WHEEL/RAIL INTERACTION DUE TO THE POLYGONAL WHEEL

Transcription

1 U.P.B. S. Bull., Sere D, Vol. D, I. 3, ISSN WHEEL/RAIL INTERACTION DUE TO THE POLYGONAL WHEEL Traan MAZILU, Măălna DUMITRIU, Crna TUDORACHE 3, Mrea SEBEŞAN 4 Arolul e faţă ee onara uer neraţun nre o oe elaă ş o ale alaaă auzaă e roţle polgonale. Oa ee oneraă o grnă Tmoheno avân orpur rge fae e ea reprezenân ule e oe, roţle ş urle e frână. Moelul ă nlue un nou moel al reazemulu pero al şne onân n ouă eme Kelvn-Vog r-reţonale penru uporul e şnă ş ala, ş un em m Kelvn-Vog/Mawell penru eraamen. Prnpalele araer ale vraţe roaă-şnă aoră roţ polgonale un analzae ulzân o nouă raare a meoe mare Green a ă. The paper am a uyng he neraon eween an ela wheele an allae ra ue o he polygonal wheel. The wheele onere a Tmoheno eam wh aahe rg-oe a ale oe, wheel an rae. The ra moel nlue a new moel of he ral pero uppor onng n wo hree-reonal Kelvn-Vog yem for he ral pa an he alla, an a me Kelvn-Vog/Mawell yem for he ugrae. The man feaure of he wheel/ral vraon ue o he polygonal wheel are analyze va a new approah of he Green mar of he ra meho. Key wor: allae ra, wheele, polygonal wheel, Green mar. Inrouon Upon uyng he ynam ehavour eween he ralway vehle an he ra, we may fn he repone o many ue uh a he paenger omfor, he rollng noe, he ruure mehanal reane he vehle an he ra an he propagaon of he vraon nue y he ran o he ulng loae n he ra envronmen [-3]. One of he ommon efe n he ralway wheel he o-alle ou-ofrounne [4] ha may ae he hape of he orrugaon wheel ue o rea rang [5], he wheel fla [6-8] or even he polygonal wheel [9-]. The former wheel efe on n he pero ameer varaon aroun he mean value, o Ao. Prof., Depar. of Ralway Vehle, Unvery POLITEHNICA of Buhare, Romana, e- mal: rmazlu@yahoo.om A. Prof., Depar. of Ralway Vehle, Unvery POLITEHNICA of Buhare, Romana 3 A. Prof., Depar. of Ralway Vehle, Unvery POLITEHNICA of Buhare, Romana 4 Eng., METROREX, Buhare, Romana.

2 96 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan ha he wheel rumferene mulple neger of he wavelengh of h efe. The perum of he polygonal wheel omnae y a few wave ha orrepon o -5 wavelengh aroun he wheel rumferene, an he amplue are n he orer of mm. The frequeny range of he vraon nue y he polygonal wheel uae eween 5 an 5 Hz when ran veloe are n he nerval of 5-5 m/h []. All ralway wheel ype are omnae y he ommon wheel eenry one harmon. A a general rule, ee h harmon, he anar ol eel wheel are omnae y he hr harmon, whle he ruer prng wheel prevalenly eh he eon harmon [3]. Many paper ugge ha he polygonal wheel of he hr orer our eaue of he wheel lampng n a hree-jaw hu [5, 4-5]. Th paper heren ere he ynam ehavour eween a wheele movng along a allae ra ue o he polygonal wheel n orer o pon ou he a feaure of h parular eaon mehanm of he wheel/ral yem. To h purpoe, he wheele moelle a a Tmheno eam wh aahe rg-oe a ale oe, wheel an rae [6]. The ral onere a an nfne Tmoheno eam reng on a pero uppor, nlung ral pa, em-leeper, alla lo an he u-grae effe. The uppor moel mprove ompare o he alreay nown moel [7-8] an allow mulang he ra repone for an eene frequeny range [9]. The Green mar of he ra meho, prevouly ue o mulae he rg wheel/ral neraon [] an he mple vehle/ra neraon [], evelope here for he ela wheele/ra neraon.. The mehanal moel an he governng equaon One oner a wheele wh polygonal wheel unformly runnng on a mooh, angen allae ra. Conerng he wo u-yem, he wheele an he ra a ymmer ruure, nlung he wheel efe an he wheel/ral ona fore, he moel of he wheel/ral neraon may e reue o a wheel runnng on a reely uppore ral Fg.. The ral ere ung an nfne unform Tmoheno eam on equan uppor wh a pan eween hem. The parameer for he ral are a follow: he ma per lengh un m, he ro-eon area S, he area momen of nera I, he eny ρ, heyoung moulu E, he hear moulu μ an he hear oeffen κ. The ane from he ro-eon neural fre o he ral foo h. The lo faor of he ral neglee. The moon of he ral ere y he olumn veor q [w, θ, ] T, where w, an θ, an for he veral plaemen, repevely he roaon of he ro-eon; he oornae along he ral an an for me.

3 Wheel/ral neraon ue o he polygonal wheel 97 In erm of he relave plaemen eween he ral ro-eon an he ral pa, he moel of he ral pa ha hree Kelvn-Vog yem wh he ela onan, r an α, an he vou ampng onan, r an α. Fg.. Mehanal moel of he wheel/ral neraon ue o he polygonal wheel: polygonal wheel; wheel/ral ona; 3 ral; 4 ral pa; 5 em-leeper; 6 alla; 7 ugrae. The em-leeper are aen a rg oe wh hree egree of freeom,.e. he longunal ranlaon, he z veral plaemen an he α roaon aro he ral, where he leeper numer, uae a he ane from he referene. The parameer for he em-leeper are: he ma M, he ma momen of nera J an he ane h eween he em-leeper an he ral pa. The alla moel on n a rg oy n he u veral ranlaon uner he em-leeper va he Kelvn-Vog yem. The longunal an roaonal reane of he alla are aen no aoun y ung wo Kelvn- Vog yem onnee o he em-leeper. The alla moel ha he ma M, he ela onan, l an β an he vou ampng onan, l an β. Fnaly, he ugrae nfluene mulae ung a me Kelvn- Vog/Mawell yem [9]. Th oluon val a long a he ale veloy

4 98 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan muh lower han he veloy of he wave ee n he ugrae y he ale ravel. The equaon of moon for he ral an peroal uppor may e wren a,,, δ Q q B A q q T T q B q C where [ ] T V Q δ Q 3 he olumn veor of he veral fore ang on he ral, δ. he Dra ela funon, q q, he olumn veor of he ral plaemen aove he uppor, [ ] T v u z α q repreen he plaemen elongng o he uppor, nlung he v hen plaemen of he Mawell yem, T, he mar fferenal opperaor of he Tmoheno eam ρ κμ κμ κμ κμ, I S EI S S m S T 4 an A, B, C an for he followng mar fferenal θ θ r r A ; 5 Δ Δ α α h r r B ; 6 α v u z D D D h D h D C 7

5 Wheel/ral neraon ue o he polygonal wheel 99 wh l l M D ; r r z M D, v D ; h h J D β α β α α ; u M D ; h α θ, h α θ, hh Δ α α, hh Δ α α. All he nal onon an he ounary onon are null [ ] T, q, [ ] T q, 8 [ ] T, lm V q, [ ] T lm ± q. 9 The wheele moel preene n Fg., where a wheele of a paenger oah wh four rae eng looe a. Fg.. Mehanal moel of he wheele: an 8 ale o; an 7 wheel; 3-6 rae.

6 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan The ale moelle a a unform Tmoheno eam wh aahe rg oe a wheel, ale oe an rae. The plaemen of he ale are ere y he olumn veor q a y, [uy, ϕy, ] T, where uy, an ϕy, are he veral plaemen an he roaon of he ro-eon, y an for he oornae along he ale. The parameer of he ale are: he Young moulu E, he hear moulu μ, he eny ρ, he lengh l, he ma per lengh un m a, he ro-eon area S a, he area momen of nera I a an he hear oeffen κ a. The oy 8 aahe o he ale a he ane e from he lef en of he ale an ha he ma M an he ma-momen nera J. Aumng he wheele moon aroun he equlrum poon an negleng he gyroop effe an he a an ynam malane of he wheel e, he governng equaon of moon may e wren a follow 8 a a a a T y, q F, q δ y e Q, where T a y, ş F, are he mar fferenal operaor, q a q a e, he olumn veor of he ale plaemen a he e eon an a Q [ Q Q [ ] T ] δ y e δ y e7 he olumn veor of he fore ang on he wheel; Q he wheel/ral ona fore, whle Q he a loa. The mar fferenal operaor appearng n he equaon of moon of he ale are a follow κaμsa m κ μ a a Sa a y y T y, κaμsa EIa κ μs ρi y a a a y M F,. 3 J The ounary onon free-free are he ame for oh en of he ale y an y l u ϕ ϕ ;, 4 y y whh mean ha he hear fore an he momen are null. Alo, he nal onon for he ale are null.

7 Wheel/ral neraon ue o he polygonal wheel The wheel/ral neraon moel ha o onan a reron, ue o he herzan ona eween he wheel an he ral. Aorng o Herz heory, one rea /3 [ Q / CH ] zδ σ[ zδ ], 5 where z δ he wheel/ral efleon, C H he Herzan onan an σ[.] he unary ep funon of Heave. The wheel/ral efleon may e wren a zδ z w V, Δr, 6 where z ue, ue 7, he wheel plaemen an Δr an for he wheel/ral relave plaemen ue o he polygonal wheel. Negleng he nfluene of he evaon offe eween he ona pon an he wheel enre, he wheel/ral relave plaemen may e wren a Δ r Δr o n V / R, 7 where Δr he amplue R he wheel rau an n an neger orreponng o he orer of he polygonal wheel. 3. Soluon of equaon of moon The oluon of he equaon of moon an e oane y ung he Green mar of he ra meho [-]. To h en, he Green funon of he wheele a he wheel ha o e alulae followng he meho uggee y he auhor [6]. Th funon ere he wheel repone when wo mpule fore are ymmerally apple on he wheel an ha he form elow N ep ζ ω h n ω ζ, 8 M M ω ζ where M he half-ma of he wheele, M he moal ma of he vraon moe, ω he reonane angular frequeny an ζ he moal ampng faor; he fr N moe are aen no aoun. Alo, one nee he Green mar of he ra nlung he repone of he ral along a leeper ay ue o a un movng mpule fore. Th mar aemle y he Green funon of he ral. In vrue of he onvoluon heorem, he wheel plaemen an he ral plaemen a he movng ona pon are gven a [ Q Q ] z h τ τ 9 w w w V, g V, ξ, τ Q τ δ ξ Vτξτ g V, Vτ, τ Q ττ, where g w V,ξ,-τ he Green funon of he ral a he movng pon.

8 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan Ne, hee plaemen are nroue n he ona equaon 5 an one oan an negral nonlnear equaon wh he ona fore a unnown [ / ] / 3 w Q C h τ [ Q Q ] τ g V, Vτ, τ Q ττ Δr H. The equaon val a long a he wheel an he ral are n ona an may e olve followng a numeral approah. 4. Numeral applaon In h eon, reul are erve from he prevou wheel/ral moel for a parular wheele ha unformly move along a allae ra, onerng he ymmer polygonal wheel of he hr orer. The moel parameer are le n Tale for he wheele an n Tale for he ra. Tale Parameer for he wheele Parameer Deny of he ale eel Young moulu of he ale Shear moulu of he ale Lengh of he ale Ma per lengh un of he ale Cro-eon area of he ale Area momen of nera of he ale ro-eon Shear oeffen of he ale ro-eon Ma of he ale o Ma of he wheel Ma of he rae Ma momen nera of he ale o Ma momen nera of he wheel Ma momen nera of he rae Poon of he ale oe Poon of he wheel Poon of he fr wo rae Sa loa Value ρ 785 g/m 3 E GPa μ 8 GPa l. m m a 57 g/m S a. m I a m 4 κ a.9 M M 8 8 g M M g M 3 M 4 M 5 M 6 7 g J J 8 gm J J 7 3 gm J 3 J 4 J 5 J 6 g m e.8 m; e 8. m e.35 m; e 7.85 m e 3.65 m; e 4.95 m Q 7 N For he numeral mulaon, he ra moel lengh of 5 leeper ay; h lengh afe he manaory reron o eep he pero feaure of he ra n he enral zone of he moel 4 leeper ay. Only he ral reepane from h zone ue, wh he purpoe o alulae Green mar of he ra.

9 Wheel/ral neraon ue o he polygonal wheel 3 Tale Parameer for he ra Parameer Value Ma per lengh un of he ral m 6 g/m Cro-eon area of he ral S m Area momen of nera of he ral ro-eon I m 4 Shear oeffen of he ral ro-eon κ.4 Dane eween he ro-eon neural fre an he ral foo h.8 m Longunal ral pa ffne 5 MN/m Veral ral pa ffne r 8 MN/m Roaonal ral pa ffne α 597 Nm/ra Longunal ral pa vou ampng onan N/m Veral ral pa vou ampng onan r 5 N/m Roaonal ral pa vou ampng onan α 7 Nm/ra Ma of em-leeper M 45 g Ma-momen of nera of em-leeper J.8 gm Sleeper ay.6 m Dane eween he em-leeper enro an he ral pa h.6 m Longunal alla ffne l 4 MN/m Veral alla ffne MN/m Roaonal alla ffne β 676 Nm/ra Longunal alla vou ampng onan l 5 N/m Veral alla vou ampng onan 7 N/m Roaonal alla vou ampng onan β 394 N/m Ma of alla lo M 5 g Veral ugrae ffne Kelvn-Vog yem 6 MN/m Veral ugrae ffne Mawell yem MN/m Veral ugrae vou ampng onan Kelvn-Vog yem 5 N/m Veral ugrae vou ampng onan Mawell yem 6 N/m One oner ha he wheel efe ha he wavelengh of 96 mm, orreponng o a wheel rumferene of 88 mm wheel ameer aou 9 mm. The efe amplue μm an h value urrenly oane from he epermenal aa [5]. Fgure 3 how he wheel plaemen an he ral plaemen a he ona pon urng he runnng a he pee of 6 m/. The yem vraon ha wo omponen, one gven y he paramer eaon ue o he leeper wh he wavelengh of 6 mm, an he oher one nale y he wheel efe wavelengh of 96 mm. Conequenly, he wheel/ral vraon moulae wh he wavelengh of 48 mm, orreponng o he frequeny of.9 Hz. Noe he paramer frequeny of 3.33 Hz an he frequeny of Hz ue o he polygonal wheel. One may oerve ha he vraon omnae y he omponen n he polygonal wheel.

10 4 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan Fg. 3. Wheel/ral plaemen a 6 m/ n he preene of he hr orer polygonal wheel amplue of. mm:, ral plaemen a ona pon;, wheel plaemen. Fg. 4. Wheel/ral ona fore a 6 m/ n he preene of he hr orer polygonal wheel amplue of. mm: a he me hory; he perum. The me hory an he perum of he ona fore are preene n Fg. 4. The ona fore ha hgh ollaon aroun he a loa value; he mamum value N an he mnmum 39.7 N. Alo, he effeve ona fore.6 N, reulng from he numeral mulaon. The hghe omponen of he ona fore perum ha he amplue of 8.4 N an ome from he polygonal efe of he wheel a frequeny of Hz. The omponen ervng from he paramer eaon of he leeper gnfan, amplue reahng.53 N frequeny of 3.33 Hz. Apar from h, all

11 Wheel/ral neraon ue o he polygonal wheel 5 omponen are very low, eepng he eon omponen of he paramer eaon wh he amplue of.96 N a frequeny of 6.7 Hz. Fg. 5. Wheel/ral effeve ona fore veru he veloy n he preene of he hr orer polygonal wheel he amplue of. mm. Fg. 5 play he effeve ona fore veru he wheele veloy, ang no aoun he ame polygonal wheel. The ona fore ha wo pea a 6 an 76 m/ orreponng o he reonane frequene of he wheele/ra yem he reonane ue he fr ymmer enng moe an he reonane gven y he rg-oy moe of he wheele. In fa, he vraon level urn hgh for he veloy eween m/h an 335 m/h. When he efe amplue hgher han a parular value, he ona fore lea o he ona lo. Suh uaon epe n Fg. 6 where he reul from he numer mulaon of he polygonal wheel/ral neraon are preene for he amplue of 6 μm an he veloy of 6 m/. A ona pon he wheel an he ral vrae almo n phae. One remar he very hgh level of he ona fore howng a pero ho. The mamum value wo me hgher han he a loa. Fnally, he la ue ue here hown n Fg. 7, where he agram of he mamum an mnmum ona fore veru he polygonal wheel amplue

12 6 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan Fg. 6. Wheel/ral neraon ue o he polygonal wheel amplue of.6 mm a pee of 6 m/: a, ral plaemen a ona pon;, wheel plaemen; ona fore. Fg. 7. Mamum an mnmum ona fore a pee of 6 m/:, mamum ona fore;, mnmum ona fore. preene. The wheele veloy of 6 m/ ha een aen no aoun for. Ovouly, when he efe amplue nreae, he mamum ona fore nreae a well, whle he mnmum ona fore ereae. Thee ren are lnear eaue he elay of he ona muh lower han he ral an wheel reepane, an, uequenly, he nonlnear ona ha a very margnal nfluene. When he efe amplue hgher han 5 μm, he ona lo our an he mamum ona fore nreae lnearly u lope maller.

13 Wheel/ral neraon ue o he polygonal wheel 7 6. Conluon The ynam ree ue o he wheel/ral vraon nue y he rregulare of he rollng urfae affe he mehanal ruure of oh vehle an ra. Th paper how a uy mean for he ynam ehavour of he wheel/ral neraon when he wheel eh he hr-orer polygonal efe. Th n of wheel efe ome from he mahne wor of he wheel an prevalen. To h en, one onere he ae of a wheele wh four rae n a paenger oah movng along a allae ra. The ale aen a a Tmoheno eam wh aahe rg-oe a ale oe, wheel an rae. Only he ymmer enng moe are aen no aoun. The meoman Green funon a he wheel oane va he reepane an he moal analy. The moel ra reue o a reely uppore ral, ne he ra onere a ymmer ruure. The ral moelle a an nfne unform Tmoheno eam. The uppor of he ral moel mprove o een he reul range aorng o he meauremen, oh for low an hgh frequene. In fa, h moel of he pero uppor on n wo hree-reonal Kelvn-Vog yem for he ral pa an he alla, an a me Kelvn-Vog/Mawell yem for he ugrae. Alo, he nera of he leeper an he alla lo nroue. The numeral mulaon how ha he ynam ehavour ue o he polygonal wheel omnan, whle he one from he paramer eaon of he leeper play a margnal role. The nfluene of he wheele veloy on n wo pea of he effeve ona fore orreponng o he enng reonane of he wheele on he ra an he reonane of wheele/ra yem ue o he rg-oy moe of he wheele. The ona fore nreae wh he amplue of he wheel rregulary. However, when ona lo, he ona fore lower. Anowlegemen Th wor wa uppore y CNCSIS-UEFISCSU, proje numer 684 PN II IDEI oe 699/8: Reearhe on he wheel/ral paramer vraon ung he ra Green mar meho. R E F E R E N C E S [] I. Seeşan, Dnama vehulelor ferovare, Dynam of he ralway vehle, E. Tehna, Buureş, 995. [] I. Seeşan, T. Mazlu, Vraţle vehulelor ferovare, Vraon of he ralway vehle, MarRom, Buureş,. [3] T. Mazlu, Vraţ roaă-şnă, Wheel/ral vraon, MarRom, Buureş, 8.

14 8 Traan Mazlu, Măălna Dumru, Crna Tuorahe, Mrea Seeşan [4] J. C. O. Nelen an A. Johanon, Ou-of-roun ralway wheel - a leraure urvey n Pro. Inn Meh. Engr, Par F: J. Ral an Rap Tran, 4,, pp [5] A. Johanon, Ou of roun ralway wheel-aemen of wheel rea rregulare n raff n Journal of Soun an Vraon 93, 6, pp [6] R. Dupa, R. Dong, Impa Loa ue o Wheel Fla an Shell n Vehle Syem Dynam, 3, 999, pp. -. [7] T. Wu an D. J. Thompon, A hyr moel for he noe generaon ue o ralway wheel fla, n Journal of Soun an Vraon, 5 no.,, pp [8] M. J. M. M. Seenergen, The role of he ona geomery n wheel-ral mpa ue o wheel fla, n Vehle Syem Dynam, 45, 7, pp [9] A. Johanon, C, Aneron, Ou of roun ralway wheel-a uy of wheel polygonalzaon hrough mulaon of hree-menonal wheel-ral neraon an wear, n Vehle Syem Dynam, 43, 5, pp [] T. Mazlu, Ineraon eween a movng wo-ma ollaor an an nfne homogeneou ruure: Green funon meho, n Arhve of Apple Mehan, 8,, pp [] P. Mene, S. Mene, Polygonalzaon of wheel rea aue y a an ynam malane, n Journal of Soun an Vraon 7, 999, pp [] J.C.O. Nelen, R. Lunen, A. Johanon, T. Verneron, Tran-Tra Ineraon an Mehanm of Irregular Wear on Wheel an ral Surfae, n Vehle Syem Dynam, 4, 3, pp [3] G. Palgen, Unrune Räer an Eenahnfahrzeugen, n Eenahnngeneur 49, 998, pp [4] W. Roe, D. Mller an J. Vllman, Reul of DB AG Invegaon Ou of Roun Wheel, n Proeeng of he Corrugaon Sympoum-Eene Ara, IFV Bahnehn, Tehnhe Unverä Berln, Germany, 997. [5] W. Momre an P Olnger, Da Unrunweren von Eenahnräern au Weroffehnher Sh-en Üerl n ZEVDET Glaer Annalen 5,, pp [6] T. Mazlu, M. Dumru, C. Tuorahe an M. Seeşan, Ralway wheele enng flely n Proeeng of he WSEAS Inernaonal Conferene on Auomaon & Informaon ICAI, G. Eneu Unvery, Ia, Romana, June 3-5,. [7] W. M. Zha, K. Y. Wang, J. H. Ln, Moellng an epermen of ralway alla vraon, n Journal of Soun an Vraon 7, 4, pp [8] J. C. O. Nelen, Hgh-frequeny veral wheel-ral ona fore-valaon of a preon moel y fel eng, n Wear, 65, 8, pp [9] T. Mazlu, M. Dumru, C. Tuorahe an M. Seeşan, On veral analy of ralway ra vraon, n Proeeng of he Romanan Aaemy Sere A, vol. no.,, pp [] T. Mazlu, Green funon for analy of ynam repone of wheel/ral o veral eaon, n Journal of Soun an Vraon, 36, 7, pp [] T. Mazlu, Preon of he neraon eween a mple movng vehle an an nfne peroally uppore ral - Green funon approah, n Vehle Syem Dynam vol. 48, no. 9,, pp. -4.