SOLUTIONS OF RECURRENCE AND SUMMATION EQUATIONS AND THEIR APPLICATIONS IN SLIDE BEARING WEAR CALCULATIONS

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1 Jourl o KOES Powertri d Trsport Vol 9 o 0 SOLUTIOS OF EUEE A SUMMATIO EQUATIOS A THEI APPLIATIOS I SLIE BEAIG EA ALULATIOS Krzyszto ierzcholsi Techicl Uiversity o Koszli Istitute o Mechtroics otechology d Vcuum Techique idecich Street Koszli Pold tel: x: e-mil: rzysztowierzcholsi@wppl Astrct The ivestigtios uder the pper iclude derivtio o o-homogeous recurrece equtio o the secod d higher orders with vrile coeiciets whose prticulr solutios with the oudry coditios set d otied rom experimetl mesuremets will e the sequeces o the wer vlue o the erig i the successive yers o opertio Owig to these ivestigtios it will e possile to predict or exmple the wer vlues o slide erigs Moreover this pper presets the some prticulr pplictios o recurrece equtios with regrd to the clcultio progosis o micro-erig prmeters such s rictio orces rictio coeiciets d wer ecurrece equtios re preseted i orm o dierece equtios where the uow uctios occur s the mi terms o the sequece o iquired vlues I this pper the properties o the prticulr vlues o recurrece equtios re deied Possiility o modultio d cotrol o metioed prolem elog to the rtiicil itelligece o H microerig Preseted prolem descries ot cotiuous reltios hece determies the mthemticl d umericl solutios i discrete spces Properly i the cse o cotiuous uctios the metioed recurrece equtios hve the sme meig s dieretil equtios ecurret equtios or discrete uctio correspod to dieretil equtios or the cotiuous uctio I il coclusios the pplictio o preseted theory i this pper cotis the umericl solutios reerrig the wer vlues o H erig system i the idicted period o opertig time Keywords: wer progosis H micro-erigs recurrece o-homogeeous equtio Iitil iormtio out recurrece tools i triology The wer progosis o two coopertig surces durig the opertio time o vrious techicl devices hs very importt meig i cotemporry techologicl processes [ 6 7] Estimtio ticiptio d clcultio o the wer vlue o coopertig surces ter my moth or exploittio yers o the mchie or device deotes the very importt prmeter o mteril qulity I this pper is cosidered the determitio o wer vlue { } or successive yers = e ssume tht wer + o coopertig surces i + time uity ie moth or yer or =0 is the uctio G o wer vlues i oregoig yers To solve this prolem we must ow the wer vlues i = successive time uits ie Our uow prticulr solutio is the sequece o wer vlues { } ie or exmple the vlues o decrese i pm o H micro-erig jourl dimeter i successive = yers [4] Metioed prolem c e writte i the ollowig order o lier o-homogeeous recurrece equtio with vrile ree term: Geerl orm o recurrece equtio c e expressed i ollowig orm [ 5]: G () where G is the complex uctio o operds Symol is the uow wer uctio turl umer determies the order o the recurrece equtio or =

2 K ierzcholsi I G is the o lier uctio t lest o the oe o operds: + the recurrece equtio () is o lier I G is lier respect to the ll operds: + the recurrece equtio () is lier For exmple lier -order recurrece equtio with coeiciets hs the ollowig homogeeous d o-homogeeous (with ree term ) orm [ 5]: 0 0 () I ll coeiciets re costts ie re idepedet o the ormul () descries the recurrece equtio with costt coeiciets I whichever oe o coeiciets depeds o the ormule () () deote recurrece equtios with vrile coeiciets I ree term equls zero the ormul () descries the homogeeous recurrece equtios I ree term is dieret o zero the ormul () descries the o-homogeeous recurrece equtios Experimetl coeiciets: 0 deped o the surces mteril dymic coditios o the movle coopertig surces requecies o existig surce virtios evirometl coditios This sectio descries the methods o the determitio o the geerl d prticulr solutios i the iiite sequece orm or lier -order recurret equtios with costt coeiciets ecurrece equtios () () will e solved i we oud the ollowig depedece [ 5]: s s s s (4) where experimetl vlues: s s s s deote oudry wer vlues o serched solutio o recurrece equtios () i plces or = It is esy to see tht ormul (4) ie solutio o recurrece equtio () or () eles to clculte the vlues o the wer uctio or ritrry without o the ecessity o determiig o remiig succeedig wer vlues j or j= Lier homogeeous -order recurrece equtios () or =0 hs -lier idepedet prticulr solutios o wer cretig the ollowig udmetl system [7]: (5) Lier homogeeous recurrece equtios hs s my prticulr solutios s order o the equtios Upper idex i rcets preseted i uctios (5) idictes the umer o succeedig prticulr solutios sorti determit creted or prticulr solutios (5) c e preseted i ollowig orm []: det 0 or j ; (6) j Geerl wer solutio o the lier homogeeous recurrece equtios () or =0 is the lier comitio o -lier idepedet prticulr solutios o recurret equtio Hece the geerl solutio o wer o metioed recurret equtio hs the ollowig orm: 0 where = Geerl solutio (7) cotis ritrry costts: Geerl solutio o the homogeeous recurrece equtios hs s my costts s the order o equtios To determie the metioed costts we eed oudry coditios For exmple such oudry () () (7) 544

3 Solutios o ecurrece d Summtio Equtios d Their Applictios i Slide Berig er lcultios coditios re s ollows: or = we hve coditio =s (8) or = we hve coditio =s Symols s i prcticl pplictios deote cocrete wer vlue o looed prticulr solutio o homogeeous recurrece equtios i time uit = Imposig coditio (8) o geerl solutio (7) we oti the ollowig system o - lier o-homogeeous lgeric equtios [7]: s s The system o equtios (9) determies the ollowig uow costts: It is esy to see tht the sorti determit is the determit o the lgeric system (6) Becuse such determit is dieret o zero or lier idepedet prticulr solutios hece the system o lgeric equtios (9) determies o-zero costt vlues s solutios Solutio o recurrece equtios is sed o the prticulr d geerl wer solutio (7) determitio s lier comitio o prticulr wer solutios Ad ext i we ow oudry coditios ie i we hve experimetl otied wer vlues s the we loo to oti cocrete prticulr solutios Such solutio we oti puttig costts determied rom the lgeric system (9) ito geerl solutio (7) I my triologicl wer prolems [5] re occurrig very ote the ollowig lier prticulr cses o wer recurrece equtio (): 4 where = ; 0<< e hve experimetl dimesioless vlues [] [] d dimesiol experimetl wer vlues [pm] d dimesiol oudry vlues [pm] [pm] [pm] 4 [pm] or the irst Eq(0) ; [pm] [pm] [pm] or the secod Eq(0) ; d [pm] [pm] or third Eq(0) ll with steris I the ext sectios o this pper we re goig to the presettio o the methods o solutios o lier homogeeous d o-homogeeous recurrece equtios with costt coeiciets Moreover will e imposed oudry coditios o the vrious geerl wer solutios hece will e determied the vrious prticulr wer vlues Properly i the cse o the spce o cotiuous uctios the ovemetioed equtios hve the sme meig s dieretil equtios The mthemticl tools o solutios o recurrece equtios re ormulted d demostrted usig Lemms d Theorem To the est o the uthor s owledge ew chievemets tht relte to the uiied discrete orms o solutios re coveiet orms or umericl clcultios d the seprtio o the lier idepedet solutios er uctio descried y lier -order homogeeous recurret equtio with costt coeiciets ow exmples re preseted o geerl d prticulr solutios or vrious types o lier homogeeous recurret equtios d or rel d complex roots o chrcteristic equtios (9) (0) 545

4 K ierzcholsi Theorem I the lier homogeeous -order recurrece equtios with costt coeiciets 0 hs the ollowig orm [ 7]: 0 0 () where deotes the uow wer uctio the the geerl solutio o this equtio is s ollows: r s s s m 0 sm m () where sm ritrry costts = The symol s where s= r; r occurrig i the ormuls (c) descries the succeedig vrious roots o the ollowig chrcteristic equtio: p( ) 0 with multiplictio ctor s ordered to the roots s wheres the sum o multiplictio ctors o the roots is equl to the order o recurrece equtio mely: 0 () r + r = () ommettor or Theorem Successive roots o the chrcteristic equtios () with multiplictio ctor idicted y the rrows re s ollows []: r (4) r er process exmple etermie the wer ormul o the micro-erig jourl i wer process is descried y the recurrece equtio: 0 or (5) I irst succeedig three yers o exploittio the vlues o decrese i m o H microerig jourl dimeter equl to ollowig wer vlues s oudry coditios: =M m = m =P m P>M (6) Solutio o exmple For Eq(5) chrcteristic equtios () hve the ollowig orm d solutios: 0 e hve s doule root = d s sigle root = Hece we hve oly two vrious roots thus r= Geerl solutio () o the recurrece equtios (5) is s ollows: ( ) m0 m m ( ) m0 m m 0 0 ( ) (7) (8) 546

5 Solutios o ecurrece d Summtio Equtios d Their Applictios i Slide Berig er lcultios Formul (8) presets the geerl solutio o recurrece equtio (5) wheres 0 0 re the ritrry costts d = ow we go to the determitio o the cocrete prticulr wer solutio o the recurrece equtio (5) stisyig oudry coditios (6) To determie such prticulr solutio we impost oudry coditios (6) o the geerl solutio (8) Ad we get the ollowig vlues: 0 =5M P; =050P050M; 0 = 05M+05005P Isertig such vlues ito geerl solutio (8) we oti the cocrete prticulr solutio o recurrece equtio (5) ie ollowig wer vlues uctio: 0 50P 0 50M 0 5M P ( ) 5M P (9) i succeedig yers = For exmple i teth yers o exploittio the wer ttis vlue: P M P M (0) ow we ssume the prticulr cse where chrcteristic equtios o -degree () or -order recurrece equtios () hve =r/ pirs mutully cojugted complex roots with multiplictio ctor deoted y the symols i successio The metioed pirs we rrge i the ollowig orm [7]: multiplictio ctor mod Arg 4 orollry multiplictio ctor mod Arg I chrcteristic equtio () hs ot rel roots d ssumig oly the complex roots () the the geerl solutio o recurrece equtios teds to the ollowig orm []: s () s () () m s sm cos s sm si s () m 0 () () sm or s= ; m=0 s ; = where sm 5 emrs ritrry costts Becuse geerl solutio () vlid oly or pir o complex mutully cojugted roots o chrcteristic equtios where ech pir hs multiplictio ctor with ollowig successio d chrcteristic equtio () hs ot rel roots the: () I this cse symol deotes eve order o recurrece equtio It is worth otice tht i ll roots o chrcteristic equtios re rel umers ie i or ech s the mplitudes o complex umer s re equl zero the geerl solutio () teds to the orm o geerl solutio () wheres =r er uctio descried y lier -order o-homogeeous recurret equtio oclusio The lier o-homogeeous -order recurrece equtio with costt coeiciets d vrile expoetil ree term [ 5]: 547

6 K ierzcholsi j0 j j 0 (4) hs the ollowig geerl solutio: or q q! Q q where uctio deotes geerl solutio o the homogeeous -order recurrece equtio creted rom the o-homogeeous Eq (4) or =0 d moreover presets certi prticulr solutio o the o-homogeeous equtio wheres umer q is the multiplicity o the root = o chrcteristic equtio p()=0 see Eq() The ssocited polyomil is deied i ollowig orm [ 7]: p( ) Q( ) (6) q ( ) (5) 4 Applictio i triology 4 Prolem The sequece o wer vlues {} ie the vlues o decrese i pm o H microerig jourl dimeter equl to sum o wer i two oregoig successive moths multiplied y dimesioless verge coeiciet 0< plus some expoet dimesiol uctio oeiciets deped o the microerig mteril the jourl gulr velocity d the requecies o virtios I the two irst moths the wer ttis dimesiol vlues i pm etermie the uow lyticl ormul {} or sequece o wer vlues umered or = moth i we ow dimesioless vlues [] [] d dimesiol vlues [pm] [pm] [pm] 4 Geerl solutio o the prolem The prolem is deied y the ollowig dierece equtio: ( ) or From ormule () (5) it ollows tht the geerl solutio o recurrece equtio (7) or two ritrry costts hs the ollowig orm: (7) (8) hrcteristic equtio () or = d recurrece equtio (7) hs the ollowig rel roots: 5/ or 0 (9) 4 By imposig the oudry coditios system: o the geerl solutio (8) we oti (0) 548

7 Solutios o ecurrece d Summtio Equtios d Their Applictios i Slide Berig er lcultios System o Eqs(0) determies ollowig uow costts : () 4 Prticulr cse o solutio e ssume ollowig ssumptios: ie 0 4 () By virtue o (5) () the o-homogeeous solutio o Eq(7) hs the orm: or () The sum o solutio (8) ie sum o wer vlues i the time uits hs the ollowig orm: (4) where ; = For dditiolly ssumptios: 0 (5) the or iiitely my time uits wer (4) teds to the ollowig limit vlue: 0 (6) 44 Prticulr clcultio exmple I two irst successive time uits H micro-erig jourl ttis dimeter decreses d m etermie the wer ter iiite time uits usig mesuremets d stochstic dt preseted y []=/6 []=/ or ritrry [m] From ormule (9) we oti ollowig exploittio prmeters: (7) Puttig dt (7) i ormul (6) the wer tti the ollowig orm: (8) 549

8 K ierzcholsi 5 oclusios A) The preseted exmple descries o-cotiuous reltios hece it determies the mthemticl d umericl solutios i discrete spces ecurret equtios or the discrete uctio correspod to dieretil equtios or the cotiuous uctio B) The pplictio o the preseted theory i metioed exmples cotis the umericl solutios reerrig the wer vlues o H erig system i the cosidered period o the opertig time ) Uiied lgorithm preseted costitutes useul tool or the solutio o recurrece equtios pplied or the wer progosis o H micro-erigs [5 7] eereces [] Kci E Prtil ieretil Equtios i Physicl d Techicl Prolems T rsw 989 [] Kosm Z umericl Methods I Egieerig Applictios Politechi doms 999 [] Koiews I ecurrece Equtios P rszw 97 [4] owici A ecurrece Sequeces ydwictwo yszej Olsztysiej Szoy Iormtyi i Zrzdzi Olszty-Toru 00 [5] ierzcholsi K Summtio equtio tools or slide microerig systems wer progosis Jourl o KOES Powertri d Trsport Vol 8 o pp rsw 0 [6] ierzcholsi K Meg lgorithm or prtil recurrece eyolds equtio XIV Itertiol oerece System Modellig d otrol pp -8 [7] ierzcholsi K Summtio equtios d their pplictios i mechics XIV Itertiol oerece System Modellig d otrol pp - 550