SOLUTIONS OF RECURRENCES WITH VARIABLE COEFFICIENTS FOR SLIDE BEARING WEAR DETERMINATION

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1 Jourl o KONE Powertri Trsport Vol. 0 No. 03 OLUTION OF REURRENE ITH VARIABLE OEFFIIENT FOR LIDE BEARING EAR DETERMINATION Krzyszto ierzcholsi Techicl Uiversity o Koszli Istitute o Techology Euctio iecich treet Koszli Pol tel.: x: e-mil: rzyszto.wierzcholsi@wp.pl Astrct The umerous methos o umericl clcultios occurrig i power-tri triology trsport cocerig wer erig etermitio prolems em the more more iormtio reerrig the slie erig wer ticiptio i succeeig yers o mchie opertios. Thereore this pper presets the methos o solutios o some speciic clss o oriry o-homogeeous recurrece equtios o seco higher orer with vrile coeiciets occurrig i hyroymic theory o erig wer prolems. otrry to lier recurrece equtios with costt coeiciets lier recurrece equtios with vrile coeiciets rrely hve lyticl solutios.. Numericl solutios o such equtios re lwys prcticle. I umerous lyticl methos o solutios o lier recurrece equtios with vrile coeiciets there re usully three reserch irectios. The irst o them epe upo the successive etermitio o the lier iepeet prticulr solutios o the cosiere recurrece equtio. The seco irectio to e chrcterize y the reuctio o the orer o recurrece equtio to oti lwys solve irst orer recurrece equtio. The thir irectio o solutios o recurrece equtios with vrile coeiciets cotis the methos o lyticl solutios y mes o summtio ctor. The mjority o the geerl methos o lyticl solutios o lier recurrece equtios with vrile coeiciets costitute pttio o the methos pplie i solutios o suitle ieretil equtios I il coclusios the pplictio o presete theory i this pper cotis the the exmples reerrig the wer vlues etermitio o HDD erig system i the iicte perio o opertig time. Keywors: wer ticiptio ter exploittio slie erigs recurrece equtio vrile coeiciets. Iitil iormtio out micro-erig wer progosis IN: e-in: DOI: / The presete pper escries the stochstic clcultio metho o wer ticiptio i HDD micro-erig tig ito ccout the ymic ehvior urig the opertig time. The wer progosis o two coopertig HDD micro-erig surces urig the opertio time hs very importt meig i cotemporry sotwre techologicl processes. olutio o presete prolem c e possile o the sis o recetly otie mesuremets o micro-erig wer urig the irst perios my e moth or yers o exploittio regrig the prmeters escriig the rom wer eects complete y the imesiol str evitio. Ater experimetl AFM mesuremets ollows tht iscrete wer vlues + o the sequece {} or = 3 i.e. the wer vlues icreses i mm 3 o HDD micro-erig jourl sleeve equl to sum P ++ Q o wer i two oregoig successive time uits my e moths where es re multiplie y imesioless vrile stochstic wer coeiciet P Q plus some imesiol vrile stochstic vlues uctio o mterils R. Vrile coeiciets P Q R epe o the otie i experimetl wy imesiol wer str evitio o micro-erig mteril the jourl gulr velocity the requecies o virtios. Vrile is umere y turl umers 3 The imesioless epeet i.e. vrile rom prmeters P Q eote y coeiciets which verge the wer i two succeeig oregoig time uits. tr evitio epeet

2 K. ierzcholsi term R verge term P ++ Q o two oregoig wers me up the sequece vlues o rel wer i succeeig time uit. I this cse wer o HDD micro-erig c e escrie y the ollowig recurrece equtio [5 7]: P Q R or Recurret equtio etermies lyticl ormul { } presetig sequece o wer vlues umere or = 3 time uits i we ow imesioless vlues P [] Q [] imesiol vlue R [mm 3 ]. To solve metioe prolem it is ecessry to the oury coitios. Hece we ssume tht i two irst time uits my e moth the wer y virtue o experimets ttis imesiol vlues [mm 3 ] [mm 3 ] <.. er etermitio i power tri triology y mes o the replcemet o vriles The metho presete i this itersectio relys o the ew vrile itrouctio which eles us to reuce the orer o the recurrece equtio. The exmple illustrtes the ove-metioe metho [ 4]. EXAMPLE Determie geerl solutio = 3 i.e. the wer or the ollowig recurrece equtio: 0 where positive costt vlues 0 re iepeet o. how the prticulr wer solutio or ow oury coitios i.e. wer vlues = = i two successive time uits = =. OLUTION OF EXAMPLE Equtio c e escrie i the ollowig orm: 0. 3 e ssume the ollowig ew vrile: y or hece rom the ormul 4 ollows: y. 5 Puttig epeeces 4 5 ito recurret equtio 3 we oti the ollowig lier homogeeous irst orer recurret equtio with vrile coeiciet: y y 0 y y 0. 6 Followig irst orer homogeeous recurrece equtio with vrile coeiciet A: hs the geerl solutio i the orm: y y A y 0 A or j j! y 8 48

3 olutios o Recurreces with Vrile oeiciets or lie Berig er Determitio where eotes the irst ritrry costt. olutio 8 i presete orm y is itrouce ito sustitutio 4 hece we oti the ollowig lier o-homogeeous irst orer recurrece equtio: B or ! Aovemetioe irst orer o-homogeeous recurrece equtio with ollowig vrile ree term B: B! or hs the geerl solutio i the orm: or! Ater terms orerig i Eq. we oti: or !. ymol eotes seco ritrry costt i the presete solutio o recurrece equtio 9. To etermie the prticulr solutio o recurrece equtio or oury coitios = = i plces = = we ought to ote tht ssumptio 4 or = y virtue o 8 we oti:. 3 y e put costt rom Eq. 3 ito Eq. the the prticulr solutio o recurrece Eq. hs the il prticulr wer vlue solutio i ollowig orm: or / 4!. 4 By virtue o 4 ollows tht or wer vlues = = i two irst time uits = = the wer vlues urig the ext time uits tti the ollowig vlues: 4!! 3 5 3!!!... 3!!...! urig the iiity time uits the wer tti the vlue: 5 49

4 K. ierzcholsi 0. e e e exp 5 The wer vlues process urig the prticulr time uits is coverget. The sum o wer vlues ter iite N < iiite or N time uits i.e. is iverget:. F N F F F N N N!! 4 3!! 0 6 The sums o wer FN FN+ FN+ lwys icrese ter successive N N+ N+ time uits. Now we show the proo tht geerl solutio stisies recurrece Eq.. PROOF e show the ollowig steps o the sequece presetig the geerl solutio : or! or!. 8 From ormule 7 8 the ollowig expressios re true:!!!!! 9.!!! 0 ustitutig the r.h.s. right h sie o Eq. 9 0 equlity ito recurrece equtio we oti:.!!!!! Ater term reuctio term orerig i expressio we oti illy: 0!!!. Hece recurrece equtio is stisie. This ct completes the PROOF. 430

5 olutios o Recurreces with Vrile oeiciets or lie Berig er Determitio 3. er process etermitio i power tri triology y mes o the summtio ctor The l.h.s. let h sie o the ollowig recurrece equtio with coeiciets A B ree term D: A B D P Q R 3 is the totl sum or the totl ieretil. I this cse we c choose such coeiciets L M tht the ollowig equlity is true: L M P Q R. 4 A o-homogeeous reuce equtio hs the orm: M R L 5 where is the ritrry summtio costt. I the equlity 4 is ot vli we c i the summtio ctor U or equtio 3. Ater multiplictio o oth sies o the equtio 3 y this ctor we c lwys i such coeiciets L tht y virtue o the totl sum the ollowig equlity is true: M M U UP L U Q U R. 6 A i this cse reuce o-homogeeous equtio hs the ollowig irst orer recurrece orm: M UR L. 7 ymol eotes reciprocl uiie opertor o summtio with the sis. Reciprocl UO regrig the uiie opertor o summtio occurrig i Eq.7 is eote y the ollowig escriptio:... 8 The reciprocl uiie opertor o summtio will e eie i the ollowig orm: x X ecuse X x 9 where X x re the uctios etermie or the turl umers = 3. Reciprocl uiie opertor o summtio is eote y the upper iex is ot lwys uivocl. EXAMPLE Determie geerl solutio = 3 i.e. the wer or the ollowig recurrece equtio: 30 where 0 eotes ritrry coeiciet iepeet o. how the prticulr wer solutio or ow oury coitios i.e. wer vlues = = i two successive time uits = =. OLUTION OF EXAMPLE L.h.s. o recurrece equtio 30 oes ot preset the totl sum. ymol is the summtio ctor ecuse y multiplyig y oth sies o equtio 30 we oti: 3 43

6 K. ierzcholsi the ollowig equlity is true:. 3 Hece equtio 3 c e writte i the ollowig orm:. 33 he reciprocl opertor is impose o the oth sies o equtio 33 we oti the ollowig:. 34 By virtue o reciprocl opertor properties we oti: J J J. 35 ymol eotes the irst summtio costt. Equtio 34 hs the orm:. 36 Diviig oth sies o Eq. 36 y we oti:. 37 The geerl solutio o recurrece equtio 37 hs the ollowig orm: 3... or s j s j ymol eotes the seco ritrry costt o summtio. Ater trsormtios o ormul 38 we oti: 3... or!! O solutio 39 we impose oury coitio = or = = i plce = hece we oti the ollowig summtio costts:. 40 I presete ormul 39 we c ow show lier comitio o two lier iepeet prticulr solutios o the homogeeous equtio plus the prticulr solutio o o-homogeeous recurrece ccorigly with Eq. 30. Aovemetioe lier comitio o prticulr solutios is writte i ollowig orm: 43

7 olutios o Recurreces with Vrile oeiciets or lie Berig er Determitio [] [] [ ] [] []! ! [] []! c! [ ]!! [ ] oclusios A The pplictio o the presete theory cotis the lyticl methos o solutios o seco orer recurret equtios with vrile coeiciets reerrig the wer vlues etermitio o erig system i the cosiere perio o the opertig time. B The recurreces equtios etermiig the wer vlues i power tri triology re solve y mes o the replcemet o vriles summtio ctor. Reereces [] Kci E. Prtil Dieretil Equtios i Physicl Techicl Prolems NT rsw 989. [] Kosm Z. Numericl Methos i Egieerig Applictios Politechi Roms 999. [3] Koiews I. Recurrece Equtios PN rszw 973. [4] Nowici A. Recurrece equeces ywictwo yszej Olsztysiej zoy Iormtyi i Zrzzi Olszty-Toru 00. [5] ierzcholsi K. ummtio Equtio Tools or lie Microerig ystems er Progosis Jourl o KONE Powertri Trsport Vol. 8 No. 3 pp [6] ierzcholsi K. Meg Algorithm or Prtil Recurrece Reyols Equtio XIV Itertiol oerece o ystem Moellig otrol pp [7] ierzcholsi K. ummtio Equtios Their Applictios i Mechics XIV Itertiol oerece o ystem Moellig otrol pp

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