Title. Author(s)ANG, K. K.; THI, T. M.; HAI, L. V. Issue Date Doc URL. Type. Note. File Information.

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1 il RACK VIBRAIONS DURING ACCEERAING AND DECEERAIN Auhor(s)ANG, K. K.; HI,. M.; HAI,. V. Issu Da Doc UR hp://hdl.handl.n/115/54479 yp procdings No h hirnh Eas Asia-Pacific Confrnc on Sruc 1, 01, Sapporo, Japan. Fil Informaion asc1-i--1.pdf Insrucions for us Hokkaido Univrsiy Collcion of Scholarly and Aca

2 RACK VIBRAIONS DURING ACCEERAING AND DECEERAING PHASES OF HIGH-SPEED RAIS K. K. ANG 1*,. M. HI, and. V. HAI 1, Dparmn of Civil and Environmnal Enginring, Naional Univrsiy of Singapor, Singapor Dparmn of Civil Enginring, Ho Chi Minh Ciy Univrsiy of chnology, Vinam Naional Univrsiy - Ho Chi Minh, Vinam ABSRAC In his papr, a compuaional sudy using h moving lmn mhod (MEM) was carrid ou o invsiga h dynamic rspons of a high-spd rail (HSR). A nw formulaion for calculaing h gnral mass, damping and siffnss marics of h moving lmn is proposd. Rsonanc in h vibraion rspons of h rail rack is found o occur during boh h acclraing and dclraing phass of h HSR. As o b xpcd, rack vibraion paks whn h HSR ravls a a spd in h viciniy of h rsonanc vlociy during h ransiion acclraing or dclraing phass rahr han a h consan vlociy phas. A paramric sudy is carrid ou o undrsand h ffcs of various facors on h rspons of h rain-rack sysm such as h rain acclraion/dclraion magniuds, h svriy of railhad roughnss and h magniud of whl load. Kywords: moving lmn mhod, whl-rail inracion, rack irrgulariy 1. INRODUCION Railway ransporaion is on of h ky mods of ravl oday. h advancmn in rain chnology lading o fasr and fasr rains is wihou doub a posiiv dvlopmn, which maks HSR ravl mor araciv as a viabl alrnaiv o ohr mods of ransporaion for long disanc ravl. In daling wih moving load problms, such as ha ncounrd in HSR, h fini lmn mhod (FEM) ncounrs difficuly whn h moving load approachs h boundary of h fini domain and ravls byond h boundary. hs difficulis can b ovrcom by mploying a larg nough domain siz bu a h xpns of significan incras in compuaional im. o ovrcom h complicaion ncounrd by FEM, Krnk al. (1999) proposd h us of FEM in convcd coordinas o obain h rspons of an lasic half-spac subjc o a moving load. h ky advanag njoyd by his approach is is abiliy o ovrcom h problm du o h moving load * Corrsponding auhor: kk_ang@nus.du.sg Prsnr: ranminhhi@nus.du.sg 1

3 ravlling ovr a fini domain. Andrsn al. (001) mployd h sam approach o solv h problm of a bam on a Klvin foundaion subjc o a harmonic moving load. Koh al. (00) adopd h ida of convcd coordinas for solving rain-rack problms, and namd h numrical algorihm as h moving lmn mhod (MEM). h mhod was subsqunly applid o h analysis of in-plan dynamic rspons of annular disk (Koh al. 006) and moving loads on a viscolasic half spac (Koh al. 007). Rcnly, Ang al. (01) applid h MEM o invsiga h jumping whl phnomnon in high-spd rain moion a consan vlociy ovr a ransiion rgion whr hr is a suddn chang of foundaion siffnss. Various rsarchrs hav invsigad h problm of loads ravlling a non-uniform vlociis. Suzuki (1977) mployd h nrgy mhod o driv h govrning quaion of a fini bam subjc o ravling loads involving acclraion. By using analyical soluions, Yadav (1991) has invsigad h vibraion rspons of a rain-rack-foundaion sysm rsuling from a vhicl ravlling a variabl vlociis. Andrs Karlsrom (006) also usd an analyical approach o invsiga ground vibraions du o acclraing and dclraing rains. Safy concrns during h acclraing and dclraing phass of a high-spd rain journy hav no bn adqualy addrssd in h liraur. On major concrn is h possibl occurrnc of rsonanc of h sysm whn h frquncy of h xrnal forc, in his Figur 1: h HSR modl cas h rail corrugaion, coincids wih h naural frquncy of a significan vibraion mod of h sysm. Whn his happns, h rspons of h sysm is dynamically amplifid and bcoms significan larg. his papr is concrnd wih a compuaional sudy of h dynamic rspons of HSR sysms involving acclraing/dclraing rains using h MEM. A nw formulaion for calculaing h gnral srucural marics of h moving lmn is proposd. A paramric sudy is prformd o undrsand h ffcs of various facors on h rspons of h rain-rack sysm, such as h magniuds of rain acclraion/dclraion, h svriy of railhad roughnss and h magniud of whl load.. FORMUAION AND MEHODOOGY h HSR sysm, as shown in Figur 1, compriss of a rain modld as a moving sprung-mass ravrsing ovr a rail bam in h posiiv x -dircion. h origin of h fixd x -axis is arbirarily locad along h bam. Howvr, for convninc, is origin is akn such ha h rain is a x 0 whn 0. h vlociy and acclraion of h rain a any insan ar v and a, rspcivly. h railhad is assumd o hav som imprfcions rsuling in h so-calld rack irrgulariy. In h

4 moving sprung-mass modl, h opmos mass m rprsns h car body whr h passngrs 1 ar. h car body is suppord by a bogi of mass m hrough a scondary suspnsion sysm modld by h spring k and dashpo 1 c 1. h bogi is in urn suppord by a whl-axl sysm of mass m hrough a primary suspnsion sysm modld by h spring k and dashpo c. h conac bwn h whl and rail bam is modld by h spring k and dashpo c. h conac forc bwn h whl and rail bam is dnod by F c. h rail bam rss on a viscolasic foundaion comprising of vrical springs k and dashpos c. h vrical displacmn of h rail bam is dnod by y, whil h vrical displacmns of h car body, bogi and whl-axl ar dnod by u 1, u and u, rspcivly. h govrning quaion of moion of h rail bam, which is modld as an Eulr-Brnoulli bam rsing on a viscolasic foundaion subjc o a moving load, is givn by 4 y y y EI m c ky F ( ) 4 c x s x (1) whr E, I and m ar h Young s modulus, scond momn of inria and mass pr uni lngh of h rail bam, rspcivly; s h disanc ravld by h rain a any insan im ; and h Dirac-dla funcion. h moving lmn mhod adops h ida in which h origin of h spaial coordinas sysm is aachd o h applid poin of h moving load. Figur 1 shows a ravlling r-axis moving a h sam spd as h moving load. h rlaionship bwn h moving coordina r and h fixd coordina x is givn by r x s () In viw of quaion (), h govrning quaion in quaion (1) may b rwrin as 4 y y y y y y y EI m v v a c v ky F ( ) 4 c r r r r r r () By adoping Galrkin s approach and procdur of wriing h wak form in rm of h displacmn fild, h formulaion for gnral mass M, damping C and siffnss K marics of h moving lmn can b drmind as follows M m N N 0 dr, r d d 0 0 C mv N N r c N N r, rr, rr rr r 0, 0, 0 0 K EI N N dr mv NN dr ma cv NN dr k NN dr (4)

5 whr h subscrip r dnos parial drivaiv wih rspc o r. For bam lmns, i is common o us h shap funcion N basd on Hrmiian cubic polynomials. Considring h spcial cas in which h rain ravrss a a consan vlociy V, i.. a 0, v V, quaion (4) rducs o M m N N 0 dr, r d d 0 0 C mv N N r c N N r, rr, rr rr r 0, 0, 0 0 K EI N N dr mv NN dr cv NN dr k NN dr (5) which ar nod o b idnical o h marics drivd by Koh al. (00). In gnral, h whl conac forc F c may b wrin as F c c y k y (6) whr h ovrdo opraor dnos diffrniaion wih rspc o im and h conac surfac, can b xprssd as y y y u r y, h indnaion a in which y r and u dno h displacmns of h rail and whl, rspcivly, and y h magniud of h rack irrgulariy a h conac poin. No ha rack irrgulariy is a major sourc of h dynamic xciaion. According o h rcommndaion by Nilsn and Abrahamsson (199), h rack irrgulariy profil can b wrin in rms of a sinusoidal funcion as follows (7) y x asin (8) whr a and dno h ampliud and wavlngh of h rack irrgulariy, rspcivly. As h dynamic rspons of h rain-rack sysm dpnds significanly on h accuracy in modling h conac bwn h whl and rack, Hrz conac hory (Esvld 001) is mployd o accoun for h nonlinar conac forc F bwn h whl and rail as follows c F KH y for y 0 c 0 for y 0 (9) E RwhlRrailprof whr K H (10) (1 ) 4

6 in which K H dnos h Hrzian spring consan; R whl and R railprof h radii of h whl and railhad, rspcivly, and h Poisson s raio of h marial. h govrning quaions for h vhicl modl ar m u k u u c u u c 1 u 1 u c 1 u 1 u m 1 u 1 k 1 u 1 u m 1 g k 1 u 1 u m g m u k u u c u u m g F c whr g dnos graviaional acclraion. Upon combining quaion (11) wih h govrning quaions for h rail bam givn in quaion (), h quaion of moion for h rain-rack sysm may b wrin as (11) Mz Cz Kz P (1) whr z, z, z dno h global acclraion, vlociy and displacmn vcors of h rain-rack sysm, rspcivly; M, C and K h global mass, damping and siffnss marics, rspcivly; and P h global load vcor. h abov dynamic quaion can b solvd by any dirc ingraion mhods such as Nwmark- mhod (Bah 1996).. NUMERICA RESUS o vrify h accuracy of h proposd MEM approach in obaining h dynamic rspons of a HSR considring variabl rain spd, h prsn soluions ar compard agains soluions obaind by Koh al. (00) using h so-calld cu-and-pas FEM. h lar involvs updaing h forc and displacmn vcors in accordanc wih h posiion of h vhicl whil kping h srucur mass, damping and siffnss marics consan. For h purpos of comparison only, h sam rain spd profil adopd by Koh al. (00) is mployd. his spd profil is shown in Figur whr i can b sn ha hr ar phass of ravl. h spd profil paramrs for his cas ar prsnd in abl undr Cas 1. h iniial phas considrs h rain o b moving a a consan acclraion of ravl and raching a maximum spd of 0 m/s afr s. During h scond phas of ravl, h rain movs a h maximum consan spd for anohr s. In h final phas, h rain dclras a a consan magniud o com o a compl hal afr anohr s of ravl. Rsuls obaind using h proposd mhod ar found o b in xclln agrmn wih hos obaind by h cu-and-pas FEM. In h following scions, h ffcs of ampliuds of rain acclraion/dclraion, rack irrgulariy and whl load on dynamic rspons of h rain-rack sysm during h acclraing or dclraing phass using h proposd MEM approach ar prsnd. h paramrs for various rain spd profils considrd ar prsnd in abl undr cass o 4. h proposd MEM modl adopd in h sudy compriss of a runcad railway rack of 50 m lngh uniformly 5

7 discrizd ino 50 moving fini lmns. Valus of paramrs rlad o h propris of rack and foundaion ar summarizd in abl 1 (Koh al. 00). h quaions of moion ar solvd using Nwmark s consan acclraion mhod mploying a im sp of s. his small im sp siz is ncssary in viw of h inhrn high naural frquncy of h rain-rack sysm. In analyss involving h Hrz nonlinar conac modl, Nwon-Raphson s mhod (Bah 1996) is mployd o solv h rsuling nonlinar quaions of moion. h radii of h whl R, whl railhad R railprof and h Poisson s raio of h whl/rail marial usd in drmining h nonlinar Hrz spring consan ar akn o b 460 mm, 00 mm and 0., rspcivly. abl 1: Paramrs for rack-foundaion modl Figur : Profil of rain spd Paramr Valu Flxural siffnss N m rack scion UIC 60 (60 E1) Siffnss of foundaion N/m Damping raio 0.1 Cas Maximum vlociy V (m/s) max abl : Profils of rain vlociis Ampliud of acclraion/dclraion a (m/s ) im paramrs 1 (s) (s) (s) Effc of ampliuds of rain acclraion/acclraion As h siffnss marix of h moving lmn and h rack irrgulariy dpnd on h rain acclraion or dclraion ampliuds, i is ncssary o invsiga h ffc of h ampliuds on h dynamic amplificaion facor (DAF) in whl-rail conac forc. No ha hr ampliuds of rain acclraion/dclraion (Cass, and 4), as shown in abl, and hr rack irrgulariis such as smooh (0.01 mm), modra (0.5 mm) and svr ( mm) ar considrd. No ha h wavlngh of all rack irrgulariis is chosn o b 1 m. Figur shows h ffc of rain acclraion/dclraion ampliuds on h DAF. h rsuls obaind show ha h magniud of acclraion/dclraion has ngligibl ffc on h DAF for all cass considrd. In viw of his finding, all ohr rsuls o b subsqunly prsnd shall prain o cas, considrd o b h ypical spd profil of oday s HSR ravls... Effc of rack irrgulariy ampliud h ffc of rack irrgulariy ampliud on h DAF is nx invsigad. No ha h wavlngh of all rack irrgulariis is chosn o b 1 m. In ordr o compar h DAF obaind a rsonan 6

8 spd during acclraing/dclraing phass, i is ncssary o drmin h DAF whn h rain ravls a h consan rsonan spd compud as, whr is h priod of h sysm. Figur 4 shows how rack irrgulariy ampliud affcs h DAF. For a nar smooh rack ( a 0.01 mm), h DAF is found o b approximaly 1, as o b xpcd. Whn h ampliud of rack irrgulariy incrass, h DAF is nod o incras gradually and hn significanly. I is inrsing o no ha h DAF is found o b highr during boh h acclraing and dclraing phass as compard o h cas in which h rain ravls a a consan spd qual o h rsonan spd. h DAF is also nod o b slighly largr during h dclraing phas as compard o h acclraing phas. Figur : Effc of ampliuds of rain acclraion/dclraion on h DAF. Figur 4: Effc of rack irrgulariy ampliud on h DAF... Effc of rack irrgulariy wavlngh As h priod of HSR sysm is fixd, h magniud of rack irrgulariy wavlngh is a significan facor on conrolling h occurrnc of h rsonan phnomnon during h acclraing and dclraing phass. Rsonanc occurs brifly during acclraing/dclraing phass whn h rain spd rachs h magniud of h rsonan spd. his occurs whn h maximum rain spd is highr han h rsonan spd. Figur 5: Effc of rack irrgulariy Figur 5 shows h ffc of rack irrgulariy wavlngh on h DAF. wavlngh on h DAF. No ha all irrgulariy ampliuds considrd ar mm. I is xpcd ha shorr irrgulariy wavlngh would lad o largr vibraions as can b obsrvd whn h wavlngh is small a 0.5 m. Whn h 7

9 wavlngh of rack irrgulariy incrass, h DAF is found o dcras. I is inrsing o no ha h DAF is highr during boh h acclraing and dclraing phass as compard o h cas in which h rain ravls a a consan spd qual o h rsonan spd..4. Effc of whl load h ffc of whl load on dynamic rspons of h sysm is nx invsigad. wo whl loads of 41 kn and 81 kn ar considrd. h smallr load corrsponds o h cas of a vry lighly loadd vhicl (Koh al. 00), and h highr load corrsponds o a ypical passngr vhicl. Figur 6 shows h ffc of whl load on h DAF for various valus of rack irrgulariy ampliud. No ha h wavlngh of all rack irrgulariis is chosn o b 1 m. I is xpcd ha h lighr whl load is, h largr DAF is, spcially for h largr rack irrgulariy ampliud. Using various valus of rack irrgulariy wavlngh, Figur 7 shows h ffc of whl load on h DAF. In his cas, h ampliud of all rack irrgulariis is chosn o b mm. I is also xpcd ha h DAF is largr whn h whl load is lighr, spcially for h smallr rack irrgulariy wavlngh. Figur 6: Effc of whl load and rack irrgulariy ampliud on h DAF. Figur 7: Effc of whl load and rack irrgulariy wavlngh on h DAF. 4. CONCUSIONS In his papr, a numrical sudy on h dynamic rspons of HSR sysm using h moving lmn mhod was carrid ou. A nw and gnral formulaion for calculaing h srucural marics of h moving lmn was proposd. h ffcs of magniuds of rain acclraion/dclraion, rack irrgulariy and whl load o rspons of HSR sysm during a rain journy ar invsigad. h rsuls obaind using h proposd MEM is found o agr wll wih rsuls found in h liraur using h cu-and-pas FEM. I is found ha h magniud of acclraion/dclraion has ngligibl ffc on h DAF. h DAF is mor incrasing whn h irrgulariy ampliud/wavlngh is mor incrasing/dcrasing. As o b xpcd, h DAF ar largr whn h HSR ravls a a rsonan spd during h acclraing/dclraing phass rahr han h cas 8

10 in which h rain ravls a a consan spd qual o h rsonan spd. Whn h whl load is lighr, h DAF is largr, spcially for h svr rack irrgulariy. 5. ACKNOWEDGMENS his rsarch is fundd by Vinam Naional Univrsiy Ho Chi Minh Ciy (VNU-HCM) undr gran numbr B REFERENCES Suzuki SI (1977). Dynamic bhavior of a fini bam subjcd o ravlling loads wih acclraion. Journal of Sound and Vibraion. 55(1), pp Yadav D (1991). Non-saionary dynamics of rain and flxibl rack ovr inrial foundaion during variabl vlociy. Journal of Sound and Vibraion. 147(1), pp Andrs Karlsrom (006). An analyical modl for ground vibraions from acclraing rains. Journal of Sound and Vibraion. 9, pp Krnk S, Kllzi, Nilsn SRK, and Kirkgaard PH (1999). Fini lmns and ransmiing boundary condiions for moving loads. Procdings of h 4h Europan Confrnc on Srucural Dynamics, Eurodyn 99, Praha, Jun 7-1, Vol. 1, pp Andrsn, Nilsn SRK, and Kirkgaard PH (001). Fini lmn modlling of infini Eulr bams on Klvin foundaions xposd o moving loads in convcd co-ordinas. Journal of Sound and Vibraion. 41(4), pp Koh CG, Ong JSY, Chua DKH, and Fng J (00). Moving Elmn for rain-rack Dynamics. Inrnaional Journal for Numrical Mhods in Enginring. 56, pp Koh CG, Sz PP, and Dng (006). Numrical and analyical mhods for in-plan dynamic rspons of annular disk. Inrnaional Journal of Solids and Srucurs. 4, pp Koh CG, Chiw GH, and im CC (007). A numrical mhod for moving load on coninuum. Journal of Sound and Vibraion. 00, pp Ang KK, Dai J, and hi M (01). Analysis of high-spd rail accouning for jumping whl phnomnon. h Inrnaional Confrnc on Advancs in Compuaional Mchanics (ACOME), Augus 14-16, Ho Chi Minh Ciy, Vinam. Nilsn JCO and Abrahamsson JS (199). Coupling of physical and modal componns for analysis of moving non-linar dynamic sysms on gnral bam srucurs. Inrnaional Journal for Numrical Mhods in Enginring., pp Esvld C (001). Modrn Railway rack (nd Ediion). MR Producions: Duisburg. Bah KJ (1996). Fini Elmn Procdurs. Prnic-Hall, Englwood Cliffs, N.J. 9