ON VERTICAL ANALYSIS OF RAILWAY TRACK VIBRATIONS

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1 THE PUBLISHING HOUSE PROCEEINGS OF THE ROMANIAN ACAEMY See A OF THE ROMANIAN ACAEMY Volume Nume / pp ON VERTICAL ANALYSIS OF RAILWAY TRACK VIBRATIONS Taan MAZILU Măălna UMITRIU Cna TUORACHE Mea SEBEŞAN Unvey Polena of Buae epamen of Ralway Vele Romana METROREX Buae Romana E-mal: mazlu@yaoo.om Te pape een peen a veal vaon analy of e allae a ung a new moel of e peo uppo of e al a mpove e peon of e al epone fo o low an g fequene. Te moel of e peo uppo on of wo ee-eonal Kelvn-Vog yem fo e al pa an e alla an a me Kelvn-Vog/Mawell yem fo e ugae. Alo e nea of e leepe an e alla lo ae noue. Te al epone ue o a aonay an movng amon loa analyze y applyng e Geen funon meo. Te nfluene of e al pa ffne an of e opple effe on alway a vaon ae peene. Key wo: Ral; Tmoeno eam; Peo uppo; Geen funon; opple effe.. INTROUCTION Te uy of e veal vaon of e allae a al wen analyng e weel/al neaon n oe o gve paal oluon fo ean ue u a fomaon of weel an al ougaon [] geneaon of ollng noe [] o a amage. Uually e a moel eue o a al eng on ee uppo nlung e ela al pa e nea of e leepe an e elay of e alla [3]. Howeve even f e Tmoeno eam moel ue fo e al w a geae auay an e Eule-Benoull eam moel a e ea an e oay nea of e o-eon ae aen no aoun wo fequene ange ll eman al. On e one an e al eepane uneemae a low fequene ellow 5 Hz. On e oe an fo e fequene a ae uae wn e pnne-pnne fequeny ange e al eepane a m pan ove-emae. To oan a goo ageemen eween e peon an e epemenal eul Za e al. [4] an Nelen [5] ave noue e nea of e alla an e elay of e ugae. epe of ee mpovemen eem a e ugae moel popoe a ngle Kelvn-Vog yem [4] o wo Kelvn- Vog yem onnee n ee [5] ae no a appopae a o mulae e al epone a low fequene. I a o e empaze a Knoe an Wu [6] ave oane an neae of e al eepane wen e fequeny eeae ellow 5 Hz aong o e meauemen upon ung a opae alla-ugae moel ang fom e eoy of e ela alf-pae. Howeve e me-oman analy of e weel/al neaon eque a mple oluon fo e a moel. In ega o e peon of e al epone a e pnne-pnne eonane e eal eul ae oane wen e lmaon of e al o-eon oaon y e al pa onee an e eeeonal al pa moel ue [7]. Te ynam epone of e peo uue une movng foe a een eenvely analyze n e pa 3 yea an ffeen meo ave een popoe. Floque eoem wa apple y Kzyzyn an Popp [8] an Kzyzyn [9] e Foue-ee appoa a een ue y Seng e al. [] wle Noog [] a evelope e Geen funon meo. Te pape een peen an analy of e veal vaon of e allae a oneng a new moel fo e al uppo a gve a goo ageemen eween e peon an e meauemen. Te Geen funon meo pevouly popoe y e peen auo [ 3] apple o pon ou e epone of e al une a aonay an movng amon foe. Alo e nfluene of e al pa ffne an e loa veloy e opple effe ae peene.

2 On veal analy of alway a vaon 57. MECHANICAL MOEL One peume a e a a ymme uue an e wo al vae nepenenly ue o e elay of e leepe. In u umane e a moel eue o an nfne Tmoeno eam e al eng on ee pa ea of em nlung e al pa em-leepe alla lo an ugae Fg... A movng foe P avel along e al ea w onan pee V; an fo me. Fg... Te moon of e al ee y e olumn veo {q} [w θ] T wee w an θ an fo e veal plaemen epevely e oaon of e o-eon; e oonae along e al. Te paamee fo e al ae a follow: e ma pe leng un m e o-eonal aea S e aea momen of nea I e eny ρ e Young moulu E e ea moulu µ an e ea oeffen κ e ane eween e o-eon neual fe an e al foo. Te al pa aen a a ee-eonal Kelvn-Vog yem wong n e longunal an veal eon an e oaon n e veal plane of e al. Te ela onan z an an e ampng onan z an ae elae o e al pa. Te em-leepe ae equan y e leepe ay of e leng; e em-leepe plae a e ane fom e efeene fame. I aume a e em-leepe a g oy avng ee-egee of feeom: e veal z an longunal anlaon an e oaon n e veal plane. Ea em-leepe a e ma M an e ma-momen of nea J. Te ane eween e em-leepe eno an e al pa. Te alla nfluene mulae y neng a e of epaae mae une em-leepe va Kelvn- Vog yem. In aon e longunal an oaonal eane of e alla ae moelle ung wo Kelvn-Vog yem. Aumng a e loa veloy mu lowe an e veloy of e wave ee n ugae y e movng loa an eue e ugae moel une one uppong pon o a me Kelvn- Vog/Mawell yem. Te popoe oluon ffeen fom e famla one gven y Za e al. [4] o Nelen [5] an avanageou ue o e goo ageemen eween e peon an meauemen wn e low fequene. Te paamee fo e alla an e ugae ae a follow: e vang ma of alla M e ela onan u an an e oeponng ampng onan u an.

3 Taan Mazlu Măălna umu Cna Tuoae Mea Seeşan 3 58 Te plaemen of e alla lo une e leepe u wle v e plaemen fom e ugae moel. All plaemen of e uppo moel ae onane n e olumn veo {q } [ z u v ] T. Te a ffeenal equaon of moon may e fomulae n e ma fom a T {} { } { }δ {} { } { } T q A q B q P C q B q. wee {P} [ Pδ V ] T e olumn veo of e foe on e al an δ. e a ela funon T an fo e Tmoeno ma opeao an A B C ae e ma ffeenal T S m S S EI S I κµ κµ κµ κµ ρ A θ θ B v u z C.3 w l l M z M v J β β u M θ θ..4 All nal onon ae null an e ounay onon ae null a well [ ] T } { q [ ] T } { q [ ] T } { lm V q [ ] T } { lm ± q..5 Te oluon of e polem an e fomulae ung e Geen funon meo [ 3]. Aong o e onvoluon eoem an ang no aoun e aualy onon of e a moel e oluon of e polem may e gven a

4 4 On veal analy of alway a vaon 59. {q } {g ξ τ} P τ δ ξvττ ξ {g V τ τ} P ττ wee e olumn veo {q } {g ξ τ} P τ δ ξvττ ξ {g Vτ τ} P ττ w θ T {g ξ τ } g ξ τ g ξ τ.7 T {g } z u v ξ τ g ξ τ g ξ τ g ξ τ g ξ τ g ξ τ onan e Geen funon of e a fo e me-oman analy w gve e epone of e al n e eon an epevely e epone of e uppo a τ momen ue o an un mpule foe apple a e τ momen n al eon ξ. Te plaemen of e al a e loang pon of paula nee fom e pepeve of e a epone. T plaemen eul fom equaon.6 eng V.6 w wv { g VV τ τ} Q ττ.8 wee e funon g w V Vτ τ epeen e me-oman Geen funon of e al a e loang pon. Te me-oman Geen funon g w V Vτ τ gve e epone of e al n e V movng eon a -τ momen ue o a movng un mpule foe plae n e eon Vτ a e momen τ. Coneng a ean me paon e funon g w V Vτ τ along a leepe ay an e oganze a a ma e Geen ma of e a. Fnally e al plaemen a e loang pon alulae numeally ung e popee of e Geen funon of e al o e peo an aenuae. Seng * τ e eon V fe an we ave e funon g w V V V* * a ee e al epone a e * momen ue o a movng un mpule w a fom e fe eon V a e nal momen an avel fom e g o e lef a e veloy of V. Aually funon e me-oman Geen funon of e al une movng un mpule loa. Pefomng e Foue anfom w epe o * e al eepane fo e movng amon foe eul. In way e amplue of e al aen a a omple value oane fo e paula ae only wen a movng un amon foe pae ove e fe eon of e al. 3. NUMERICAL APPLICATION In eon numeal eul fom a paula allae a moel une a movng amon loa ae poue ung e peene appoa owae. Te ae of e aonay amon loa aen a e efeene ae. Te paamee fo e al ae: m 6 g/m S m I m 4.8 m ρ 785 g/m 3 E GPa µ 8 GPa an κ.4. Te paamee fo e ee uppo ae oen a: M 45 g J.8 gm M 5 g.6 m.6 m 5 MN/m 8 MN/m /3 597 Nm/a 6 mm al pa w u 39.6 MN/m MN/m 6 MN/m MN/m N/m 3 N/m /3 64 Nm/a u 5 N/m 7 N/m 5 N/m an 6 N/m. Fo e numeal mulaon e a moel leng of 5 leepe ay; leng afe e manaoy eman o peeve e peo feaue of e a n e enal zone of e moel. Only e al eepane fom zone ulze n oe o alulae e Geen ma of e a. Fgue 3. ow e eepane of e al a loang pon ue o e aonay amon eaon fo o eaon ang a m pan an aove a leepe n e oman of 6 Hz. One an oeve a e al eepane neae elow 5 Hz ue o e nfluene of e alla an e ugae. T ape onen w e eul eve fom e meauemen peene y Knoe an Wu [6]. Wn e ange of e m fequene e al an e leepe vae le a yem w wo-egee of feeom an eaue of e al eepane omnae y wo eonane fequene aou

5 6 Taan Mazlu Măălna umu Cna Tuoae Mea Seeşan 5 an 48 Hz. Te leepe play e ole of a vaon aoe w eplan wy e al a an aneonane fequeny a a 4 Hz eween e wo eonane. A lowe fequene e al an e leepe ae n pae an en eyon e an-eonane fequeny ey vae n an-pae. Fg. 3.. A we an ee e effe of e peo uppo of e al eome moe even wen e fequeny ge an e an-eonane fequeny an e al vae moe nenely a m pan an aove leepe. Te ge ffeene appea a e pnne-pnne eonane fequeny epevely a 74 Hz wen e enng waveleng equal we e pan. Alo one a o oeve e an-eonane zone loae a Hz wen e amon loa apple aove e leepe. Fg. 3.. Fg Te al pa ffne epeen a ual paamee fo e al ynam n geneal an n paula fo e wea of e ollng ufae. To lluae ape e al pa ffne of 4 MN/m a een aen no aoun ela al pa an e ao eween e al eepane fo e ela an e efeene al pa a een playe n Fg. 3.. Te al eepane ge a low fequene ue o e ela al pa

6 6 On veal analy of alway a vaon 6 epeally aove leepe. Howeve a fequene aove e eon eonane 48 Hz e ela al pa a a onay effe an e al eome moe g paulaly a e m pan. A e pnne-pnne eonane 74 Hz eem a e ffne of e al pa a a wea nfluene on e ynam eavou of e al. A e an-eonane fequeny of e al aove leepe e al eepane ge wen e al pa moe ela. Wn e pnne-pnne eonane ange e al eepane epen on e veloy of un amon loa ue o e opple effe. To onep e ea one one e veloy of e wave a popagae o e g an o e lef fom e aonay amon loa. Wen e amon loa move along e al e wave a popagae n fon of e loa a a ge veloy V wle e veloy of e wave a popagae en of e loa lowe V. Une u umane ea wave geneae a paula eonane fequeny fom e eonane onon e waveleng equal we e pan. T ape peene n Fg. 3.3 wee e al eepane aove e leepe an a e m pan fo e ae of e movng un amon foe own. A one an ee a e m pan e pea geneae y e aonay amon loa eplae y e wo pea poue y e opple effe ue o e movng amon foe. Te fequene of e ee pea ae n ame pogeon w a e ao of V/. Fo nane e fequene of 7 an 4 Hz oepon o e eonane fequene ue o e opple effe a 8 m/ an e pan of.6 m. Te level of e wo pea eeae a e amon foe pee neae. Te al eepane aove leepe eman paally unange. Fg Fgue 3.4 play e plaemen of e al a e loang pon fo m/ an 8 m/ an wo value of e loa fequeny 48 Hz an 74 Hz. Geneally e moon moulae ue o e peoally uppo. A 48 Hz ee no ffeene eween e magnue of e loang pon plaemen fo e wo veloe an e ajeoe of e loang pon ave e ame envelope. On e onay a e fequeny of 74 Hz pnne-pnne eonane e foe veloy gnfanly nfluene e moon of e loang pon. Te plaemen of e loang pon lowe fo e veloy ae of 8 m/ ompae o e one of m/. 6. CONCLUSIONS T pape a eal w e veal vaon of e allae a ung a new moel fo e al peo uppo. Te moel of e peo uppo on of wo ee-eonal Kelvn-Vog yem fo e al pa an e alla an a me Kelvn-Vog/Mawell yem fo e ugae. Alo e nea of e leepe an e alla lo noue. Te analy of e a vaon a een pefome n o fequeny an me-oman y applyng e Geen funon meo. Te peon of e al epone oepon o e eul eve fom e meauemen. Te omple nfluene of e al pa ffne on e al epone a een peene fo e ae of e aonay amon loa.

7 6 Taan Mazlu Măălna umu Cna Tuoae Mea Seeşan 7 Fo fequene wn e pnne-pnne ange e eepane of e al a wo pea nea of one ue o e opple effe. In aon e me-oman analy eveal a e al vaon e moulae one an e loa veloy n e paula pnne-pnne ange ongly nfluene level. All ee feaue an ave an mpa on e weel/al neaon an e o-p fomaon. ACKNOWLEGEMENTS T wo uppoe y e NURC Te Naonal Unvey Reea Counl une poje 699: Reeae on e weel/al paame vaon ung e a Geen ma meo. REFERENCES. SHENG X. THOMPSON. J. JONES C. J. C. XIE G. IWNICKI S.. ALLEN P. HSU S. S. Smulaon of ougne naon an gow on alway al Jounal of Soun an Vaon 93 pp REMINGTON J. P. Weel/al ollng noe: wa o we now? Wa on we now? Wee o we go fom ee? Jounal of Soun an Vaon pp GRASSIE S. L. GREGORY R. W. HARRISON. JOHNSON K.L. Te ynam epone of alway a o g fequeny veal eaon Jounal Meanal Engneeng Sene 4 pp ZHAI W. M. WANG K. Y. LIN J. H. Moellng an epemen of alway alla vaon Jounal of Soun an Vaon 7 pp NIELSEN J. C. O. Hg-fequeny veal weel-al ona foe-valaon of a peon moel y fel eng Wea 65 pp KNOTHE K. WU Y. Reepane eavou of alway a an ugae Ave of Apple Mean 68 pp MAZILU T. Popagaon of amon veal wave n a al UPB Senf Bullen See : Meanal Engneeng 67 pp KRZYZYNSKI T. POPP K. On e avellng wave appoa fo ee-onnuou uue une movng loa ZAMM Zef fu Angewae Maema un Mean 76 Suppl.4 pp KRZYZYNSKI T. On ynam of a alway a moelle a a wo-menon peo uue Heavy Vele Syem 6 pp SHENG X. LI M. JONES C. J. C. THOMPSON. J. Ung e Foue-ee appoa o uy neaon eween movng weel an a peoally uppoe al Jounal of Soun an Vaon 33 pp NORBORG A. Weel/al noe geneaon ue o nonlnea effe an paame eaon Jounal of e Aoual Soey of Amea pp MAZILU T. Geen funon fo analy of ynam epone of weel/al o veal eaon Jounal of Soun an Vaon 36 pp MAZILU T. Weel-al vaon n Romanan MaRom 8. Reeve Ooe 9