PRACE NAUKOWE POLITECHNIKI WARSZAWSKIEJ z. 116 Elektryka 2001

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1 PRACE AUKOWE POLITECHIKI WARZAWKIEJ z. 6 Elekyka Romual Małek Zespół Maemayk Fzyk Woe Ubańsk Insyu Maszyn Elekyzny PROBLEM OF TEMPERATURE FIELD MODULATIO OF IDUCTIO MOTOR COOLED BY THE FIRT ORDER OLID LIUID PHAE TRAITIO Temal penomena asng n some ele moos ae esbe pysal moel an an aequae maemaal moel s onsue. Te moel n e fom of a ona poblem s appomae n a numeal ay. Te ffeene poblem e unaon eo s pesene an e sably of e ffeene seme s nesgae. Te sysem of lnea algeba equaons obane by full sezaon s sole by Gauss elmnaon meo fo ee agonal maes. Te numeal soluon s pesene.. ITRODUCTIO One of e bas equemens mus be me befoe e suessful ompleon of e esgn poess of an opmal ele mane s a eale an auae analyss of all e emal saes an onons of su a mane. A e pesen leel of eelopmen of ele mane alulaons nfomaon s only aalable fo e emal seay saes s n geneal nsuffen. Ts s of paula mpoane fo e eoy of manes esgne fo so-me opeaon fo eample o open o lose anal loks o lf o loe abges o opeae ales n ppelnes e. Anpang ne ens n e esgn eelopmen of ele manes speal-ype ele moos fo so-me ues ee bul a e Insue of Eleal Manes of Wasa Unesy of Tenology. By usng e maemaal ools of poenal eoy paal ffeenal equaons an Volea negal equaons a meo o eemne e emal paamees of su ele manes as foun.

2 Pysal moels base on e seal pouon of ee-pase nuon moos of lose onsuon oolng sysems mofe an ange fom fan-nue enlaon sysems o a sysem usng a sol-lqu pase anson ae been manufaue. A speally selee lo melng pon sol pase oolng meum l see Fg. l flls a ambe nlues en onneons of e sao nng as ell as empy spaes n e slos. Dung e opeaon of e moo e oolng meum s eae gaually angng s pase fom sol o lqu. Te eang of e meum n s sol an lqu pases an manly! e anson of e meum fom e sol no e lqu pase esul n e absopon an aumulaon of a onseable pa of e geneae ea. Ts ea s sspae lae a s en opeang e mane une eue loa onons o en s se off a esulng eun of e oolng meum o s sol pase. Te lages emal sesses n an nuon moo ae foun n s sao nng an n paula n e en onneons. Tese appen o be ose pas of e moo e oolng meum as e bes aess o. Oe pas of e moo ae gly essan o een ey lage ses n e empeaue. Te bas aanages of e ne esgn oe e aonal one ae: ey g oeloang apay lo leels of nose an baon. Fo a eale espon of e esgn an e onsuon of e ne moo ogee e esuls of e measuemens aken ung s als see [ 9].. THE MATHEMATICAL MODEL An ele mane onsee fom e epon of emal analyss s a sysem of ae boes neangng emal enegy among em an e enonmen by means of onuon o aaon. Takng no aoun all e emal penomena a ake plae n su a sysem as been mpossble so fa. Teefoe seems neessay o make ean smplfyng assumpons e mos mpoan of s e assumpon of e epeaably of e emal penomena n ea slo p of e sao. Ts paula assumpon lassal n e eoy of emal alulaons of ele manes makes possble o eae a smple one-mensonal moel of e mane. Is pysal espon as been pepae by employng e meo of emal balane a s by usng all e sgnfan ea flues elae o e eange an aumulaon of emal enegy n e sysem. Depenng on e eque egee of auay of e alulaons e follong one-mensonal ea ansfe pas an be snguse: sao nng sao sees fame oo nng oo sees an saf. In e eample pesene only e sao nngs an e sao sees ae been onsee. We pon ou a e empeaue sbuons an be foun n a smla manne. Due o e emal symmey of bo ens of e moo e ogn of e oonae sysem use s plae a e geomeal ene of e moo e OX as onng e longunal as of e mane. In su a oonae sysem e ea ansfe pas an be e no o sepaae subses: D { : l} an D { : l } l ee D onans e sees an e slos an D onans e en onneons.

3 Dung one full yle of e mane's opeaon e oolng meum unegoes a pase anson a esuls n ea ansfe onons e angng ompleely. Tese anges ae abup n naue neessang a sepaae espon of ea of e peos of e mane's opeaon. Fg.. A oss-seon of e nuon ele moo: - oolng meum; - ambe; - en onneons; 4 - fame; 5 - sees; 6 - ole Fg.. Te poess of e nease n e empeaue of e sao nngs of e nuon ele moo n e peo e oolng meum sol n K

4 4 Te follong peos an be snguse: e peo fom e sa-up of e mane o e begnnng of e pase anson of e meum e meum's pase anson peo e peo fom e omplee melon of e oolng meum o e momen e nng eaes s lm empeaue fo a lass of nsulao. Beause of e aboe assumpons a espon of e emal sae of e boes fomng e sysem mus onss of s ses of equaons o geomeal egons n ee me neals. Te emal sae of e elemenay olumes n e sysem nng sees e oolng meum an be esbe usng e pnple of onseaon of enegy. Te soluons e ae yng o fn ae me-epenen empeaue sbuons n e sao nngs an an n e sao sees. Fo an eleal engnee e sbuon of empeaue n e olume of e melng oolng meum s of no paal mpoane. Te ea balane n egon D of e slo aea of e nng of e sao s p ee p s e ea elease by nenal ea soues ung a me neal n a olume elemen of oppe: p p [l β ] ee p s e amoun of oppe los pe un olume β s e empeaue oeffen of e essane of oppe s e nng's oss-seonal aea an. Te onsans an enoe e spef ea an spef ensy of oppe espeely. Beause eefoe. Te flu an be foun usng Foue s la q λ ga ee λ s emal onuy of e maeals n e appopae emal pas an

5 5. As λ an λ eefoe λ Te las em of e ea balane equaon epesens e emal enegy eange beeen an elemenay olume of oppe an an elemenay olume of e sao sees by means of ea onuane oug e slo solaon. Ts emal enegy an be epesse as follos: ] [ g λ ee: s e nng u n a slo g s e kness of e slo solaon λ s e emal onuy of e slo solaon. Fnally as a esul of e ea balane one obans a seon-oe paal ffeenal equaon an be en n e fom. p g g g p λ β λ λ Inoung a ne noaon fo e pysal onsans pesene n e aboe equaon e follong fom s obane:. b a. Te ea balane fo e oe emal pa a s oug e sao sees s as follos: p

6 6 ee e em epesens e emal enegy eange beeen a sufae elemen of e fame oesponng o e sao's slo p along e leng an e enonmen n aoane eon's la k[ ] ee: s e of a fame seon oesponng o e slo p k s e oeffen of ea ansfe oug e moo asng an e a laye aaen o e asng s e amben empeaue. Te equaon esbng e empeaue sbuon n e sao sees s λ λ g λ k g g k p F Te ne pysal quanes appeang n e aboe equaon ae: p F s e ossseonal aea of a segmen of e see pakage; s e spef ea of e see pak an λ s e emal onuy of sao sees aoss e pakage. Fnally. a b.. mulaneously e eang of e slo pa of e nng an e sao sees e empeaue of e en onneons fom egon D ses. Te ea balane fo e en onneons akes e fom belo: p 4. Wn e me neal no onsee e oolng meum s sll n s sol pase. Te ea quany 4 nely appeang n e emal balane of e one-mensonal moel an be ealuae only f e assumpon s mae a e laye of e oolng meum s a pa of a paalleleppe plae ee: 4 s e spef ea of e oolng meum 4 s e spef ensy of e oolng meum 4 s e sufae aea of e oss-seon oug e ambe onanng e oolng meum. Afe efnng n a smla manne fo egon D e ems of e emal balane e follong equaon s obane:

7 p k p k β λ an fnally subsung e pysal onsans e oban e follong equaon:. b a. As a esul of e ok one by e moo e empeaue of e nngs ses onnuously afe eeeng e melng pon of e oolng meum auses a anson of e meum fom e sol no e lqu sae. In e egon D ee ae no anges n e onfguaon of e boes akng pa n e ea eange poess bu e maeal popees of e nngs an sees fo eample e essanes ae sube o onnuous anges. Fo ea neal of e moo's ok see en of seon e aeage alues of e paamees appeang n e aboe equaons an be assume an as a esul e oeffens of e equaons ae eal numbes a ae ffeen n ffeen neals. Dung e melon peo.e. en e oolng meum goes fom e sol no e lqu pase e sbuon of e empeaue n egon D s eefoe esbe by e follong equaons:. b a. b a Te emal balane n egon D ll ffe fom e balane alulae n e peeng peo sne o e ea flues aleay aken no aoun a ne one namely e flu esponsble fo e melon of e oolng meum s aually ae. Te amoun of ea absobe by e melng boy s popoonal o a pa of s olume uenly angng s pase. Te kness of e melng laye s a funon of e eloy ε e fon of e pase anson poess aels n e oolng meum an ene s also epenen on e empeaue of e en onneons. Te nesgaon of e epenene ε f eques an analyss of e ea eange poess n sysems a aable pase bounay. Te espon of e penomena onsss of o equaons of ea onuon n e sol an lqu pases an e equaon of enegy balane on e pase bounay. Base on e esuls of [] an usng a ompue o alulae e so-alle solfaon oeffen b f one obans e follong elaon: 4 4 m b m ε ee: s e ea of fuson of e oolng meum m s e aeage of an elemenay pa of e ambe oesponng o one slo p 4 s e spef ensy of e oolng meum.

8 8 By makng e emal balane n aoane e pnples esbe fo e egon D ung e melng peo e follong equaon s obane: a b ne e fnal empeaue of e moo eees e melng pon of e oolng meum e moo an also opeae ung e peo e oolng meum s n s lqu pase. Wn D e naue of e emal balane oes no ange. I an be esbe analogously as n e o peeng peos. Te bas ffeene n e espon of e penomena akng plae n e negbouoo of e en onneons n D n e lqu pase of e oolng meum s e appeaane of naual oneon as one of e ea ansfe moes. Wen alulang e ompose ea eange a onuon an naual oneon e same appoa s employe as en analysng only ea onuon. Insea of e λ -alue anoe subsue fo e ea onuane oeffen λ s noue nlues bo e onuon as ell as e oneon of ea. Te alue of λ λ an be eemne usng e epemenal esuls of Kaussol an Bekmann quoe n [5] n a fom onenen fo makng paal use of e elaon obane. Te analyss of e emal balane esuls n e follong ffeenal equaon: a b. Te nal empeaue of all e pas of e moo s equal o e nal empeaue of s suounngs eefoe..4 Te bounay onons fo e nual emal pas ae /.5 an /..6 Tese bounay onons esul fom e emal symmey of bo ses of e moo so l / l /.7 an l l..8 Te ona onon on e nefae beeen D an D n e nng a s on e fs emal pa an e emal pa s.

9 9 l /..9 Te fom of e onon esuls fom e esgn onseaons. On e nefae beeen e sees an e ambe ee s a laye of epoy lamnae lo ea onuy. Aonally ee s a onseable ffeene beeen e emal onues along an aoss e muually nsulae sees l /.. Te onon esulng fom e assume epeaably of e emal penomena n ea slo p s a e en l of all en onneons e empeaue s e same. Te one-mensonal moel pesene enables a smple espon of e emal penomena akng plae n an ele mane n e egon a e oolng meum mels. Only a o-mensonal moel an emoe e ffules neen o a onemensonal moel. Ts emak also apples o e eemnaon of e momens en e pase anson sas o ens. ne empeaue ffeenes n e ny of e en onneons ae small s assume a e melon of e oolng meum an ene e neessy o nlue n e emal balane akes plae en e aeage empeaue of bo ens of e egon D eaes e melng pon [ l l ]. Te en of e poess s eemne by e momen e asng of e moo eaes e empeaue. If no sepaae emal pas ae esablse n e moel fo e asng of e moo s e ase ee an be assume suffen auay a e aeage empeaue of e asng s equal o e aeage empeaue of e sao sees: [ l ]. Ts fa esuls fom e ey lo alue of e emal essane a e nefae beeen e sees an e asng. Te empeaue sbuon eemne a e en of ea eang peo along e nual emal pas also sees as e nal onon fo e ne peo of opeaon. Le us obsee a a soluon of e aboe poblem an be obane n a paula ase fom e esul of e follong pape onens a non-lnea ona poblem [7].. A UMERICAL OLUTIO An llusaon of e moel pesene aboe s e numeal alulaon of e emal saes of e sees n e moo. I onens e lnea poblem appeang n e fs peo of e mane s opeaon unl e momen a e pase anson of e meum begns. Fo s pupose e appomae e poblem.-. by e appopae ffeene seme an en poe s sably.

10 We efne on e lae { ; o l fo } P : R. ee l / l l / T/R. naual an R. Le us noue e follong noaon fo e alues of on P : ν ν fo. ν ν l K; KR.4 Replang e eaes n e sysem.-. by e fne ffeenes appea n e equales O fo O O fo a e lae pons { ; o l fo } P : R.8 espeely e oban e follong sysem of equaons: a a a b b b Assumng e bouneness of / an / n P an base on.5-.7 e an onlue a e eo of e appomaon of.-. by.9-. s O.

11 Le us noue e noaon fo a e a e.. fo b e g g.4.5 an..6 As a esul sysem.9-. an be en n e follong eo fom: E G E.7 ee: R. Le us onse e follong non-omogeneous bounay onons oespon o e onons.5-.:.8 l.9 l. l l.. l l. G g g g E e e e

12 In e sequel e assume a an ae boune fo an. T Replang n.8-. e eaes of e funons by e fne ffeenes e oban ee: fo an R. Le us noue e follong maes: H.8 H.9 H. an eos..

13 As a esul e an e e onons.-.7 n e abbeae fom H H - H. fo R. Jonng e equaons.7. e oban e ffeene seme an en n e fom H H - E G E M E G E - H H. fo... R. Le us onse e sysem of equaons A f.4 ee H H... E G E... E G E... A E G E... H H s e ma of sysem. an M.6 f f f M f f f f f.7 gen by.6 fo. Te follong lemma s al.

14 4 Lemma. Le us assume a sasfes e follong onon see.4 mn g.8 fo an se a ma[ b b b]..9 If... ; R; ae e soluons of e sysem.4 an a> en e nequaly ma ma f a.4 fo... R - s sasfe fo </a. Te smple poof of e lemma esulng fom e efnon.9 of e onsan a ll be ome. In oe o eamne e sably of e ffeene seme le us noue e nom of e eo see.6 n e follong ay: ma fo -. ma.4 Le be a soluon of sysem.4. By e efnon.4 of e nom an Lemma e ae e esmae a ma f.4 fo ; an a> an </a Lemma. If s a soluon of sysem. sasfes e onon.8 an a> en fo </a e follong nequaly ols: a ma ma.4 fo.

15 5 Poof. Le us noe a e soluon of sysem. an be en n e fom see [6] p. 67 z.44 ee M.45 s a soluon of e sysem A M.46 an z z z z M.47 s a soluon of e sysem Az M.48 Te nequaly of.4 as e esul a e soluons of e sysem.46 an.48 ae e follong esmaes: [ ] ma ma ma a.49 ma ma z a.5 fo... - an

16 6 Takng no aoun onons.-.7 fo an efnon.4 of e nom of eo - e an e nequaly.49 n e fom a.5 Fnally ombnng.5 an.5 e oban.4 o e an fomulae e eoem on e sably of e ffeene seme. Teoem. Le be a soluon of e ffeene seme efne by. an le sasfy onon.8. If a > see.9 en fo < /a e ffeene seme s sable an e follong esmae s al: ma e at ma ma.5 fo... R an Poof. Le b : ma ma.5 fo... R an. Fom e nequaly.4 e ae b a fo... R. Hene e esmae.54 b a k a k.55 s sasfe. Takng < /a e ae /-a a an fnally R R at ma a R a b e Tb..56 fo... R. Remak. By e La eoem see [] p. 6] e onegene of e soluon of e see poblem o e soluon of e poblem.-. n e nom.4 esuls fom s sably. Te ffeene poblem omogeneous bounay onons fo... R as been sole. Elmnang an fom e fs an las equaons of. espeely e oban

17 7 - G - E - E - E G E M E G E E G EH - ee H H H s e ma of e fom H.58 Usng e Gauss elmnaon meo see [4] p. 4 e ae e follong algom fo solng eple eae seme. Fo... R e ealuae e soluon k : α : G E E β α β : G E k k k : β : α k : G Eα : G Eα k k k β E k k Eβ k k - k Te numeal soluon s pesene n Fg...59 Te aae of ese esuls obane by e numeal meo s n ageemen a of e measuemens pefome fo e pysal moel e nuon ele moo of e ype f -4 of poe P 5.5 kw - measung sysem an esuls [7] n e me neal oesponng o e seay sae of e oolng meum. Le us pon ou a fo e oe neals of me e ffeene beeen e esuls of e numeal alulaons an e measuemens of e pysal moel s geae ue o e ffules n eemnng e pysal paamees of e oolng meum. Cuenly e sues on onsuons of o-menson moo fo so-me opeaon ae onnue [8]. Te effos ae also onenae on numeal soluon of paal ffeenal equaons obane fom e moel. To-menson moel ae pesely eemne emal paamees of e oolng meum ll appomae sgnfanly e esuls of ompuaon o e esuls of laboaoy measuemens. Te ne sues on mpong e onsuon of moo oole by e sol-lqu pase anson ae planne [9].

18 8 Refeenes [] Calsla H. A. Jaege J. C.: Conon of Hea: ols. Ofo Unesy Pess Ofo 959. [] Dya M. Jankoska J. Jankosk M.: uey of umeal Meos an Algoms. Pa. PW Wasa 98 n Pols. [] Koze E. Ubańsk W.: ues on e nuonal ele mane oole e fs-oe pase anson sysem. Po. In. Conf. on Eleal Manes Vol. Buapes 98. [4] amask A. A. kolae E..: Meos of solng Lae Equaons. PW Wasa 988 n Pols. [5] Wśnesk.: Hea Eange. Wasa Unesy of Tenology Wasa 979 n Pols. [6] Gouno. K. Rabenk W. : Dffeenes semes. auka Moso 977 n Russan. [7] Ubańsk W.: Usng a sol-lqu pase anson sysem fo oolng so-me uy nuon moo. Doo s ess. Insue of Eleal Manes Wasa Unesy of Tenology 984 n Pols. [8] Ubańsk W.: To-menson moel of ee-pase moo oole by e sol-lqu pase-anson. Insue of Maemas Wasa Unesy of Tenology 94/ 988 n Pols. [9] Ubańsk W.: Pelmnay sues on onsuon of nuon moos aape o oolng by sol-lqu pase anson. PA Insue of Eleal Manes Wasa Unesy of Tenology 5/4/5/6 99 n Pols. ZAGADIEIA OBLICZEŃ CIEPLYCH ILIKÓW IDUKCYJYCH CHŁODZOYCH PRZEMIAĄ FAZOWĄ I RODZAJU CIAŁO TAŁE-CIECZ eszzene W aykule pzesaono oygnalną konsukę slnka nukynego ófazoego pzeznazonego o pay oyze opaoanego Insyue Maszyn Elekyzny PW. Cłozene maszyny zealzoano opau o akumulaę epła pozas pzemany fazoe ało sałe-ez meum łoząego. Zapezenoano posaoe ey konsuk. Zbuoano enoymaoy moel eplny la sanu neusalonego meoą bezpośeną. ukłay ónań ząskoy zęu ugego z aunkam bzegoym II III IV ozau. Zefnoano algoym numeyznego ozązana ónań. Poano pozye leauy uzupełnaąe poblemaykę baań laboaoyny oblzeń numeyzny moel slnkó.

The sound field of moving sources

The sound field of moving sources Nose Engneeng / Aoss -- ong Soes The son el o mong soes ong pon soes The pesse el geneae by pon soe o geneal me an The pess T poson I he soe s onenae a he sngle mong pon, soe may I he soe s I be wen as