body for settng up the clearance model of the pston-cylnder assembly for RBF neural classfer Wykorzystane hstogramów wdma cepstrum drgań korpusu slnka do budowy wzorców luzu w układze tłok-cylnder dla klasyfkatora neuronowego RBF* The paper presents an attempt to evaluate the wear of pston-cylnder assembly wth the ad of vbraton sgnal recorded on spark gnton (SI) engne body. The subject of the study was a four-cylnder combuston engne. dm 3. Dagnosng combuston engnes wth vbraton methods s specfcally dffcult due to the presence of multple sources of vbraton nterferng wth the symptoms of damages. Dagnosng engnes wth vbro-accoustc methods s dffcult also due to the necessty to analyse non-statonary and transent sgnals [,7]. Varous methods for selecton of usable sgnal are utlsed n the dagnosng process. Changes of the engne techncal condton resultng from early stages of wear are dffcult to detect for the effect of mechancal defect maskng by adaptve engne control systems [5]. Accordng to the studes carred out, t s possble to utlse artfcal neural networks for the evaluaton of the clearance n pston-cylnder assembly. Keywords: dagnostcs, combuston engnes, artfcal neural networks. W artykule przedstawono próbę oceny zużyca złożena tłok-cylnder za pomocą sygnału drgań rejestrowanego na kadłube slnka ZI. Obektem badań był czterocylndrowy slnk spalnowy o pojemnośc, dm 3. Dagnozowane slnka spalnowego metodam drganowym jest szczególne utrudnona ze względu na występowane welu źródeł drgań, co jest przyczyną wzajemnego zakłócana symptomów uszkodzeń. Dagnozowane uszkodzeń slnków metodam wbroakustycznym jest trudne także ze względu na koneczność analzy sygnałów nestacjonarnych mpulsowych [,7]. W procese dagnozowana stosuje sę różne sposoby selekcj sygnału użytecznego. Zmany stanu techncznego slnka wywołane wczesnym fazam jego zużyca są trudne do wykryca ze względu na maskowana usterek mechancznych przez adaptacyjne układy sterowana slnka [5]. Z przeprowadzonych badań wynka, że stneje możlwość wykorzystana sztucznych sec neuronowych do oceny luzu w układze tłok-cylnder. Słowa kluczowe: dagnostyka, slnk spalnowe, sec neuronowe.. Introducton For the mprovement of the level of safety, the devces enablng current observaton of vehcle movement parameters are becomng more and more mportant. The fast development of technques nfluence on more and more wdespread uses of such devces [0, 2]. Modern IC engne mantenance programmes ncorporate varous methods and technques for early fault detecton to mantan effcency and hgh relablty [3, 5]. Dagnostc systems used n modern combuston engnes are ntended to localse the component or system whch, due to natural wear or damage, can no longer perform as specfed by ts manufacturer. For engnes, the hghest effcency of on-board dagnostcs has been acheved n the feld of toxc emsson control. Some defects, however, such as the wear of cylnder bearng surface above the admssble lmts, for a gven engne, n many cases cause no reacton of the dagnostc system. In most cases, ths s attrbutable to the algorthms for adaptaton controls of combuston engnes [8]. One of the methods for dagnostc data acquston s to montor the level of vbraton generated by engne components. The major ssue referred to n the lterature related to methods of artfcal ntellgence s the method for creatng data used n the process of neural network operatons. The ablty to set up models s the guarantee for a successful classfyng process usng neural networks [4, 9,, 3-6]. Data n the experments carred out s derved from tme runs of the vbraton acceleratons n the engne body. The subject of tests was a Fat Panda wth SI engne. dm 3. The tests (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 5

NAUKA I TECHNIKA were carred out n Bosch FLA 203 roller bench. The vbraton acceleraton sgnal of the engne body was measured perpendcularly to the cylnder axs wth a sensor placed at the 4th cylnder. Vbraton acceleraton transducer type ICP 320C5 by PCB Pezoelectroncs and data acquston card NI PCI-643 controlled by a program developed n LabVew envronment were used for the measurements. The sgnals were recorded at the velocty of 2500 rpm, at the samplng frequency of 40 khz. Durng the tests, 23 runs of acceleratons of the engne body vbraton were recorded before the repar, and 27 runs of acceleratons of the engne body vbraton were recorded after the repar, ncludng full operatng cycles wthn the rotaton angle of 0-720º. The engne repar nvolved the replacement of worn pstons whch reduced the clearance n the pston-cylnder assembly. The present paper descrbes an attempt to detect clearance n the pston-crank assembly by measurng the acceleratons of body vbratons and, based on that, settng up models for radal artfcal neural networks. 2. Settng up the model by use the hstogram of vbraton spectrum The analyss of tme runs excluded the possblty to use them drectly as the data for neural classfers. Refer to fgure for examples of vbraton sgnals recorded before and after the repar. The repar of the engne dd not explctly affect the character of changes n local measurements derved from the vbraton sgnals. Both, the measurements of average poston, dfferentaton, the group of slope measure and the dstrbuton kurtoss of measurable varants of vbraton acceleratons n tme doman dd not allow the clearance n the pston-cylnder assembly to be explctly dentfed. Refer to fgure 2 for examples of spectrum derved from the vbraton sgnal for two dfferent states of the engne. Accordng to the studes, sgnal analyss n two selected representatve frequency ranges s requred to evaluate the pston-assembly wear. Therefore, n the next stage of model constructon, the spectrum range acheved was dvded nto 40 sub-ranges, everyone equals 500 Hz. A hstogram was prepared to enable the descrpton of the character of spectrum changes for each sub-range. The lmts of the hstogram ranges were assumed by dvdng the ampltude of spectrum (determned for the maxmum value of a gven subrange) nto 5 equal parts. The procedure was shown n fgure 3. The hstogram ranges assumed for further experments are as follows: --range : 0 to 20 % maxmum spectrum ampltude n a gven sub-range, --range 2: 20 to 40 % maxmum spectrum ampltude n a gven sub-range, --range 3: 40 to 60 % maxmum spectrum ampltude n a gven sub-range, --range 4: 60 to 80 % maxmum spectrum ampltude n a gven sub-range, --range 5: 80 to 00 % maxmum spectrum ampltude n a gven sub-range. Refer to fgure 4 for an example of spectrum hstogram for acceleratons of engne body vbratons wth varous clearance values n the pston-cylnder assembly. a) b) Fg.. Vbraton acceleraton runs recorded on the body before (a) and after (b) the engne repar a) b) Fg. 2. Spectrum of vbraton acceleratons recorded o the body before (a) and after (b) the engne repar 6 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology Fg. 3. Procedure of spectrum hstograms determnaton a) b) Fg. 4. Sample spectrum hstogram of vbraton acceleratons recorded o the body before (a) and after (b) the engne repar For all recorded tme runs of the acceleratons of vbraton measured pror to and after the engne repar, spectrum hstograms were determned accordng to the procedure descrbed. Another stage of the modellng process was to select only those ranges of spectrum ampltude (range 5) and only such spectrum sub-ranges (sub-range 40) for whch the separaton for classes referrng to the worn and new pstons was vsble. Refer to fgure 5 for a sample comparson of spectrum sub-range and hstogram range for correct and ncorrect classes separaton. As a result of selectons whch best separate the states pror to and after the engne repar, 7 comparsons between the spectrum sub-range and hstogram range were selected. Percent share of selected comparsons served as the nput data for artfcal neural networks. For the studes carred out, artfcal neural networks of RBF type were utlsed (Radal Bass Functon). The radal neural networks are used as the neural classfers dvdng the set of data nto a determned number of output categores. They are of three-layer structure: nput, hdden and output layer. The number of output neurons equals the number of classfcaton categores. Whle usng such network type, proper smoothenng coeffcent γ should be selected. It represents the radal devaton of Gauss functons and s a measure of the range of neurons n the hdden layer [3, 5]. In the experments carred out, the radal neural networks had the followng structure: --number of nput neurons: 7, --number of output neurons: 2, --number of neurons n the hdden layer: 50. Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 7

NAUKA I TECHNIKA a) b) Fg. 5. Sample comparson between the spectrum sub-range and hstogram range for the correct (a) and ncorrect (b) separaton of wear classes of the pston-cylnder assembly ( engne pror to repar, engne after repar) The neural network was expected to assgn the recorded vbraton sgnal to one of two classes correspondng to the engne pror to and after ts repar. 50 tme runs of the engne body vbraton acceleratons were dvded nto two equal parts and utlsed for teachng and testng the performance of neural networks. In the experments amed at the constructon of a proper neural classfer of RBF type, the performance of the network for 86 varous values of γ coeffcent were checked. Wth the experments carred out, t was possble to set up properly operatng neural classfer. Such result was obtaned for γ coeffcent value wthn the range 2 50000. For γ coeffcent n the value range 0.000 2 sgnfcant ncrease of the classfcaton error was notceable. 3. Settng up the model by use the hstogram of vbraton cepstrum Accordng to the studes carred out to date, wherever the vbro-accoustc sgnal can be presented as a seres of elementary events, the cepstrum analyss can be useful: 2 ( ) ( )= ( ()) C τ F log F x t Occurrence of nose, especally of perodc nature, s possble to be dentfed wth that analyss [2]. It allows to separate the seres components respectvely to the pulse response of the system and the ntaton [7]. For the dagnostcs of techncal facltes, the cepstrum analyss s used for all applcatons n whch the change of state results n the appearance or dsappearance of harmoncs. Cepstrum was determned from the recorded acceleratons of engne body vbratons. 2 () Refer to fgure 6 for examples of cepstrum derved from the vbraton sgnal for two dfferent states of the engne. Accordng to the studes, sgnal analyss n two selected representatve frequency ranges s requred to evaluate the pston-assembly wear. Therefore, n the next stage of model constructon, the cepstrum range acheved was dvded nto 5 sub-ranges: --sub-range I: 0 to 0.0 s, --sub-range II: 0.0 to 0.02 s, --sub-range III: 0.02 to 0.03 s, --sub-range IV: 0.03 to 0.04 s, --sub-range V: 0.04 to 0.05 s. A hstogram was prepared to enable the descrpton of the character of cepstrum changes for each sub-range. The lmts of the hstogram ranges were assumed by dvdng the ampltude of cepstrum (determned for the maxmum value of a gven sub-range) nto 5 equal parts. Based on the ntal experments, t was found that better results are obtaned by assumng the value of the maxmum cepstrum ampltude range, separately for sgnals recorded pror to and after the engne repar, than for the range determned based on all recorded sgnals, ether before, or after the repar. The hstogram ranges assumed for further experments are as follows: --range : 0 to 20 % maxmum cepstrum ampltude n a gven sub-range, --range 2: 20 to 40 % maxmum cepstrum ampltude n a gven sub-range, --range 3: 40 to 60 % maxmum cepstrum ampltude n a gven sub-range, --range 4: 60 to 80 % maxmum cepstrum ampltude n a gven sub-range, --range 5: 80 to 00 % maxmum cepstrum ampltude n a gven sub-range. Fg. 6. Cepstrum of vbraton acceleratons recorded o the body before (a) and after (b) the engne repar 8 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology a) b) Fg. 7. Sample cepstrum hstogram of vbraton acceleratons recorded o the body before (a) and after (b) the engne repar Refer to fgure 7 for an example of cepstrum hstogram for acceleratons of engne body vbratons wth varous clearance values n the pston-cylnder assembly. For all recorded tme runs of the acceleratons of vbraton measured pror to and after the engne repar, cepstrum hstograms were determned accordng to the procedure descrbed. Another stage of the modellng process was to select only those ranges of cepstrum ampltude (range 5) and only such cepstrum sub-ranges (sub-range I V) for whch the separaton for classes referrng to the worn and new pstons was vsble. Refer to fgure 8 for a sample comparson of cepstrum subrange and hstogram range for correct and ncorrect classes separaton. As a result of selectons whch best separate the states pror to and after the engne repar, 5 comparsons between the cepstrum sub-range and hstogram range were selected. Percent share of selected comparsons served as the nput data for artfcal neural networks. In the experments carred out, the radal neural networks had the followng structure: --number of nput neurons: 5, --number of output neurons: 2, --number of neurons n the hdden layer: 50. The neural network was expected to assgn the recorded vbraton sgnal to one of two classes correspondng to the engne pror to and after ts repar. 50 tme runs of the engne body vbraton acceleratons were dvded nto two equal parts and utlsed for teachng and testng the performance of neural networks. In the experments amed at the constructon of a proper neural classfer of RBF type, the performance of the network for 86 varous values of γ coeffcent were checked. The results of the effect of the γ coeffcent on the classfcaton error value are presented n fgure 9. Wth the experments carred out, t was possble to set up properly operatng neural classfer. Such result was obtaned for γ coeffcent value wthn the range 2 50000. For γ coeffcent n the value range 0.000 2 sgnfcant ncrease of the classfcaton error was notceable. 4. Summary Wth the studes carred out, t was proven that t s possble to set up a properly operatng neural classfer able to dentfy the degree of wear n the pston-cylnder assembly, based on the sgnal of vbraton acceleraton n the engne body. Faultless classfcaton was successfully obtaned wth the use of radal neural network wth properly selected value of γ coeffcent. At the same tme, based on the experments carred out, the crucal role was confrmed for the selecton of proper method for pre-treatment of data ntended for neural network teachng. The results obtaned confrmed the usefulness of the cepstrum hstogram and the spectrum hstogram of the acceleraton of engne body vbraton for that purpose. The effcency of OBD systems allowng the detecton of engne mechancal defects masked by electronc controls n modern vehcles can be ncreased by the development of systems utlsng radal artfcal neural networks. a) b) Fg. 8. Sample comparson between the cepstrum sub-range and hstogram range for the correct (a) and ncorrect (b) separaton of wear classes of the pston-cylnder assembly ( engne pror to repar, engne after repar) Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 9

NAUKA I TECHNIKA Fg. 9. The effect of the γ coeffcent on the correctness of the RBF neural network classfcaton (γ the radal devaton of Gauss functons) 5. References. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. 3. 4. 5. 6. 7. Batko W, Dąbrowsk Z, Engel Z, Kcńsk J, Weyna S. Modern methods of examnaton of vbroacoustc processes. Radom: Wydawnctwo ITeE, 2005. Cempel C. Vbroaccoustc dagnostcs of machnes. Warszawa: Państwowe Wydawnctwo Naukowe, 989. Charchals A. Operatonal supervson of the shp s power plant equpped wth gas turbne engnes. Problemy Eksploatacj - Mantenance Problems 200; 4: 05-3. Czech P, Łazarz B, Wojnar G. Detecton of local defects of gear teeth usng artfcal neural networks and genetc algorthms. Radom: Wydawnctwo ITeE, 2007. Dąbrowsk Z, Madej H. Maskng mechancal damages n the modern control systems of combuston engnes. Journal of KONES 2006; 3(3): 53-60. Droźdzel P. The nfluence of the vehcle work organzaton condtons on the engne start-up parameters. Eksploatacja Nezawodnosc - Mantenance and Relablty 2008; (37): 72-74. Heywood J. B. Internal combuston engnes fundamentals. New York: McGraw Hll Book Company, 988. Isermann R. Dagnoss methods for electronc controlled vehcles. Vehcle System Dynamcs 200; 36(2-3): 77-7. Korbcz J, Koścelny J, Kowalczuk Z, Cholewa W. (collectve work) Process dagnostcs. Models. Methods for artfcal ntellgence. Applcatons. Warszawa: Wydawnctwa Naukowo-Technczne, 2002. Merksz J, Tarkowsk S. Selected aspects of usng deck recorders n automotve vehcles n automotve vehcles. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 2(50): 50-58. Moczulsk W, Przystałka P. Applcaton of Neural Networks for Dagnostcs of Dynamc Processes. Glwce: AI-METH, 2004. Newczas A, Borowec M, Sen A K, Ltak G, Huncz J, Koszałka G. Vbratons of a vehcle excted by real road profles. Forsch Ingeneurwes 200; 74: 99-09. Osowsk St. Neural networks for nformaton processng. Warszawa: Ofcyna Wydawncza Poltechnk Warszawskej, 2000. Tadeusewcz R. Neural networks. Warszawa: Akademcka Ofcyna Wydawncza, 993. Tadeusewcz R, Lula P. Introductn to neural networks. Krakow: StatSoft, 200. Shao Y, L X, Mechefske C. K, Chen Z. Rear axle gear damage predcton usng vbraton sgnal preprocessng coupled wth RBF neural networks. Eksploatacja Nezawodnosc - Mantenance and Relablty 2009; 4(44): 57-64. Żółtowsk B, Cempel C. (collectve work) Machne dagnostcs engneerng. Warszawa-Bydgoszcz-Radom: Wydawnctwo ITeE, 2004. Dr nż. Potr CZECH Prof. dr hab. nż. Henryk MADEJ Faculty of Transport Slesan Unversty of Technology Krasńskego Street 8, 40-09 Katowce, Poland e-mal: potr.czech@polsl.pl 20 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: LIUDVINAVIČIUS L, LINGAITIS L P, BUREIKA G. Investgaton on wheel-sets slp and slde control problems of locomotves wth ac tracton motors. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 2-28. Longnas LIUDVINAVIČIUS Leonas Povlas LINGAITIS Gntautas BUREIKA INVESTIGATION ON WHEEL-SETS SLIP AND SLIDE CONTROL PROBLEMS OF LOCOMOTIVES WITH AC TRACTION MOTORS BADANIE PROBLEMÓW BUKSOWANIA KOŁOWEGO I STEROWANIA PROCESU ŚLIZGANIA W LOKOMOTYWACH Z SILNIKAMI TRAKCJI ASYNCHRONICZNEJ The artcle s dedcated to the soluton of drve parameter adjustment and correcton problems of AC tracton motors. Havng assessed dfferent ral adheson ratos and AC electrcal machne operaton peculartes, there have been proposed new dynamc slp and slde process control methods to DC/AC, AC/AC current system locomotves. There s descrbed the nfluence of mechancal characterstcs of AC tracton motors to the formaton of wheel slp process and there are provded structural control schemes of the dynamc slp process. The slp process formaton and control condtons for wheel pars are provded n mathematcal and graphcal forms. There are provded automatc control parameters of dynamc slp and slde process for AC/AC current locomotves. Keywords: desel-electrc locomotves, AC tracton motor, slp-slde control system, adheson coeffcent, tracton generator characterstcs, ant-slp drve. Artykuł jest przeznaczony dla rozwązana problemów regulowana korekcj parametrów pracy przekładn lokomotywy z slnkem trakcyjnym AC. Ocenając różne czynnk przyczepnośc koła lokomotywy z torem osoblwośc maszyn elektrycznych typu AC, w artykule przedstawono nowe metody buksowana dynamcznego sterowana procesu ślzgana przekładn lokomotywy dla systemów prądu AC/DC, AC/AC. Ocenając wpływ rodzaju charakterystyk mechancznych slnków trakcyjnych AC na formowane procesu buksowana kół, w artykule przedstawono strukturalne schematy sterowana procesów buksowana dynamcznego. Formowane procesu buksowana par kół warunk jego sterowana przedstawono w wyrażenu matematycznym forme grafcznej. W artykule zaprezentowano parametry buksowana dynamcznego sterowana automatycznego procesu ślzgana dla lokomotywy systemów prądu AC/AC. Słowa kluczowe: lokomotywa elektryczna z slnkem Desla, slnk trakcj AC, system sterowana buksowana ślzgana, współczynnk przyczepnośc, przekładna antyślzgowa.. Introducton The man ncome (over 90%) of the Lthuanan state ralway transport company Letuvos geležnkela Plc. comes from freght transportaton. Thus t s very mportant to ensure contnuous and stable freght tran traffc by transportng the bggest possble cargo amounts. The most ntensve freght flows as well as the heavest freght rollng stocks cross Lthuanan ralways from Belorussa to the harbors of the Baltc sea,.e. Klapeda and Kalnngrad. When carryng heavy rollng stocks, especally f t s necessary to move standstll rollng stocks or get over steep and long slopes, locomotves face the problem how to ensure stable tracton mode,.e. the phenomenon of locomotve wheel set slp and slde. The locomotve nteracton wth rals and the causes of ther slp have been analyzed by many researchers all over the world. In case of drve wheel slp and slde there s a sudden drop of locomotve tracton force as well as tran speed. The tran may be stopped at the uphll, thus dsturbng (or even nterruptng) sde track traffc. The phenomena of tran stable drvng uphll were descrbed by Bureka [4], by Ludvnavčus et al []. In case of movement of standstll trans and sldng of locomotve wheels, there s no suffcent tracton force to move the locomotve. However f sand s put under the sldng wheels the adheson of drve wheels wth the rals shall ncrease several tmes and, moreover, there shall be a rapd ncrease of the tracton force. Ths force has a negatve effect on rollng stocks as a huge amount of shock force s transferred to the whole rollng stock length through automatc couplng pont. Ths shock force affects rollng stock structures (frames, bodes, suspensons, carrages, etc.) and can harm cargos and as well as ther packagng. These problems were analyzed by Bąk et al [2], Jastremskas et al [7]. There have been theoretcal assumptons of locomotve wheel set slp and slde causes when carryng out the research. The artcle dscusses theoretcal and practcal aspects of formaton of wheel set rollng surface concalty, ral tlt angle and wheel par poston n the track, redstrbuton of locomotve drve wheel set axs force to ral n tracton mode at slp condtons. There are analyzed peculartes of locomotve adheson Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 2

NAUKA I TECHNIKA weght optmum usage, axs force change n tracton mode. The nfluence of locomotve adheson rato change to tracton characterstc and tracton force change are descrbed. There are examned varous locomotve constructon schemes wth AC tracton motor drve. Mechancal characterstcs of AC tracton motors are provded. New generaton desel locomotves, e.g. SIEMENS ER- 20CF use AC motors for wheel set tracton motor that are sgnfcantly more relable and unsophstcated f compared to the prevous DC motors used for drves. The three-phase asynchronous tracton motor s the motor used most frequently for modern locomotve drves because of ts excellent relablty, ts smple and thus low-cost constructon and low mantenance costs. The three-phase asynchronous tracton motor s also known as squrrel-cage rotor. On the other hand, AC motor rotaton speed adjustment dffers essentally from DC motor rotaton speed change. Thus, amng to control and adjust slp and slde processes of drve wheel sets t s necessary to apply dfferent prncples of electrcal machne operaton parameter trackng, measurement and adjustment. These ssues are analysed by Lngats and Ludvnavčus [8]. Herenafter the authors descrbe the causes of locomotve drve wheel slde and provde measures for the correcton and control of slp and slde processes of locomotve wheel sets wth AC tracton motors. Theoretcal presumptons for locomotve slp formaton and wheel set slde on ral consequences The locomotve wheel set slppage s evdent when the locomotve tracton force exceeds adheson force. The man locomotve tracton law s the followng: F tr Ψ adh P () adh where: Ψ adh locomotve wheel-sets adheson to ral coeffcent; P adh locomotve adheson weght, N. In order to mantan stable tracton mode (to avod wheels slp) t s necessary to mantan formula () nequalty condton. The change of locomotves wheel-sets adheson coeffcent value n dfferent seasons s provded n fgure The estmaton of adheson coeffcent of a newly made locomotve s carred out by means of practcal tests n dfferent seasons and dfferent day tme. In order to obtan objectve results several hundred of practcal tests are carred out. Fgure shows that adheson coeffcent values are hghly dsspated as there are many factors that nfluence n case of equal tran mass, ncludng ran, snow, ral contamnaton wth ol products, etc. [6]. Tracton characterstcs of desel-electrc powered locomotve are provded n fgure 2. The desel-electrc powered locomotve adheson coeffcent value n the season change tme (dsspated, e.g. 0.33 to 0.2) s as shown n fgure. However, when operatng the sad desel locomotves n season change tme, the rollng stock mass s not adjusted due to the changes of adheson condtons. Caused by the decrease of the adheson coeffcent the tracton power becomes nsuffcent for ths rollng stock towng. In such a case sand s sprnkled ntensvely to ncrease the adheson coeffcent. However locomotve startng tracton tme s ncreased due to ntensve slppng, thus causng accelerated wheel set and ral wear. 3. Slppng process of locomotve drve wheel sets 3.. Peculartes of locomotve tracton and tran movement resstance force operaton The wheel to ral force redstrbuton scheme (n tracton mode) of locomotve drve wheel set axle s provded n fgure 3. The scheme llustrates the rsng moments M o that emerge n the tracton mode that decrease the wheel set force to the ral of the frst wheel set by one value ΔP (rse up) and ncrease the axle load to the ral of the second wheel set ΔP (press down). Such torques are formed due to the tracton force F tr of locomotve drve wheel sets that acts as a wheel set tangent and the moment of tran resstance force W j that acts n the heght of the automatc couplng pont h. Locomotve tracton force at wheel rm and locomotve movement resstance fore at automatc couplng pont h form the force par or the rse torque M 0 that decreases the load of locomotve front wheel set and ncreases the fnal wheel set to ral force. Thus, the least loaded wheel set shall be the frst to lose the necessary adheson wth the rals. The most frequent slp occurs namely at the frst wheel set. Locomotve tracton force s also nfluenced by wheel set tracton force unevenness. If less loaded wheel set produces bgger tracton force, t starts slppng earler (when the tracton force t produces s smaller) and ths decreases the total locomotve tracton force. Fg.. Locomotve adheson coeffcent dsspaton dependence on locomotve speed n dfferent seasons Fg. 2. Desel-electrc powered locomotve characterstcs n tracton mode: tracton characterstcs of one locomotve F k = f(v); 2 tracton characterstcs of two locomotves; 3 tracton characterstcs of three locomotves; 4 locomotve power at wheel rm P k = f(v); 5 Desel engne power characterstcs N D = f(v) 22 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology Fg. 3 Wheel to ral force redstrbuton scheme (n tracton mode) of locomotve drve wheel set axle: F adh and F 2 adh drve wheel tracton forces; P o wheel-to-ral force; W J resstance force at automatc couplng pont; h heght from the ralhead to automatc couplng pont. 3.2. Peculartes of engne operaton and locaton of wheel set to ral statc force Tracton engne operaton and the nfluence of ts locaton to statc wheel-to-ral force are especally sgnfcant when the suspenson of tracton motor s axal-supportng. Actve forces and torques of the sad stuaton are provded n fgure 4. Tracton motor acheves torque M p that rotates wheel set. Tracton motor shaft s connected to the wheel set va reducer. If the tracton motor s located to the wheel set as shown n fgure 3 and locomotve moves n the defned drecton, force Z s drected upward. Moreover, motor stator s nfluenced by torque M s =M v that s appled to motor supports by force P 3 calculated as follows: M P3 = (2) L Thus by locatng the tracton motor behnd a wheel set and wth supportng-frame suspenson of the tracton motor, the wheel to ral force s decreased by one value P. Fg. 4. Dfferent schemes of electrc tracton motor dslocaton nfluence on the wheel force-to ral: P 3 electrc tracton motor force to ral; M p motor torque appled to the motor shaft; M c resstance torque appled to tracton motor shaft. When locomotve changes ts movement drecton, the wheel to ral force ncreases by the same value. 3.3. Peculartes of locomotve axle force change n tracton mode The boge to ral force redstrbuton scheme (n tracton mode) of the sx-axle locomotve s provded n fgure 5. Frames of boges are affected by drve wheel set tracton forces F a,..., F a6 and automatc couplng pont of rollng stock s affected by movement resstance forces W j that are equal W j = n F aš. Due to the operaton heght dfference of the tracton force F T generated by locomotve drve wheel sets and tran movement resstance force W j that acts at couplng pont there s torque M T : ( ) M = n F h r (3) T a r where: n wheel set number, unts; F a tracton force of one drve wheel set, N; h couplng pont heght (dstance from ralhead to couplng pont longtudnal axs ), m; r r wheel set rollng radus, m. Torque M T tlts locomotve body at axs Y that crosses locomotve weght center C and t changes the value of vertcal force T actng on boges and t s calculated as follows: T = M T / B (4) where: B dstance between boges (bass), m. Wheel set force change P s calculated as follows: T MT P = RT = = m m B where: m boge wheel set number. At torque M T the frst boge s less loaded and the second one s more loaded. (5) Fg. 5. Redstrbuton scheme of locomotve boge to ral force: F a, F a2, F a3, F a4, F a5, F a6 tracton forces of drve wheel sets; W j movement resstance forces that act on the rollng stock; q a statc wheel to ral force of a wheel set; h heght from the ralhead to automatc couplng pont; B dstance between boge centers (bass). A locomotve s movement drecton. Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 23

NAUKA I TECHNIKA 3.4. Peculartes of locomotve adheson coeffcent changes Locomotve tracton force s nfluenced by nequaltes of tracton forces of every wheel set that are formed due to the uneven dstrbuton of the locomotve weght between wheel sets. If less loaded locomotve wheel set produces too hgh tracton force compared to the adheson condtons, slppng shall be started at lower tracton force thus lowerng the total locomotve tracton force. Adheson coeffcent s also nfluenced by rm eroson, ral wear and dfferences n wheel dameters of wheel sets. At road curves of small radus, wheel set adheson to ral s worse due to wheel set slde caused by uneven dstance that s traveled by a wheel of the wheel set. Adheson coeffcent depends on the locomotve electrc drve type (AC/DC or AC/AC), tracton motor connecton type, nature of tracton motor mechancal characterstcs. The locomotve tracton force may be decreased by the force due to wheel set force to ral redstrbuton and t s calculated as follows: ΔP + ΔP + ΔP 2. Force of the least loaded wheel set s calculated as follows: 2Pmn = qa ( 003. ) P (6) where: 0.03 the tolerance of the statc force to ral of wheel set accordng to techncal locomotve characterstcs wth load q a less than 225 kn. Rato of the least loaded wheel set and calculable force s called locomotve adheson use coeffcent β k and t s expresses as follows: qamn qa( r 003, ) P P β K = = = 097,. (7) q q q a Locomotve adheson use coeffcent shows the part of the tracton force that s actually acheved by the locomotve. Havng analyzed the sad theoretcal presumptons t can be sad that locomotve adheson use coeffcent depends on wheel set and tracton motor locaton, suspenson type, nature of the tracton motor mechancal characterstcs. a 4. Locomotve DC and AC tracton motor speedtorque characterstcs a Fg. 6. Speed-torque characterstcs of the DC and AC tracton motor: synchronous tracton motor, 2 asynchronous tracton motor speed-torque characterstcs, 3 ndvdually excted shunt-wound DC tracton motor, 4 seres wound DC tracton, speed-torque characterstcs; Δn asynchronous tracton motor speed varaton; Δn 2 separately excted shunt - wound DC tracton motor speed varaton; Δn 3 seres wound DC tracton speed varaton Locomotve tracton motor mechancal characterstcs are n fgure 6. Speed-torque characterstcs of the DC and AC tracton motor are descrbed by Fuest and Dörng [6]. The speed - torque natural characterstcs are dfferent [5]. Varaton n load moment causes varaton n speed. For locomotve tracton force and wheel-sets slp requrements n startng mode the best s synchronous tracton motor speed- torque characterstc (fg. 6, st curve). Fgure 6 shows the speedtorque characterstc () curve of a synchronous tracton motor. In the speed-torque characterstc the torque s usually shown dependent on the speed. Asynchronous tracton motor and separately excted shunt - wound DC tracton motor speed - torque characterstcs a varaton n load moment s smlar-szed tolerance (fg. 6, st and 3 rd curves). Locomotve wth ths tracton motor speed- torque characterstcs rarely comprse wheel-set slp condtons. The 4 th curve (n fg. 6) shows that for locomotve wth DC seres-wound tracton motors ant-slp systems must be used because when t s operated wthout load torque speed ncreases very quckly. Fgure 6 shows the speed-torque characterstc 2 nd curve of a three-phase asynchronous tracton motor. In the speed-torque characterstc, the torque s usually shown dependent on the speed. There M = 0 when the motor s dlng. If a load s appled to the motor, the speed drops and the torque ncreases. The maxmum torque whch a motor can produce s known as the pull-out torque. 5. Parallel operaton and control peculartes of AC tracton motors Prncpal scheme of AC/AC electrcal system locomotve electrcal drve when three asynchronous tracton motors of one boge are powered from the common frequency converter FC s shown n fgure 7. Peculartes of parallel operaton of tracton motors are analyzed at the same statc resstance moment M st. Due to the nevtable geometrcal devatons of parts (wheel set, ral), dfferences of electrcal and magnetc propertes of materals, the characterstcs of the same type tracton motors are dfferent. In the nomnal load mode the tracton motor rotor speeds may dffer ±3 %. When asynchronous tracton motors are powered from one source the currents among parallel tracton motors are redstrbuted due to the natural mechancal characterstcs of the same type asynchronous tracton motors. Tracton motors acheve dfferent respectve torques M, M 2, M 3, rotor rotaton speeds n,n 2,n 3 and develop dfferent tracton forces of ndvdual motors. Natural mechancal characterstcs of the same type asynchronous tracton motors (ATM) at the same statc resstance torque M st at the wheel set force pont A are shown n Fg. 8. Natural mechancal characterstcs of the same type ATMs may be formed at the pont where synchronous speed s n o. When analyzng natural mechancal characterstcs of the same type asynchronous torque motors wth the same statc resstance torque M st at the wheel set force pont A t can be seen that the rotaton speed of the frst ATM rotor n, rotaton speed of the second ATM rotor n 2 and rotaton speed of the thrd ATM rotor n 3 do not synchronze. When the wheel set dameter s equal and the speed of tracton motor rotor s dfferent, the dstance traveled by wheel sets s dfferent. Ths phenomenon causes wheel set slp condtons and redstrbuton of tracton forces. 24 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology Fg. 7. Crcut dagram of desel-electrc powered locomotve (AC/AC current system) drve, when one frequency converter s used to power three AC tracton motors: DM desel engne; G synchronous tracton generator; FC frequency converter; M, M2, M3 asynchronous tracton motors; WS, WS2, WS3 wheel-sets; M st statc resstance torque U, f - asynchronous tracton motor drve parameters Fg. 8. The same type natural mechancal characterstcs of asynchronous tracton motors at the same statc resstance torque 6. Control systems of asynchronous motors wth ndvdual frequency converters Control systems of asynchronous motors wth ndvdual frequency converters are shown n fgure 9. When usng ATM control va ndvdual frequency converters, the slppng and sldng process s easer to control as the speed of every tracton motor can be adjusted separately. However, ths control system s often more complex and less relable. 7. Frequency control peculartes of locomotves wth asynchronous tracton motors For locomotves wth frequency converter mode of ATM that satsfes tracton theory requrements, there s a need to change the nature of natural mechancal characterstcs artfcally by developng there a part of hyperbolc functon. Thus ATM areas of the natural characterstcs are shown as A, A 2, A 3 (fg. 0). The nature of ATM artfcal mechancal characterstcs corresponds to DC seres exctaton DC tracton motor mechancal characterstcs. Therefore locomotves wth ATMs should have nstalled control systems to control wheel set slppng process. Durng locomotve wheel set slppng they temporarly change the nature of the ATM artfcal mechancal propertes and recover the condtons of wheel set adheson wth rals. Computer aded control system for locomotve wheel set slppng and sldng process control that s proposed by the authors wll change the nature of ATM artfcal mechancal propertes and t wll automatcally recover good condtons of wheel set adheson to rals. Adjusted ATM dynamc mechancal propertes wth appled frequency speed control method are provded n fg. 0. Fg. 9. Control system of asynchronous engnes wth ndvdual frequency converters: FC,, FC6 frequency converters; WS,, WS6 wheel-sets of locomotve Fg. 0. Artfcal asynchronous mechancal characterstcs of asynchronous motor when ATM frequency rotaton speed control method s appled Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 25

NAUKA I TECHNIKA 8. Automatc control of slppng process of locomotves wth AC tracton motors Fgure shows AC/AC system that s proposed by the authors for computer aded control of AC/AC system of locomotve wheel set slppng and sldng process that s comprsed: wheel set speed sensors BR- BR6 that are mounted n tracton motors, A and B boge wheel set speed tolerance sgnal comparson blocks SCBI, SCBII [9]. When tolerance of one of ndvdual wheel sets of boges (A or B) s exceeded, there are generated control sgnals A and B that adjust sgnals sent by computer aded control system (n Fg. marked respectvely as X A and X B ) to nverter elements IGBT transstors that have changng on and off cycles [3]. The automatc control system of slppng process that s proposed by the authors shall decrease the value of supply power and ts frequency of one boge from the three asynchronous engnes for a short tme. Thus the rotaton speed of one boge of tracton motor-wheel set shall be equalzed and wheel to ral adheson shall be restored. Such dynamc characterstcs are shown n fgure n green as E -Y wth supply power frequency f (). E 2 -Y 2 at supply power frequency f (2), E 3 -Y 3 wth supply power frequency f (3). When adheson condtons are restored, there s automatc restoraton of ATM supply power and ts frequency value,.e. t returns to any pont of the prevous operatng characterstc (fgure 0). Tracton forces are not dsrupted n locomotve control process as slppng process s controlled automatcally. 9. Algorthm for slppng process parameter adjustment and control The authors provde computer aded control system of locomotve wheel set slppng and sldng process control. Varants of the control algorthm: a wheel set speed tolerance sgnals are compared wth the tolerance sgnals n blocks SCBI, SCBII and they are adjusted by changng parameters of that nverter to control sgnals Y and Y 3 (thereof the respectve three asynchronous tracton engnes (ndvdual controlled) are powered). b n case wth controlled parameters of nverters I and II the process of boge wheel set slppage s contnued and boge wheel set speeds are not equalzed, the generator voltage of synchronous tracton motor s lmted by decreasng exctaton current wth sgnal Y 2 (general parameter control). Computer aded control system of AC/AC current system desel locomotve wheel set slppng-sldng process conssts (fgure ): DM desel engne; G synchronous tracton generator; I, II frequency converters; UCR-uncontrolled rectfer; M,, M6 asynchronous tracton motors; speed sensors of BR tracton motors; LD locomotve drver; SCBI boge A wheel set speed tolerance sgnal comparson unt; SCBII-B boge wheel set speed tolerance sgnal comparson unt; Y control sgnal that adjusts nverter I electrcal parameters; Y 2 synchronous tracton motor sgnal that adjusts generator exctaton current; Y 3 control sgnal that adjusts nverter II electrcal parameters; R exctaton current regulator of synchronous tracton generator; G E exctaton wndng of synchronous tracton generator; Δn,..., Δn 6 speed tolerance sgnals of tracton motor; X A total compared boge A wheel set speed tolerance sgnal; X B total compared boge B wheel set speed tolerance sgnal. Authors propose to nstall an encoder n ATM rotaton speed and locaton coordnate measurement sensors BR that are nstalled n the tracton motor (see fgure 2). The encoder s analogous or dgtal converter wth analogous sgnal at the output or a certan number of pulses that s proportonal to rotaton speed or turn angle [2]. Fg. 2. Asynchronous tracton motor wth nternal optcal encoder: encoder; 2 stator; 3 - termnal box; 4 stator wndngs; 5 ferromagnetc core; 6 clamps; 7 coolng channel; 8 shaft Fg.. Crcut dagram of desel-electrc powered locomotve (AC/AC electrc system) automatc wheel-sets ant-slp and slde control system process parameters computer drve: SCBI tracton motors speed control block (boge A); SCBII tracton motors speed control block (boge B) 26 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology To use an encoder n ATM vector control system s suggested by Ludvnavčus et al [0]. The encoder conssts of a lght source, mask, code dsk and sensors. The code dsc contans artfcal spaces that make lght permeable and mpermeable segments. Lght senstve sensors are mounted behnd the code dsc. The scheme of optcal encoder s provded n fgure 3 [5]. When encoder s nstalled n the ATM control system that s proposed by the authors, there can be receved analogous or dgtal nformaton, thus developng analogous-dgtal (hybrd) locomotve electrcal drve control system [, 4]. Pursuant to the purpose, the encoder may contan varous codes: bnary code, Gray code, Gray-express code [3]. All of them are sutable for computer system of AC/AC current system desel locomotve wheel set slppng-sldng process control that s proposed by the authors. 0. Tests of locomotve wheel set slppng and sldng process computer aded control and management system 0.. Locomotve tests descrpton The software performs the several tasks. It ensures hghlyeffcent wheel-to-ral force transmsson by means of contnuous wheel force at ts rm; slp control; lmtaton of wheel acceleraton; relable determnaton of the reference speed, whch should represent the true tran speed. The software protects the mechancal components aganst excess stress and reduces wear on the rals and wheel-set by avodng: wheel blockng (flat spots on the runnng surface); synchronous drftng of the wheel speeds (worn ral heads); nherently stable rotatonal vbratons n the drve tran. The software contnuously montors the movement of the vehcle and the runnng wheels. If the movement varables devate from the tolerance values, the tractve effort demanded by the overall control level s automatcally reduced to a level whch can be physcally transmtted from the wheel to the ral. Fg. 3. Optcal encoder scheme Due to contnuous montorng of the movement varables relatng to the vehcle and the wheels, t s ensured that tracton s kept under control under dfferent track condtons. Power mode: The test tran s accelerated from standstll to maxmum speed. All wheel speeds are montored va the data logger to confrm the proper functonalty. Dynamc brake only: The test tran s decelerated from maxmum speed to standstll wth the dynamc brake. Dynamc and pneumatc brake: The test tran s decelerated from maxmum speed to standstll wth the dynamc and the ndependent brake. Ths s not a normal servce operaton. It shows, that the wheel slde s corrected va sgnals transmtted from the pneumatc wheel slde system to the Locomotve Computer Unt for the lmtaton of electrc brake n case of actve pneumatc brake. 0.2. Valuaton of tests results AC/AC system desel-electrc powered locomotve wheels slp and slde computer control results are presented n Fg. 4. Fg. 4. AC/AC system desel-electrc powered locomotve test characterstcs:upper lnes left scale: st boge torque reference before slp and slde control; 2 st boge torque after slp and slde control; 3 2 nd boge torque reference before slp and slde control; 4 2 nd boge torque after slp and slde control; Lower lnes rght scale: st wheel-set speed; 2 2 nd wheel-set speed; 3 3 rd wheel-set speed; 4 4 th wheel-set speed; 5 5 th wheel-set speed; 6 6 th wheel-set speed Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 27

NAUKA I TECHNIKA. Conclusons. The non-tradtonal computer aded slppng an sldng control system of locomotve AC tracton motors that s proposed by the authors allows automatc contnuous control of electrcal parameters of the nverter and tracton generator n tracton mode (dynamc mode) wth the smultaneous slppng process control. 2. When the provded slp and slde control system of locomotves wth AC tracton motors s used, tran drver does not need to nterrupt tracton mode control and ths way there are no condtons for the formaton of the rollng stock longtudnal tensle and compresson forces. 3. 4. 5. 6. When the provded slp and slde control system of locomotves wth AC tracton motors s used, there s optmal tracton force control. When the provded slp and slde control system of locomotves wth AC tracton motors s used, there are less energy transformaton losses n nternal combuston engnes. When the provded slp and slde control system of locomotves wth AC tracton motors s used, the adheson coeffcent s lttle dependant on the season. When the provded slp and slde control system of locomotves wth AC tracton motors s used, the adheson coeffcent s recovered automatcally and there s no need to put sand under the wheels to ncrease the adheson. 2. References. Antonak J. Urządzena systemy transportu podzemnego w kopalnach. Katowce: Wydawnctwo Śląsk, 990. 2. Bąk R, Burczyńsk T. Wytrzymałość materałów z elementam ujęca komputerowego. Warszawa: WNT, 200. 3. Braess H H, Seffert U. Veweg Handbuch Kraftfahrzeugtechnk. Fredrch Veweg & Sohn Verlagsgesellschaft GmbH, Wesbaden. 2000. 4. Bureka G. A Mathematcal Model of Tran Contnuous Moton Uphll. Transport 2008; 23(2): 35-37. 5. Encoder catalogue. Kübler Catalogue. Germany. 2003. 7 p. 6. Fuest K, Dörng P. Elektrsche Maschne und Antrebe. Lehr- und Arbetsbuch. Veweg, Wesbaden. 7. Jastremskas V, Vačūnas G, Černašėjus O, Rudznskas V. Investgaton nto the mechancal propertes and metal creaks of a desel locomotve wheel. Transport 200; 25(3): 287-292. 8. Lngats L P, Ludvnavčus L. Electrc drves of tracton rollng stocks wth AC motors. Transport 2006; 2(3): 223 229. 9. Ludvnavčus L, Lngats L P. New locomotve energy management systems. Eksploatacja Nezawodnosc - Mantenance and Relablty 200; 45(): 35-4. 0. Ludvnavčus L, Lngats L P, Dalydka S, Jastremskas V. The aspect of vector control usng the asynchronous tracton motor n locomotves. Transport 2009; 24 (4): 38-324.. Ludvnavčus L, Lngats L P. Electrodynamcs brakng n hgh-speed ral transport. Transport 2007; 22(3): 78-86. 2. Precson nductosyn poston transducers for ndustral automaton, aerospace and mltary applcaton. Farrand Controls nformaton Issue. USA. 996. 5 p. 3. Sen P C. Prncples of Electrc Machnes and Power Electroncs. New York-Chchester-Brsbane-Toronto-Sngapore-Wenhem. John Wley&Sons. 996. 4. Stöltng H-D. Elektronsch betrebene Klenmaschne. Vorlesungsmanuskrpt. Unverstät Hannover, 2002. 5. Strekopytov V V, Grshchenko A В, Kruchek V A. Electrc drves of the locomotves. Moscow: Marshrut. 2003; 305 p. 6. Vačūnas G, Lngats L P. The nfluence of ralroad curves on the wear of lubrcated and unlubrcated wheel flangs. Eksploatacja Nezawodnosc - Mantenance and Relablty 200; 47(3): 40-43. Longnas LIUDVINAVIČIUS, M. Sc., Eng. Prof. dr hab. Leonas Povlas LINGAITIS Prof. dr Gntautas BUREIKA Deptartment of Ralway Transport, Vlnus Gedmnas Techncal Unversty, J. Basanavčaus 28, LT-03224 Vlnus, Lthuana e-mal: long@vgtu.lt; leonasl@vgtu.lt; gntautas.bureka@vgtu.lt 28 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: TOMASZEK H, JASZTAL M, ZIEJA M. A smplfed method to assess fatgue lfe of selected structural components of an arcraft for a varable load spectrum. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 29-34. Henryk TOMASZEK Mchał JASZTAL Marusz ZIEJA A smplfed method to assess fatgue lfe of selected structural components of an arcraft for a varable load spectrum Uproszczona metoda oceny trwałośc zmęczenowej wybranych elementów konstrukcj statku powetrznego dla zmennego wdma obcążena* The assessment of fatgue lfe of an arcraft s structural component operatng under varable load spectrum causes many and varous problems, hence the need for smplfed methods that facltate t. The presented study covers the queston of rearrangng an actual spectrum wth varable values of cycles nto a homogeneous spectrum wth weghted cycles. A method for the evaluaton of fatgue lfe of some selected arcraft s structural component wth an ntal crack has been presented usng a rearranged spectrum. To model an ncrement n the crack length a dfference equaton has been appled whch, after rearrangement, resulted n a partal dfferental equaton of the Fokker-Planck type. A densty functon of the crack length s a partcular soluton to ths equaton. Usng the densty functon of a crack length, fatgue lfe of the structural component has been determned for the crack that keeps growng up to the permssble value l d lower than the crtcal value l kr. What has been gven consderaton n ths study s the case when the exponent of the Pars equaton m 2. Keywords: load cycle, weghted load cycle, relablty, durablty, load spectrum. Ocena trwałośc zmęczenowej elementu konstrukcj pracującego pod wpływem zmennego wdma obcążena przysparza welu trudnośc. Stąd potrzeba poszukwana uproszczonych metod umożlwających tą ocenę. Przedstawona praca obejmuje przekształcene wdma rzeczywstego o zmennych wartoścach cykl w wdmo jednorodne o cyklach ważonych. Wykorzystując wdmo przekształcone przedstawono metodę oceny trwałośc zmęczenowej wybranego elementu konstrukcj statku powetrznego z początkowym pęknęcem. Do modelowana przyrostu długośc pęknęca wykorzystano równane różncowe z którego po przekształcenu otrzymano równane różnczkowe cząstkowe typu Fokkera-Plancka. Rozwązanem szczególnym tego równana jest funkcja gęstośc długośc pęknęca elementu. Wykorzystując następne funkcję gęstośc długośc pęknęca określono trwałość zmęczenową elementu konstrukcj dla pęknęca narastającego do wartośc dopuszczalnej l d mnejszej od wartośc krytycznej l kr. W pracy rozpatruje sę przypadek, gdy wykładnk równana Parsa m 2. Słowa kluczowe: cykl obcążena, ważony cykl obcążena, nezawodność, trwałość, wdmo obcążena.. Introducton The assessment of fatgue lfe of an arcraft structural component operatng under varable load spectrum causes many and varous problems, however t proves essental to flght safety. The present study s an effort to fnd a smplfed method of fatgue lfe determnaton. Ths smplfcaton conssts n the rearrangement of an actual load spectrum nto a homogeneous spectrum wth weghted cycles. The rearrangement has been outlned n Secton 2. It has been assumed that an ntal crack n the structural component s l 0. As affected by the load of a varable spectrum the crack grows up to some permssble length l d (safe) shorter than the crtcal length l kr. The crack growth rate, approached n a determnstc way, has been descrbed wth the Pars formula of the followng form []: where: ΔK the range of changes n the stress ntensty factor, C, m materal constants, N a varable that denotes the number of structure-affectng load cycles. () (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 29

NAUKA I TECHNIKA 2. How to fnd the crackng rate for the load n the form of a rearranged load spectrum cal coeffcents [4, 5]. 8) The range of stress varatons has been shown n Table 3 The followng assumptons have been made for the need of rearrangng the actual spectrum wth varable load values nto the spectrum wth weghted cycles: ) An arcraft structural component keeps operatng under varable loads whle performng ts functons. 2) The load spectrum that affects the component n the course of a standard flght of the arcraft s known. 3) We assume that ths spectrum of loads allows of the determnaton of: -- the total number of load cycles Nc n the course of one flght, -- there s the L number of thresholds of maxmum load values σmax, σ2max,..., σlmax. 4) The number of repettons of maxmum threshold values n the assumed spectrum s as follows: σmax occurs n tmes, σ2max occurs n2 tmes,, σlmax occurs nl tmes. The number of repettons of specfc threshold values of load n the course of one flght s: 9) Account has been taken of the effect of overload cycles upon the crack growth rate (table 4): Δσ,ef = CP Δσ where: CP - factor of crack growth retardaton after overload cycles occurred [3]. In the case gven consderaton t has been also assumed that the rate of crack growth n the structural component, approached n a determnstc way, follows the Pars law wrtten down wth formula (). For the above specfed assumptons, n ths case for the -th type of a load cycle (ganed from the descrpton of the spectrum of loadng n a standard cycle), formula () takes the followng form:, where =,2,...,n. 6) Table gves maxmum σmax and mnmum stress values n the cycles, and frequences of ther appearng n the spectrum P 7) Table 2 gves the statement of stress ratos and emprcal coeffcents of nfluence on crack growth U where: Table. Maxmum σmax and mnmum Table 2. Stress ratos where: =,2,...,L. stress values n the cycles, and frequences of ther appearng n the spectrum P and emprcal coeffcents of nfluence on crack gr owth U Table 3. Range of stress Δσ and frequences of ther appearng n the spectrum P Table 4. Range of effectve stress Δσ,ef whch takes nto consderaton effect of overload cycles 30 (2) where Mk specfy nfluence of crack locaton and dmensons wth relatons to structural element dmensons on crack growth velocty []. Wth account taken of all types of load cycles, the relatonshp (2) takes the form: 5) The mnmum value n thresholds s determned wth the followng relatonshp: ; α, α2, α3 empr-, Eksploatacja N ezawodnosc - Mantenance and Relablty nr 4/20 (3)

Scence and Technology Formula (3) can be expressed as a functon of tme, or more precsely, of the arcraft s flyng tme. Therefore, we assume that: N = λt (5) where: T the duraton of the arcraft s standard flght to determne load spectrum, Nc the number of load cycles durng the standard flght. After these rearrangements, formula (3) takes the followng form: (4) where: λ the rate of load cycles, N the number of load cycles, t flyng tme of an arcraft. In our case λ=/δt, where Δt denotes duraton of the fatgue loadng cycle that affects the structural component. We can assume the followng formula for Δt: where: Snce λδt =, the above wrtten equaton takes the form: (2) Let (3) (4) Substtuton of relatonshp (4) nto equaton () gves: (6) Formula (6) descrbes the crack growth rate for a homogeneous load spectrum wth weghted cycles. 3. Fndng the densty functon of the crack length (5) The result of equaton (6) should be substtuted for the crack length l n equaton (5). What we get s: Let Ul,t denote probablty that for the arcraft s flyng tme t the structural component s crack length s l. The dfference equaton for the above lsted assumptons takes the followng form [2, 6]: (7) where: Δ l the crack length ncrement durng one equvalent load cycle. The value of the crack-length ncrement calculated on the bass of (6) wll be: (8) Equaton (7) expressed n terms of functon notaton takes the followng form: (6) (9) Wth account taken of (6), coeffcents of equaton (5) can be wrtten down as: where: U(l,t) the densty functon of the crack length after the arcraft s flyng tme t expressed n terms of flght hours has elapsed; (- λ Δt) the probablty that no equvalent load cycle occurs n tme Δt; λδt - the probablty that an equvalent load cycle occurs n tme Δt. Equaton (9) can be rearranged nto a partal dfferental equaton usng the followng approxmatons: (7) (8) (0) Havng substtuted equaton (0) nto equaton (9) the followng s arrved at: () Eksploatacja N ezawodnosc - Mantenance and Relablty nr 4/20 3

NAUKA I TECHNIKA Equaton (5) wth coeffcents (7) and (8) takes the followng form for m 2: (9) A partcular soluton of equaton (9) takes the form [2, 6]: where: (20) (2) Now, the ntegral (2) s calculated: (22).e. A(t) s a varance of the crack growth for the flyng tme t. The computatonal formula takes the form: (24).e. B(t) s an average of the crack length for the flyng tme t of the arcraft. The computatonal formula takes, therefore, the followng form: Calculaton of the ntegral (22) follows: (23) 4. Determnaton of fatgue lfe of the selected structural component of the arcraft For the densty functon of the crack length versus the flyng tme of the arcraft the structural component s relablty can be found from the relatonshp [2, 6]: (25) where the form of the densty functon of the crack length U(l,t) has been determned by the relatonshp (20). The permssble crack length l d can be found usng the stress ntensty factor of the followng form: (26) When the crack length and the stress reach ther crtcal values, l kr and σ kr respectvely, the factor determned by the relatonshp (26) also becomes a crtcal value K c and s then called fracture toughness of the materal: 32 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology (27) Ths relatonshp together wth the safety factor allow of fndng the permssble crack length: where: k safety factor. Normalzaton of the ntegral n equaton (25) results n: (28) (29) where: B(t) and A(t) are determned wth relatonshps (23) and (24), respectvely. Wth the relablty level found we take values of the upper lmt of the ntegral (29) from the normal dstrbuton table. Hence the relatonshp: (30) Where: value of the upper lmt of the ntegral (29), for whch value of the ntegral equals. From relatonshp (30) we can fnd value of the flyng tme such that makes the assumed relablty level reached. 5. Fnal remarks and a computatonal example To llustrate the above descrbed method, a computatonal example has been presented. The example covers the rate of growth of an average-length fatgue crack n a structural component made from the steel of specfed materal propertes, subjected to an actual load spectrum. Computatons have been performed for the spectrum of varable- ampltude loads, whch represents an actual component-loadng spectrum and has been rearranged n the way dscussed n Secton 2 [2]. Table 5 below shows quanttes that descrbe the rearranged loadng spectrum used n our study. The table 5 ncludes: values of ranges of changes n stress n cycle σ for assumed load factors and frequences of ther occurrence P, and factors wth both load cycle asymmetry and how t affects the crack growth taken nto account. For some specfed materal of the pattern component, the followng values of materal constants have been used n the computatons: m = 3,5 C = 3,2 0-2 In our example the followng values have been used for the computatons: the ntal crack length n the component assumed to be l 0 = 0mm, and permssble crack length found from the relatonshp (28) l d = 25mm. It has been also assumed that the crack P growth retardaton factor after overload cycles C =, whereas the factor wth the load cycle asymmetry and how t affects the crack growth taken nto account has been defned by the emprcally formulated equaton. In numercal calculatons account has been also taken of the change n the M k coeffcent n the course of the crack growth. Then, the rearranged equaton (23) for the average crack length has been used to make t depend on the number of loadng cycles N, on the bass of equaton (4). (3) Usng the above wrtten relatonshp, the ncrement n the average crack length aganst the number of loadng cycles N over the range from the ntal crack length l 0 to the permssble crack length l d = 25mm has been found. Fgure shows the change n the average crack length aganst the number of loadng cycles. Fg.. Increment n the average crack length aganst the number of loadng cycles Table 5. Quanttes whch descrbe the rearranged loadng spectrum Load factor 2 3 4 5 6 7 Number of cycles 5 4 0 30 50 40 σ max [MPa] 86 59 4 29 2 93 72 mn σ sr, [MPa] -28-3 8 7 23 27 27 Factor ˆR -0,505-0,088 0,0567 0,37 0,2053 0,2903 0,375 Range of stress Δσ,ef [MPa] 24 72 33 2 89 66 45 Emprcal functon U 0,5030 0,5238 0,569 0,5955 0,6228 0,6559 0,6906 Share of load factor n the spectrum P (frequency of occurrence) 0,0042 0,0208 0,067 0,047 0,25 0,2083 0,5833 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 33

NAUKA I TECHNIKA On the bass of computatons of growth of an averagelength fatgue crack B(N) one can fnd that the permssble crack length l d = 25mm wll be reached after loadng cycles. To fnd fatgue lfe of the structural component gven consderaton, wth probablstc approach adopted, one should also take account of the crack length scatter A(N) defned wth equaton (24). Then, for the already found densty functon of the crack length aganst the number of loadng cycles the structural component s relablty can be determned: A great advantage of the presented method s that t takes account of physcal phenomena accompanyng the varable loadng spectrum. Values of materal constants used n ths method, and of other types of factors as well, all of them ndspensable for the computatons, are to be found expermentally, whereas some of them (e.g. C,m n Pars equaton) can be estmated usng servce data on the crack growth. The method of moments or the functon of lkelhood prove applcable. (32) 6. References. Kocańda S, Szala J. Podstawy oblczeń zmęczenowych, PWN, Warszawa 985. 2. Kocańda D, Tomaszek H, Jasztal M. Predctng fatgue crack growth and fatgue lfe under varable ampltude loadng, Fatgue of Arcraft Structures - Monographc Seres Issue 200, Insttute of Avaton Scentfc Publcatons, Warsaw 200: 37 5. 3. Rama Chandra Murthy A, Palan, Nagesh R, Iyer G.S. An mproved Wheeler model for remanng lfe predcton of cracked plate panels under tensle-compressve overloadng, SID, 2005; 3: 203-23. 4. Schjve J. The sgnfcance of fractography for nvestgatons of fatgue crack growth under varable-ampltude loadng, Fatgue Fract. Eng. Mater. Struct. 999; 22: 87 99. 5. Schjve J, Skorupa M, Skorupa A, Machnewcz T, Gruszczyńsk P. Fatgue crack growth n alumnum alloy D6 under constant and varable ampltude loadng. Int. J. Fatgue, 2004; 26: 5. 6. Tomaszek H., Żurek J., Jasztal M. Prognozowane uszkodzeń zagrażających bezpeczeństwu lotów statków powetrznych, Wydawnctwo naukowe ITE, Radom 2008. Prof. dr hab. nż. Henryk TOMASZEK Ar Force Insttute of Technology ul. Ksęca Bolesława 6, 0-494 Warsaw Dr nż. Mchał JASZTAL Mltary Unversty of Technology ul. Kalskego 2, 00-908 Warsaw E-mal: mjasztal@wat.edu.pl, Dr nż. Marusz ZIEJA Ar Force Insttute of Technology ul. Ksęca Bolesława 6, 0-494 Warsaw E-mal: marusz.zeja@twl.pl 34 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: DZICZKOWSKI L, BUCHACZ A. Modfcaton of a calbraton method for conductometers. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 35-40. Leszek DZICZKOWSKI Andrzej BUCHACZ Modfcaton of a calbraton method for conductometers Modyfkacja metody skalowana konduktometru* The paper presents an nnovatve and mproved method for doubled calbraton of an eddy-current conductometer. Implementaton of the proposed method makes t possble to acheve ndependence of the nstrument ndcatons on surface condton of the examned parts. The calbratng functon s determned on-lne when the measurements are n progress on the bass of parameters that are assgned to the contact probe col and calculated wth use of general mathematc equatons. The calculated correcton parameter can be consdered as a measure of the surface roughness. Applcaton of the modfcaton as descrbed n ths paper can contrbute to effcency mprovement of operatonal examnatons for components made of non-ferrous materals durng ther lfetme. Keywords: eddy currents, conductometry, non-destructve testng, surface roughness. Zaproponowano orygnalną, ulepszoną metodę podwójnego skalowana konduktometru wroprądowego. Realzując tę metodę można uzyskać nezależność wskazań przyrządu od stanu powerzchn badanych elementów. Funkcja skalująca konduktometr jest wyznaczana dynamczne w czase trwana pomaru na podstawe parametrów przypsanych eksperymentalne sondze-cewce stykowej oraz unwersalnych równań matematycznych. Oblczona poprawka może być traktowana jako mara chropowatośc powerzchn. Stosując opsaną modyfkację można znaczne usprawnć badana eksploatacyjne elementów wykonanych z metal neżelaznych. Słowa kluczowe: prądy wrowe, konduktometra, badana nenszczące, chropowatość powerzchn.. Introducton An eddy-current conductometer s an nstrument that can be used for non-destructve measurements of conductance (specfc electrcal conductvty) of materals that the examned part s made of. The measurng probe, made as a contact col and suppled wth alternate current, s approached to the surface under test. As the col conducts electrc current, t generates prmary magnetc feld, whch, n turn, entals nducton of eddy currents n a conductve materal beneath the col. Accordng to the Lentz rule the secondary magnetc feld produced by such eddy currents s a negatve feedback,.e. the drecton of ts nducton vector s opposte to the one of the exctng magnetc feld. Consequently, a resultng magnetc feld s orgnated wthn the area of the col probe that amends the col mpedance. The feld component produced by eddy currents depends on the conductance parameter of the materal the examned part s made of. Therefore the dfferental mpedance of the probe shall also depend on the measured conductance. Eddy-current conductometers measure mpedance components of the col probe and use the measurement results to calculate the actual conductance. Conductometers, beng nstruments desgned to measure absolute values of conductance, not only are meant to measure specfc electrcal conductvty of non-ferrous materals or electrolytes, but also can be used to determne purty of metals, montor structures and homogenety of non-ferrous alloys and for ndrect tests of strength and hardness, estmaton of phosphor content n copper, control of castng operatons (polarzaton of copper), montorng of separaton processes, e.g. Cu-Cr, sortng and separaton of metal scrap, detecton of fatgue and thermal defects n alumnum alloys (operaton tests of arcrafts) and many other applcatons. Conductometers are calbrated aganst dedcated patterns, where the determnable parameter..e. conductance s assgned to physcal varables that can be measured n a drect manner (col resstance and nductance. But the measurement results are accurate and relable only when structures and shapes of nvestgated workpeces are smlar to recprocal parameters of patterns that were already used for calbraton. Unfortunately, examnatons wth use of eddy currents s assocated wth two detrmental effects that make the measurements more troublesome and dffcult. Applcaton lmts for the eddy current method are determned by consequences of the followng phenomena:. Surfaces of examned workpeces are not always flat, wth frequent buckles and cavtes. It s possble to place a spacng rng below the measurng col to elmnate swvelng of the probe on the surface, but anyway t s mpossble to approach the probe col to the examned surface so that the dstance between the probe and the surface would be always constant durng the calbraton process. Varatons of the dstance between the col and the examned surface consderably affect nstablty of the col mpedance, even more than varaton of the conductance to be determned. (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 35

NAUKA I TECHNIKA 2. The problem s usually resolved by ntroducton of the compensaton mechansm that counterbalances varatons of the dstance between the col and the surface. The conventonal mechansm was ntally proposed by F. Förster and s descrbed n detals n the paper of Dzczkowsk [5]. The compensaton method s based on a brdge crcut, where one branch of the brdge ncorporates the measurng col connected n seres wth a capactor. The second branch represents also a seral resonance crcut that s made up of a reference col and an adjustable capactor. The measurement conssts n undmensonal balancng of the brdge by approprate varatons of the capactor value. Characterstc parameters of the resonance crcut vary n pace wth alteratons of col resstance and nductance, whlst these parameters, n turn, depend smultaneously on conductance of the materal under test and dstance between the probe and the examned surface. By crafty tunng of the resonance crcuts t s possble to acheve growth of ndcatons for the measured conductance when the col s moved away to a small dstance. Farther retreatng of col lead to a rapd drop of ndcatons coupled wth the measurement result. Fnally a range of certan dstances between the col probe and the surface s obtaned where the result of measurements fts nto lmts of the assumed devaton. Consequently, measurement accuracy s sacrfced n favour of the possblty to take measurements for uneven workpeces. The authors strongly beleve that t was the most mportant achevement n technques of measurements carred out wth use of eddy currents n the area of non-destructve tests. The amount of data processng s not a problem as modern measurng nstruments are provded wth mcrocontrollers wth massve computaton performance and capable to carry out calculatons n the real tme mode when the measurements are stll n progress. Therefore t s necessary to seek for such computaton algorthms, where varatons n the dstance between the measurng col and the examned surface can be compensated wthout notceable ncrease of errors n determnaton of conductance. The magnetc feld nduced by eddy currents s drected opposte to the exctng feld. Therefore a specfc barrer s formed that prevents from penetraton of the prmary feld deeply nwards the examned workpeces. Consequently, eddy currents are nduced exclusvely wthn the surface-adjacent layer of examned workpeces and propertes of only that layer affect results of measurements. Penetraton depth of eddy currents substantally depends on ther frequency. Research studes are descrbed, for nstance, n the collectve study that deals wth modelng and detecton of surface flaws [8]. Ths study, smlarly to other lterature references, uses the smplfed defnton for penetraton depth of eddy currents that has been sourced from the eddy current heatng technology. Somewhat more detaled descrpton of that phenomenon can be found n the paper of Dzczkowsk [2]. Snce eddy currents flow only on the surface of examned workpece, any surface roughness makes flow of these currents more dffcult. If so, measurements of conductance wth use of an eddy current nstrument make t possble to dentfy surface roughness wth apparent growth of conductance and apparent ncrease of dstance between the col and the workpece surface. The measurng nstrument was ntally calbrated aganst polshed specmen, thus one should expect consderable devatons of measurement results. The wdespreadly accepted approach, descrbed n techncal gudelnes and lterature references, assumes applcaton of low frequences as n such a case eddy currents are capable to penetrate much deeper and possble roughness only slghtly dsturb measurements. More detaled analyss leads also to the concluson that frequency of eddy currents affects not only penetraton depth but also senstvty of the measurng nstrument [4, 6]. For a specfc test t s frequently more convenent to adjust the most sutable frequency that s best for the desred senstvty. But such processes as selecton of the best frequency, elmnaton of the mpact from surface roughness and compensaton of devatons are always nseparably nterconnected. Impedance of a measurng col s always a complex parameter, therefore measurng of ts two components enables ndependent determnaton of two parameters for the workpece under test or makes t possble to fnd out one parameter and compensate effect of another one. The manner, how results of two measurements are used to determne a snge, dscretely selected parameter depend on type of the measurng nstrument. The nstruments that can be possble used for measurements,.e. flaw detectors and conductometers substantally dffer from one another. The measurement process carred out wth use of a conductometer s carred out as a sequence of operatons amed at determnaton of the absolute conductance value for the materal the workpece s made of. The fnal result should be ndependent on other factors that unfortunately also nfluence varatons of components for the col mpedance. These factors nclude poston the measurng col aganst a workpece under tests and condton of the workpece surface. For nvestgatons wth use of flaw detectors the nformaton about absolute value of the conductance for the materal the workpeces s made of s rrelevant as the tests are focused on detecton of any possble dscontnutes wthn the structure under test [9] as for flaw detecton t s necessary to fnd out all possble cracks or excessve surface roughness. Therefore t s the nstrument that should be used to measure apparent varatons of conductance as opposed to a conductometer that, by ts nature, s nsenstve to such apparent varatons. 2. Propertes of the desred mathematcal model The calbraton process wth use of standard patterns results n establshng the nterrelatonshp between the parameters that can be measured n a drect way and the value to be detected. Each pattern that has been used for calbraton s meant to fnd out a sngle pont for the calbraton scale, but between the calbraton ponts the functon relatonshp must be approxmated. The more accurate the approxmaton functon s, the less number of calbraton patterns s necessary to fnd out the calbraton scale that guarantees acceptable devatons wthn the assumed measurement range. When an eddy current devce s used to perform measurements for varous frequences of the exctng feld or even ths frequency s automatcally adjusted n course of the test wth mandatory compensaton for varatons n the dstance between the probe (col) and the examned surface, the calbraton process s equvalent to settng up a functon of four varables. Such a calbratng functons has the followng arguments: frequency of eddy currents, dstance between the probe and the 36 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology examned surface, measured dfferental resstance of the col (real component of mpedance) and the measured dfferental nductance of the col (the parameter that depends on the magnary component of mpedance). The determned conductance value was unambguously assgned to each set of four foregong arguments. The calbraton process wth use of standard patterns should be carred out for a set that s made up of a measurng nstrument and a col probe. After a probe s replaced wth another one the calbraton process must be repeated. To set up the calbraton functon one must have a general but suffcently accurate mathematcal model that must enable to take account for all the foregong provsons. 3. The mathematc descrpton of the effect how a conductve workpece affects varatons of a measurng col The avalable lterature offers a number of mathematcal models that can be sutable to carry out the foregong operatons. These models have been obtaned after resolvng the Maxwell equatons. They dffer between each other n terms of the resolvng method and other approxmatons appled. The most popular one s the Fnte-Element Method (FEM). Qute a lot of computatons s carred out wth use of the Boundary Element Method (BEM) and the Fnte Dfference Tme Doman (FDTD) methods. The foregong methods are sutable for flaw detecton technology to establsh models of flaws and can be used to fnd out how a structure of any shape affects the parameters that can be measured drectly. Under assumpton that the structure to be nvestgated s homogenous, suffcently large and ts shape can be naturally descrbed wthn a cylndrcal coordnate system, the avalable analytcal methods seem to be more convenent. Complete systems of equatons capable to descrbe the these phenomena were publshed by Dodd C. D., Deeds W. E and Luqure J. W. n early 70 s []. For the Dodd and Deed s model the functons that are the result for the dfferental equatons provded adopt the form of an ntegral of the Bessel functon wthn unlmted boundares. It s convenent to present the vector of magnetc potental as a sum of a seres. It enables to relatvely easly to take account for fnte, but cylndrcal dmensons of examned workpeces and avod the burdensome process assocated wth computaton of the ntegral. That method, commonly known as TREE (Truncated Regon Egenfuncton Expanson) was subjected to numerous modfcatons and has become a convenent and fast soluton sutable for practcal applcatons [2]. The reference study provdes the mathematcal expressons n the form that s sutable for practcal applcatons. The left-had sde of fgure presents the model ar-flled col [2] wth n turns encapsulated wthn a rng wth the followng dmensons: r nternal radus of the wndng, r 2 external radus of the wndng, z dstance between the outermost part of the col and the surface of the examned workpece, z 2 dstance between the nner part of the col and the surface of the examned workpece. It s assumed that a thck cylnder wth the radus of b and made of a conductve materal wth the conductance of σ s postoned below the col provded that the cylnder materal s not a ferromagnetc stuff. The mpedance varaton of an ar-flled col wth already known dmensons and approached to a conductve workpece s expresses by means of the formula [2]: where: 2 jω2πµ 0n ZT = 2 2 ( r2 r) ( z2 z) 2 qz 2 Int ( qr, qr exp( ) exp( qz 2 ) 2) q p 2 5 = qb J qb + q q p ( ) ( ) 0 x2 () (, 2) = ( ) (2) Int x x xj x dx x The x stand for roots of the equaton: and q coeffcents are expressed as: J (x ) = 0 (3) q x b = (4) It turn, p can be calculated from the relatonshp: 2 p = q + jωµ σ (5) The formula () makes t possble to calculate varatons (dfferentals) of the resstance and nductance parameters for an ar-flled col wth consderaton to ts geometrcal parameters: r = Re(ΔZ T ) (6) l Im Z = ( T ) A substantal nconvenence of the foregong model that prevents from applcaton of the same for desgn of calbraton equpment s the nfeasblty to take account for all dmensons of the col. When addtonal factors are ntroduced nto computaton the foregong equatons become much more soph- ω 0 (7) Fg.. Dmensons of contact cols to be used for modellng Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 37

NAUKA I TECHNIKA stcated, the computaton tme s nadmssbly prolonged and calculaton of the calbraton factors n the real tme mode turns out to be nfeasble hen the test s stll n progress. Regardless to the fact that two cols wth the same dmensons and the same number of turns were prepared, the spoolng process for them durng manufacturng =s not controlled, therefore the probes desgned wth use of these cols demonstrate slghtly dfferent senstvtes. Consequently, a separate calbraton procedure must be carred out for each of the cols. The calbraton method that s proposed by authors of ths paper uses a smplfed model that consders a col as a 2D structure. It s assumed that all n turns are encapsulated by a crcle wth the radus of r 0 and postoned n parallel to the workpece surface wthn the dstance of h. It s also assumed that the conductve workpece s a half-space. The rght-hand sde of Fg. presents a schematc dagram of a model 2D col. A smlar soluton was appled n [] wth further modfcaton n [3]. The prmary reason for such smplfcaton was the wsh to develop a mathematcal model that would be suffcently fast for computatons and would enable to fnd out the optmum eddy current frequency wth the crteron of mnmum errors (devatons). It s why applcaton of the followng generalzed parameters proves to be convenent: α = 2 h r0 (8) β = r 0 ωµσ 0 (9) If so, varatons (dfferental) of col mpedance s descrbed by means of the equaton below: 2 + j Z = ωπ r µµ rn jβ λ λ αβλ 2 0 0 e J ( βλ) dλ 2 λ + λ + j 0 2 (0) Upon extracton of the real and magnary parts t was possble to fnd out varaton (dfferental) of the col resstance: where: 2 r = R R = n ωπµ r φ( αβ, ) () 0 φαβ β λ λ 2 + j αβλ (, ) = Re 2 j ( βλ) λ e J d 2 λ + λ + j 0 and also the col nductance: where: 00 (2) l = L L = n πµ r χαβ (, ) (3) 0 2 χαβ β λ λ 2 + j αβλ (, ) = Im 2 j ( βλ) λ e J d (4) 2 λ + λ + j 0 Expressons (2) and (4) serve as formulas for generalzed descrpton of varatons (dfferentals) exercsed by the col mpedance due to presence of a conductve half-space. 4. Determnaton of the calbraton functon for conductometers Implementaton of the proposed methods needs to use standard patterns wth already known conductance values, wth polshed surfaces and thckness values much exceedng 00 the expected penetraton depth of eddy currents. The contact col s approached to a standard pattern and varatons of the col resstance (r) and nductance (l) are measured at a specfc frequency o eddy currents. The system of equatons, obtaned after substtuton of (8) and (9) to () and (3), enables to calculate values of r 0 and h=h 0 when the measured varatons of r and l are already known. The calculated values of r 0 and h 0 are equvalent parameters of the col, thus they are stored for further use each tme when the conductance parameter s to be found out. Durng each workng cycle varatons (dfferentals) of the col resstance and nductance are measured and then, wth use of an embedded controller, necessary numercal computatons are performed. The computaton consst n resolvng of the equaton system () and (3) wth determnaton of the α and β values. Then the equaton (9) together wth the modfed equaton (8): α = ( h + h) r 2 0 0 (5) are used to calculate the parameters of σ and h. Calculaton of two parameters at a tme s a natural soluton used to compensates effect of the dstance between a probe and a workpece surface onto the result for conductance measurement. When surface of the examned workpece s smooth and flat, the h value should be zeroed, as the probe was approached to the examned surface wthout a spacng rng. When the measured dstance ncreases and amounts to h=h p, t serves as the nformaton about the surface roughness. The experence acqured durng operatonal examnatons suggests that two types of surface unevenness should be taken nto account. The frst type represents convex or concave surfaces that form the workpece shape and can be observed down the dstances comparable wth horzontal dmensons of the col. Such unevenness shall not be measured as apparent growth of the col conductance. Applcaton of the proposed method of measurements and computatons makes t possble to compensate effects of such convex and concave unevenness due to compensaton of the dstance between the probe and the surface of the examned workpece. In such a case the result of conductance measurements s burdened wth merely an addtonal error that s caused by drop of the nstrument senstvty [6]. The second type of surface unevenness s rather assocated wth surface flaws than large-szed buckles. Such flaws nclude cracks, scratches, delamnaton, remnants after machnng processes. In the theory of machnery desgn such faults are referred to as surface roughness [0]. Occurrence of surface roughness s pronounced by apparent growth of the col conductance, whlst the proposed method for measurements, calculatons and calbraton s not suffcently effectve for that applcaton. The r 0 and h 0 parameters can be consdered as establshng a correlaton between each real col and a certan dmensonless model col. If so, the smplfed mathematcal model appled to that dmensonless col can be used to determne the calbraton factor for the specfc nstrument. If the measurements condtons only slghtly dffer from the actual crcumstances for calbraton wth standard patterns, the error (devatons) for determnaton of conductance shall be nsgnfcant and acceptable. However, the ensung problem must be resolved,.e. how to determne the number of calbraton ponts and, consequently, the number of standard patterns that s necessary for calbraton. One has to keep on mnd that the overall objectve s to assure 38 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology the maxmum permssble calbraton error s never exceeded over the entre measurement range. 5. Compensaton of the roughness effect onto results of conductance measurements The measurements wth use of the proposed calbraton method lead to determnaton of two parameters: the conductance (σ) and dstance (h=h p ). The h p value serves as the measure for apparent growth of the dstance between the probe and surface of the examned workpece. Such an apparent growth provdes nformaton that the roughness really occurs. After completon of the frst calbraton cycle wth use of polshed standard samples the nstruments should be recalbrated, but wth use of standard samples wth defned roughness. It was assumed that the average roughness profle of the examned workpeces s smlar to the roughness profles demonstrated by avalable standard patterns. Now t s enough to measure the apparent dstance between the probe (col) and the sample surface and repeat the measurements for several standard patterns wth already known conductance values. One has to keep n mnd that apparent varaton (dfferental) of the mentoned dstance s caused by the surface roughness. Upon takng the measurements t s necessary to store the assocated pars of fgures: the apparent dstance h p and the apparent varaton of conductance expresses by the dfferental of the β parameter. Such a dfferental can be expressed by means of the followng formula: ( ) β = β β = r 2 π fµ σ σ (6) zo s z 0 0 s z where: σ s - actual conductance of the appled standard pattern wth defned roughness or actual conductance determned from measurements, σ z - measured conductance,.e. apparently reduced due to roughness of the workpece surface, β z - generalzed parameters specfed by the equaton (9) for σ = σ z, β s -generalzed parameters specfed by the equaton (9) for σ = σ s. Next, the followng functon can be developed by nterpolaton and stored as a calbraton curve: β zo = f ( h p ) (7) Applcaton of the double calbraton process makes t possble to fnd out the correcton factor for measurements of conductance. Effectveness of the proposed method has proved to be really good when roughness profle of standard patterns s dentcal wth recprocal profles of examned workpeces. It was, for nstance, the case, where the roughness standard patterns were sampled from the same manufacturng lne where conductance measurements were carred out. The examnaton enabled to fnd out that the apparent varaton of the dstance h p between the probe and the examned workpece, ntally measured for compensaton purposes, can serve as an eddy current measure of roughness. Although that measure s not equvalent wth the parameter commonly used for technologcal evaluaton, but a correlaton between these two parameters can be found [7]. The examples of results for measurements wth use of Ra and Rz methods as a comparson aganst the eddy current technque are summarzed n table. Table Ra [μm] Rz [μm] h p [μm] Ra [μm] Rz [μm] h p [μm] 347 579 690 26 80 30 56 747 850 5 48 60 96 347 250 6.2 22 30 35 98 250.4 7.4 0 7. Verfcaton of the calbraton process Implementaton of the foregong calbraton method that conssts n calculaton of equvalent parameters r 0 and h 0 makes t possble to fnd out the calbraton functon wthn the vcnty of the pont defned by parameters of the standard patterns and the appled frequency. The experments revealed that bas of the calbraton pont,.e. alteraton of frequency or conductance, faled to ental any changes determned by means of the method nvolvng r 0 and h 0 parameters. The recorded alteratons of these parameters were not hgher than the estmated errors (devatons) that could have resulted from propertes of the electronc measurement path. Therefore t s enough to calbrate the nstrument only n one pont, wth use of a sngle standard pattern. To verfy ths hypothess some addtonal computatons were carred out. Applcaton of the aforementoned TREE method enabled to calculate varatons (dfferentals) of resstance and nductance for specfc, already assumed dmensons. Then the system of equatons () and (3) was resolved wth consderaton to (8) and (9). Upon the varatons of resstance and nductance were known, t was possble to calculate generalzed parameters r 0 and h. It turned out that the value of the r 0 parameter s ndependent on eddy current frequency, col conductance and the dstance between the col and surface of the examned work- The measurements comprse determnaton of the dstance value h p and the value for the generalzed parameter β=β z that s burdened by errors due to the surface roughness. The β z value correspond to the ncorrectly measured value of conductance σ z. Therefore the equaton (7) s used to fnd out the correcton factor Δβ zo for the generalzed parameter. Fnaly, the desred value of σ s can be calculated by means of the equaton (6). 6. Roughness measure Fg. 2. Effect of the dstance between a 3D col and the surface of the examned workpece onto the correspondng dstance of the equvalent model col Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 39

NAUKA I TECHNIKA pece. Also the value of h proved to be unbounded wth frequency and conductance values. To check relatonshp between h and the dstance between the col and surface of the examned workpece some further computatons were performed. It was assumed that dmensons r and r 2 are constant alke to the value of the dfferental z 2 - z. The value of z was made varable and the equvalent parameter of h was computed. Obtaned results are presented on fgure 2. It was observed that the relatonshp between the values h and z represents a lnear functon wth devatons from lnearty even smaller that the expected errors resultng from naccuracy of numercal calculatons. 8. Conclusons When the calbraton method proposed n ths study s appled to calbraton of conductometers t s possble to effectvely determne conductance n the real tme mode and to smultaneously compensate effects of varatons n the probe (col) dstance from surface of the examned workpece. Implementaton of the double calbraton method enables effcent compensaton of the effect exercsed by surface roughness of the examned workpeces provded that the roughness profle of standard patterns already used for calbraton s smlar to the correspondng roughness profle of examned workpeces. The determned compensatng factor ntended to elmnate effect of surface unevenness s actually the measure of roughness. Owng to equvalent parameters of the col determned durng the calbraton process t s possble to substtute each real col wth a model one, not only for the calbraton procedure but also for further numercal calculatons. It partcular, t s possble to use a smple mathematcal model to fnd out the optmum frequency of eddy current, the best sutable for any specfc applcaton. 9. References. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. Dodd C V, Deeds W E, Luqure J W. Integral solutons to some eddy current problems. Internatonal Journal of Nondestructve Testng 969; : 29 90. Dzczkowsk L. A defnton of eddy current penetraton depth useful for flaw detecton and conductvty measurement. Journal of Appled Mathematcs and Mechancs 2008; 8(): 0205-0206. Dzczkowsk L. A mathematc model to determne optmum condtons for measurements of materal conductance by means of the eddy current method applcable to large structures. Mašnostroene Technosfera XXI weka. Sbornk Trudow XV Meždunarodnoj Naučno-technčeskoj Konferencj. Doneck 2009; 4: 69-73. Dzczkowsk L, Dzczkowska M. Effect of naccurate settng for the exctng feld frequency onto results of conductance measurements wth use of the eddy current method. Avtomatzacja: Problemy, Ide, Rešenja, Materaly Meždunarodnoj Naučnotechnčeskoj Konferencj. Sevastopol: Vdavnctvo SevNTU, 200; 2: 3-5. Dzczkowsk L. Effect of eddy current frequency on measurng propertes of devces used n non- destructve measurements of non-ferromagnetc metal plates. Archves of Materals Scence and Engneerng 2008; 32(2): 77-84. Dzczkowsk L. Errors n conductance measurement of materals that are used for constructon of thck plates. Mašnostroene Technosfera XXI weka. Sbornk Trudow XV Meždunarodnoj Naučno-technčeskoj Konferencj. Doneck 2009; 4: 66-69. Dzczkowsk L. Surface roughness and eddy current. 6th Internatonal Conference Mechanc Systems and Materals. MSM Opole Poland 5 8 July 200: 58 59. Gernhas Ramos H M, Postolache O, Corrêa Alegra F, Lopes Rbero A. Usng the Skn Effect to Estmate Cracks Depths n Mettalc Structures. Internatonal Instrumentaton and Measurement Technology Conference. I2MTC Sngapore 5-7 May 2009: -6. Jasńsk W, Zawada P. The nvestgaton of dscreteness n materal of catalytc ppes by the eddy current method. Eksploatacja Nezawodnosc - Mantenance and Relablty 2003; 2: 28-30. PN-87/M-0425. Struktura geometryczna powerzchn. Chropowatość powerzchn. Wartośc lczbowe parametrów. Norma Polska 30.06.987. Smankowa L. Mathematcal presentaton of mpedance varaton of a col cause by the measured object. TESLA electroncs 97; 4: 2-7. Theodoulds T, Krezs E. Seres expansons n eddy current nondestrutve evaluaton models. Journal of Materals Technology 2005; 6(-2): 343-347. Leszek DZICZKOWSKI, Ph.D., Eng. Insttute of Electroncs Slesan Unversty of Technology n Glwce 6, Akademcka St., 44-00 Glwce, Poland e-mal: ldzczk@wp.pl Prof. Andrzej BUCHACZ D.Cs., Eng. Insttute of Engneerng Processes Automaton and Integrated Manufacturng Systems Slesan Unversty of Technology n Glwce 35, Konarskego St., 44-00 Glwce, Poland e-mal: andrzej.buchacz@polsl.pl 40 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: ŚWIĆ A, MAZUREK L. Modelng the relablty and effcency of flexble synchronous producton lne. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 4-48. Anton ŚWIĆ Lech MAZUREK Modelng the relablty and effcency of flexble synchronous producton lne Modelowane nezawodnośc wydajnośc synchroncznej elastycznej ln produkcyjnej* The paper ntroduces a mathematcal model of operaton of a flexble synchronous producton lne (FSPL) of multfunctonal CNC machnes that ncludes one redundant multfunctonal CNC machne whch can take over the functons of every FSPL machne. The graph of FSPL state, relatons and equatons used to calculate relablty and productvty are shown. Maple, the software used for relablty and productvty calculatons and modellng, as well as the mathematcal results are presented. Keywords: model, relablty, effcency, synchronous flexble producton lne, redundant technologcal cell. Przedstawono model matematyczny funkcjonowana synchroncznej elastycznej ln produkcyjnej (SELP) z obrabarek welozadanowych CNC, w skład której wchodz maszyna technologczna rezerwowa. Maszyna technologczna rezerwowa może przejmować funkcje każdej z obrabarek SELP. Przedstawono graf stanu SELP, zależnośc, równana do oblczana nezawodnośc wydajnośc SELP. Opracowano program oblczeń nezawodnośc wydajnośc (Maple) oraz zaprezentowano wynk modelowana optymalzacj lośc obrabarek. Słowa kluczowe: model, nezawodność, wydajność, synchronczna elastyczna lna produkcyjna, maszyna technologczna rezerwowa.. Introducton Mult-role CNC machnes are manly desgned for processng frame type parts whch have many holes wth dfferent dameters and precson (class 5 to ), on whch resstance ponts are based, and addtonal tools are connected to the frame and to mountng connectons (to attach the part usng screws or pegs and to facltate processng, establshng datums, and assembly). The dmensons of the man hole dameters vary wthn a wde range (from 6 to 50 mm) and depend on the type of part [, 2, 5, 7]. The work [8] ntroduces specfcaton of processng and classfcaton of holes for the system of automated desgn of technologcal processes. Modern market condtons requre producton charactersed by quck start and quck change of the assortment of produced parts. CNC machne tools and Flexble Producton Systems (FPS), combng the hgh flexblty of tradtonal equpment and the hgh effcency of machne tools, are the most effectve equpment for mult-nomenclature producton [, 2, 3, 0, ]. 2. Methodology of modelng FSPL relablty and effcency Every mult-role CNC machne tool can be consdered as a complex system. If the system contans n number of seral connected elements, damage of any of them leads to the falure of the whole system and can be descrbed by graph fg.. Fg.. The condtons graph of the multrole CNC machne tool: a) system elements from to n; b) elements condtons; c) machne as the sum of all elements States on the graph: S 0 all n elements of the system are operatng, S the frst element faled and the system s non-operatonal, S 2 the second element faled and the system s out of order;.., S n nth element faled and the system s not workng. Indcatons on the graph: (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 4

NAUKA I TECHNIKA λ,( =, n) the ntensty of falure stream of to n elements. µ,( =, n) the ntensty of restoraton stream of workng ablty of to n elements. Because of the fact that after falure of any element the rest of the elements cannot functon properly untl the tme of restoraton of ts work ablty, t s consdered that only one element can fal at a tme. All falure and restoraton streams are consdered as smple. The system of equatons for the qualfcaton of fnal probabltes s presented below: n P0 Σλ = Σ Pµ ; = = Pµ = P0λ; P2µ 2 = P0λ2;... Pµ = P0 λ ;... Pµ Pλ. n n n = 0 n The standardzaton condton: n j= 0 () Pj = (2) After change of the frst equaton of system () to the standardzaton condton (2) and solutons, every probablty P, = ( n s, expressed ) by P 0 : λ P = P 0 µ. (3) The set of numbers s marked as I ( I). Let us ntroduce the j, belongng to ths set: j I. Wth regard of new letters, after the substtuton of 3 to the standardzaton condton 2 the followng formula s receved: P 0 = n j + Σ λ j = µ j After substtuton of (4) to (3): P λ where: ρ µ ρ λ j =, j =. µ λ = = ( + ) j ρ n λ n j Σ + j= µ µ Σ j= j The output system (fg.) s replaced wth the smple twostate element: workng and non-work (n the damage condton; non-operatonal). The dagram of such an element or new system s ntroduced n fg. 2. where λ Σ s defned as: Fg. 2. Dagram of the system element n Σ = = λ Σ λ ρ j (4) (5) (6) The value μ Σ s defned from dependence: P0 µ Σ = λσ P After substtuton of (4) to (7) the followng formula s receved: λσ µ Σ = n (8) Σ ρ j= Obtaned dependences allow defnng the total ntensty of falure stream and total ntensty of stream of restorng t to work for the system presented n fgure 2, therefore allow modelng of effcency of system work. 3. The mathematcal model of functonng of FSPL wth redundant technologcal cell At present we use the structure of the flexble synchronous producton lne (FSPL) from the mult-role CNC machne tools wth a redundant technologcal cell (RTC) whch can replace any mult-role machne technologcal cell (TC) [4, 6]. Fgure 3 ntroduces the structure of such a flexble system (FSPL). The redundant technologcal cell (RTC) can replace only one damaged machne (TC), so whole system (FSPL) stops workng after falure of two machnes (TC). Fg.3. The FSPL structure The graph of states (FSPL), ncludng RTC, s ntroduced on fg. 4. States on the graph: S 0 - all multrole machnes (TC) are operatng; S - st TC does not operate; S 2-2 nd TC 2 does not operate;, S n - n - е TC n does not operate; S, - second TC 2 falure whle TC does not operate;.. S,2 - thrd TC 3 falure whle TC does not operate; ; S,n- - the n e TC n falure whle TC does not operate; S 2, - frst TC falure whle TC 2 does not operate; S 2,2 - thrd TC 3 falure whle TC 2 does not operate; ; S 2,n- the n - e TC n falure whle TC 2 does not operate; S 3, - frst TC falure whle TC 3 does not operate; S 3,2 - second TC 2 falure whle TC 3 does not operate; S 3,3 - (not shown on graph) the TC falure whle TC does not 4 3 operate;... ; S 3,n- - the n e TC falure whle TC does not operate; S, n 3 - frst TC falure whle TC does not operate; S - second,2 TC 2 falure whle TC does not operate; ; S,n- - falure n-e TC n whle TC does not operate; S n, - frst TC falure whle TC does n not operate; S n,2 - second TC falure whle TC does not operate; 2 n ; S n,n- - n - e TC falure whle TC does not operate; States n n S 0, S, S 2, S 3, S 4,..., S n - able to work and remanng for an emergency. Graph clarfcaton: λ ( =, n); µ ( =, n) - ntensty of the falure stream and restorng the workng ablty of technologcal devces TC =, n). The number of states s consderable (eg. for n = 0 the number of states s N = n 2 +=0, whch makes model constructon and analyss dffcult). That s the approach based on the ncrease of states s proposed. 0 j (7) 42 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology Fg. 4. Graph of FSPL condtons, ncludng one reserved RTC place We solate the followng subsets n E set (power N): E = S, S,, S, 2,..., S, n E = S, S, S,..., S n { } 2 { 2 2, 2,2 2, } ; {,,,..., n } E = { S, S,, S, 2,..., S, n } ; ; ; ; En = { Sn, Sn,, Sn, 2,..., Sn, n } E = S S S S 3 3 3, 3,2 3, We wll qualfy the probablty of system exstence n these subsets. In ths case we wll consder a dagram of equvalent enlarged system shown n fgure 5. Fg. 5. Dagram of condtons equvalent of enlarged system States on the dagram (fg. 5): S 0 - all multrole CNC machne tools are operatng; S Σ - the system s n one of states of the E subset; S 2Σ - the system s n one of the states of E 2 subset; ; S nσ - the system s n one of states of the E n subset. On the graph: λ,( =, n) - s the ntensty of the falure streams UT, ( = n, µ 0,( ) = ;, n) - the ntensty of the stream restorng the system workng ablty from subsets E, = ( n, ). The task conssts n defnng µ 0,( =, n). If all dagram (Fg. 4) condtons probabltes are known, then µ 0,( =, n) can be determned from the dependence: P µ 0 = µ n P + P j= ( j ) j ; (9) where P j - the states probablty of S j E, the rate before μ n P (), equal, then condtonal probablty, that t s n P + Pj j= ( j ) nclude n the subset of states E, the system s n the state S. We wll mark the component of Е set as Sk, k( = N, ) (S k E). Dvdng the numerator and the nomnatve () by the probablty of system beng n the E { } n P = P S E = P + P subset, we wll receve: z k j j= ( j ) P µ 0 = µ yµ P = P (0) where P y - the condtonal probablty of system beng n the S state. We wll qualfy the condtonal probablty of the elements of E subsets: P = P S = S / S E ; P = P S = S / S E { } { } y k k jy k k They are equal: P jy P Py = P Pj = = P P + P j n P j j= ( j ) () (2) To determne the probabltes P y and Pjy, ( =, n j; = n ) we should consder subsets E, = ( n as, ndependent ) subsets. For comfort, the set of numbers j s marked as J, (j J). Let us ntroduce numbers m, also belongng to ths subset (m J). Wth regard of the new numbers of dependence to determne P y and P jy we get: P y P jy = + = + n j= ( j ) ρ j n m= ( m ) ρ ρ j m (3) (4) j λ j where ρ µ ρ λm j =, m = - the mported ntenstes of stre- µ ams. Substtutng (3) n (0) we wll receve: m µ = ( + ρ ) µ 0 n j= ( j ) j (5) All ntenstes n the dagram (Fg.5) are known, and the probablty of states P 0,P Σ,... P Σ,..., P nσ, s defned accordng to well-known dependences [9]: n n P0 = + ρ( + ρj ) = j= ( j ) (6) n n n P = + + ρ ( ρj ) ρ( + ρj) (7) = j= ( j ) j= ( j ) Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 43

NAUKA I TECHNIKA After calculatons accordng to relatons (6) and (7), the graph probablty condtons, ntroduced n Fg. 5, can determne the probablty of states S( = n, and ) Sj( =, n ; j =, n ) of the dagram, ntroduced n fgure 3. Accordng to (), (2) and (3), (4): n P = Py P = ( + ρ P j ) (8) j= ( j ) ρ j Pj = PjyP = P n + ρ m= ( m ) After substtutng (7) n (8) and (9) : P j P n n = + ρ ( + ρj ) ρ = j= ( j ) m n n = + ρ ( + ρj ) ρρ j = j= ( j ) (9) (20) (2) The whole ntal structure of the flexble synchronous lne (FSPL) pf mult-role CNC machne tools, ncludng reserve workng place (RTC), s replaced through one smplest equvalent element for whch the ntenstes of the falures streams λ Σ and the restoraton of effcency μ Σ are known. An element wth two states s consdered as the smplest: the standby and the workng state. A dagram of condtons of such an element s shown n Fg.6. Fg. 6. Graph of FSPL condtons, referred to the smplest element States n the dagram (fg. 6): S 0Σ - able to the work; S Σ - broken (unable to work). We wll ntroduce two new subsets of states for the dagram n fgure 4: U - able to work, encrcled wth dashed lne, and V- ncapable of workng: {,,...,,..., } U = S S S S 0 {,,...,, j,...,, n,...,,,..., j,...,, n,..., n,,..., n, j,..., n, n } V = S S S S S S S S S The subset U answers state S 0Σ ntroduced n fgure 6, and the subset V - state S Σ. The probablty of the system beng n states S 0Σ and S Σ s equal to: n n 0 = 0 + = (22) P P P n n = = 0 j (23) = j= ( j ) P P P Intenstes λ Σ and μ Σ for graph ntroduced on fg. 6 are equal to: n n n P P λ = λ = λ ( j ) ( ) n j = P0 j= ( j ) = j= ( j ) P + P n n Pj µ = µ = j = j= ( j ) P n 0 = P n n j n n = j= ( j ) j j= j= ( j ) (24) µ j (25) P The developed model for determnng the relablty and effcency of FSPL allows replacng any technologcal machne n lne wth redundant technologcal machne. 4. The software for defnng effcency of FSPL The program for defnng the parameters of functonng of synchronous FSPL was wrtten n the mathematcal software for analytc calculatons Maple. Ths envronment s a powerful computer tool, able the solve complex mathematcal tasks. It contans tools related to many mathematcal felds (algebra, dscreet mathematcs, dfferental and ntegral mathematcs, numercal and dfferent methods) and also allows graphcal representaton, and connecton to external modules and programmng tools. The components of the program: --block pattern of nput data, --block of calculaton of requred parameters functons of the synchronous lne wth (wthout) the reserve place, --block of formattng results of the experment and output for these results, Input data to the executon of research: --maxmum number of cells n the lne - N, --ntensty of the stream of damage λ and restorng the workng ablty μ of every unt ( = N ), --average tme of servce for every producton ndvdual cell t = N, --step of calculatons n (total number equal to the dfference between the values of two of the current number of cells n lne n of neghbourng cycles). The block of calculatons comprsed the followng operatons: λ --defnng ntenstes of streams ρ =, = N, µ --qualfcaton of the ntensty μ 0 accordng to dependence (5), --calculaton of the probablty P 0 accordng to dependence (6), --calculaton of the probablty P Σ, P, P j accordng to dependences (7), (8), (2), respectvely, --qualfcaton of the rate of readness of the lne K g = P 0Σ accordng to dependence (22), --calculaton of the effcency of the lne: Q =, where t t max maxmal tme among average tmes of servce for every producton cell t ( = N, --defnng the parameters of functonng of the synchronous lne not ncludng the reserve place: ' -- rate of readness of the lne K g =, + Σρ Q = K g. t ' ' -- effcency of the lne max K g max --calculaton of current values -- ncrease of the coeffcent of readness of the lne as absolute value ΔK g =K g -K g K g n percentages δ Kg = 00% ' K, K gmax gmax 44 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology -- ncrease of the effcency of the lne as absolute value Q = Q - Q Q n percentages δq = ' Q, Q 00% These calculaton are taken cyclcally untl the condton n = N s not met. After that the programme works out the results of the experment (ncrease of effcency) and presents the results on the screen as a matrx and a chart. 5. Results of calculatons of relablty and effcency parameters of FSPL Above presented software for mathematcal calculatons (Maple) was used to determne the ncrease of effcency of FSPL. Calculatons were conducted for sets of dfferent nput data of FSPL contanng 0 machne tools. The algorthm of calculatng parameters of FSPL that contans multpurpose CNC machne tools was worked out. The followng parameters were defned as nput parameters for modelng n all consdered cases: --ntensty of falure stream λ, --ntensty of restorng to work stream µ, --average servce tme of machne tool t. Calculatons of ncrease of lne effcency were conducted for maxmum number of TC equal to 0: max max Case : Calculatons were conducted for constant parameters of relablty and servce: λ = 0,2 [h - ], µ = 5 [h - ], t = 0, [h] Results of the ncrease of effcency calculatons conducted n Maple are presented as matrxes and graph n fgure 7. The ncrease of effcency ΔQ s almost lnear n whole range of numers of machnes tools ( to 0) n FSPL. Case 2: Calculatons were conducted for constant parameters of relablty and servce: λ = 0,25 [h - ], µ = 4 [h - ], t = 0, [h]. Results of the ncrease of effcency calculatons conducted n Maple are presented as matrxes and graph n fgure 8. Sgnfcant and almost lnear ncrease of effcency of FSPL ΔQ was observed when ncreasng number of machne tools (up to 8), after further ncreasng number of machne tools the effcency decreases. Case 3: Calculatons were conducted for constant parameters of relablty and servce: λ = 0,3 [h - ], µ = 3 [h - ], t = 0, [h]. Fg. 7. Dependence of quantty of machne tools on effcency of lne Fg. 8. Dependence of quantty of machne tools on effcency of lne Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 45

NAUKA I TECHNIKA Results of the ncrease of effcency calculatons conducted n Maple are presented as matrxes and graph n fgure 9. The effcency ncreases steeply rght up to the maxmum, then decreases together wth ncreasng the number of TM n Sgnfcant and almost lnear ncrease of effcency of FSPL ΔQ was observed when ncreasng number of machne tools (up to 5), after further ncreasng number of machne tools the effcency consderably decreases. Fg. 9. Dependence of quantty of machne tools on effcency of lne 6. Defnng the optmal number number of machnes n FSPL contanng redundant machne tool The process of runnng the flexble synchronous producton lne whch conssts of consecutvely connected technologcal machnes wth one redundant technologcal machne was examned (Fg. 3). The lne conssts of technologcal machnes of one type n number of n (TM,, TM n ) and one redundant technologcal machne whch s able to replace every sngle machne of TM system. The model of functonng the FSPL s presented n [4] but the whole lne s structure s changed, accordng to relablty parameters, nto the smple equvalent element wth two states (workng and emergency), defnng ts functonng ndexes: ) the ntensty of streams of falure and restoraton of workng ablty; 2) expected value of the producton unt servce tme; 3) avalablty factor; 4) effcency rato takng nto consderaton relablty parameters. By creatng the model t was assumed that all streams whch carry the system from one state to another are smple and servce tmes are dsposed exponentally. However the ntensty of falure stream λ I, restoraton stream of workng ablty μ and servce tmes t are dfferent for every TM. The gven model dffers from the one n references [6] where those factors were equal. The am of the model was to defne the productvty gan that s the dfference of productvty of FSPL wth redundant machne tool and productvty wthout t: Q = Q - Q The analyss of the results obtaned by applyng the model shows, that by ncreasng the number of machne tools n the lne the dagram of the productvty gan s smlar to fgure 0. Fg. 0. The theoretcal graph of dependence of quantty of the TM s n lne on the effcency the lne and the dagram can practcally reach zero. It s obvous because by sgnfcant ncreasng the number of TM n FSPL, one redundant machne tool can not manage to replace the fxed number of TM, well the productvty of FSPL wth redundant machne tool s equal the productvty of FSPL wthout redundant machne tool. --That optmzaton of the task where t s necessary to defne the number of TM n FSPL wth redundant machne tool wll allow obtanng the maxmum productvty gan. Ths s the task of nteger programmng wthout reservaton (takng nto consderaton that number of TM s even). The soluton doesn t requre any specally developed algorthm, t s better to choose the quck choce process whch conssts of repeatable procedure n cycles, n each step j of the mathematcal model [6] for the current number of technologcal machnes TM j t defnes the avalablty factor of FSPL wth redundant machne tool and wthout t and also ts productvty (Q j and Q accordngly). The productvty gan Q s defned. If j j Q j > Q j-, then the current number of TM n the lne - n j s assumed as the optmum pont n îïò. Otherwse the cycle can be stopped. The condtonng of unmodular target functon (there s the only extreme that s the global extreme) s confrmed by fndngs of research wth dfferent parameters. In that procedure 46 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology the quantty j s changed from to n îïò and at the begnnng of the cycle n îïò =. The program of searchng for optmal number of technologcal machnes s realzed n the mathematcal software Maple 9. The ntensty of falure and restoraton stream of workng ablty and also the servce tmes for every sngle machne tool are assumed to be equal. The presented below research s connected wth defnng the nfluence of relablty parameters on optmal number of machne tools n the lne by fxed servce tmes [7].. Relablty parameters were analyzed dependng on the ntensty of falure of every sngle machne tool λ = 0,25 0,35 h - wth constants μ = 3 h - and t = 0,05 h (Fg. ). Maxmum productvty gan and optmal number of machne tools are: --case : ΔQ max = 27,90 pcs /h, n opt = 9 machne tools, --case 2: ΔQ max = 27,50 pcs /h, n opt = 0 machne tools, --case 3: ΔQ max = 27,08 pcs /h, n opt = 2 machne tools. 2. Relablty parameters were analyzed dependng on the ntensty restoraton stream of workng ablty of every sngle machne tool μ = 3, 4, 5 h -, wth constant λ = 3 h - and t = 0, h (Fg. 2). Maxmum productvty gan and optmal number of machne tools are: --case : ΔQ max = 27,0 pcs /h, n opt = 0 machne tools, --case 2: ΔQ max = 26,50 pcs /h, n opt = 2 machne tools, --case 3: ΔQ max = 26,0 pcs /h, n opt = 4 machne tools. The analyss of obtaned results proved that all graphs do have one extremum. The effcency gan n all cases ntally ncreases when ncreasng number of TMs n FSPL, reaches maxmum and decreases after that. 7. Conclusons A methodology for modellng CNC machne tools and FSPL s presented. A mathematcal model of machne tools and FSPL as a structure of elementary technologcal cells has been developed. The obtaned results relate to machnng n flexble systems for Markov chans. All calculatons were conducted for unversal CNC machnng centre KORRADI VH 000, ncluded n the producton lne for machnng engne blocks. Smulatons were performed for FSPL that ncorporates a vertcal machnng centre CINCINNATI SABRE 000 and a vertcal machnng centre CINCINNATI ARROW 000. After smulaton followng results were obtaned: --maxmal effcency gan ΔQ max = 27,90 pcs./h optmal number n opt = 9 machne tools, for followng parametes of ntensty of falure stream of λ = 0,25 0,35 h - and constant μ = 3 h - and t = 0,05 h. --maxmal effcency gan ΔQ max = 27,50 pcs./h optmal number n opt = 0 machne tools, for followng parametes of ntensty of restorng to work stream of λ = 3 5 h - and constant μ = 3 h - and t = 0, h. The presented values of ntensty of streams of each machne tool damage λ, restoraton to work µ and t were obtaned n ndustry condtons. The obtaned results of smulatons of gan n productvty and the optmum numbers of machne tools ndcate that wth deteroraton n relablty parameters there s a decrease n the optmum number of machne tools, but for a specfc number of machne tools the gan n productvty s hgher than for a lne wth analogous parameters of mantenance and better ndces of relablty. Fg.. The graph of effcency vs. number of TM s by followng parameters: ) λ = 0,35 h -, µ = 3 h -, t = 0,05 h, 2) λ = 0,30 h -, µ = 3 h -, t = 0,05 h,3) λ = 0,25 h -, µ = 3 h -, t = 0,05 h. Fg.2. The graph of effcency vs. number of TM s by followng parameters:) µ = 5 h -, λ = 3 h -, t = 0, h, 2) µ = 4 h -, λ = 3 h -, t = 0, h,3) µ = 3 h -, λ = 3 h -, t = 0, h Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 47

NAUKA I TECHNIKA 8. References. Ba Y, Ja X, Cheng Z. Group optmzaton models for mult-component system compound mantenance tasks. Eksploatacja Nezawodnosc Mantenance and Relablty 20; (49): 42 47. 2. Burduk A. Próba adaptacj szeregowych struktur nezawodnoścowych do analzy oceny ryzyka systemów produkcyjnych. Eksploatacja Nezawodnosc Mantenance and Relablty 200; 3(47): 85 96. 3. Farooq S, O Bren C. Rsk calculatons n the manufacturng technology selecton process. Journal of Manufacturng Technology Management 200; vol. 2, : 28 49. 4. Flpowcz O, Mazurek L, Taranenko V, Śwć A. Model matematyczny funkcjonowana elastycznej ln produkcyjnej. Pomary. Automatyka. Robotyka 2007; 2:. 5. Halas W, Taranenko V, Swc A, Taranenko G. Investgaton of nfluence of grndng regmes on surface tenson state. Lecture Notes In Artfcal Intellgence, Vol. 5027. Berln, Hedelberg: Sprnger Verlag 2008: 749 756. 6. Mazurek L, Flpowcz O, Taranenko W, Śwć A. Model procesu przezbrajana welozadanowych obrabarek NC w elastycznym systeme produkcyjnym. Przegląd Mechanczny 2007; 5/S: 04 06. 7. Mazurek L, Śwć A, Flpowcz O, Taranenko W. Zwększene efektywnośc pracy obrabarek welozadanowych w elastycznych systemach produkcyjnych. Lubln: Poltechnka Lubelska, 200. 8. Mazurek L, Swc A, Taranenko V. Holes processng and classfcaton n automated technologcal process projectng system. Acta Mechanca Slovaca, Journal publshed by Faculty of Mechancal Engneerng, the Techncal Unversty n Kosce, Kosce 2006; 2-A, 325 330. 9. Śwć A, Taranenko V. Projektowane technologcznych systemów produkcyjnych. Lubln: Wydawnctwo Poltechnk Lubelskej, 2003. 0. Taranenko G, Taranenko W, Śwć A, Szabelsk J. Modelowane układów dynamcznych obróbk skrawanem wałów o małej sztywnośc. Eksploatacja Nezawodnosc Mantenance and Relablty 200; 4 (48), s. 4 5.. Wang Z, Kang R, Xe L. Dynamc relablty modelng of systems wth common cause falure under random load. Eksploatacja Nezawodnosc Mantenance and Relablty 2009; 3(43): 47 54. Prof. Anton Swć, D.Sc., Ph.D., Eng. Insttute of Technologcal Systems of Informaton Lubln Unversty of Technology Nadbystrzycka Str. 36, 20-68 Lubln, Poland e-mal: a.swc@pollub.pl Lech Mazurek, Ph.D., Eng. The State School of Hgher Educaton n Chelm Pocztowa Str.54, 22-00 Chelm, Poland e-mal: lmazurek@pwsz.chelm.pl 48 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: CHENG Z H, BAI Y S, CAI L Y, WANG L C, LI P Y, CHEN L. Research on warranty nterval of mult-component system wth falure nteracton. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 49-55. Zhong Hua CHENG Yong Sheng BAI L Yng CAI Lu Chao Wang Peng Ju LI Lng CHEN Research on Warranty Interval of Mult-component System wth Falure Interacton Badana okresu gwarancyjnego dla systemu weloskładnkowego, w którym zachodzą nterakcje uszkodzenowe Based on the analyss of falure nteracton, mperfect preventve warranty polcy s adopted for the mult-component system. Average falure rate of each warranty nterval s studed and warranty cost model and avalablty model are bult as vewed from nteractve falure rate. Then Warranty perod project s brought forward as an example, whch can valdate the feasblty of model and show the advantage of the project. The research can provde technque and methods for determnng Warranty Perod of mult-component system, whch further enrches and perfects the warranty theory. Keywords: warranty perod, cost, avalablty, falure nteracton, mult-component. W oparcu o analzę nterakcj uszkodzenowych, przyjęto dla systemu weloskładnkowego poltykę gwarancyjną obejmującą nepełną odnowę proflaktyczną.zbadano średną ntensywność uszkodzeń dla każdego okresu gwarancyjnego oraz skonstruowano modele kosztów obsług gwarancyjnej oraz dostępnośc borąc pod uwagę ntensywność uszkodzeń nterakcyjnych. Jako przykład podano projekt okresu gwarancyjnego, który może potwerdzć poprawność przyjętego modelu oraz przedstawono zalety takego projektu. W badanach opracowano technkę metody ustalana okresu gwarancyjnego dla systemów weloskładnkowych, które stanową stotny wkład do teor gwarancj.. Słowa kluczowe: okres gwarancyjny, koszty, dostępność, nterakcje uszkodzenowe, weloelementowy.. Introducton In order to prevent product falure or ts serous results and keep t n a prescrbed state, a seres of actvtes performed by the manufacture alone or jontly wth the user are called preventve warranty. It prmarly ncludes trouble shootng, perdc perfect mantenance and perdc mperfect mantenance etc. The paper researches on preventve warranty whch manly contans perdc perfect mantenance polcy and perdc replacement polcy. Chun [2] ntroduced perdc preventve warranty n pror tme when he studed product warranty. Jack [4] further studes the model and the product can be made to repar as good as new after preventve warranty, whch permt preventve warranty nterval varable. In order to acheve the lowest warranty cost, Yeh [3] mproves the model as to make the degree of the preventve warranty reach some requred level. On the base of the updated warranty polcy, many scholars set up preventve warranty cost model n warranty nterval and study the optmal perdc preventve warranty nterval, whch am at gettng the lowest warranty cost [, 4, 9]. The document [5] balances the saved and added cost by warranty products preventve warranty, and studes optmal preventve warranty strategy regardng product s long term average scale of charges n mnmum as goal, and determnes the best preventve nterval and provdes the effcent algorthm. The above mentoned studes whch research on warranty nterval amed at ndependent components and mult-component wth separate falure, whch affects the practce of the applcatons to some extent. Along wth technology development, the product wth more complex and ts varous components wth more nteracton between certan parts of system, each of the falure of ts own abrason or agng, or some other unts s the falure of the product, whch make s not enough for warranty research to only pay attenton to sngle component or mult-component system wth separate falure. Therefore, based on the analyss of mult-component wth falure nteracton, ths paper wll establsh warranty cost model and avalablty model under the mperfect preventve warranty polcy, analyze cost effectveness at unt nterval, decde warranty decson-makng project of mult-component system wth falure nteracton and valdate ths project. Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 49

NAUKA I TECHNIKA 2. Falure nteracton analyss Thomas [2] thnks that system of the mantenance nteracton among the nternal components can be dvded nto three categores: economy nteracton, falure nteracton, and structure nteracton. The so-called economy nteracton s that mantenance costs of a few parts by reparng together were lower than separately. The structure nteracton s a body conssted by a number of components, whch reparng some one means to repar other parts. Falure nteracton s that falure of some component of the system wll cause falure dstrbuton of other parts of the system changng, so falure nteracton was beleved to be exsted between the two parts. In early relevant documents, two condtons of falure nteracton [8]: (a) a component (affectng components) falure resulted n malfuncton of other components (affected components) at the rate of p ( p ). (b) The falure of the affected components wll ncrease the agng degree, but wll not cause the mmedate problems. The results of the two cases are that the falure rate of the affected components s accelerated. The components nteracton wll ncrease the falure rate, therefore the system falure rate s called relevant falure rate. Falure nteracton may be stable or not []. When the falure nteracton s stable, the affected components falure rate s hgher than the ndependent rate, but remanng on some certan level. When t s unstable, the affected components falure rate wll ncrease rapdly n a very short tme. Accordng to the analyss model of the falure nteracton n lterature [0], for the system consstng of q components, the components falure nteracton rate ncludes ntal falure rate and new addton falure rate, and expressed as follows: {λ(t)}=[i]{λ0(t)}+[θ(t)]{λ(t)}b () In whch, {λ(t)}s the vector of q,whch shows the falure nteracton rate, and{λ(t)}b s the falure vector of the falure nteracton q. {λ0(t)}s the ndependent falure vector of q. [I] s the unt matrx of q q, and [θ(t)] s the relevant coeffcent matrx. the elements of θab(t)(a, b=,2,,q) s the relevant coeffcent, whch shows the affected degree of component b towards component a. when θab s equal to zero, there s no nfluence among the components; when θab s equals to one, whch shows that component a wll cause falure of component b. The relevant coeffcent can be decded by the followng methods: a) Get t by probablty theory. b) Accordng to the experence estmaton of desgner, the manufacturer and mantenance personnel. c) Based on the estmaton of mechancal and knetcs. d) Based on laboratory testng. ure of key components, and the whole system needs warranty after whch the falure rate wll reman. To facltate the research, as to mult-components we have the followng hypothess: a) Imperfect preventve warranty s adopted n warran- ty nterval. When falure of each component occurred, warranty must be adopted. Falure rate after warranty s between as good as new and as bad as old. Falure rate of subsystem wll changed when warranty of key component s carred. b) The system has the characterstcs of agng, and the falu- re rate wll ncrease wth tme ncreases. c) The mprovement n mperfect preventve warranty of the system s a constant. d) The devoted preventve warranty to the system s a con- stant, whch s stable n despte of the varaton of warranty frequency and tme. The tme for machne halt s also a constant. e) The falure type belongs to sngle falure model, whch has the characterstcs of falure nteracton wthout the consderaton of multple falures. f) The study object s Mult-component seres System wth Falure Interacton composed by key components and subsystems. 3.2. Cost model There are assumptons that mperfect preventve warranty s adopted n warranty nterval, T s warranty nterval, each whole preventve warranty cost C p s the functon of preventve warranty expected cost C pr ; loss of unt tme for shutdown C d ; and the tme of each preventve warranty T p (C p =C pr +C d T p ). So, warranty cost of system n nterval s expressed as followed: CTW (, ) = nc + EC ( T) + EC( W n( T + T )) p n j= j In whch, n s the number of mperfect preventve warranty n warranty nterval W, n=nt[w/(t+t p )]. EC j (T) s the expected cost of j th (0 j ) mperfect preventve warranty nterval of the system. EC(W-n(T+T p )) s the expected cost of the tme between n(t+t p ) to W of the system. The falure rate of key component of the j th mperfect preventve warranty nterval s as followed: p (2) λ () t = λ ( t ( j ) αt) (3) If falure happens n key component, falure rate of subsystem λ sb (t) wll ncrease. Based on the falure nteracton, the average falure rate nteracton of subsystem n j th mperfect preventve warranty nterval s as followed: 3. Warranty nterval decson-makng model j 3.. Model descrpton and hypothess λ jsb() t = λsb( t ( j ) αt) + θ [ nkλk( t ( ) αt)] njkλk() t = 2 j Ths paper manly studes the two components system (4) λ jsb() t = λsb( t ( j ) αt) + θ [ nkλk( t ( ) αt)] njkλk() t composed by one key component and subsystems, the system = 2 wll be carred wth the mperfect preventve warranty, wthout Falure quanttes of key component n j th mperfect preventve warranty nterval can be expressed by falure rate. It s as consderaton of falure nteracton n subsystems. In each mperfect preventve warranty nterval, the key components wll followed: have ts least warranty when t occurs falures. The falure rate remaned after warranty, but wll ncrease the subsystem falure jt + ( j ) Tp rate λsb; on the contrary, subsystem falure wll cause the fal- njk = λ jk () t dt (5) jk k ( j )( T+ Tp ) 50 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology Falure quanttes of subsystem n j th mperfect preventve warranty nterval can be expressed by average falure rate. It s as followed: n jsb = jt + ( j ) Tp λ jsb() t dt ( j )( T+ Tp ) Each falure warranty cost C f s the functon of falure warranty expected cost C fr ; loss of unt tme for shutdown C d ; and the tme of each falure warranty T f (C f =C fr +C d T f ). Accordng to the falure number of key component and subsystem n j th mperfect preventve warranty nterval, the expected warranty cost of the system n j th mperfect preventve warranty nterval s followng: (6) EC ( T) = ( n + n ) C (7) j jk jsb f In the same way, the falure number of key component of the tme between n(t+t p ) to W as follows: n W = ( n+ ) k ( n+ ) k nt ( + TP ) The falure number of subsystem of the tme between n(t+t p )to W as follows: The functon of warranty cost n warranty nterval can be gotten by takng formula (7) and (0) nto(2): In whch, ET j (T) s the expected shutdown tme of j th (0 j ) mperfect preventve warranty nterval T of the system. ET(Wn(T+T p )) s the expected shutdown tme of the tme between n(t+t p ) to W of the system. Accordng to the analyss method of warranty cost of system, the expected falure warranty shutdown tme of the system n j th mperfect preventve warranty nterval s followng: ET ( W n( T + Tp) = ( n( n ) k + n( n ) sb) Tf = + + ( n ) k ( t) + ( n ) sb λ + λ + ( t nt ( + TP ) W (4) ET ( W n( T + Tp) = ( n( n ) k + n( n ) sb) Tf = + + ( n ) k ( t) + ( n ) sb λ + λ + ( t) dtt f nt ( + TP ) n n ATW CTW (, ) = ncp + ECj( T) + EC( W n( T + Tp)) W DTW (, ) = (, ) = ntp + ETj( T) + ETW ( nt ( + Tp)) W j= j= n n ATW = nc p + ( njk + njsb) C f + ( n( n+ ) k + n( n+ ) sb) C W DTW (, ) = (, ) = ntp + ETj( T) + ETW ( nt ( + Tp)) W f j= j= + = nc p + jt ( j ) T n p () jt + ( j ) T n p W(6) [ λjk ( t) + λ jsb( t)] dtc = nt + + f p [ λjk ( t) λ jsb( t)] dttf W + λ () t + λ j= ( j )( T+ Tp ) ( j )( T+ Tp ) j= nt ( + TP ) W + + + + jt + ( j ) T n p W λ( n ) k() t λ( n ) sb() t dtc = f nt + + p nt ( + TP ) W [ λjk ( t) λ jsb( t)] dttf + + λ( n+ ) k() t λ( n+ ) sb() t dtt f ( j )( T+ Tp ) j= nt ( + TP ) The expected falure warranty shutdown tme of system of the tme between n(t+t p ) to W as follow: ET ( W n( T + Tp) = ( n( n ) k + n( n ) sb) Tf = + + ( n ) k ( t) ( n ) sb λ + + λ + ( t W nt ( + TP )(5) ET ( W n( T + Tp) = ( n( n ) k + n( n ) sb) Tf = + + ( n ) k ( t) ( n ) sb λ + + λ + ( t) dtt f λ () t dt (8) nt ( + TP ) The functon of warranty shutdown tme n warranty nterval can be gotten by takng formula (4) and (5) nto (3): jt + ( j ) T n p W W DTW (, ) = ntp + [ jk () t + jsb()] t dttf n( n+ ) sb = λ ( n+ ) sb() t dt (9) λ λ + λ( ) () t + λ( j= ( j )( T+ Tp ) nt ( + TP ) nt ( + TP ) (6) jt + ( j ) T n p W So, the expected falure warranty DTW cost (, of ) = system nt of the tme p + [ jk () t + jsb()] t dttf between n(t+t p ) to W as follows: λ λ + + λ( n+ ) k() t λ( n+ ) sb() t dtt f j= ( j )( T+ Tp ) nt ( + TP ) EC( W n( T + Tp) = ( n( n+ ) k + n( n+ ) sb) C f (0) The functon of avalablty n warranty nterval can be gven takng formula (6) to (2). W W n+ k n+ ( n+ ) k ( n+ ) 3.3. Avalablty model Expected avalablty n warranty nterval can be expressed as follow: W D( T, W ) A( T, W ) = (2) W D(T, W) and C(T, W) has the same expresson, C p and C f s replaced by T p and T f. So expected shutdown tme n warranty nterval wth mperfect preventve warranty nterval T s as follows: jt j + ( )T p ETj( T) = ( njk + njsb) Tf = jk () t + jsb λ λ () t dttf (3) ( j )( T+ Tp ) 3.4. Model resoluton Unt cost-effectve of system s derved from cost and avalablty quantfcatonally. And scentfc warranty needs to control warranty cost, at the same tme to guarantee avalablty. So, models are analyzed by unt cost-effectve, as follows: C( T, W ) V = (8) W A( T, W ) The functon of Unt cost-effectve can be gven takng formula (5) and () to (8). Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 5

NAUKA I TECHNIKA 4. Case analyss Desel, as a complex equpment, s core and the key part, and the the advantages and the dsadvantages of whose performance take effect on the output of energy and the tracton of power. Desel manly ncludes: pressure booster, ol pump, and movement components etc. The pressure booster falure caused by other components malfuncton and falure; the movement components falure s fatgue-type falure, the proporton of the relevant falure s relatvely small [3] ; the ol pump falure s manly of fatgue-type falure almost wthout relevant falure, whch wll also lead relevant falure to the pressure booster. Accordng to research and analyss, the desel engnes may be consdered as mult-component system wth falure nteracton, whch composed by the pressure booster and subsystems (all the rest of the components). And the dscplne of the booster falure obeys webull dstrbuton: j jt + ( j ) T 2 p jt + ( j ) Tp nk [ t 08. ( ) T ] 4 2 njsb = λ jsb() t dt = 498. 0 + 05. = 000 dt m ( j )( T+ T m t p ) 2 n [ t 08. ( j ) T] ( j )( T+ Tp ) λ() t = (20) jk 2 2 000 η η (25) j jt + ( j ) T 2 p jt + ( j ) Tp In whch, m s 2, η=000. Falure rate λ sb of subsystem s nk [ t 08. ( ) T ] 4 2 n 4.98 0-4 jsb =. Average tme T f of falure warranty s 3. Average λ jsb() t dt = 498. 0 + 05. = 000 dt ( j )( T+ Tp ) 2 n [ t 08. ( j ) T] ( j )( T+ Tp ) warranty cost C fr of each falure s 300. Average warranty loss jk 2 2 000 C d of unt tme s 900. Assumptons of desel: So, the expected warranty cost of the system n j th mperfect Imperfect preventve warranty polcy s adopted n warranty preventve warranty nterval s followng: nterval. Improve factor α s 0.8. In each mperfect preventve warranty nterval, falure rate λ k of supercharger wll not change warranted, but falure rate λ sb EC j( T) = ( njk + njsb) Cf of subsystem wll ncrease. jt + ( j ) Tp 2 Relevant coeffcent θ s 0.5. Whereas, falure of supercharger [ t 08. ( j ) T] dt 2 ( j )( 000 T+ T happens mmedately f falure of subsystem happen. The tme p ) of each preventve warranty T p s. preventve warranty expected cost C pr = 3000 j 2 (26) jt + ( j ) T nk t T s 300. [ 08. ( ) ] p 4 2 + 498. 0 + 05. 000 = dt 2 4.. Calculate process ( j )( T+ Tp ) njk [ t 08. ( j ) T] 2 2 000 The number of mperfect preventve warranty n warranty nterval W s as follows: n=nt[w/t+] (2) The whole falure warranty cost of ths system 2 n C f =300+900 3=3000. The whole preventve warranty cost ( n+ ) k = λ( n+ ) k() t dt = [ t 08. nt] dt 2 (27) nt ( + TP ) nt ( + T 000 P ) C p =00+900 =000. The falure rate of booster n the j th mperfect preventve The falure number of subsystem of the tme between warranty nterval s as follows: n(t+t 2 t 08.( j ) T 2 p ) to W as follows: λjk () t = λk ( t ( j ) αt) = = [ t 08. ( j ) T ] 2 W 000 000 000 2 t 08.( j ) T 2 (22) n( n+ ) sb = n sb t dt λjk () t = λk ( t ( j ) αt) = = [ t 08. ( j ) T ] λ ( + ) () = 2 000 000 000 nt ( + TP ) Then, falure number of booster n j th mperfect preventve warranty nterval s as follows: jt + ( j ) Tp 2 njk = λ jk () t dt = [ t 08. ( j ) T ] 2 dt 000 ( j )( T+ Tp ) V = jt + ( j ) T n p W nc p + jk t + jsb t dtc f λ () λ () + n+ k t + n+ sb t λ( ) () λ( ) () dtc f j= ( j )( T+ Tp ) nt+ T ( P ) jt + ( j ) T n p W W ntp + jk t + jsb t dtt f λ () λ () + + + + λ( n ) k() t λ( n ) sb() t dtt f j= ( j )( T+ Tp ) + nt ( TP ) jt + ( j ) Tp ( j )( T+ Tp ) 52 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 (23) The average falure rate nteracton of subsystem n j th mperfect preventve warranty nterval s as follows: λ jsb() t = λsb( t ( j ) αt) + j + θ nkλk ( t ( ) αt) njkλk () t = 2 = j 2 nk [ t 08. ( ) T] 4 2 = 498. 0 + 0. 5 = 000 2 njk [ t 08. ( j ) T] 2 2 000 (24) Consequently, falure number of subsystem n j th mperfect preventve warranty nterval s as follows: The falure number of booster of the tme between n(t+t p ) to W as follows: W n+ 2 W nk [ t 08. ( ) T] 4 2 = 498. 0 + 05. = 000 dt 2 nt ( + T n( n+ ) k [ t 08. nt ] P ) 2 2 000 W (9) (28)

Scence and Technology The expected falure warranty cost of system of the tme between n(t+t p )to W as follows: EC( W n( T + Tp)) = ( n( n+ ) k + n( n+ ) sb) C f W 2 [ t 08. nt ] 2 dt 000 n( T+ T P ) = 3000 n+ 2 W nk [ t 08. ( ) T] 4 2 + 498. 0 + 0. 5 = 000 dt 2 n n+ k t nt n( T+ TP ) ( ) [ 08. ] 2 2 000 (29) Above all, warranty cost functon C(T, W) can be gven. In the same way, the expected falure warranty shutdown tme of the system n j th mperfect preventve warranty nterval s followng: ETj( T) = ( njk + njsb) Tf jt + ( j ) T p 2 [ t 08. ( j ) T] dt 2 ( j )( T+ T 000 p ) = 3 j 2 (30) jt + ( j ) T nk [ t 08. ( ) T] p 4 2 + 498. 0 + 05. = 000 dt 2 ( j )( T+ Tp ) jk 08 2 2 n [ 000 t. ( j ) T ] The expected falure warranty shutdown tme of system of the tme between n(t+t p ) to W as follows: ET ( W n( T + Tp)) = ( n( n+ ) k + n( n+ ) sb) Tf W 2 [ t 08. nt ] 2 dt 000 n( T+ T P ) = 3 n+ 2 (3) W nk [ t 08. ( ) T] 4 2 + 498. 0 + 05. = 000 dt n n 2 ( T T n+ k t nt + P ( ) 08 2 2 000 [. ] ) Then, avalablty functon A(T, W) n warranty nterval can be gven too. Unt cost-effectve functon can be gven as follows: C( T, W ) C( T, W ) V = = (32) W A( T, W ) W D( T, W ) Warranty cost, avalablty and unt cost-effectve s smulated as fgure, fgure 2 and fgure 3. Accordng to fgure, fgure 2 and fgure 3, optmal warranty nterval and mperfect warranty nterval are not exsted. However, warranty nterval of general system s defned to 3 year. So, 3 projects of the desel are as followng: () When mperfect preventve warranty nterval s the same to warranty nterval, whch s 3 year, falure rate relevant coeffcent s zero, and the ntalzed warranty cost, avalablty and unt cost-effectve are as follows: T=W=080 (3 years), C mn= 46683, A=0.8657, V=49.94. (2) When warranty nterval s 3 year, mperfect preventve warranty polcy and falure rate relevant coeffcent are consdered. Choosng best unt cost-effectve T, the ntalzed warranty cost, avalablty and unt cost-effectve are as follows: T=20, W=080 (3 years), C mn= 39236, A=0.9699, V=37.46. (3) When warranty nterval has dfferent years, mperfect preventve warranty polcy and falure rate relevant coeffcent are consdered, the best warranty data are as follows: 4.2. Result analyss () The proposal one and two are the operaton outcome when usng general correctve mantenance warranty polcy and mperfect preventve warranty polcy respectvely. After a comparatve analyss of the two proposals, when the warranty nterval s three years, the warranty cost s relatvely hgh and the avalablty low based on the general correctve mantenance warranty polcy. Compared wth not adoptng the mperfect warranty polcy not consderng falure nteracton, the warranty cost of the desel wll have a decrease of sxteen percent, and avalablty an ncrease of twelve percent, when adoptng the mperfect warranty consderng falure nteracton. (2) Table ncludes varous correspondng data for the warranty cost and avalablty and unt cost effcent n dfferent Fg.. Cost model Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 53

NAUKA I TECHNIKA Fg.3. Cost-effcent functon of unt tme Fg.2. Avalablty model warranty ntervals, through whch we can easly fnd that, n the premse of the guarantee for pump for ntal unt effcent cost (that s the unt effcent cost wthout preventve warranty),the warranty nterval can be extended to nne years f usng mperfect preventve warranty. (3) Table ncludes the balance of the needs and nterests between the manufacture and the users of dfferent warranty ntervals, the correspondng nformaton data as guarantee cost, the avalablty and unt effcent cost, whch can provde a avalable scentfc nformaton for equpments usng department about warranty cost and the avalablty n addton, the procurement department can also select the standard warranty nterval based on the reference nformaton data and actual needs. Tab.. Correspondng project of dfferent warranty ntervals Seral number W/day T/day C/yuan A V 080(3 years) 20 39236 0.9699 37.46 2 440(43years) 26 52859 0.950 38.64 3 800(53 years) 32 65954 0.9302 39.39 4 260(63 years) 36 83223 0.9098 42.35 5 2520(73 years) 40 99822 0.8978 44.2 6 2880(83 years) 46 8337 0.8777 46.8 7 3240(93 years) 46 36738 0.864 48.99 8 3600(03 years) 50 52833 0.8485 50.03 54 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology 5. Concluson Ths artcle manly amed at the study of mult-component wth falure nteracton, and analyss on the falure nteracton n mult-component system wth the pont of relevant coeffcent on falure rate, based on the mprovng mperfect preventve warranty polcy. A model for cost and avalablty s establshed and make analyss on the model. At last combned wth cases, proposal on the mult-component system wth falure nteracton s put forward, for whch the artcle provde analytcal valdaton. ********** The research work s supported by the Natonal Natural Scence Foundaton of Chna under contract number 709735. ********** 6. References. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. 3. Chen J A and Chen Y H. Renewng warranty and preventve mantenance for products wth falure penalty post-warranty. Qualty and Relablty Engneerng Internatonal 2007, 23: 07-2. Chun Y H. Optmal Number of Perodc Preventve Mantenance Operatons under Warranty. Relablty Engneerng and System Safety 992, 37(3): 223-225. Gao Png. The Research on Preventve Mantenance Decson of Complex Equpment Based on Relablty Analyss. Tsnghua Unversty, 2008. Jack N and Dagpunar J S. An Optmal Imperfect Mantenance Polcy over a Warranty Perod. Mcroelectroncs and Relablty 994, 34(3): 529-534. Jang Guo, Hu Fe. An Optmal Preventve Mantenance Polcy for Product Sold wth Warranty. Journal of Mathematcs 2009, 29(4): 546-550. Jung G M and Park D H. Optmal Mantenance Polces durng the Post Warranty Perod. Relablty Engneerng and System Safety 2003, 82: 73-85. Km C S, Djamaludn I and Murthy D N P. Warranty and Dscrete Preventve Mantenance. Relablty Engneerng and System Safety 2004, 84: 30-309. Murthy D N P and Nguyen D G. Study of 2-component system wth falure nteracton. Naval Res. Logst. 985, 32: 239-47. Pascual R and Ortega J H. Optmal Replacement and Over haul Decsons wth Imperfect Mantenance and Warranty Contracts. Relablty Engneerng and System Safety 2006, 9: 24-248. Sun Y, Ma L and Mathew J. Falure analyss of engneerng systems wth preventve mantenance and falure nteractons. Computers & Industral Engneerng 2009, 57: 539-549. Sun Y, Ma L and Mathew J. In J. Mathew (Ed.), Proceedng of the 0th Asa-Pacfc Vbraton Conference. Gold Coast: Queensland Unversty of Technology 2003: 664-668. Thomas A M. Analyss of F/A-8 Engne mantenance costs usng the boeng dependablty cost model. Monterey CA: Naval Postgraduate School 994. Yeh R H and Lo H C. Optmal Preventve Mantenance Warranty Polcy for Reparable Products. European Journal of Operatonal Research 200, 34(): 59-69. Prof. CHENG Zhong Hua, Ph. D. BAI Yong Sheng, Ph. D. CAI L Yng, Ph. D. Wang Lu Chao, M. E. LI Peng Ju, M. E. CHEN Lng, M. E. Department of Management Engneerng Mechancal Engneerng College Shjazhuang, Hebe, 050003, P.R. Chna E-mal: bays820708@sna.com Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 55

Artcle ctaton nfo: BRZOZOWSKI K, NOWAKOWSKI J. Toxcty of exhaust gases of compresson gnton engne under condtons of varable load for dfferent values of engne control parameters. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 56-62. Krzysztof BRZOZOWSKI Jacek NOWAKOWSKI Toxcty of exhaust gases of compresson gnton engne under condtons of varable load for dfferent values of engne control parameters TOKSYCZNOŚĆ SPALIN SILNIKA O ZAPŁONIE SAMOCZYNNYM W WARUNKACH ZMIENNEGO OBCIĄŻENIA DLA RÓŻNYCH WARTOŚCI PARAMETRÓW REGULACYJNYCH* Ths work presents a procedure leadng to emprcal dentfcaton of the dependency of smoke level and emsson of harmful compounds on selected engne control parameters n an compresson gnton engne. In the course of experments dentfcaton of emsson of ntrogen oxdes, carbohydrates, carbon oxde and smoke level of exhaust gases was carred out, dependng on: rotatonal speed, fuel amount, exhaust gas recrculaton factor and njecton tmng. Usng artfcal neural networks has been proposed to generalze the results of experments. The computed values of coeffcents used to evaluate approxmaton errors and predcton of smoke level and emsson of harmful compounds confrm effectveness of the proposed method of generalzng the dentfcaton procedure. Keywords: compresson gnton engne, engne control parameters, emsson of harmful compounds. W pracy przedstawono postępowane prowadzące do eksperymentalnej dentyfkacj zależnośc zadymena emsj zwązków szkodlwych od wybranych parametrów regulacyjnych slnka o zapłone samoczynnym. W ramach badań przeprowadzono dentyfkację emsj tlenków azotu, węglowodorów, tlenku węgla oraz zadymena spaln w zależnośc od: prędkośc obrotowej, dawk palwa, stopna recyrkulacj spaln kąta wyprzedzena wtrysku. Do uogólnena wynków badań eksperymentalnych zaproponowano wykorzystane sztucznych sec neuronowych. Oblczone wartośc wskaźnków wykorzystanych do oceny błędów aproksymacj oraz predykcj zadymena emsj zwązków szkodlwych spaln potwerdzają skuteczność zaproponowanej metody uogólnena badań dentyfkacyjnych. Słowa kluczowe: slnk o zapłone samoczynnym, parametry regulacyjne, emsja zwązków szkodlwych.. Introducton Development of pston combuston engnes s currently determned to a large extent by actons amed at lmtng envronmental mpacts of ther usage. Ths drecton of research makes fulflment of future requrements on emsson of harmful compounds n exhaust gases as well as fuel consumpton a necessty. Introducton of electroncally controlled hgh pressure njecton systems enabled fulflment of exstng legal requrements and mprovement of algorthms controllng these systems wll enable further decrease of negatve mpact on the natural envronment. It s therefore an mportant research problem to develop sutable (optmal) algorthms for controllng engne operaton under varable condtons of usage takng nto account basc engne control parameters, such as [2, 5, 9]: --rotatonal speed, --fuel amount, dvded and njecton pressure, --njecton tmng, --exhaust gas recrculaton factor. In the case of compressor supercharged engnes the control algorthms may also take addtonal parameters on nput, such as [2, 2]: --pressure of load n the ntake manfold, --temperature of load n the ntake manfold. The standard procedure n order to defne the engne control algorthm ncludes expermental studes upon whch so called dscrete control maps are determned. Havng been nput to the controller, for gven ranges of rotatonal speed and load these maps defne the values of other control parameters and correcton coeffcents for transent states [7]. Such approach though leaves a possblty to apply more precse control based on contnuous dependences,.e. wthout averagng expermental results for ranges of rotatonal speed value and range of load value gven a pror. Such task had already been accomplshed by the authors n papers [6, 3] among others. In the context of mplementng control accountng for emsson of harmful compounds dentfcaton of engne exhaust gases toxcty under condtons of varable load for arbtrary techncally admssble (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl 56 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology values of control parameters s a basc element n desgnng sutable algorthms. Smlar problems related to searchng for contnuous approxmatons of emsson were treated among others n papers [7, 8, 9, 0,, 6, 20]. In [9] the authors used a neural network of a smple archtecture for modelng emssons of partculate matter, presentng the results as an approxmatng surface for selected ranges of values of rotatonal speed and torque. Smoke level modelng usng neural networks was also dealt wth by the authors of [0], mappng the smoke levels n partcular modes of the ECE-R49 test. Neural networks for predcton of emsson of exhaust gases from desel-powered engne, bodesel and blends of both fuels was proposed n [7]. However, expermental studes were carred out only for one selected pont correspondng to engne operaton at full load at a speed of 400 rpm, whch were then used both n the learnng process and verfcaton of the neural network. The authors used the neural network to determne the emsson for blends of desel and bodesel varyng n percentage. Smlar problem was consdered n [8], dong research for maxmum load at dfferent rotatonal speeds. In turn, n the paper [6] emsson from the engne usng neural networks dependng on the compresson rato, the percentage of bodesel and njecton tmng was modeled. Presented results were lmted only to the condtons correspondng to full load. In [20], the last one of the prevously mentoned works, the neural network was appled to modelng torque, fuel consumpton and emsson of exhaust gases of compresson gnton engne fueled doubly by desel and CNG. Data for network tranng were obtaned only under condtons of maxmum load at a gven rotatonal speed. The problem of modelng emsson usng artfcal neural networks was also the subject of paper [], agan only under condtons of maxmum load at a gven rotatonal speed. A common feature ndcated by the authors of cted works s the concluson that neural networks reproduce wth suffcent accuracy the emsson of harmful compounds n exhaust gases thus provdng an alternatve to full, tme-consumng expermental studes. The paper presents an own method of dentfyng the dependency between emsson and values of partcular control parameters as well as a way of generalzng them usng artfcal neural networks. In ths extent the present work comes as a supplement and expanson upon concepts descrbed n [8]. 2. Summary of dentfcaton experments Identfcaton experments searchng for dependency between emsson of ntrogen oxdes, carbohydrates and carbon oxde and the values of control parameters were carred out for an compresson gnton engne, dedcated and used for propellng automobles, whose techncal detals are lad out n Table. Research was conducted on a dynamc engne test stand consstng of the followng devces: --dynamometrc brake of AVL Dynoexact APA 202 type, --control system AVL PUMA OPEN.3, --stand control panel AVL EMCON 400. Emsson of harmful components of the exhaust gases was regstered usng an AVL CEB 200 set of analyzers equpped wth CLD, HFID and NDIR analyzer modules and to measure smoke level of exhaust gases a fltraton opacmeter of AVL 45S was used. Realzaton of set forth dentfcaton experments requred varyng standard engne operaton parameters. Wthn the present work dentfcaton of emsson of ntrogen oxdes, carbohydrates, carbon oxde and exhaust gases smoke level was performed dependng on: rotatonal speed, fuel amount, exhaust gas recrculaton factor and gnton tmng, whereas varyng the three latter parameters requred alteraton of engne controller software whch was done usng the INCA V6.2 software and a reprogrammng devce ECU - ETAS ES590. The range of dentfcaton experments carred out s shown n fg.. It should be underlned here that n the entre range of realzed experments shown n fg. the fuel amount was suppled dvded nto a constant ntal amount ndependent of the pont of engne operaton and a varable man amount. For the standard algorthm of controllng njecton amounts n the tested engne contans dfferent strateges of njecton sequence dependng on the pont of engne operaton. Ths renders unambguous dentfcaton of emsson of ntrogen oxdes, carbohydrates, carbon Fg.. Plot of the range of dentfcaton experments Tab.. Techncal detals of the engne Engne CI engne supercharged by a turbo compressor wth drect njecton equpped wth an electroncally controlled Common Ral system Layout of cylnders 4 n lne Number of valves per cylnder 4 Bore 69.6 mm Stroke 82 mm Total dsplacement 248 cm 3 Compresson rato 6.8 Maxmum power 55.2 kw / 4000 rpm Maxmum torque 90 N m / 500 rpm Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 57

NAUKA I TECHNIKA oxde and smoke level of the exhaust gases dependng on the engne control parameters mpossble. The next stage of dentfcaton work encompassed generalzaton of results of engne tests for arbtrary techncally admssble values of control parameters. Realzaton of ths stage requred formulatng an approprate approxmaton task of a multvarate functon. Two approaches are possble here: --approxmaton for a form of approxmaton functon gven a pror whose coeffcents are determned by means of the least squares method, --approxmaton by means of feedforward artfcal neural networks wth radal or sgmodal neuron actvaton functons. The frst of the mentoned approxmaton methods was used among others n the paper [] assumng a polynomal of the form: ( ) n n j nr = 0 j= 0 r= 0 j r,..., j,..., r j r F x...... a x... x... x () = where: a0,...,0,...,0,..., a n,..., nj,..., n - polynomal coeffcents, r n,...,n r - nteger and non-negatve maxmum powers of correspondng elements of the vector x of control parameters. The polynomal () on applyng larger exponents enables obtanng a very good ft to test results. Its dsadvantage however les n the strongly local character of the ft and consequently weak generalzaton propertes outsde of the closest neghbourhood of the approxmaton nodes. Therefore n the papers [2, 3] approxmatons of the results of toxcty tests were realzed usng power functons, F j r ( x)= 0x... xj... xr (2) α α α α where α 0 α r are coeffcents wth real values. These functons, although worse fttng at approxmaton nodes than polynomals, ensure acceptable generalzaton propertes. Modfcatons of power functons of the form (2) were also used, e.g. n [2] ntroducton of a constant term was proposed n order to ensure non-negatvty of the functon on the entre doman of arguments: α α j αr F ( x)= α x... x... x + α (3) 0 j r r + The second of the abovementoned methods usng the means of artfcal ntellgence does not requre gvng a pror the form of the approxmaton functon. Ths s the path chosen for the present work. 3. Approxmaton usng feedforward artfcal neural networks For generalzng of the results of dentfcaton experments feedforward artfcal neural networks may be used wth an archtecture featurng so called hdden layer or layers. Authors have already n ther earler papers [3, 4, 5, 6] employed artfcal neural networks n the approxmaton task of expermental results of emssons wth an actvaton functon on one or multple neurons n the output layer. In emsson approxmaton tasks t means that on the output of the network there appeared a sngle sgnal (when the network served to approxmate the emsson of one component of the exhaust gases) or multple sgnals n leveragng a sngle neural network to approxmate the emsson of all components analyzed. Summary of approxmaton errors presented e.g. n [4], ncludng ther comparson to approxmaton errors when usng power functons [2, 3], confrms ther usefulness n approxmaton tasks of emsson of harmful compounds n exhaust gases. Therefore n the present paper artfcal neural networks wth a sgmodal neuron actvaton functon n unpolar form has been used to generalze the results of dentfcaton: t t f( ) w z w z = + e (4) where: w - vector of weghts on neuron s nput connectons, z - vector of nput sgnals to the neuron. Assumng a sngle neuron n the output layer the value of the output sgnal for a last hdden layer of k elements s calculated as: k F ( x ) = f w z (5) = Fg. 2. General archtecture of the neural networks used n the approxmaton task Network learnng, whch conssts n modfyng the weghts of ndvdual connectons between the neurons formng the network, was performed untl the mean relatve error for the entre tranng set has reached at least the expected value. For generalzaton of the results of dentfcaton experments usng four networks wth general archtecture as n fg. 2 was proposed. The networks proposed n the current paper have four varable sgnals on nput correspondng to elements of the vector x of control parameters and one constant nput and on output just one sgnal correspondng to the emsson of a gven compound. The networks were traned tll the mean relatve approxmaton error for each component of the exhaust gases reached a value below 0% both n the tranng and verfyng set for the ablty to generalze the responses of each of the four networks. For tranng the networks the momentum method was used wth ncremental updatng of weghts,.e. n step n + of network learnng the weghts were modfed accordng to the formula [4]: ( ) ( n+ w ) ( n) ( n) ( n) ( n) ( n ) = w β E ( w )+ δ w w (6) where: β - learnng coeffcent, E - goal functon n the form of mean square error of network s response relatve to the values expected for a gven tranng vector, E(w) - gradent vector of the goal functon, δ - moment coeffcent from the nterval [0,]. In the learnng process a random choce of weghts was used after every teraton of network tranng and random determnaton of tranng patterns. If for a gven network archtecture after applyng many teratons the assumed value of the mean relatve error wasn t reached, the archtecture was modfed. 58 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology As the result of the descrbed scheme of acton for each compound the mnmal network archtecture was found enablng approxmaton of emsson, n each case requrng two hdden layers, albet wth dfferent numbers of neurons n partcular layers. Summary of nformaton about the resultant archtectures of the neural networks and values of coeffcents used for evaluaton of approxmaton and predcton errors are presented n table 2. Fgure 3 shows a comparson of smoke levels and emssons of the compounds concerned obtaned as the responses of artfcal neural networks (F*) for control parameters consttutng nput sgnals from the tranng and verfyng sets related to the values observed expermentally (F). The values of resultng approxmaton and predcton errors presented n Table 2 confrm effectveness of the proposed method of generalzng dentfcaton experments n the doman of mappng toxcty and smoke level of exhaust gases. Mean relatve errors of both the approxmaton of data from the tranng set and the predcton n the case of the verfyng set do not exceed 0%. Tab. 2. Archtecture of neural networks and values of coeffcents used for evaluaton of approxmaton error n the tranng set and predcton of error n the set used for the verfcaton process CO HC NO x D Archtecture The number of neurons n the frst hdden layer 7 5 4 9 The number of neurons n the second hdden 5 4 3 9 layer Evaluaton of approxmaton error (tranng set) Average relatve percentage error 8.48 8.98 8 9.85 Medan of average relatve percentage error 5.82 6.7 6.50 6.56 Coeffcent of determnaton 0.979 0.976 0.986 0.99 Evaluaton of approxmaton error (verfyng set) Average relatve percentage error 9.82 9.55 6.79 7.92 Medan of average relatve percentage error 7.24 7.33 6.86 6.00 Coeffcent of determnaton 0.983 0.983 0.972 0.97 Fg. 3. A comparson of emssons of the compounds concerned and smoke levels obtaned as the responses of artfcal neural networks (F*) related to the values observed expermentally (F) Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 59

NAUKA I TECHNIKA 4. Concluson The dentfcaton procedure presented n the current work s one of the stages n desgnng algorthms for controllng the control parameters n order to lmt smoke level and emsson of harmful compounds n exhaust gases. The obtaned surfaces presentng the approxmated values of the results of expermental research of smoke levels and emsson of harmful compounds for two selected rotatonal speeds and two values of load are shown n fg. 4 and 5. Markers vsble on the surfaces present the dscrete values obtaned n the course of dentfcaton experments. Fg. 4. Surfaces approxmatng smoke level and emsson of harmful compounds n exhaust gases determned by usng artfcal neural networks for rotatonal speed 2500 rpm and load 0.25 M max Fg. 5. Surfaces approxmatng smoke level and emsson of harmful compounds n exhaust gases determned by usng artfcal neural networks for rotatonal speed 3500 rpm and load 0.5 M max 60 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology The target realzaton of the task of emsson control smultaneously ensurng sustanment of tracton parameters requres addtonally formulatng a model of the workng cycle and solvng an optmzaton task by means of approprate choce of values of control parameters. Methodology of the acton s schemed n fg. 6. It s worth notng that n the presented scheme of acton the approxmaton task would be formulated also n relaton to the parameters of a theoretcal-emprcal model of a workng cycle of a CI engne. Authors plan to employ artfcal ntellgence methods n ths case to realze t as well. Fg. 6. Scheme of methodology of acton amed at formulatng and solvng the task of emsson control 5. References. 2. 3. 4. 5. 6. 7. 8. 9. 0. Brzozowsk K, Romanszyn K. An effectve method of creatng dynamc characterstcs usng drve tests. The Archve of Mechancal Engneerng 2003; 50(4): 403-420. Brzozowsk K. Mkroskalowe modele emsj dyspersj zaneczyszczeń samochodowych. Zeszyty Naukowe ATH - Sera Rozprawy Naukowe; 8, 2006. Brzozowsk K, Nowakowsk J. Zastosowane sztucznych sec neuronowych do modelowana emsj z slnka o zapłone samoczynnym. Journal of KONES: Internal Combuston Engnes 2005; 2(-2): 5-59. Brzozowska L, Brzozowsk K, Warwas K. Zastosowane sztucznych sec neuronowych do modelowana ekologcznych właścwośc pojazdów. The Archves of Automotve Engneerng 2005; 3: 229-247. Brzozowska L, Brzozowsk K, Nowakowsk J. An applcaton of artfcal neural network to desel engne modellng. Proceedngs of the Thrd IEEE Workshop Intelgent Data Acquston and Advanced Computng Systems - Technology and Appllcaton 2005: 42-46. Brzozowsk K, Nowakowsk J. The model of compresson gnton engne wth artfcal neural networks. Combuston Engnes 2008; : 44-49. Canakc M, Erdl A, Arcakloğlu E. Performance and exhaust emssons of a bodesel engne. Appled Energy 2006; 83: 594-605. Canakc M, Ozsezen A.N, Arcakloğlu E, Erdl A. Predcton of performance and exhaust emssons of a desel engne fueled wth bodesel produced from waste fryng palm ol. Expert Systems wth Applcatons 2009; 36: 9268-9280. de Lucas A, Durán A, Carmona M, Lapuerta M. Modelng desel partculate emssons wth neural networks. Fuel 200; 80: 539-548. Ghazkhan M, Mrza I. Soot emsson predcton of a waste-gated turbo-charged DI desel engne usng artfcal neural network. Neural Comput. & Applc. 20; 20: 303-308. Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 6

NAUKA I TECHNIKA. Ghobadan B, Rahm H, Nkbakht A.M, Najaf G, Yusaf T.F. Desel engne performance and exhaust emsson analyss usng waste cookng bodesel fuel wth an artfcal neural network. Renewable Energy 2009; 34: 976-982. 2. Nowakowsk J. Model cyklu roboczego slnka o zapłone samoczynnym jego zastosowane do doboru parametrów regulacyjnych. Zeszyty Naukowe ATH - Sera Rozprawy Naukowe; 5, 2005. 3. Nowakowsk J, Brzozowsk K. Numercal model and programme for smulatng workng process n the compresson-gnton engne wth EGR. Proceedngs of the 2th European Automotve Congress EAEC 2009: -8. 4. Osowsk S. Sec neuronowe w ujęcu algorytmcznym. Warszawa: WNT, 996. 5. Pont F, Cort E, Serra G, De Cesare M. Common Ral Mult-Jet Desel Engne Combuston Model Development for Control Purposes. SAE Paper 2007-0-0383. 6. Shvakumar, Srnvasa Pa P, Shrnvasa Rao B.R. Artfcal Neural Network based predcton of performance and emsson characterstcs of a varable compresson rato CI engne usng WCO as a bodesel at dfferent njecton tmngs. Appled Energy 20; 88: 2344-2354. 7. Sobeszczańsk M, Petras D, Knefel T. Dobór algorytmów sterowana rozruchem, nagrzewanem, wolnym obrotam oraz recyrkulacją spaln slnka o zapłone skrowym zaslanego w systeme wtrysku MPI. The Archve of Automotve Engneerng 2002; 2-3: 53-67. 8. Sobeszczańsk M, Brzozowsk K, Nowakowsk J, Tekelak K. The mpact of SI engne controllng parameters on the content of toxc components and smokness of exhaust gases. Eksploatacja Nezawodnosc - Mantenance and Relablty 2008; 3: 53-6. 9. Tomshma H, Matsumoto T, Ok M, Nagata K. The Advanced Desel Common Ral System for Achevng a Good Balance Between Ecology and Economy. SAE Paper 2008-28-007. 20. Yusaf T.F, Buttsworth D.R, Saleh K.H, Yousf B.F. CNG-desel engne performance and exhaust emsson analyss wth the ad of artfcal neural network. Appled Energy 200; 87: 66-669. 2. Wang E. Control System Desgn for Varable Nozzle Turbocharger. SAE Paper 2009-0-668. Dr hab. nż. Krzysztof BRZOZOWSKI, prof. ATH Department of Transport and Appled Computer Scence Unversty of Belsko-Bała Ul. Wllowa 2, 43-309 Belsko-Bała, Poland e-mal: kbrzozowsk@ath.eu Dr hab. nż. Jacek NOWAKOWSKI, prof. ATH Department of Combuston Engnes and Vehcles Unversty of Belsko-Bała Ul. Wllowa 2, 43-309 Belsko-Bała, Poland E-mal: jnowakow@ath.belsko.pl 62 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: S. Shakuntla, A. K. Lal, S. S. Bhata: The comparatve study of the subsystems subjected to ndependent and smultaneous falure. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 63-7. S. SHAKUNTLA A. K. LAL S. S. BHATIA COMPARATIVE STUDY OF THE SUBSYSTEMS SUBJECTED TO INDEPENDENT AND SIMULTANEOUS FAILURE BADANIA PORÓWNAWCZE PODSYSTEMÓW ULEGAJĄCYCH USZKODZENIOM NIEZALEŻNYM I JEDNOCZESNYM The paper dscusses the comparson between avalablty of a ppe manufacturng ndustry when the sub systems are subjected to smultaneously and ndependent falure. The falure rates of the sub-systems are constant and the repar rates are varable. The governng dfferental equatons of the system are solved usng Lagrange s Method. The performance evaluaton of system s done by means of long run avalablty makng use of software package Matlab 7.0.4. The tables for varous parameters are gven whch can be useful to the plant management for mprovng and plannng the mantenance schedule. Keywords: Supplementary Varable Technque, Chapman- Kolmogorov, MAT-LAB, Lagrange s Method, Steady State Avalablty. W artykule porównano dostępność zakładu przemysłowego produkującego rury w przypadkach występowana jednoczesnych nezależnych uszkodzeń podsystemów. Badana prowadzono przy stałych ntensywnoścach uszkodzeń podsystemów zmennych ntensywnoścach napraw. Konstytutywne równana różnczkowe systemu rozwązano przy użycu metody Lagrange'a. Oceny wydajnośc systemu dokonano na podstawe długotrwałej dostępnośc z wykorzystanem paketu oprogramowana Matlab 7.0.4. Przedstawono tabele dla różnych parametrów, które mogą być wykorzystywane przez osoby zarządzające produkcją przy poprawanu planowanu harmonogramów przeglądów. Słowa kluczowe: technka dodatkowej zmennej, Chapman Kołmogorow, MAT-LAB, metoda Lagrange'a, gotowość stacjonarna.. Introducton Avalablty analyss of a system can beneft the ndustry n terms of hgher productvty and low mantenance cost. It s possble to mprove the avalablty of the plant wth proper mantenance plannng and montorng. Relablty analyss helps us to obtan the necessary nformaton about the control of varous parameters. The polytube ndustry nvolves many processes.e. Mxture, Extruder, De and Cutter. The De and Cutter machne can also work n reduced state. The process starts from the Mxture secton n whch ppe mxture s prepared wth the help of PVC rsng, CaCo 3, ctrc acd and wax whch s heated up to 30 0 C. The heated materal s then cooled up to 00 0 C and transported to the Extruder by conveyors. Wth the help of De and Extruder, ppe s prepared. After smoothng the ppe, sortng process take place. In ths process, the ppe carred to Cutter s cut nto dfferent szes as per the need and requrement. The mechancal systems have attracted the attenton of several researchers n the area of relablty theory. Sngh [7] frst tme appled relablty technology to analyze the workng of process ndustres( sugar, fertlzer and paper ndustres). Zhang [8] studed the stochastc behavor of an (N+) standby system under preemptve prorty repar and obtan the expresson of transent and steady state of the system usng supplementary varable and Laplace Transform. Dyal and Sngh [] studed relablty analyss of a system n a fluctuatng envronment. Sngh and Mahajan [6] examned the relablty and long run avalablty of a Utensls Manufacturng Plant usng Laplace transforms. Gupta et. al. [2] studed the behavor of Cement manufacturng plant. Kureghan and Dtlevson [3] analyzed the avalablty, relablty and downtme of system wth reparable components. Kumar et.al. [4] dscussed the behavor analyss of Urea decomposton n the Fertlzer ndustry under general repar polcy. Kumar et.al. [5] analyzed the desgned and cost of a refnng system n the sugar ndustry usng supplementary varable technque. 2. System descrptons The Polytube ndustry conssts of four subsystems, namely: Sub-system A (Mxture): It mxes raw materal such as PVC rsng, calcum carbonate, wax and other chemcals n approprate proporton for manufacturng ppe. It conssts of a heater by whch the raw materal s heated up to 30 0 C and transported Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 63

NAUKA I TECHNIKA to the extruder by conveyors. It conssts of blades and a motor whose falure cause complete falure of the system. Sub-system B (Extruder): Raw materal obtaned from mxer s heated n ths secton. It conssts of a heater to heat the raw materal at dfferent temperatures. The qualty of the product depends upon heatng process. Its falure causes the complete falure of the system. Sub-system C (De): It s used to make dfferent szes of ppe. Mnor falure of the sub-system reduced the capacty of the system and hence loss n producton. Major falure results n complete falure of the system. Sub-system D (Cutter): Ths sub-system has two unts arranged n seres. Frst unt s blade whch cut the ppe and the second unt s motor whch cut the ppe n dfferent sze. Falure of blade reduces the capacty of the system whle the falure of motor causes the complete falure of the system. 4. Mathematcal formulatons The dfferental-dfference equatons obtaned from the state transton s as follows (Fg. ): () (2) (3) (4) (5) (6) 3. Sub-systems subjected to ndependent falures 3.. Notatons A, B, C, D : Indcates that the sub-system s workng n full capacty. : Indcate the reduced state of the sub-system C and D. a, b, c, d : Indcate the faled state of the sub-system. λ, λ 2 : Transton rate of subsystem C and D. α : Falure rate of the sub-system A, B, C, D. ϕ(x)ψ(y) : General repar rates of A, B, C, D respectvely. μ(z)σ(s) P o (t) P (x,t) P j (y,t) P 2 (s,t) P (z,t) P 4 (z,t) P 3() (s,t) : The system s workng n full capacty. : Probablty that there s falure n subsystem A at tme t and has an elapsed repar tme x. = 5,7,0,37 : Probablty that there s falure n subsystem B at tme t and has an elapsed repar tme y. j = 6,8,,4,8. : Probablty that there s falure n subsystem D at tme t and system remans n reduced state and has an elapsed repar tme s. : Probablty that there s falure n subsystem C at tme t and system remans n reduced state and has an elapsed repar tme z. : Probablty when the sub system D s n reduced state and has an elapsed repar tme s then the sub system C comes n reduced state and has an elapsed repar tme z. 3.2. Assumptons : Probablty when the sub system C comes frstly n reduced state and has an elapsed repar tme z then the sub system D comes n faled state and has an elapsed repar tme s. The assumptons used n developng the performance model are as follows [5, 6]:. Falure rates are constant over tme and statstcally ndependent. 2. A repared unt as good as new, performance wse, for a specfed duraton. 3. Suffcent repar facltes are provded. 4. Servce ncludes repar and/or replacement. Where, (7) (8) (9) (0) 0 () (2) (3) (4) (5) (6) (7) (8) (9) (20) (2) 64 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology 3 a B C D 2 A b C D a B C D 5 a B C D Φ(x) α α 2 Ψ(y) Φ(x) α α Φ(x) 4 A b C D 5 A B C d σ(s) α 2 Ψ(y) α 4 3 AB C D λ λ 2 4 A B C D 0 ABCD σ(s) µ(z) α 3 Φ(x) µ(z) σ(s) α 4 α 3 Ψ(y) α 2 µ(z) 6 A b C D 8 A B c D 2 a B C α 7 A B C D α 4 σ(s) 9 A B C d α 2 Ψ(y) α 3 µ(z) A b C D 0 AB c D Fg.. Transton dagram of Poly Tube Industry when Sub-System smultaneously 4.2. Boundary condtons: Equaton () s lnear dfferental equaton of frst order and other equatons (2-2) are partal dfferental equatons of frst order. Usng the boundary and ntal condtons, the equatons (-2) are solved to gve the followng soluton: 4.. Intal condtons: (0)= otherwse 0 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 65

NAUKA I TECHNIKA (, ) = ( ) () (, ) = ( ) () (, ) = () (, ) (, ) = () (, ) (, ) = μ() (, ) (, ) = () (, ) (, ) = () (, ) (, ) = () (, ) (, ) = () () (, ) (, ) = () () (, ) (, ) = μ() () (, ) (, ) = () () (, ) (, ) = () () (, ) (, ) = () () (, ) (, ) = μ() () (, ) (, ) = () () (, ) It s evdent that all probabltes are obtaned n terms of P 0 (t) whch s gven by (). The tme dependent avalablty A(t) s: 4.3. Steady State Avalablty (22) In the process ndustry, we requre long run avalablty of the system, whch s obtaned by puttng =0, 0 and takng probabltes ndependent of "t" then lmtng probabltes from (-2) are: Wth ntal condtons On solvng these equatons recursvely, we have the steady state probabltes: Now usng normalzng condtons Where 5. Sub- system subjected to smultaneously falure 5.. Notatons A, B, C, D : Indcates that the sub-system s workng n full capacty. : Indcate the reduced state of the sub-system C and D. a,b,c,d, : Indcate the faled state of the sub-system. α : Falure rate of the sub-system A,B,C,D. ϕ(x)ψ(y) : General repar rates of A,B,C,D respectvely. μ(z)σ(s) P o (t) P (x,t) P j (y,t) P 4 (z,t) P 7 (s,z,t) P k (z,t) P l (z,t): : The system s workng n full capacty. : Probablty that there s falure n subsystem A at tme t and t s repared wthn tme nterval (x,x+δ). For =,5,2,3. : Probablty that there s falure n subsystem B at tme t and t s repared wthn tme nterval (yyx+δ). For j = 2,6,,4. 3 (s,t) : Probablty that there s falure n subsystem D at tme t and system remans n reduced state tll t s repared wthn tme nterval (s,s+δ). : Probablty that there s falure n subsystem C at tme t and system remans n reduced state tll t s repared wthn tme nterval (z,z+δ). : Probablty that there s falure n subsystem C and D at tme t and system remans n reduced state tll t s repared wthn tme nterval (z,z+δ) and (s,s+δ) respectvely. : Probablty that there s falure n subsystem C at tme t and t s repared wthn tme nterval (z,z+δ). For k = 8,0. 5.2. Assumptons : Probablty that there s falure n subsystem D at tme t and t s repared wthn tme nterval (s,s+δ). For l = 9,5.. Falure rates are constant over tme and statstcally ndependent. 66 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology 2. 3. 4. 5. 6. A repared unt as good as new, performance wse, for a specfed duraton. Suffcent repar facltes are provded. Servce ncludes repar and/or replacement. System may work at reduced capacty. There are smultaneous falures 6. Mathematcal formulatons The dfferental-dfference equatons obtaned from the state transton s as follows (fgure 2): (23) (24) (25) (26) (27) (28) (29) (30) (3) (32) (33) (34) 6.. Intal condtons: (0) = otherwse 0 For =,5,2,3 For j=2,6,,4 For k =4,7,8,0 For l=3,7,9,5 P 0 (0) = otherwse 0 P (x,0) = 0 For =,5,2,3 P j (y,0) = 0 For j=2,6,,4 P k (z,0) = 0 For k =4,7,8,0 P l (s,0) = 0 For l=3,7,9,5 6.2. Boundary Condtons: (35) (36) (37) (38) Where, Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 67

NAUKA I TECHNIKA abcd 0 AbCD 6 abcd 5 ab 7 Ab 4 AbC ABCd 2 ab 3 ABc 5 5 σ(s) µ ( z ) µ(z) ) µ ( z ) AB µ(z) 4() ) ABC 2 µ(z) µ(z) ) µ (z) σ(s) σ(s) µ(z) ) AB d 6 ABCD 0 µ(z) µ(z) ab 7 ) AB D AB ) Ab 8 µ ( σ(s) z µ(z) ) Fg. 2. Transton dagram of Poly Tube Industry when Sub-System faled ndependently µ(z) ) Ab 8 ABcD 9 AB 20 σ(s) µ(z) ) µ (z) ABc Equaton (23) s lnear dfferental equaton of frst order and other equatons (24-38) are partal dfferental equatons of frst order. Usng the boundary and ntal condtons, the equatons (23-38) are solved to gve the followng soluton: It s evdent that all probabltes are obtaned n terms of P 0 (t) whch s gven by (23). The tme dependent avalablty A(t) s: (39) 6.3. Steady state avalablty In the process ndustry, we requre long run avalablty of the system, whch s obtaned by puttng =0, 0 and ta- 68 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology kng probabltes ndependent of t then lmtng probabltes from (23-38) are : Under the avalable facltes the concern management mnmze the falure tme of each sub-system adoptng the followng measures I. Gettng the nformaton of falure of each equpment at the earlest moment. II. Startng the repar work at the earlest moment. III. Provdng traned workers. IV. Provdng the specal tools requred. V. Makng avalable the spare parts and specal parts Where: On solvng these equatons recursvely, we get: Now usng normalzng condtons: (40) 7. Performance analyss Table shows that wth the ncrease n falure rate of mxture α from 0.0057 to 0.0063, by keepng other parameters constant (λ =0.00, λ 2 =0.002, α 2 =0.007, α 3 =0.033, α 4 =0.05, ψ=2, µ=0.33, σ=0.02), the avalablty of system decreases approxmately.3%. Whereas, the avalablty ncreases approxmately 0.63% wth the ncrease repar rate of mxture from 0.5 to.. Smlarly, table 5 shows that avalablty decreases approxmately by.2% wth the ncrease of falure rate of mxture α from 0.0057 to 0.0063 keepng other parameters constant (λ =0.00, λ 2 =0.002, α 2 =0.007, α 3 =0.0, α 4 =0.05, ψ =2, µ =0.33, σ =0.02) whereas, the avalablty ncreases approxmately 0.63% wth the ncrease repar rate of mxture ϕ from 0.5 to.. Tables 2, 3, 4, 6, 7 and 8 shows that there s a almost neglgble change n the avalablty of the Polytube extruson system wth the ncrease the repar rate of Extruder, De and Cutter Subsystems. 8. Conclusons The comparatve study of the table and table 5 shows that when the Sub-Systems De and Cutter faled smultaneously or ndependently, then n both cases, the Sub- System A.e. Mxture affects the Avalablty apprecably. We, thus make an nference that management should take paramount care of sub system A n order to mprove overall relablty. ********** Author would lke to thank Er. Sanjay Kajal, NIT Kurukshetra and Dr Ja Sngh, Punjab College of Engneerng, Lalru, Punjab for ther support and cooperaton whle analyss of the data of the ndustry and Sh. Sudarshan Aggarwal, Managng Drector, Polytube Industry, Kurukshetra for provdng possble data of the ndustry. ********** Tab.. Effect of falure and repar rate of the Sub-System Mxture (A) on Avalablty when both Sub-Systems faled ndependently. ϕ α 0.0057 0.0059 0.006 0.0063 Constant values 0.5 0.9769 0.9732 0.9694 0.9656 α 2 =0.007, ψ=2, 0.7 0.9802 0.9775 0.9748 0.972 λ =0.00, λ 2 =0.002 0.9 0..982 0.9796 0.9770 0.9757 α 3 =0.0, µ =0.33. 0.9832 0.985 0.9797 0.9780 α 4 =0.05, σ =0.02 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 69

NAUKA I TECHNIKA Tab. 2. Effect of falure and repar rate of the Sub-System Extruder (B) on Avalablty when both sub-systems faled ndependently. ψ α 2 0.007 0.009 0.0 0.03 Constant values 2 0.9769 0.9760 0.975 0.974 α =0.0057, ϕ =0.5 4 0.9786 0.978 0.9777 0.9772 λ =0.00, λ 2 =0.002 6 0..9792 0.9789 0.9785 0.9782 α 3 =0.0, µ =0.33 8 0.9795 0.9792 0.9790 0.9786 α 4 =0.05,σ =2 Tab. 3. Effect of falure and repar rate of the Sub-System De (C) on Avalablty when both sub-systems faled ndependently. α 3 µ 0.0 0.02 0.03 0.04 Constant values 0.33 0.9769 0.962 0.9382 0.9067 α =0.0057, ϕ=0.5 0.53 0.9788 0.9673 0.99486 0.9236 α 2 =0.007, ψ =2 0.73 0..99799 0.9705 0.9552 0.9344 α 4 =0.05, σ =2 0.93 0.9807 0.9728 0.9597 0.9420 λ =0.002, λ 2 =0.002 Tab. 4. Effect of falure and repar rate of the sub-system Cutter (D) on Avalablty when both sub-systems faled ndependently. σ α 4 0.05 0.030 0.045 0.060 Constant values 2 0.9769 0.9763 0.9756 0.9749 α =0.0057, ϕ=0.5 4 0.982 0.9809 0.9809 0.9802 α 2 =0.007, ψ =0.04 6 0..9825 0.9823 0.982 0.988 α 3 =0.33, µ =0.02 8 0.983 0.9829 0.9827 0.9826 λ =0.00, λ 2 =0.002 Tab. 5. Effect of falure and repar rate of the Sub-System Mxture (A) on Avalablty when both Sub-Systems faled smultaneously. α ϕ 0.0057 0.0059 0.006 0.0063 Constant values 0.5 0.9853 0.9849 0.9845 0.984 α 2 =0.007, ψ =2 0.7 0.9884 0.9882 0.9879 0.9876 α 3 =0.0, µ =0.33 0.9 0..9902 0.9900 0.9898 0.9896 α 4 =0.05, σ =0.02. 0.993 0.992 0.990 0.9908 λ =0.00, λ 2 =0.002 Tab. 6. Effect of falure and repar rate of the sub-system Extruder (B) on Avalablty when both sub-systems faled smultaneously. ψ α 2 0.007 0.009 0.0 0.03 Constant values 2 0.9853 0.9843 0.9832 0.9823 α =0.0057, ϕ=0.5 4 0.9869 0.9865 0.9860 0.9855 α 3 =0.0, µ =0.33 6 0..9875 0.9872 0.9869 0.9865 α 4 =0.05, σ =2 8 0.9878 0.9876 0.9873 0.987 λ =0.00, λ 2 =0.002 Tab. 7. Effect of falure and repar rate of the Sub-System De (C) on Avalablty when both Sub-Systems faled smultaneously. µ α 3 0.0 0.02 0.03 0.04 Constant values 0.33 0.9853 0.9853 0.9853 0.9853 α =0.0057, ϕ=0.5 0.53 0.9853 0.9853 0.9853 0.9853 α 2 =0.007, ψ =2 0.73 0..9854 0.9854 0.9854 0.9855 α 4 =0.05, σ =2 0.93 0.9854 0.9854 0.9854 0.9855 λ =0.00, λ 2 =0.002 Tab. 8. Effect of falure and repar rate of the Sub-System Cutter (D) on Avalablty when both Sub-Systems faled smultaneously. σ α 4 0.05 0.030 0.045 0.060 Constant values 2 0.9853 0.9853 0.9852 0.9852 α =0.0057, ϕ=0.5 4 0.9853 0.9853 0.9853 0.9852s α 2 =0.007, ψ =0.04 6 0..9853 0.9852 0.9852 0.9852 α 3 =0.33,µ =0.02 8 0.9854 0.9853 0.9853 0.9852 λ =0.00, λ 2 =0.002 70 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology 9. References. Dayal B, Sngh J. Relablty analyss of a system n a fluctuatng envronment. Mcroelectron. Relablty 992; 32: 60-603. 2. Gupta P. Lal A K, Sharma R K Sngh J. Behavoral Study of the Cement manufacturng Plant A Numercal Approach. Journal of Mathematcs and Systems Scences 2005; (): 50-69. 3. Kureghan A D, Dtlevson O D. Avalablty relablty & downtme of system wth reparable components. Relablty Engneerng and System Safety 2007; 92(2): 66-72. 4. Kumar D, Pandey P C, Sngh J. Behavour analyss of Urea decomposton n the Fertlzer ndustry under general repar polcy, Mcroelectron. Relablty 99; 3(5): 85-854. 5. Kumar D, Pandey P C, Sngh J. Desgn and cost analyss of a refnng system n a Sugar ndustry. Mcroelectron. Relablty 990; 30(6): 025-028. 6. Sngh J, Mahajan P. Relablty of Utensls Manufacturng Plant A Case Study. Opsearch, 999; 36(3): 260-269. 7. Sngh,J.Relablty consderaton of Indus tro agrcultural problems usng a heurstc approach va dfferental dynamc programmng CSIR New Delh. Inda Project Report, 983. 8. Zhang. Relablty analyss of an (N+)-unt stand by system wth preemptve Prorty Rule. Mcroelectron. Relablty 996; 36(): 9-26. Mrs. S. SHAKUNTLA Dr. A. K. LAL Prof. S. S. BHATIA School of Mathematcs and Computer Applcaton T.I.E.T. Patala, Punjab, Inda-47004 E-mal: shakus25@gmal.com E-mal: aklal@thapar.edu E-mal: ssbhata63@yahoo.com Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 7

Artcle ctaton nfo: LATALSKI J: Modellng of macro fber composte pezoelectrc actve elements n ABAQUS system. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 72-78. Jarosław LATALSKI Modellng of macro fber composte pezoelectrc actve elements n ABAQUS system Modelowane w systeme ABAQUS pezoelektrycznych elementów aktywnych typu MFC* The paper presents an approach to effectve modellng of pezoelectrc transducers n fnte element method based software. A macroscopc model of an actve element made of macro fber composte (MFC type) exhbtng d 33 effect s developed n ABAQUS system. Next, a multlayer composte beam wth the dscussed pezoelectrc actuator s analysed. Both a drect and a converse pezo effects are analyzed numercally, calculatng respectvely charge on transducer s poles subject to forced dsplacements and beam statc deflectons wth respect to assumed supply voltage. The outcomes of the numercal smulatons are compared to the laboratory test results. Next, the worked-out FEM model of MFC actuator/sensor s used to detect the smulated defect n composte materal. Keywords: pezoelectrc transducer, actve elements, FEM, ABAQUS. W pracy przedstawono sposób efektywnego modelowana kompozytowych elementów pezoelektrycznych metodą elementów skończonych. W systeme ABAQUS przygotowano makroskopowy model elementu aktywnego typu MFC wykorzystującego efekt pezoelektryczny d 33. W dalszej kolejnośc przeanalzowano welowarstwową belkę kompozytową z naklejonym badanym elementem aktywnym. Numeryczne zbadano prosty odwrotny efekt pezoelektryczny, wyznaczając odpowedno wartośc napęć na zacskach elementu aktywnego przy wymuszonym odkształcenu układu oraz ugęca statyczne przy różnych wartoścach napęca zaslającego. Rezultaty tych analz porównano z wynkam rzeczywstych pomarów przeprowadzonych na stanowsku laboratoryjnym. Opracowany model numeryczny wykorzystano do próby wykryca symulowanego uszkodzena materału kompozytowego. Słowa kuczowe: przetwornk pezoelektryczny, elementy aktywne, metoda elementów skończonych, ABAQUS.. Introducton Snce 80s one observes a sgnfcant ncrease of nterest n a class of systems that started to be called ntellgent. Ths group contans shape memory alloys (SMA), magneto- and electrorheologcal fluds (MRF, ERF) and pezoelectrcs. Common examples of the last ones are lead zrconate ttanate PZT, barum ttanate BaTO 3 and polyvnyldene fluorde polymer PVDF. As the pezoelectrcs gradually started to become wdespread, n md-90s the research actvtes were amed at usng PZTs for structural health montorng and structural relablty assessment. One of the most promsng drectons s related to an applcaton of pezoceramcs [3], [4] and mcro fber compostes. The results of already carred out projects confrm a hgh sutablty of MFC type elements not only for structural health montorng but for vbraton suppresson, energy harvestng etc. as well [9], []. It should be emphaszed, that most of the current research projects are focused not only on structural testng n laboratores or n real operatng condtons. An attenton s gven to the analyss and desgn of structural health montorng systems by means of numercal smulatons [5]. The crucal problem n analyss and desgn of structures wth pezoelectrc elements are nteractons between mechancal and electrcal phenomena see relatons () and (2). Therefore, the exact analytcal soluton of consttutve equatons s possble only for the smplest models e.g. symmetrc dscs under statc load or at resonance condtons. The analytcal solutons for more complex geometres and loadngs requre radcal smplfcatons. Ths leads to serous dscrepances between predcted and observed systems responses [7]. The complexty of the dscussed phenomena nvolves the usage of approxmate approaches to the desgn. Numercal methods, and especally a fnte element method seems to be a natural choce. The man advantage of ths approach s a possblty of modellng complex phenomena by the dvson of a doman nto smple elements, where the soluton of equlbrum equatons s possble. Although the frst lbrares of pezo-elements were mplemented nto fnte element software n late eghtes, these are stll avalable only n selected FEM systems. Followng the above observatons the am of ths paper s to develop an effectve macroscopc fnte element modellng of pezoelements made of macro fber compostes. It s expected, (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl 72 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

SCIENCE ANd TECHNoLogy that the proposed method wll enable further smulaton research on usng MFC transducers for structural health montorng task e.g. detectng delamnaton (as seres of experments presented n [8]), crack propagaton etc. Moreover, numercal smulatons of changng machnes operatng condtons, lke vbraton restranng, mght be possble. 2. Mathematcal models of pezoelectrc materals Implementaton of pezoelectrc transducers n engneerng structures requres not only detaled nformaton about ther strength data but characterstcs of ther behavour under mechancal and electrcal loads as well. One of the frst researches amng at developng lnear consttutve equatons of pezoelectrc materals (LPCE) are models elaborated by D.A. Berlncourt and D.R. Curran and later by H. Jaffe [2]. Several phenomena lke coercon effect and changes n crystallographc structure are left out of account durng the dervaton, but because of lnearty of pezoelectrcty phenomena wthn the range of small and medum electrcal felds, these models descrbe the behavour of real systems qute accurately and are stll n common usage today. The consttutve equatons descrbng the pezoelectrc property are based on the assumpton that the total stran n the transducer s the sum of mechancal stran nduced by the mechancal stress and the stran caused by the appled electrc voltage. The model s formulated as a par of two matrx relatons (accordng to Mandel-Vogt notaton), descrbng the nteractons of mechancal and electrcal phenomena [6]: where: ε stran vector ( ), S E complance matrx (m 2 /N) at constant electrc feld, σ stress vector (N/m 2 ), d matrx of pezoelectrc stran constants (m/v), E electrc feld matrx (V/m), D vector of electrc dsplacements (C/m 2 ), ξ σ matrx of permttvty coeffcents (F/m = C/m V) at constant stress. The frst of the above relatons represents the converse pezoelectrc effect.e. the stran of the sample under the electrc feld load. Ths corresponds to the devce beng used as an actuator. The second equaton (2) deals wth the opposte case, when the transducer s beng used as a sensor (drect pezoelectrc effect). The gven above notaton s named to be a dsplacement-based formulaton the dependent varable s stran vector (tensor). The system of equatons () (2) mght be wrtten also n another form the so called force-based formulaton: where: C E elastc stffness matrx (N/m 2 ) calculated at constant electrc feld, e pezoelectrc stress coeffcents matrx (N/m V = C/m 2 ) (T denotes matrx transposton), and ξ ε matrx of delectrc permttvty coeffcents (F/m) calculated at constant stran. The force based formulaton s frequently used for e.g. problems solved by fnte element method, where the dsplacements (stran ε) are a prmary varable and all the remanng () (2) (3) (4) quanttes are ε dependent. In other problems e.g. n case of analyss of thn plates n plane stress state the dsplacement based formulaton mght be more reasonable. Therefore there are two matrces of pezoelectrc coeffcents d j e j defned as follows: E D j dj = = ε σ j E E D j ej = = σ (5) ε ε j E whch are related to each other by materal constants matrx: σ E d = es (6) It should be underlned that the pezoelectrc coeffcents matrces d and e are determned not only at constant electrc feld and stran or stress respectvely (E, ε and σ), but also at a constant temperature. Although detaled tests show that pezoelectrc coeffcents are temperature dependent, n practcal applcatons t s assumed that ths nfluence on pezoelectrc coeffcents, as well as on other materal data, s neglgble. For 3-D systems equatons (-2) after expanson are: ε S S2 S3 S4 S5 S6 d d2 d3 σ ε 2 S2 S22 S23 S24 S25 S26 d2 d22 d32 σ 2 ε 3 S3 S32 S33 S44 S35 S36 d3 d 23 d 33 σ 3 ε 4 S4 S42 S43 S44 S45 S46 d4 d24 d 34 σ 4 ε 5 = S5 S52 S53 S54 S55 S56 d5 d25 d ε 35 σ 5 6 S D 6 S62 S63 S64 S65 S66 d6 d26 d 36 σ d d2 d3 d4 d5 d6 ξ ξ2 ξ 6 3 E D2 d2 d22 d23 d24 d25 d26 ξ2 ξ22 ξ23 E2 D3 d3 d32 d33 d34 d35 d36 ξ3 ξ32 ξ33 E3 The terms (S,S 22,...ξ 33 ) located at the man dagonal represent strctly mechancal and electrcal effects. The recprocal couplng of these two domans s gven by d j parameters located outsde of the man dagonal. These values correspond to the effect of deformaton caused by electrc feld and charge nducton resultng from materal s stress. Therefore these coeffcents are often used to compare the power of dfferent pezoelectrc materals between each other. The gven above system of equatons s sgnfcantly smplfed for materals exhbtng 4mm crystallne class (e.g. lead zrconate ttanate PZT, barum ttanate BaTO 3 ) or 6mm one [6]. Ths s related to the symmetry of elastc, electrc and electro-mechanc propertes. Assumng accordng to ANSI IEEE 76 standards that the devce s poled along the 3 axs and that the pezoelectrc materal s a transversely sotropc one (-2 beeng the plane of transverse sotropy) the consttutve equatons (7) mght be smplfed to: ε S S2 S3 0 0 0 0 0 d3 σ ε 2 S2 S S3 0 0 0 0 0 d3 σ 2 ε 3 S3 S3 S33 0 0 0 0 0 d 33 σ 3 ε 4 0 0 0 S55 0 0 0 d5 0 σ 4 ε 5 = 0 0 0 0 S55 0 d5 0 0 σ 5 (8) ε 6 0 0 0 0 0 2( S 0 0 0 D S2 ) σ 6 0 0 0 0 d5 0 ξ 0 0 E D2 0 0 0 d5 0 0 0 ξ22 0 E2 D3 d3 d3 d33 0 0 0 0 0 ξ33 E3 Most of the commercally avalable actve elements exhbt a d 33 effect or d 3 one. The d 33 effect corresponds to the deformaton of a specmen n the drecton of drvng electrc feld (also (7) Eks p l o a t a c j a N E z a w o d N o s c - Ma N t E N a N c E a N d RElablty N R 4/20 73

NAUKA I TECHNIKA polng drecton); whereas n the second case the deformaton occurs n the plane perpendcular to the electrc feld vector. 3. Numercal model of the pezoelectrc element and ts verfcaton For numercal and laboratory tests M-8503-P element made by Smart Materal Corp., Sarasota (FL) USA s used see fgure. Ths s the transducer of d 33 effect type. Fg.. Actve element M-8503-P made by Smart Materal Corp., Sarasota (FL) USA Despte of modular structure of the transducer under consderaton (70 sectons of electrocouples dstant by 0.5 mm from each other see zoomng area n fgure ) the pezoelement hasn t been modelled n mcro scale. The am of the paper s to develop a smplfed but effectve method of modellng, assurng results consstent wth manufacturer s data and laboratory experments. Therefore, n the presented approach a supplementary body made of orthotropc, homogenous pezoelectrc materal s proposed, where voltage s appled to opposte specmen faces. The transducer s doman s modelled n ABAQUS software by the sold contnuum elements C3D20E,.e. 20-nodal second order, havng four degrees of freedom n each node. Three DOF are translatonal and the fourth one s an electrc charge drectly related to the pezoelectrc propertes of the materal. Accordng to the data provded by the manufacturer the value of d 33 coeffcent n M-8503-P element s not constant wth respect to the electrc feld. For E < kv/mm the parameter d 33 equals 400 0-2 m/v, for hgher magntude felds ( E > kv/ mm) the d 33 ncreases to 460 0-2 m/v (all these data are gven for a sngle couple of electrodes). Followng ths nformaton the provded d 33 values have to be multpled by the number of sectons n M-8503-P element (Fgure ) to get the effectve value to be put nto the FEM model for prelmnary calculatons. Ths has been done to set tral values of pezoelectrc coeffcents. The remanng elastc and ferroelectrc propertes are taken from the manufacturer catalog: --elastc orthotropc materal gven by Young s modul E = E 2 = 5857 MPa, E 3 = 30336 MPa, Krchhoff s modul G 2 = G 3 = G 23 = 555 MPa and Posson s ratos v 2 = v 3 = v 23 = 0,3, --sotropc ferroelectrc materal wth constant permttvty ξ = 8 0-9 F/m coeffcents on the man dagonal. All the out-of-dagonal terms n ξ matrx are neglected (see equaton (8)). Numercal model valdaton Verfcaton of the pezoelectrc element model s done by means of two tests gven n manufacturer s documentaton. In the frst numercal experment strans for a free element under specfed drvng voltage are calculated and compared to catalog data. In the second experment called blockng force test the force exerted by the pezoelement wth excluded deformatons s calculated and compared to the reference value. Mechancal and electrcal boundary condtons for the conducted tests are set as follows: --free stran test nodes located at the bottom face of the pezoelectrc specmen are allowed to move only n xy plane, nodes located at one of the lateral longtudnal faces are restrcted to move n yz plane only. Fnally, one of lateral transversal faces gets the support along y drecton and, addtonally, constant electrc potental 0 V s kept there. Charge potental of 500 V s appled along the specmen length. These condtons are presented n fgure 2. --blockng force test same boundary condtons as n the prevous experment wth addtonal constrant u y = 0 for the nodes on the opposte, transverse (short) face. Results of ntal numercal tests shows a serous dscrepancy comparng to the catalogue data. In free stran test the error s 6-3% axal strans 206-363 0-6 wth respect to reference value ε y = 050 0-6 ; n blockng force test the error s 7-33%.e. +32,7-+37,58 N wth respect to 28.00 N. Therefore, n order to acheve a satsfactory accuracy the correcton of the effectve pezoelectrc coeffcent s necessary. Mnmsng the relatve error n both tests as a functon of d 33 parameter ts optmal value for the M-8503-P element s found to be 59 0-9 m/v. After ths correcton strans calculated n the frst test are 040 0-6 (vs 050 0-6 by manufacturer); force for blocked element s R y = 28.46 wth respect to 28.00 N n documentaton. Therefore, Fg. 2. Boundary condtons n computatons valdatng the model of an examned transducer 74 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology the results of the fnal smulatons for the updated d 33 parameter prove the pezoelectrc patch model and modellng technque for MFC transducers to be correct. 4. Analyss of a composte beam wth pezoelectrc patch To present the potental of the proposed effectve pezoelement modellng technque two experments are performed. In the frst one a reference composte beam wth bonded patch of M-8503-P actve element s analysed. Experments representng both drect and converse pezoelectrc effects are run. Wthn the frame of these voltage at transducer s poles s recorded for forced beam bendng and next dsplacements of a specfed pont on specmen resultng from voltage appled to PZT patch. Results of laboratory experments are compared to the outcomes of numercal calculatons n ABAQUS system. In performed smulatons the prevously derved and verfed fnte element model of the transducer s used. Second stage of research comprses the analyss of the composte beam wth a smulated materal defect. Damage correspondng to broken composte fbers (or ther pull-out from matrx resultng from adhesve jont falure) s smulated as a local change n composte materal stffness. Specmen wth ths smulated defect s tested numercally. Wthn the tests programme voltage nduced on transducer poles wth respect to forced specmen bendng s recorded. Outcomes are related to the data for an undamaged (healthy) sample. 4.. Model and sample specmen testng The verfed model of the pezoelectrc transducer s used to smulate the statc response of composte cantlever beam wth M-8503-P actve element. The basc beam s made of undrectonal glass fbers tape and epoxy Prme 20 (Scomn 800 + hardener 8824, fbers rato 50±2% accordng to manufacturer Macro Compostes, UK data). The subsequent layers of the composte are set n the followng order: 0 /90 /+45 / -45 /+45 /90 /0 (wth respect to Oy axs pontng along the beam length). Pezoelement M-8503-P s bonded on the upper face of a specmen, drectly at the clamped end. Draft vew of a structure under consderaton s gven n fgure 3 (all dmensons n mm). Composte model of the beam s defned as a lamna type one. Ths approach enables modellng of a composte as a set of orthotropc layers n plane-stress state. In numercal smulatons the followng data provded by the manufacturer of a composte s used: Young modulus along fbers E = 20 000 MPa, transversal Young modulus E 2 = 2 000 MPa, shear modul G 2 = G 3 = G 23 = 9 800.7 MPa and Posson s rato ν 2 = 0.26. The ABAQUS fnte element model based on shell elements s made accordng to Layup-Ply technque []. In the analyss S8R elements are used.e. second order ones wth reduced ntegraton. The pezoceramc materal propertes are defned as descrbed n prevous secton of the paper; the already verfed pezoelectrc coeffcent d 33 = 59.0 0-9 m/v s used n further smulatons. All the remanng (.e. out-of-dagonal) coeffcents of d matrx are set to be 0. Accordng to manufacturer data and actual measurements the actve area of an element s set to 85 3 mm. Next, the model of a pezoelement s glued to master structure by TIE constrants method, whch results n jonng approprate DOF of both bodes. The mechancal boundary condtons are set by encastrng the nodes on shorter sde face of a beam. The fnal FEM model conssts of 37 elements (beam 300, pezoelectrc element 7) and 243 nodes; ths results n 7034 degrees of freedom. The fnte element model of the structure and composte stackng sequence s presented n fgure 4. Durng the statc analyss of the pezo-composte system a seres of smulatons s run. Subsequent loadngs wthn the range of 0-500 V are appled to the postve termnal of the z 05 powerzchna czynna / actve area (85 x 3 mm) A M-8503-P 300 x y 2,9 2,6 Fg. 3. Scheme of the composte beam under consderaton Fg. 4. Dscrete model of the composte beam and layout of ndvdual layers (orentaton of fbers n drecton corresponds to Oy axs Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 75

NAUKA I TECHNIKA transducer, keepng the constant value of 0 V at a negatve one (see fgure 2). The deflectons of a beam are calculated by reportng the vertcal dsplacement of a pont located at the edge of the pezoelement (see pont A n fgure 3). A converse pezoelectrc effect s also tested.e. calculatng electrcal potental on transducer termnals wth respect to forced beam bendng. 4.2. Laboratory tests To verfy the presented above approach to modellng systems of composte structures wth pezoelements an expermental stand has been prepared n the Laboratory for Dynamcs and Strength of Materals at the Lubln Unversty of Technology. The setup, shown n fgure 5, conssts of a unform composte cantlever beam wth a sngle pezoelectrc transducer lamnated onto the upper face. The transducer s connected to D.C. power unt wth a controllable resstance dvder. The requested voltages are provded by a hgh-voltage amplfer model PA05039 made by Smart Materals company. At the second stage of tests a converse problem s examned. Durng these a mcrometer screw s used to deflect the composte beam, meanwhle voltages nduced by the pezoelement are recorded. 4.3. Results statement Results of numercal calculatons are presented n the followng fgures. Fgure 6 presents a shape of the beam n bendng calculated by ABAQUS for three dfferent drvng voltages: 500 V, 000 V and 500 V. Horzontal axs s amed along beam length (Oy left end clamped) and vertcal axs corresponds to beam transversal dsplacements; all data are gven n mm. Fgure 7 presents the statement of numercal and laboratory tests for the converse pezoelectrc effect. The vertcal translatons of a pont located at the external edge of the transducer (see pont A n fgure 3) are compared. The deflectons are collected for several drvng voltages. Fgure 8 presents the results of drect pezoelectrc effect testng. The assumed a pror beam deflecton has been forced by mcrometer screw, meanwhle the voltage at transducer s poles s recorded. These expermental results are compared to smu- Fg. 5. General vew of the test stand [mm] 0,0 Oy -0,5 -,0 -,5-2,0-2,5-3,0-3,5 500 V 000 V 500 V 0 50 00 50 200 250 300 [mm] Fg. 6. Deflecton of the tested specmen at dfferent drvng voltages (numercal sumulaton) [mm],00 0,90 0,80 0,70 0,60 0,50 0,40 kompozyt szklany/glass epoxy composte Fg. 7. Vertcal translatons of pont A obtaned from numercal smulatons and laboratory tests of composte beam wth pezoelement for dfferent drvng voltages 0,30 0,20 0,0 MES/FEM pomary laboratoryjne laboratory experments 0,00 0 200 400 600 800 000 200 400 600 [V] 76 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology [V] 250 200 kompozyt szklany/glass epoxy composte 50 00 Fg. 8. Comparson of numercal smulatons and laboratory tests of a converse pezoelectc effect for the specmen under consderaton 50 MES/FEM pomary laboratoryjne laboratory experments 0 0 0,2 0,4 0,6 0,8,0,2,4 [mm] latons n ABAQUS system run on models prepared accordng to proposed technque. In all dscussed cases a hgh accuracy of smulatons and laboratory test s acheved. For the examned ponts the calculated dfferences do not exeed 6%. 4.4. Numercal test of a damaged composte beam Results of the above tests nduced an dea of usng the proposed macroscopc modellng technque of MFC type tranducers for detectng composte materal damage. The smulated materal defect corresponds to composte fbers breakng (or ther pull-out from matrx resultng from adhesve jont falure). Macroscopc, smplfed representaton of ths type of materal damage conssts n partal fber removal from consttutve equatons, and next from the formula for effectve longtudnal and shear stffness lamnae modul [2]. Assumung the fbers to be orented along Ox axs, the effectve lamna stfnesses modul are calculated accordng to the rule of mxtures as follows [0]: Exx = EfVf + EmVm + αv f Eyy = Em V f νxy = Vfν f + Vmν m where subscrpts x and y denote prncpal drectons, V denotes volumetrc rato of a consttutent, ndex f denotes fber, ndex m matrx. Coeffcent α s gven by: (9) E α = E f f / E / E m m + (0) Followng the above equatons t s possble to formulate the consttutve equatons for the damaged composte materal, where the altered longtudnal stffenesses (9) need to be used. Denotng the volume of broken fbers n the damaged materal by V df, value of stffness modul E are decreased accordng to the followng formula [2]: E = E V + E ( V ) E = E xx f df m df yy m + αv V df df () For fnte element calculatons the prevously tested and verfed numercal model of the composte beam wth the PZT patch s used. Composte materal damage s smulated by decreasng the lamna stffness on the 20 mm long secton, startng at 2 mm dstance from the clamped edge. It has been assumed that the volumetrc rato V df of damaged fbers n composte materal equals to 0.25. In performed numercal tests the electrc potental at transducer s poles s calculated subject to forced vertcal deflecton of pont A (see fgure 3). The outcomes of the smulaton are compared to the results of smlar tests run on the undamaged system. The relevant results are presented graphcally see fgure 9. The obtaned results ndcate, that the calculated voltages, n the whole tested range, are approxmately 8% lower whle comparng to the reference (healthy) system. These dfferences exceed the error resultng from the comparson of a reference system and laboratory tests outcomes. [V] 250 200 kompozyt szklany/glass epoxy composte 50 00 50 układ bez uszkodzena reference system (no defect) układ z symulowanym uszkodzenem system wth smulated defect Fg. 9. Results of numercal test of a composte beam wth smulated materal defect 0 0 0,2 0,4 0,6 0,8,0,2,4[mm] Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 77

NAUKA I TECHNIKA 6. Conclusons The paper presents the possbltes of effectve modellng of MFC type pezoelectrc transducers exhbtng d 33 effect. A hgh correlaton of numercal results and the outcomes of laboratory tests s acheved. In none of the examned cases the error exeeded 6%. Ths observaton confrms the valdty of derved macroscopc approach to fnte element modellng of MFC type transducers and also valdty of suggested dscretsaton of the specmen doman. The outcomes of these ntal nvestgatons encourage for further numercal smulaton research usng the derved and already verfed modellng technque. These mght focus on fnte element method structural health montorng by means of dynamc methods especally modal analyss and wave propagaton. The next stage mght be modellng of plates and shells. Multple pezoelements used as sensors and actuators seems to be of partcular nterest too. ********** The fnancal support of the Structural Founds n the Operatonal Programme Innovatve Economy (IE OP) fnanced from the European Regonal Development Found research project Modern materal technologes n aerospace ndustry number POIG.00.02-0005/08 s gratefully acknowledged. ********** 7. References Abaqus 6.9 Documentaton. 2 Adams D E. Health Montorng of Structural Materals and Components: Methods and Applcatons. Hoboken (NJ): John Wley & Sons, 2007. 3 Kabeya K. Structural Health Montorng Usng Multple Pezoelectrc Sensors And Actuators. Blacksburg (VA): Vrgna Polytechnc Insttute and State Unversty, 998. 4 Kessler S S. Pezoelectrc-Based In-Stu Damage Detecton of Composte Materals for Structural Health Montorng Systems. PhD thess. Boston (MA): Massachusetts Insttute of Technology, 2002. 5. Klepka T, Dębsk H, Rydarowsk H. Characterstcs of hgh-densty polyethylene and ts propertes smulaton wth use of fnte element method. Polmery 2009; 54(9): 668-672. 6 Leo D J. Engneerng Analyss of Smart Materal Systems. Hoboken (NJ): John Wley & Sons, 2007. 7 Lowre F, Can M, Stewart M. Fnte Element Modellng of Electroceramcs. Techncal Report, A50. Teddngton: Centre for Materals Measurement and Technology, Natonal Physcal Laboratory, 999. 8 La Saponara V, Horsley D A, Lestar W. Structural Health Montorng of Glass/Epoxy Composte Plates Usng PZT and PMN-PT Transducers. ASME Journal of Engneerng Materals and Technology 20; 33(): 00. 9 Schönecker A J, Daue T, Brückner B, Freytag Ch, Hähne L, Rödg T. Overvew on macrofber composte applcatons. Proceedngs of the SPIE Smart Structures and Materals 2006 (ed. Armstrong WD) 2006; 670: 408-45. 0 Kollar L, Sprnger G S. Mechancs of Composte Structures. Cambrdge: Cambrdge Unversty Press, 2003. Varadan V K, Vnoy K J, Gopalakrshnan S. Smart Materal Systems and MEMS: Desgn and Development Methodologes. Chchester (NH): John Wley & Sons, 2006. 2 Yousef-Koma A. Intellgent Materals. W: Pezoelectrc Ceramcs as Intellgent Multfunctonal Materals: 23-254. Cambrdge: The Royal Socety of Chemstry Publshng, 2008. Jarosław LATALSKI, Ph.D. Eng. Department of Appled Mechancs Faculty of Mechancal Engneerng Lubln Unversty of Technology 36 Nadbystrzycka Str., 20-68 Lubln, Poland e-mal: j.latalsk@pollub.pl 78 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Artcle ctaton nfo: NOWAKOWSKI T: Problems of relablty modellng of multple-phased systems. Eksploatacja Nezawodnosc - Mantenance and Relablty 20; 4: 79-84. Tomasz NOWAKOWSKI Problems of relablty modellng of multple-phased systems Problemy modelowana nezawodnośc systemów welofazowych* Ths artcle dscusses the basc problems connected wth modellng the relablty of multple-phased systems. Operaton and mantenance of such systems s assocated wth executon of varous tasks whch together lead to the achevement of the fnal goal. Such systems nclude logstc and transport systems. Two types of models are dscussed: synthetc models, whch cover a system s entre operatng lfe and models n whch the partcular phases are consdered separately. As an example, a k out of n system s used wth dfferent k parameters for each phase. The advantages and dsadvantages of three models are dscussed: the conservatve model whch s based on an analyss of the block relablty model, a Markov model wth fxed duraton of each phase and a Markov model wth random phase duraton. Keywords: relablty, modelng, multple-phased system. W artykule omówono podstawowe problemy zwązane z modelowanem systemów welofazowych. Eksploatacja takch systemów zwązana jest z realzacją różnych zadań, które składają sę na osągnęce celu końcowego. Do takch systemów można zalczyć systemy logstyczne systemy transportowe. Omówono dwa rodzaje model: modele syntetyczne ujmujące cały okres eksploatacj systemu modele, w których poszczególne fazy są rozpatrywane oddzelne. Wykorzystano przykładowy system o strukturze progowej zmennej w kolejnych trzech fazach eksploatacj. Przedstawono zalety wady korzystana z modelu konserwatywnego bazującego na analze modelu blokowego model Markowa z ustaloną losowo zmennych czasem trwana poszczególnych faz. Słowa kluczowe: nezawodność, modelowane, system welofazowy.. Introducton There are many systems whose operatng lfe conssts of a seres of separate tme ntervals. In each of those ntervals, the system carres out dfferent tasks, the results of whch contrbute to the achevement of the fnal goal. Such systems are referred to n the lterature (e.g., [4]) as phased msson systems (PMS). Examples of such systems, besde transportaton and logstc systems, can be found n many areas of applcaton such as the nuclear power ndustry, avaton, shpbuldng, the telecommuncatons ndustry, the constructon ndustry, electroncs, and many others (e.g., [8, 9]). Research done currently [0] concerns, among others, a combned ral/water transport system for the transportaton of coal n the corrdor of the Oder Waterway. Because the concept of perodc executon of tasks by such complex systems as a transportaton or a logstc system s related to a much wder set of systems than suggested by the concept of msson, the term multple-phased systems was proposed n [4]. In multple-phased systems, the ndvdual phases may be characterzed by many dfferent propertes [4]: --a task executed n a gven phase may dffer from the tasks executed n the remanng phases, --the requrements regardng performance and relablty may dffer among phases, --durng some of the phases, the system may be subjected to a partcularly strong nfluence of the envronment, whch may cause a consderable ncrease n falure rate, --the structure of a system may change over tme dependng on the functonal and relablty requrements formulated for the currently executed phase, --proper executon of tasks wthn a gven phase may brng other effects for the system than those obtaned n other phases. 2. Relablty models of multple-phased systems In system relablty modellng, the use of the concept of a mult-phased system allows better approxmaton of realty on account of the followng assumptons: --a system s operatonal structure s not constant; t may change between phases dependng on the mportance / crtcalty of a gven phase, --the hstory of falures or repars of a gven component n a gven phase affects the behavour of the system n the followng phase. Hence, the state of a component at the begnnng of a gven phase depends on the state of the component at the end of the prevous phase. --the crtera defnng the level to whch the requrements related to performance and relablty are met may dffer n a gven phase from those for the next phase. These assumptons are used n varous ways n the models known from the lterature. There are two types of models: synthetc models, whch cover a system s entre operatng lfe and models n whch the partcular phases are consdered separately. (*) Tekst artykułu w polskej wersj językowej dostępny w elektroncznym wydanu kwartalnka na strone www.en.org.pl Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 79

NAUKA I TECHNIKA Synthetc models, n whch all phases are represented together (e.g., [, 2, 5, 4]), have a number of advantages as they make t possble to use smlartes among phases to obtan a compact model n whch all phases are adequately embedded. The constructon of a synthetc model lke that may be nether easy nor convenent n the cases where dfferences outwegh the smlartes among the ndvdual phases. Separate modellng of each phase (e.g., [7, 4]), n turn, enables mmedate characterzaton of the dfferences among phases wth respect to falure rate and structural requrements. Each phase can be solved separately, and the results obtaned can be combned wth those from other phases to gve total results for the system. The man weakness of the separatng approach to modellng (whch does not occur n synthetc models) s how t deals wth the relatonshps among phases, whch have to be taken nto account when dstrbutng components among the phases. Such an approach requres clearly specfed mappng of the states of a component at the end of a phase relatve to the state of the same component at the begnnng of a followng phase. A task of ths sort s conceptually smple, but may be nconvenent and s a potental source of errors for complex systems. The most dffcult decson n the process of modellng regards the way n whch the ndvdual phases are combned nto a sngle model and the way n whch the relablty characterstcs of the entre system are estmated. An analyss [] was conducted for an example system descrbed by the followng assumptons: --a system made up of three components (A, B and C), --durng ts operatng lfe, the system goes through 3 successve phases (I, II and III), --the falure rates for the ndvdual components are constant over the tme of duraton of a gven operatonal phase, but may dffer among the ndvdual phases (λ j, = A, B, C, j = I, II, III), --the components can be servced or repared; the falure rate n the partcular phases s constant, but may also change n successve operatonal phases (μ j, = A, B, C, j = I, II, III), --the system s relablty structure s a k j -out-of-n threshold structure; the parameter k s phase-dependent and s k I =, k II = 2, k III = 3, at n = 3. A system has a k-out-of-n threshold structure f the system functons f and only f at least k of the n elements functon. A block dagram of such a system s shown n fgure. In the dagram, k elements are connected n a seres and may be replaced wth any of the (k+, n) elements (there are models wth more formalzed redundancy schemes, e.g., [3]). It s also easy to show that a threshold structure s a generalzaton of a seres and a parallel structure: --a -out-of-n structure s a parallel structure, --an n-out-of-n structure s a seres structure. The relablty functon of a threshold structure system, assumng that the system s made up of dentcal components (R (t) = R 2 (t) = = R n (t) = R(t)), s expressed by formula: s ( ) n n! n ( ) ( ( )) () = k! ( n )! R t = Rt Rt For modellng a system made up of renewable components, the Markov model s most frequently used approprate state Fg.. A block dagram of a threshold structure graphs for the ndvdual operatonal phases of an example system are shown n fgures 2, 3 and 4. The state of the system has been descrbed usng the followng notaton S = <S A, S B, S C > (2) where: S = 0 faled component, S = workng component. The faled states of the system n the ndvdual phases have been shaded. Emphass should be gven to one of the adopted assumptons [] sayng that no further falures are possble durng repar of system components. 2.. A conservatve relablty model The smplest approach to combnng phase relabltes nto system relablty s to use a seres model of system relablty n whch the successve phases of system operaton represent structure components. Ths s possble f the components of the system do not show dynamc changes n propertes such as transton errors or ncomplete repar [5]. Then, a conservatve estmate of a system relablty functon (a bottom-up estmate) s obtaned. An example of ths knd of structure s shown n fgure 5. The results of calculatons of the relablty of such a system, f the components are rreparable, are precse but lead to bottom-up estmaton of the relablty of the actual system. Fg. 2. A Markov model for phase I [] 80 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology Fg. 3. A Markov model for phase II [] Fg. 4. A Markov model for phase III [] Fg. 5. A structure of system relablty a seres structure of phases [] A soluton to ths problem s to analyze a system usng the success path set or the falure cut set methods [6]. The propertes of mnmal path and mnmal cut sets are, among others, as follows [3]: --the structure of a system can be represented by means of a pseudostructure made up of mnmal path sets connected n parallel, --the structure of a system can be represented by means of a pseudostructure made up of mnmal cut sets connected n a seres, --the structure of a mnmal path set P j (j =,..., p) s a seres structure, --the structure of a mnmal cut set K j (j =,..., k) s a parallel structure. The structure of a system ϕ can thus be represented by the structures of ts mnmal paths: φ ( x)= maxmn x j p Pj whch corresponds to the pseudostructure ϕ P made up of mnmal paths, or by the structures of mnmal cuts: φ ( x)= mn max x j k Kj whch corresponds to the pseudostructure ϕ K made up of mnmal cuts. Then, the relablty of the system, whch s not worse than the relablty of a system wth the pseudostructure ϕ K and not better than the relablty of a system wth the pseudostructure ϕ P, can be estmated top-down and bottom-up. The relabltes of systems wth pseudostructures ϕ P and ϕ K are relatvely easy to determne. (3) (4) max P mn x = R mn P max x = s j p Pj j k Kj 2.2. A Markov model fxed phase duraton The Markov model may turn out to be an effectve tool for relablty analyss of complex systems whch show varable behavour durng executon of a task, such as, for example, --varyng transton probabltes between phases or --a lmted number of mantenance kts. In the proposed modellng strategy, each phase of operaton of a mult-phased system s modelled wth a separate Markov model. It s assumed that the fnal relablty state of a system n phase j s the ntal state for phase j +. A process dagram s shown n fgure 6. In phase I, executed over tme T, workng states nclude, among others, states (), (0), (0) and (0), and transton to phase II, to analogous workng states s possble. States (00), (00) and (00), n turn, wll be faled states n phase II, and the fnal probabltes of beng n these states wll add up to gve the probablty of system falure. Obvously, a system may also fal n phase I state (000). For phase III, only the state n whch all components are workng () s a workng state. Inconvenences of ths method of modellng the relablty of mult-phased systems are connected wth the dffcultes n relatng the correspondng states of the system among the ndvdual phases. Other complcatons arse (e.g., [5]) f a gven component s subject to falure n one phase but does not fal n another or when falures n one phase are not dagnosable untl the component s used n the subsequent phase. (5) Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 8

NAUKA I TECHNIKA Fg. 6. A dagram of a relablty model wth fxed phase duraton [] A synthetc model of the approach formulated earler was proposed n [5]. The phase ndex φ j was ntroduced φ j = f (T j- t < T j ) (6) 0 f t < T j-, t > T j where: T j end-of-phase-j tme pont. The φ j ndex specfes whch transton between states belongs to the gven phase j. The use of ths method does not change the state space of a system nor does t requre determnaton of new probabltes for transtons between states. The obtaned model s stll a Markov model, but s no longer homogeneous the values of transton probabltes depend on the operatng tme of a system. An example of such a model s shown n fgure 7. To smplfy the dagram, only the components falure rates were taken nto account; the notaton for repar rates s analogous. For each state number, the number of phase n whch ths state s a faled state has been gven. 2.3. A Markov model random phase duraton If determnstc duraton of the ndvdual phases of system operaton cannot be defned and these tme perods are random varables, then a non-homogeneous Markov process has to be used for modellng. The approach presented n [4] s based on soluton of a sngle non-homogeneous Markov model n whch the concept of state transtons has been generalzed to nclude phase changes. One mert of ths approach s that t can take nto account the dependence of phase change on the state of the system and the dependence of falures and repars on tme n the ndvdual phases. There s also no need to correlate the probabltes of beng n a state between phases. The dsadvantage of ths method s the hgh level of complexty of the model. The state space can be very large because t represents the sum of the states of all the submodels Because the sze of the state space n Markov models s (n the worst 82 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Scence and Technology case) an exponental functon of the number of components, ts growth may be crtcal for the possblty of modellng. 3. Conclusons From the models obtaned t was possble to estmate the basc measures of system relablty such as, for example, the probablty of accurate performance of a logstcs task, the probablty of the occurrence of an error/falure as a functon of task executon tme, or the mean tme between falures. The presented approach was used to model the relablty of ntermodal transport systems by means of Markov and sem-markov models (e.g., [2], [3] and [6]). It can be sad that the ntermodal transport system (from the pont of vew of relablty modellng) s a standard example of a system wth perodc/phased tasks (mssons). Transport of loads s dvded nto phases n whch the transportaton actvtes are performed dfferent mean of transport. Possbltes of relablty and safety modellng of combned transport of coal on Odra Rver Corrdor [0] s shown n table. For example, takng nto consderaton the whole system, for fxed phase duraton of ral and water transport we have obtaned the followng estmatons [0]: --avalablty of ral system ( ) + ( ( ) ( ) ) + At ()= µ / λ + µ P t λ t µ λ µ t 0 ( P 0 exp( ( )) /( λ + µ ) (7) t µ / ( λ µ ) + ( P( t ) λ ( t ) µ ) λ + µ t 0 ( P 0 exp( ( )) /( λ + µ ) ()= + where: P(t 0 ) probablty of up state for t = 0, --avalablty of combned system: A()= t µ / ( λ + µ ) + ( P( t0) λ ( P( t0) µ ) exp( ( λ + µ )) t /( λ + µ ) (8) µ / ( λ + µ ) ( P t0 λ ( P t0 µ ) exp( ( λ + µ )) t /( λ + µ ) + ( ) ( ) Table. Schema of analysed models of combned transport systems Fg. 7. A synthetc model of a sample mult-phased system [] where: number of phase P ( t ) A( t) (9) + 0 = The estmated statstcal data are dvded nto two groups of undesred events of ral transport and waterborne transport (on the frst level of the system decomposton). In the case of ral transport these are the events connected wth falures and hazards resultng from ral cars operaton and mantenance and falures or faults of ral system nfrastructure. Whereas for waterborne transport system these are events caused by falures and faults of nland waterway nfrastructure and restrctons come from mproper water flow and events fxed wth falures and mantenance actons of pushers and barges. The dfference n falures and hazards structure results from condtons and shares of partcular events see fgure 8. For gven results of data analyss, the combned system of coal transportaton s more relable than the ral one, whle the total 50 hours msson tme s dvded on 6 hours ral phase and 46 hours water transport phase fgure 9. Fg. 8. Rate of undesred events [0] Transport system Relablty model Ral Block dagram Markov Ral-water-ral Block dagram Markov fxed phase duraton Markov random phase duraton Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20 83

NAUKA I TECHNIKA One can see that the model correctly shows the effect of change of mean of transport n the 6 th hour of the msson and t also shows the ncrease of the system avalablty accordng to hgher relablty of water transport. Certanly uncertanly of the assesment of the combned transport avalablty depends on mentoned uncertanly of nput data. Fg. 9. Results of comparson of ral and combned transportaton systems 4. References. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. 3. 4. 5. 6. Alam M., Al-Saggaf U.M. Quanttatve relablty evaluaton of reparable phased-msson systems usng Markov approach. IEEE Transactons Relablty986; 35: 498-503. Aupperle B.E., et al. Evaluaton of fault-tolerant systems wth non-homogeneous workloads. 9th IEEE Int. Fault Tolerant Computng Symp 989. Broln A. Relablty Engneerng. Theory and Practce. Sprnger-Verlag, Berln Hedelberg 999. Bondavall A., Charadonna S., D Gandomenco F., Mura I. Dependablty Modelng and Evaluaton of Multple-Phased Systems Usng DEEM. IEEE Transactons on Relablty 2004; 53: 4. Dugan J.B. Automated Analyss of Phased-Msson Relablty. IEEE Transactons on Relablty 99; 40:. Dugan J.B., Veeraraghavan M., Boyd M., Mttal N. Bounded approxmate relablty models for fault tolerant dstrbuted systems. Proc. 8th Symp. Relable Dstrbuted Systems 989. Esary J.D., Zehms H. Relablty analyss of phased mssons. Relablty and Fault Tree Analyss. Phladelpha: SIAM; 975. Hoła B. Methodology of estmaton of accdent stuaton n buldng ndustry. Archves of Cvl and Mechancal Engneerng 2009; 9;. Kobylńsk L. System and rsk approach to shp safety, wth specal emphass on stablty. Archves of Cvl and Mechancal Engneerng 2007; 7; 4. Kulczyk J., Nowakowsk T., Restel F. Relablty analyss of combned coal transport system n Odra rver corrdor. Proceedngs of ESREL 20 (n preparaton). Nowakowsk T. Nezawodność systemów logstycznych. Wrocław Unversty of Technology Publshng House 20. Nowakowsk T. Relablty model of combned transportaton system. Probablstc safety assessment and management. PSAM7- ESREL 2004. London [etc.] : Sprnger. Nowakowsk T. Zając M. Analyss of relablty model of combned transportaton system. Advances n safety and relablty. Proceedngs of the European Safety and Relablty Conference 2005. Leden : A.A.Balkema. Smotherman M., Zemoudeh K. A nonhomogeneuous Markov model for phased-msson relablty analyss. IEEE Transactons Relablty 989; 38: 585-590. Xng L., Dugan J.B. Analyss of generalzed phased msson system relablty, performance and senstvty. IEEE Transactons Relablty 2002; 5: 99-2. Zając M. Model nezawodnośc systemu transportu ntermodalnego. Raporty Inst. Konstr. Ekspl. Masz. PWroc. 2007, Ser. Matrces 3. Doctoral dssertaton. Dr hab. nż. Tomasz NOWAKOWSKI, prof. nadzw. Wrocław Unversty of Technology Faculty of Mechancal Engneerng Insttute of Machne Desgn and Operaton Department of Logstcs and Transportaton Systems Wyb. Wyspańskego 27, 50-370 Wrocław, Poland e-mal: tomasz.nowakowsk@pwr.wroc.pl 84 Eks p l o a t a c j a N e z a w o d n o s c - Ma n t e n a n c e a n d Relablty n r 4/20

Prof. AGH Jan Szybka, dr nż. Zdzsław Bronec, dr nż. Robert Plch AGH Akadema Górnczo-Hutncza Wydzał Inżyner Mechancznej Robotyk Katedra Konstrukcj Eksploatacj Maszyn Al. A. Mckewcza 30, 30-059 Kraków E-mal: szybja@agh.edu.pl Prognozowane uszkodzena cepłocągu na podstawe oceny ryzyka eksploatacj Słowa kluczowe: cepłocąg, uszkodzene, wytrzymałość rurocągu, ryzyko eksploatacj, prognozowane uszkodzena Streszczene: W artykule przedstawono procedurę postępowana w wyznaczena okresu użytkowana cepłocągu eksploatowanego w zakładach górnczych. Na podstawe badań przeprowadzonych w warunkach rzeczywstej eksploatacj prognozowano awarę rurocągu, która może być spowodowana pęknęcem na skutek przekroczena jego wytrzymałośc. Opsano warunk eksploatacj cepłocągu czynnk wpływające na jego degradację. W ujęcu probablstycznym oszacowano ryzyko eksploatacj wyznaczono okres użytkowana, po którym pownna nastąpć wymana rurocągów.. Wstęp Zapewnene bezawaryjnej pracy badanego cepłocągu jest zadanem prorytetowym dla przedsęborstwa na terene, którego jest on eksploatowany. Nawet krótke przerwy w dostawe wody gorącej mogą spowodować poważne komplkacje w funkcjonowanu zakładu znaczące straty fnansowe. Ocena zagrożena awarą jest stotną nformacją, poneważ w systeme eksploatacj można przygotować dzałana prewencyjne, przecwdzałając uszkodzenom, które mogą wystąpć w nedalekej przyszłośc. Problemy oceny nezawodnośc ryzyka eksploatacj rurocągów analzowano w artykułach [5,8,9]. Aktualna lteratura skupa sę główne na wpływe korozj starzena sę elementów rurocągów [3,4,6]. Ważnym problemem zwązanym z prognozowanem awar cepłocągu jest równeż odnawane proflaktyczne [3], a w analzowanym przypadku wyznaczene termnu budowy dodatkowej, rezerwowej ln cepłownczej stanowącej nadmarowy rurocąg, który będze użytkowany w mejsce stnejącego. Dotychczas eksploatowany cepłocąg po wyremontowanu będze wykorzystywany jako rezerwowy w sytuacjach awaryjnych w planowanych pracach remontowych. Dwa cepłocąg wzajemne rezerwujące sę w eksploatacj stanową gwarancję wysokej nezawodnośc w dostarczanu gorącej wody. Autorzy w nnejszym opracowanu przedstawl metodykę prognozowana awar wyznaczana okresu użytkowana cepłocągu w warunkach rzeczywstej eksploatacj. 2. Charakterystyka konstrukcyjno-eksploatacyjna oraz dentyfkacja uszkodzeń cepłocągu

Analzowany obekt w postac rurocągu jest wykorzystywany do transportu wody gorącej o średnej temperaturze około 463 [K]. Wartośc cśneń roboczych przesyłanej wody wahają sę w grancach,33 2,44 [MPa] w cągu doby a maksymalny wydatek przepływu osąga wartośc 700 [t/h]. Rurocąg zbudowany jest z rur stalowych bez szwu, wykonanych ze stal nskowęglowej R35 (P235GH) PN-80/H-7429. Wymagana mnmalna wartość grancy plastycznośc materału rur w temp. 473 [K] wynos R et 85 [MPa]. Rurocąg jest zabezpeczony zolacją ceplną o grubośc 25 [cm], na którą składają sę warstwy waty szklanej, ownęte płaszczem ochronnym z papy blachy alumnowej. Na ocenę stanu techncznego występujące uszkodzena rurocągu najwększy wpływ ma zmana grubośc ścanek rur będąca skutkem występujących wżerów materałowych [3,4,0]. Badany rurocąg jest pod wpływem nekorzystnych oddzaływań środowska chemcznego transportowanego medum występujących równocześne mpulsowych obcążeń dynamcznych. Postępująca degradacja mkrostruktury stal zachodz w wynku dzałana agresywnych chemczne zwązków oraz występujących naprężeń, wysokej temperatury czasu. Podstawową przyczyną zmany grubośc ścanek rur jest przede wszystkm korozja elektrochemczna, której źródłem jest wysok stopeń zmneralzowana użytkowanej wody. Dodatkowym czynnkem szybkego utlenana sę powerzchn rur jest sprzyjająca korozj wysoka temperatura wody. Obecność w wodze welu zwązków chemcznych, cząstek stałych podwyższona temperatura przyczynają sę do przyspeszonego procesu utlenana na powerzchn metalu jak równeż do pewnego zużyca ścernego warstwy werzchnej rurocągu [0]. Z przeprowadzonych badań grubośc ścanek rurocągu wynka, że ubytk korozyjne w rurach osągają wartośc klku mlmetrów. Powoduje to obnżane pola powerzchn nebezpecznych przekrojów rurocągu wzrostu pozomu sumarycznych naprężeń złożonych. Inny rodzaj oddzaływana destrukcyjnego korozj chemcznej, to powstawane na powerzchn metalu głębokch wżerów. Skutkem lokalnych zman grubośc ścank są odkształcena rury proporcjonalne do różncy temperatur. Stosując zasadę superpozycj do odkształceń wywołanych obcążenam eksploatacyjnym (cśnene medum jego zmany w czase) należy równeż dodać odkształcena wywołane przez naprężena ceplne. Zmenny stan obcążeń rurocągu uwdaczna sę równeż dużą wartoścą przemeszczeń rur w stosunku do wykonanych konstrukcj wsporczych podpór. Na podstawe welokrotnych obserwacj potwerdza sę także zmenność przemeszczeń konstrukcj we wszystkch kerunkach x, y, z. Te dochodzące do klkudzesęcu centymetrów przesunęca rurocągu są przyczyną powstawana zman jego położena, a w jednym z odnotowanych przypadków zaobserwowano jego zsunęce z betonowej podpory. Powstające pod wpływem przemeszczeń odkształcena rurocągu są szczególne nebezpeczne dla przekrojów o osłabonych ścankach. 3. Ocena naprężeń w wytypowanych odcnkach rurocągu W rurach cenkoścennych poddanych dzałanu cśnena wewnętrznego p (warunek cenkoścennośc h/r 0,2 jest spełnony) powstają naprężena obwodowe mają one jednakową wartość na całej grubośc (w danym przekroju) rury [7]: r p h () D Z D r W (2) 4

gdze: h grubość ścank rury [mm], r średn promeń rury [mm], D z średnca zewnętrzna rury [mm], D w średnca wewnętrzna rury [mm], R średn promeń gęca rury [mm]. Dla zman cśnena określonych na podstawe weloletnch badań : p śr =,75795 [MPa], p mn =,33457 [MPa], p max = 2,44297 [MPa] oraz dla zman grubośc rury ( 508x) w wybranych punktach pomarowych wyznaczono przebeg zman naprężeń obwodowych. W przypadku przejśca rury z odcnka prostego w odcnek zakrzywony zmane ulegają naprężena obwodowe (wywołane dzałanem cśnena wewnętrznego) wzdłuż obwodu rury. Średne naprężene obwodowe w ścance rury zakrzywonej po strone wewnętrznej: oraz po strone zewnętrznej : wew R R 0,5 r r (3) Dla rury naprężeń: R 0,5 r zew (4) R r 508x oraz średnego promena gęca R = 750 [mm] otrzymano wartośc wew,247 (5) zew 0,876 (6) Naprężena obwodowe w prostym odcnku rury ( 508x) wynoszą dla podanych zman cśnena: 248,5 p (7) Uwzględnając notowane wartośc cśneń otrzymano: mn = 30,493 [MPa], śr = 39,739 [MPa], max =55,89 [MPa] Borąc pod uwagę wzór na średne naprężene obwodowe w ścance rury zakrzywonej po wewnętrznej strone otrzymano: wew, mn = 37,5962 [MPa], wew, śr = 49,5225 [MPa], wew, max = 68,8208 [MPa] Impulsowe (dynamczne) dzałane cśnena może powodować dwukrotny wzrost przemeszczena rurocągu [2] w stosunku do statycznego dzałana cśnena. Wynkem tego są przemeszczena rurocągu na podporach destablzacja jego zamocowana (podparca). Uzasadnonym rozwązanem nżynerskm jest w tym przypadku zmana konstrukcyjna w sposobe zaweszena podparca kompensatora.

Równeż skutkem dzałana mpulsowego obcążena [2] w kerunku podłużnym bądź poprzecznym jest dwukrotny wzrost naprężeń obwodowych w stosunku do naprężeń obwodowych wywołanych statycznym dzałanem obcążena. Właścwośc materałów wykazują rozrzut danych dośwadczalnych uzyskanych w czase badań własnośc mechancznych. Rozrzut ten występuje zarówno przy badanach wytrzymałośc zmęczenowej jak statycznej wytrzymałośc, np. grancy plastycznośc R e lub wytrzymałośc doraźnej R m, czyl możemy wytrzymałość materału traktować jako zmenną losową. Współczynnk zmennośc υ R charakteryzujące rozrzut własnośc mechancznych uzyskuje sę z badań statystycznych, wg Warszyńskego [2], dla stal υ R = 0,05 0, jest on wskaźnkem jakośc materału zależącym od warunków wytwarzana obróbk. Można oszacować wartość średną R e, śr wartość mnmalną R e, mn grancy plastycznośc korzystając ze wzoru: R e, mn Re, sr u vr (8) gdze u przyjmowane jest u= 2 3, tj. zapewnające wększe nż 0,98 prawdopodobeństwo, że wytrzymałość będze wększa od R e, mn. Korzystając z powyższych zaleceń wzorów w dalszej częśc wyznaczono parametry rozkładu grancy plastycznośc R e. 4. Ocena prawdopodobeństwa uszkodzena rurocągu Analzowany cepłocąg eksploatowany jest w trudnych zmennych warunkach. Zarówno cśnene jak temperatura czynnka transportowanego zmenają sę w czase. Ponadto transportowany czynnk charakteryzuje sę składem chemcznym, który w połączenu z wysoką temperaturą powoduje szybk proces korozj ścanek cepłocągu. Zmnejszające sę grubośc ścanek oraz dynamczny charakter obcążena powodują cągły wzrost ryzyka przekroczena stanów grancznych pojawana sę uszkodzena. Prawdopodobeństwo uszkodzena cepłocągu może zostać wyznaczone przy wykorzystanu wynków pomarów grubośc ścanek cepłocągu oraz wynków notowanych zman obcążena cepłocągu (cśnene temperatura). Przyblżone oszacowane prawdopodobeństwa uszkodzena konstrukcj można wyznaczyć za pomocą metody Streleckego. Prawdopodobeństwo uszkodzena P wyznaczane jest w oparcu o rozkłady obcążena oraz wytrzymałośc konstrukcj (rys. ) dane jest zależnoścą []: gdze: 2 P 2 2 (9) 2 pola powerzchn pod krzywym gęstośc rozkładu wytrzymałośc wynkającego z obcążeń rozkładu naprężeń, przedstawone na rysunku. f rozkład naprężeń rozkład wytrzymałośc 2 [MPa]

Rys.. Rozkłady wytrzymałośc naprężeń [] Na podstawe danych wytrzymałoścowych materału, z którego wykonano analzowany rurocąg (dane w punkce 2 3), opsano rozkład jego grancy plastycznośc jako rozkład wytrzymałośc rozkładem normalnym N(264; 39,5). Rozkłady naprężeń wyznaczono na podstawe zmerzonych grubośc ścanek notowanych zman cśnena. W tabel zameszczono prawdopodobeństwa uszkodzena cepłocągu wg metody Streleckego, wyznaczone dla średnej z pomarów grubośc ścanek oraz dla najgorszego zmerzonego punktu w różnych latach eksploatacj. Ze względu na newelke wartośc prawdopodobeństwa uszkodzena wyrażone zostały w procentach. Tab.. Prawdopodobeństwa uszkodzena cepłocągu wg metody Streleckego [] Średna z pomarów Najgorszy punkt Rok pomaru 999 200 200 999 200 200 Rok eksploatacj 6 8 7 6 8 7 Prawdopodobeństwo uszkodzena P [%] > 0,002476 0,00327 0,00703 0,0095 0,0895 0,04033 2 0,00656 0,00234 0,005385 0,009 0,0743 0,04704 0,0004 0,000723 0,00379 0,00023 0,033 0,9 Wyznaczone powyżej prawdopodobeństwo uszkodzena zarówno dla średnej grubośc ścanek z punktów merzonych jak dla najgorszego zmerzonego punktu daje jedyne ocenę punktową. Na badanym rurocągu mogą znajdować sę jednak punkty znaczne bardzej nebezpeczne, w których grubośc ścanek mogą być mnejsze. W zwązku z tym dodatkowo zaproponowano nną metodę oceny ryzyka. Według tej metody należy wyznaczyć rozkłady grubośc ścanek zmerzonych w danym termne odneść do wyznaczonej mnmalnej grubośc ścank cepłocągu. Na podstawe warunku wytrzymałoścowego: r p h R e,mn (0) można węc zapsać: h mn 2 D R Z e,mn p p () Uwzględnając maksymalne merzone cśnene w rurocągu oraz jego dynamczne dzałane wyznaczono h mn =6,64 [mm]. Następne wyznaczono rozkłady merzonych grubośc ścanek rurocągu w różnych termnach określono prawdopodobeństwo, że w rozkładze tym wystąp wartość mnejsza bądź równa wartośc h mn (rys. 2). Prawdopodobeństwo to

określa możlwość wystąpena na rurocągu ścank o grubośc równej bądź mnejszej od wymaganej wartośc mnmalnej. Przyjęto że jest to równeż ryzyko wystąpena uszkodzena. Rys. 2. Rozkłady grubośc ścanek cepłocągu zmerzone w różnych termnach Wyznaczone wartośc prawdopodobeństw wystąpena uszkodzena według proponowanej metody zameszczono w tabel 2. Tab. 2. Prawdopodobeństwa uszkodzena cepłocągu [] Rok pomaru 999 200 200 Rok eksploatacj 6 8 7 Rozkład grubośc ścanek Prawdopodobeństwo uszkodzena P N(2,8; 0,9) N(8,85;,46) N(7,5;,3) P 0 P=0,065 P=0,223 Otrzymane według proponowanej metody wartośc prawdopodobeństwa wystąpena uszkodzeń są znaczne wększe od wyznaczonych wg metody Streleckego. Wynka to z faktu, że zgodne z określonym rozkładam grubośc ścanek możlwe jest równeż występowane ścanek o mnejszej grubośc nż otrzymane z pomarów w wybranych punktach. Podejśce take lepej odzwercedla badany rzeczywsty przypadek cepłocągu, na którym w czase pomarów przebadano tylko znkomy procent jego powerzchn rzeczywstej. 5. Prognozowane uszkodzena cepłocągu Prawdopodobeństwo uszkodzena cepłocągu w rozdzale 4-tym wyznaczono dla punktu na rurocągu, w którym na podstawe pomarów nastąpło najwększe ścenene ścank. To mejsce, oznaczone numerem 2 (rys. 3), potraktowano jako najbardzej nebezpeczne. Pomary grubośc ścank rurocągu wykonano w szesnastu punktach