DETERMINING THE WEIBULL AND EXPONENTIALLY DISTRIBUTED SERVICEABILITY OF MACHINERY

Transcription

1 284 DETERMINING THE WEIBULL AND EXPONENTIALLY DISTRIBUTED SERVICEABILITY OF MACHINERY V. Vljasoo Asrac The deermnaon of machnery servcealy enales o ncrease s relaly and economc effcency, o arrange prevenve echnoservce ndependen from economc crera and o fx he level consderng he expenses on echnoservce and manenance. The deermned servcealy levels enale o defne qualave level of operang machnery, he rao of defecs and falures. The gven servcealy levels, calculaed y acheved echnoservce or resored servcealy manenance wll enale o prognoscae s relaly ndces. Thus a greaer sress wll e pu on he qualy of echnoservce. The gven mehods can e appled for solvng praccal prolems n relaly of work. Inroducon The use of machnery shows s proaly of servcealy s ofen raher lale o he Weull and exponenal dsruon. As usual n real workng condons a come-up falure s an even n whch case s no possle or purposeful o use machnery efore s servcealy has no een resored. In case of prevenve manenance here s o do wh a parally faled machnery. The machnery wh decreased servcealy works wh lower effcency. The knd of down sae can e permed o a ceran level. The servcealy ulmae lm of machnery s usually ased on economc ndcaors. In case of prevenve manenance an opmum manenance nerval s ofen appled only y mnmum expenses for echnoservce and manenance. In ha case he level of machnery servcealy s mosly economc ndcaors. Accordng o he gven soluon n he arcle he Weull and exponenally dsrued servcealy of machnery can e deermned ndependen from economc ndcaors. Mehods By he characersc feaures of exponenal dsruon law a full falure y s formng proaly s an even, whch consss of wo each-oher excludng pars. One par s formed y falure, he oher one relale work. In he oppose case, relale work can e regarded an even, when he pars excludng each oher are n he rao relale work and falure. Relale work nensy and s proaly P( ) characerzng relale work of machnery can e found y he formula λ P( ) = e, () where P( ) relale work proaly of -h operang me; λ parameer h of falure level; -h operang me h. If he power λ of exponenal dsruon n he formula () s equal o he falure N, n he even N = 0.368, =, he proaly of relale work P ( ) = and smooh operang me can e calculaed y he formula = ln0.692, (2) where mean operang me per falure h. Analogcally n he even whch consss of falure and relale work, smooh operang me 2 and proaly of relale work P( 2 ) can e deermned. By he formula ()

2 Deermnng he Weull and exponenally dsrued servcealy of machnery 285 calculaons he proaly P( 2 ) = relale work and he correspondng smooh operang me 2 can e calculaed y he formula 2 = ln (3) By he formulae (), (2), (3) value of relale work proaly of good and defecve servceale machnery and smooh operang me can e calculaed. Work relaly ndcaors of praccal work capacy machnery can e calculaed smlarly u no y he numer of falures, u y he proaly of relale work. Wh he proaly of an even P( ) = he oppose even has he proaly F( ) = Analogcally, y he addon heorem [] he proaly of he even P( o ) = s he oppose even proaly F( o ) = The chosen proaly P( o ) and he duraon o of smooh operang me calculaed y he formula () characerze machnery relaly workng wh paral work capaces. The laer can e calculaed y he formula o = ln (4) The crcal lm of exponenally dsrued machnery servcealy has een deermned y falure. I s presumed n case of crcal workng capacy he proaly of relale work s gger han ha wh defcen workng capacy and smaller han n he case of paral workng capacy. Thus he smooh operang me characerzng crcal servcealy mus e durng he nervals o < k < 2. Falure can e calculaed f he dfference (0.09) eween he values of he operang mes o and s suraced, n case of 2, from he resuled falure. If, n case of crcal operang me k deermned falure N k = 0.54, and y he formula () deermned proaly of relale work P( k ) = 0.582, operang me of relale work k can e calculaed y he formula k = ln (5) Thus he condon o < k < 2 s compleed. In sysemazng he resuls, he ndcaors characerzng exponenally dsrued servcealy of operang and falng machnery are he followng (Tale ). Tale. Exponenally dsrued servcealy levels of machnery Operang me, h Proaly of relale work P( ) Falure N Formula Servcealy esmaon [2] = ln.000 complee e e = ln exemplary v v = ln very good = ln good o o = ln sasfacory k k = ln poor = ln defcen = ln mssng Exponenally dsrued servcealy level of machnery s complee, f he proaly of s relale work s.0; exemplary, f P( e ) = 0.900; very good, f P( v ) = 0.800; good, f P( ) = (reak-downs can e here, whch, o ceran exen, decrease workng capacy of machnery); sasfacory, f P( o ) = (reak-downs are here, whch parally decrease, workng capacy of machnery); poor, f P( k ) = (reak-downs are here, whch essenally decrease workng capacy of machnery and make s use prolemacal) and defecve, f P( 2 ) = (reak-downs are here whch essenally decrease workng

3 286 V. Vljasoo capacy of machnery and make useless). The machnery has los s workng capacy for one reason or oher, f he proaly of s relale work P( ) = In mos cases y chosen dsruon law he deermnaon of operaon relaly can e reduced o only one condonal equpmen, a machne, a componen (par), or an elemen. In connecon wh ha a defec and a falure usually have a quanave and qualave quany rao, whch can e characerzed as servcealy (workng capacy) as well. Thus a parally faled machnery has defecs, whereas defecs whch reduce workng capacy of workng machnery, form a falure. A par of relaly heory, enalng o nvesgae relaly (change of workng capacy) y he numer of falures and reasons, deals no wh colleced defecs u wh a falure, whch can also e a quanave par of a falure. In relaly heory he proaly dsruon law of machnery servcealy s chosen y varaon facor ν [3, 4, 5]. For deermnng he Weull and exponenally dsrued servcealy level of machnery we choose he varaon facors 0.927; 0.8; 0.705; 0.605; 0.505; [3, Supplemen, Tale 4]. Proaly of relale work sujeced o he Weull dsruon law of machnery, can e found y he formula ( ) P a = e, (6) where P( ) proaly of relale work of -h operang me; -h operang me h; a and Weull s parameers. The Weull parameer a s calculaed y he formula σ a =, (7) C where σ mean square devaon h, σ = v ; C he Weull facor [3, Supplemen, Tale 4]; v varaon facor; mean operang me per falure h. Knowng he values and ν, s possle o deermne mean square devaon σ and parameer a. The power of exponen n he formula (6), n case =, equals o falure N, he proaly of whch F () s calculaed y he formula []. F( ) = P( ). (8) Accordng o he Weull dsruon law a full falure, y he proaly of s comeup, can e consdered as an even conssng of wo pars excludng each oher. One par s formed y he falure N, correspondng o a varaon facor and he second par s s oppose even, relale work M. Falure s calculaed y he formula N where N falure per -h operang me; = F ( ), (9) a F () proaly of falure per operang me. In he oppose case relale work can e an even, f s pars, excludng each oher, compared wh he prevous, are oppose y he asolue value,.e. f he asolue value of he proaly of relale work s equalzed o he proaly of falure P( ) = F( ) and he asolue value of he laer, n s urn, s equal o he proaly of relale work F( ) = P( ). Hence he servcealy y he Weull dsruon law s good, f falure

4 Deermnng he Weull and exponenally dsrued servcealy of machnery 287 N = N (Formula 9) s calculaed n he condon F( ) = P( ) and he proaly of relale work P( ) n he condon = y he formula P( ) = e N. (0) Operang me of relale work = s calculaed y he formula [ P ] = ln ( ) a. () Relaly ndcaors of he machnery workng wh paral servcealy can e deermned smlarly, u no y falure, u y he proaly of relale work. Choosng he asolue value of falure calculaed y he formulae (6) and (8), as he proaly of relale work.e. P( o ) = F ( ), we can deermne y he formula () he relale work operang me = o. The values P( o ) and o characerze he servcealy of machnery workng wh paral workng capacy and her deermnaon mehods are vald f he varaon facor s eween.0 ν In case he varaon facor s eween > ν 0.399, he relaly ndcaors for he machnery wh paral workng capacy are calculaed smlarly, y falure. The falure N o s calculaed y he formula (9), whereas F () y he formula (8). The proaly of relale work P( o ) s calculaed y he formula (0), and he correspondng relale operang me = o y he formula (). The crcal lm for he Weull dsrued servcealy of machnery, n case he varaon facor s eween.0 ν 0.765, s deermned y falure. The falure N k s calculaed y he formula (9), whereas F () y he formula (8). The proaly of relale work P( k ) s calculaed y he formula (0) and he correspondng operang me of smooh relale work = k y he formula (). The crcal lm for he Weull dsrued servcealy of machnery, f he varaon facor s eween > ν 0.399, s calculaed y he proaly of falure. Havng chosen he proaly of relale work P( k ) calculaed y he formulae (6) and (8) he proaly of falure F( ) asolue value P( k ) = F( ), y he formula () he operang me of relale work = k s deermned. The values P( k ) and k characerze machnery relaly workng wh crcal servcealy. The relaly ndcaors of machnery workng wh defecve servcealy P( 2 ) and 2 are calculaed y falure n he whole lms of varaon facors (.0 ν 0.399). If he varaon facor s eween.0 ν he falure N 2 s deermned, n case of he operang me o and he dfference of falures N = N o N s added o he resuled falure N k, n crcal servcealy. If he varaon facor s eween > ν 0.399, he falure N 2 s calculaed, n case of he operang me k and he dfference of falures N = N k N s added o he resuled falure N o, n paral servcealy. The proaly of relale work P( 2 ) s calculaed, n oh cases, y he formula (0) and he correspondng relale work operang me = 2 y he formula (). The mean servcealy of operang and faled machnery deermned y he Weull dsruon are gven n he Tale 2. The servcealy of machnery s complee, f he proaly of relale work s.0; exemplary, f P ( e e ) = 0.900; very good, f P ( v v ) = 0.800; good, f he mean value of relale work proaly P ( ) = (here can e defecs, whch, o small exen, decrease servcealy); sasfacory, f P ( o ) = (here are defecs, whch parally decrease servcealy of machnery); poor, f P ( k ) = (here are defecs, whch essenally decrease servcealy of machnery and s use s prolemac) and defecve, f P ( 2 ) = (here are defecs, whch essenally decrease servcealy and make

5 288 V. Vljasoo useless). For some reason or oher he machnery has los s servcealy, faled, f he mean value of relale work proaly P ( ) = and he mean value of falure N = Tale 2. The Weull dsrued servcealy mean levels of machnery (0.765 > ν 0.399) Operang me, h Proaly of relale work P ( ) Falure N Formula = [ ln P( 0 ) ] e e e = [ ln P( e ) ] v v v = [ ln P( v ) ] = [ ln P( ) ] o o = [ ln P( o ) ] k k = [ ln P( k ) ] 2 = [ ln P( 2 ) ] = [ ln P( ) ] Servcealy esmaon [2] a complee a exemplary a very good a good a a poor a a sasfacory defecve mssng Summary The deermnaon of machnery servcealy enales o ncrease s relaly and economc effcency, o arrange prevenve echnoservce ndependen from economc crera and o fx he level consderng he expenses on echnoservce and manenance. The deermned servcealy levels enale o defne qualave level of operang machnery, he rao of defecs and falures. Exponenally dsrued servcealy levels are: P( 0 ) =.0 complee; P( e ) = exemplary; P( v ) = very good; P( ) = good, P( o ) = sasfacory, P( ) = poor and P( 2 ) = defecve, P( ) = mssng. The Weull dsrued servcealy levels of machnery are: P( 0 ) =.0 complee, 004 P( e e ) = exemplary, P( v v ) = very good, P ( ) = good, P ( o ) = sasfacory; P ( k ) = poor, P ( 2 ) = defecve and P ( ) = mssng. The gven servcealy levels, calculaed y acheved echnoservce or resored servcealy manenance wll enale o prognoscae s relaly ndces. Thus a greaer sress wll e pu on he qualy of echnoservce. The gven mehods can e appled for solvng praccal prolems n relaly of work.

6 Deermnng he Weull and exponenally dsrued servcealy of machnery 289 KOKKUVÕTE: Tehnka Weull- ja eksponenjaousele alluva öövõme määramse meoodka. Tehnka öövõme asemee määramne võmalda õsa selle öökndlus ja majanduslkku efekvsus, korraldada õrked enneava ehnohooldus majanduslkes kreerumdes sõlumaul võ hnnaa selle eesmärgkohas ase lähuval ehnohooldusele ja hooldusremondle ehud rahalses kuluuses. Määraud öövõme asemed võmaldavad prleda ööava ehnka öövõme kvalavse ase, rkee ja õrgee vahekorda. Esaud öövõmeasemed võmaldavad ehnka ehnohooldusega saavuaud võ hooldusremondga aasaud öövõme aseme järg prognoosda selle öökndlusnäajad. Sellega aseu ka suurem rõhk ehnohooldusööde egemse kvaleedle. Esaud meoodka on rakendaav öökndlusalase praklse ülesannee lahendamsel. References. Gursk J. Tõenäosuseoora ja maemaalse saska elemendd. Valgus, Tallnn, Kaufman: Кофман А. Введение в теорию нечетких множеств. Пер. с франц. Радио и связь, Москва, Selvanov, Aremjev: Селиванов А. И., Артемъев Ю. Н. Теортические основы ремонта и надежности сельскохозяйственной техники. Колос, Москва, Aremjev: Артемъев Ю. Н. Качество ремонта и надёжность машин в сельском хозяйстве. Колос, Москва, Presman: Прейсман В. И. Основы надёжности сельскохозяйственной техники. 2-е изд. доп. и перераб., Выша шк., Киев, 988.