A Versatile Method for Analyzing the Influence of Track Irregularity on Vehicle-track-bridge Coupled System

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1 Rah Jounal of Applid Sin, Engining and Thnology 7(6): , 01 ISSN: ; -ISSN: Maxwll Sinifi Oganizaion, 01 Sumid: Mah 0, 013 Apd: Mah 9, 013 Pulihd: Fuay 15, 01 A Vail Mhod fo Analyzing h Influn of Tak Igulaiy on Vhil-ak-idg Coupld Sym 1 Zhang Hui, 1 L.Q. Yao, 1 Zhao Qian-Fng and Wang Huan-Ran 1 Shool of Uan Rail Tanpoaion, Soohow Univiy, China Shool of Mhanial Engining and Mhani, Ningo Univiy, China Aa: A vail fini lmn mhod i applid o analyz h dynami pon of ailway ak and idg und moving ailway vhil. Th whol ym i dividd ino wo uym. Th vhil and ailway ak a gadd a an ingad uym whil h ailway ak and idg a gadd a h oh uym. Th quaion of moion fo h wo lmn a dily divd y man of Hamilon pinipl. Af y amling h iffn, damping, ma mai and h vo of nodal load of all lmn, h gloal quaion of moion a oaind and olvd y Nwmak-β mhod. Numial xampl dmona ha h mhod i vail and o whil daling wih h dynami pon of vhil-ak-idg oupld ym. Th xampl alo dmona ha h influn of h ak igulaii on h ingadd ym i vy ignifian and ha h a of val diffn ak igulaii xiing a h am im play mo ignifian on h dynami pon han h a of on ak igulaiy. Kywod: Fini lmn, Hamilon pinipl, ak igulaii INTRODUCTION Th dynami pon of ak and idg uu ujd o moving vhil hav long n an ining opi in h fild of ailway ngining. Two kind of numial mhod, i.., modal uppoiion mhod and fini lmn mhod, a widly ud o akl h polm. Fo h inaion ym wn moving vhil and idg o am, h modal uppoiion ha n ud in Rfn (Fýa, 1999; Huon and Chung, 1979; Chung al., 1999; Yang and Fond, 1996; Li, 00; Lou, 005a; Zhai, 1998). On paaion modal mhod ha n ud whn h aov ah alihd h ym quaion. Th mhod dividd h whol ym ino wo uym a h infa of h idg and vhil whil h wo uym a olvd in oninuou iaion y whl-ail fo paaly. Th pon of h whol ym an oaind u h ompuaional ffiiny and onvgn of hi mhod a no vy good. Thn auming h igid onnion wn h whl and ail whn (Lou, 005) alihd h whol ym ha ignod h nomal ona wn h whl and ail and i would au a lag dviaion. Zhai (1998) popod a nw impl xplii ingaion mhod whn h alihd a whol ym oniding h paaion wn h whl and ail. Thi mhod ha n widly ud la u h i a lil oul amling h whol maix. Anoh numial mhod, ha i, h fini lmn mhod i alo vy vail y many ah (Lin and Thwy, 1990; Yang al., 1999; Yang and Wu, 001; Chng al., 001; Lou and Zng, 005; Lou, 007). Chng al. (001) ud a ypial idg-ak-vhil fini lmn o inviga h inaion among vhil and ak uu and idg uu. Th wo yp of quaion of moion of fini lmn fom fo h ni ym a divd y man of h pinipl of a aionay valu of oal ponial ngy of dynami ym in Lou and Zng (005). Lou (007) alo udid h inaion wn val vhil and guid-way ym y h fini lmn mhod. Th aov ah alihd h whol ym quaion y h fini lmn mhod u h i a lil oul amling h whol maix a am a Zhai (1998). In hi udy, h dynami pon of ailway ak and idg und a moving ailway vhil a invigad y man of fini lmn mhod. Th whol ym i dividd ino wo uym. Th vhil and ailway ak a gadd a an ingad uym whil h ailway ak and idg a gadd a h oh uym. Th quaion of moion fo h wo lmn a dily divd y man of Hamilon pinipl. Af y amling h iffn, damping, ma mai and h vo of nodal load of all lmn, h gloal quaion of moion a oaind and olvd y Nwmak (1959) -β mhod. Th whol maix an aily amld y hi mhod and h ul an alulad in l im. Thi udy alo udid h ff of val diffn ak igulaii on h dynami pon of vhil-ak-idg oupld ym. Wha mo, h pap onidd h a of val diffn yp Coponding Auho: L.Q. Yao, Shool of Uan Rail Tanpoaion, Soohow Univiy, China 1156

2 of ak igulaiy xiing a h am im whil analyzing h ingad ym. HAMILTON S PRINCIPLE I i wll-known ha Hamilon pinipl an xpd in h fom: R. J. Appl. Si. Eng. Thnol., 7(6): , 01 δ( T V ) d + W d 0 δ = (1) n 1 1 wh, T dno h kini ngy fo an ni dynami ym, V dno h ponial ngy fo an ni dynami ym, δδδδ nnnn dno h viual wok don y h nononvaiv fo fo an ni dynami ym and δ i h vaiaion ymol. Equaion (1) how ha h um of h im-vaiaion of h diffn in kini and ponial ngi and h wok don y h non-onvaiv fo ov any im inval i - qual zo. Th appliaion of hi pinipl lad dily o h quaion of moion fo any givn ym. By omining wih h fini lmn mhod and applying Eq. (1) o an lmn, h quaion of moion fo an lmn an alo alihd. Equaion of moion fo h ym of vhil and idg oniding ak uu: Modl of vhil-ak-idg ingad ym: Figu 1 how a ain oniing of a i of idnial fou-whl vhil moving on a ak uu ing on a i of muli-pan oninuou am o modl ailway idg and h wo appoah mankmn. Th ain ompi N v idnial vhil fom lf o igh and pod wih pd v and alaion a a im along h longiudinal diion. I i aumd ha ah whl of all vhil alway mainain ona wih h ail. Eah vhil in h ain i modld a a ma-ping-damp ym oniing of a a ody, wo ogi, fou whl and wo-ag upnion. I i aumd ha h downwad vial diplamn and lokwi diion oaion of vhil a akn a poiiv and ha hy a maud wih fn o hi piv ai quiliium poiion fo oming ono h ak onnd. Th ail i modld a a lina lai Bnoulli- Eul am uppod y di violai uppo, h la i modld a a lina lai Bnoulli-Eul am uppod y oninuouly violai uppo, h idg a modld a a i of muli-pan oninuou Bnoulli-Eul am, h lf and igh mankmn a modld a igid in idg pa. In h analyi, h whol ym i dividd ino wo uym. Th vhil and ailway ak a gadd a an ingad uym whil h ailway ak and idg a gadd a h oh uym. Modl and quaion of ak-idg ingad lmn: A hown in Fig., h modl of ak-idg inaion lmn oni of ail lmn, la lmn and idg lmn. Th ail lmn and h la lmn a onnd y om di violai uppo whil h la lmn and h idg lmn a onnd y oninuouly violai uppo. Aoding o Hamilon pinipl, h gna h oponding kini ngy T 1, ponial ngy V 1 and viual wok δδδδ n 1. Thn ingaion fo h vaiaion of h kini ngy, h ponial ngy and h viual wok ov any im inval 1 o i: 1 Nv T d T d T d T d () δ = δ + δ + δ 1 1 δ = δ + δ 1 V d U d U d δ δ δ + U d + U d + U d n1 n, n, n, (3) δw = δw + δw + δw () wh, T, T & T : Th kini ngy of h i h ail am lmn, h j h la am lmn and h k h idg am lmn pivly v Sla Rail Lf mankmn Bidg Righ mankmn Fig. 1: Modl of vhil-ak-idg ingad ym 1157

3 R. J. Appl. Si. Eng. Thnol., 7(6): , 01 θi yi Rail yi+1 θi+1 x wh, all maix a maix: θj yj d Sla kd yj+1 θj+1 h vo {} q = [ y θ y θ ] i i i ( i 1) ( i 1) + + θk yk y k Bidg yk+1 θk+1 Fig. : Modl of ak-idg inaion lmn Ca ody k Ra ogi m J kw yj θj w xj1 xj y1j θ1j y l Rail m yj θj kw k J m J w xj3 xj Fig. 3: Modl of vhil-ak inaion lmn v Fon ogi U, U & U : Th flxual ain ngy of h i h ail am lmn, h j h la am lmn and h k h idg am lmn pivly U : Th ping ain ngy of di ping wn h i h ail am lmn and h j h la am lmn U : Th ping ain ngy of oninuou ping wn h j h la am lmn and h k h idg am lmn δδww nnnn, : Th innal viual wok pfomd y no onvaiv fo wn i h ail am lmn and j h la am lmn δδww nnnn, : Th innal viual wok pfomd y no onvaiv fo wn j h la am lmn and k h idg am lmn δδww nnnnnn : Th innal viual wok pfomd y no onvaiv fo fo h k h idg am lmn Sin all vaiaion a aiay, on an oain h vial quaion of moion of a ak-idg inaion lmn. Th quaion an win in paiiond fom a: x (5) 1158 h vo {} q = [ y θ y θ ] i i i ( i+ 1) ( i+ 1) h vo {} q = [ y θ y θ ] i i i ( i 1) ( i 1) + + Modl of vhil-ak ingad lmn: A hown in Fig. 3, h modl of vhil-ak inaion lmn oni of vhil and fou ail lmn und fou whl-. Vhil alo oni a a ody, wo ogi, fou whl- and wo-ag upnion. Eah whl of vhil i aumd o alway in ona wih h upp ail. Aoding o Hamilon pinipl, h gna h oponding kini ngy T, ponial ngy V and viual wok δδww nnnn. Thn ingaion fo h vaiaion of h kini ngy, h ponial ngy and h viual wok ov any im inval 1 o i: = + δt d δt d δt d + δt d + 1 w 1 δt d n n, n, w (6) δw = δw + δw (7) + + δ = δ + δ V d V d V d δv d w 1 1 δu d w δu d δu d wh, T : Th kini ngy of h anlaion of h n of h a ody and h oaion aou i n T : Th kini ngy of h anlaion of h n of h wo ogi and h oaion aou hi n T w : Th kini ngy of h fou axl du o anlaion of h n onidd paaly T : Th kini ngy of h fou ail am lmn und h fou whl pivly U : Th ping ain ngy of h vhil and wo ogi U w : Th ping ain ngy of h ogi and whl U (8) : Th flxual ain ngy of h all ail am lmn und h fou whl δδww nnnn, : Th innal viual wok pfomd y no onvaiv fo wn h vhil and wo ogi

4 R. J. Appl. Si. Eng. Thnol., 7(6): , 01 δδww nnnn, : Th innal viual wok pfomd y no onvaiv fo wn h ogi and whl V, V & V w : Th ponial ngy of h gaviy of h a ody, h gaviy of h wo ogi and h gaviy of h fou whl (x) la ā x Sin all vaiaion a aiay, on an oain h vial quaion of moion of a vhil-ak inaion lmn. Th quaion an win in paiiond fom a: Fig. : Igulaiy on ak ufa.5 (9) wh, [M] w, [C] w, [K] w a 6 6 maix, [C] v, [K] v a 6 16 maix, [C] v, [K] v a 6 16 maix, [M], [C], [K] a 6 16 maix, h vo {} q = [ y θ y θ y θ ]. v j j 1j 1j j j Th quaion of moion of vhil-ak-idg ingad ym: Th onvnional amly po an mployd o fom h gloal quaion of moion fo h ni vhil-ak-idg ym, whih will appa a: [ M]{ q } + [ M]{ q } + [ M]{ q} = { P} () wh, h mai [M], [C] and [K] a h gloal ma, damping and iffn mai of h ni ym pivly, h vo {P} i h gloal vo of h ni ym, h vo {q}, {qq } and {qq } dno h diplamn, vloiy and alaion vo of h ni ym. Equaion an olvd y p y p ingaion mhod uh a h Nwmak-β mhod o Wilon-θ mhod o oain imulanouly h dynami pon of vhil, ak o idg. NUMERICAL EXAMPLES An xampl o vify h vail mhod and o analyi h influn of ak igulaiy on ym: A ingl-pan imply uppod ailway idg wih h wo appoah uppod on mankmn i onidd, a hown in Fig. 1. I i aumd ha h lngh of ak uu on ah appoah mankmn i qual. Th following daa a adopd fo h vhil: ma of a ody m =.175 kg, ma momn of inia of a ody J =.08 6 kg m, ma of on ogi m = 300 kg, ma momn of inia of on ogi J = kg m, ma of on whl m w = kg, ping iffn of h ond upnion ym k = N/m, damping offiin of h 1159 Vial diplamn (mm) mooh ufa oniding igulaiy () Fig. 5: Vial diplamn of a ody ond upnion ym = 9.0 N / m, ping iffn of h pimay upnion ym 6 k = 1.18 N / m, damping offiin of h pimay w upnion ym 3.9 N / m w =, half of hoizonal dian wn wo ogi L = 8.75m, half of hoizonal dian wn wo axl L = 1.5m, vhil vloiy v = 7.78 m/ and a = 0 m/. Th following daa a adopd fo h ak uu: h oal longiudinal lngh of ak uu m, 11 5 =.06 pa, =.037 m m = 51.5 kg / m, E E =.1 pa, I I = m, m kg / m =, iffn of 13 di ping wn ail and la k = N / m, d offiin of di damp wn ail and la = 0 N / m, iffn of oninuou ping wn d 7 la and idg k = 6.58 N / m, offiin of di damp wn la and idg = 3.1 N / m. Th following daa a adopd fo h imply uppod idg: L = 30m, E =.93 pa, I =.88m, m = 1. kg / m and damping aio of idg ζ =. Duing h analyi, h quaion of moion fo h ingad ym a olvd y h Nwmak-β mhod wih im p = 05.

5 R. J. Appl. Si. Eng. Thnol., 7(6): , 01 Vial alaion (m/ ) mooh ufa oniding igulaiy - () Fig. 6: Vial alaion of a ody Vial diplamn (mm) mooh ufa oniding igulaiy -0.5 () Fig. 7: Vial diplamn of midpoin of h ail Vial alaion (m/ ) mooh ufa oniding igulaiy -30 () Fig. 8: Vial alaion of midpoin of h ail Vial diplamn (mm) mooh ufa oniding igulaiy -0.5 () Fig. 9: Vial diplamn of midpoin of h idg Vial alaion (m/ ) 0. mooh ufa 0.3 oniding igulaiy () Fig. : Vial alaion of midpoin of h idg Cona fo (kn) mooh ufa oniding igulaiy 16 () Fig. 11: Cona fo wn fon axl of fon ogi and ail A hown in Fig., on yp of igulaiy funion fo h vial pofil of h ak i adopd: 1 ( x) = a(1 o π x/ l a ) (11) wh, x i h along-ak dian, aa i maximum dph of ak igulaiy and l a i h lngh of ak 3 igulaiy, a = 1.0 m and l = 1m a ud fo h a 1160 loal ak igulaiy. I i aumd ha h poin of maximum dph of h ak igulaiy loa a h midpoin of h ail. Coniding h ak uu igulaiy, h im-hioy pon of h n of a ody, of h midpoin of h ail, of h midpoin of idg and wo ona fo wn whl- and ail hav n plod in Fig. 5 o 1 along wih ho oniding mooh ak ufa. A hown in Fig. 5 o 1, h dynami pon wihou oniding ak igulaiy a am a ho

6 R. J. Appl. Si. Eng. Thnol., 7(6): , 01 Tal 1: Th maximum of dynami pon y val yp of ak igulaiy Th max. of dynami pon Fi yp Sond yp Thid yp Fouh yp Ca ody Di. (mm) A. (m/ ) Midpoin of ail Di. (mm) A. (m/ ) Midpoin of idg Di. (mm) A. (m/ ) Cona fo wn fon axl of fon ogi and ail (kn) Max.: Maximum Cona fo (kn) mooh ufa oniding igulaiy 16 () Fig. 1: Cona fo wn a axl of a ogi and ail in Lou (005). Thfo, h vailiy and auay of hi mhod hav n povd. A hown in Fig. 5 and 9, h dynami pon fo oniding loal ak igulaiy a almo am a ho fo oniding mooh ufa of ak. Howv, a hown in Fig. 6 o 8 and o 1, h dynami pon, i.., vial alaion of a ody, vial diplamn and vial alaion of midpoin of h ail, vial alaion of midpoin of idg and wo ona fo wn whl- and ail, fo oniding ak igulaiy hav ignifian vaiaion whil h vhil aiv a h poiion of loal ak igulaiy. Oviouly, mainaining a mooh ak ufa in ailway ngining i vy impoan. Th influn of ak igulaiy on vhil-akidg oupld ym: Th a val yp of ak igulaiy o analyi h influn of ak igulaiy on vhil-ak-idg oupld ym a follow: Th fi yp: I i aumd ha h i mooh ufa in ail. Th ond yp: I i aumd ha h i on yp of ak igulaiy a am a hi in xampl 1. Th hid yp: I i aumd ha h i a 0 m lngh of uidn und igh of h midpoin of h ail. Now h iffn of ping and offiin of damp wn la and idg a all zo Th foh yp: I i aumd ha h i aovmniond of h ond and h hid yp of ak igulaiy a h am im. Th dynami pon of vhil-ak-idg oupld ym y h influn of h aovmniond ak igulaiy a hown in Tal 1. A hown in Tal 1, ompad wih h fi yp, h dynami pon fo oniding h ond yp of ak igulaiy, i.., vial alaion of a ody, vial diplamn and alaion of h midpoin of ail, vial alaion of h midpoin of idg and ona fo wn fon axl of fon ogi and ail hav ignifian vaiaion whil h oh a am a ho in h fi yp; Compad wih h fi yp, h dynami pon fo oniding h hid yp of ak igulaiy, i.., vial diplamn and alaion of a ody, vial diplamn and alaion of h midpoin of ail, vial diplamn and alaion of h midpoin of idg hav ignifian vaiaion whil h oh a am a ho in h fi yp; Compad wih h ond yp and hid yp, vial diplamn and alaion of a ody and vial diplamn of h midpoin of idg fo oniding h fouh yp of ak igulaiy a am a ho in h hid yp whil h oh dynami pon all ina lvanly. CONCLUSION In hi udy, h dynami pon of ailway ak and idg und a moving ailway vhil a invigad y man of fini lmn mhod. Th whol ym i dividd ino wo uym. Th vhil and ailway ak a gadd a an ingad uym whil h ailway ak and idg a gadd a h oh uym. Th quaion of moion fo h wo lmn a dily divd y man of Hamilon pinipl. Af y amling h iffn mai, h damping mai, h ma mai and h vo of nodal load of all lmn, h gloal quaion of moion fo h ingad ym a oaind. Th quaion an olvd y p-yp ingaion mhod, o oain imulanouly h dynami pon of vhil, ak and idg. Th ul how ha h whol maix an aily amld y hi mhod and h ul an alulad in l im.

7 R. J. Appl. Si. Eng. Thnol., 7(6): , 01 Fom h numial ul oaind in aov wo xampl, wo onluion an ahd: Th mhod ud in hi udy an widly applid in h analyi of vhil-ak-idg oupld ym. Wha mo, h mhod ha mo vailiy and on on om diffn oj. Th ff of ak igulaiy on h dynami pon of ym a ignifian. Th aov wo xampl dmona ha vial alaion of a ody, vial diplamn and alaion of h midpoin of ail, vial alaion of h midpoin of idg and ona fo wn fon axl of fon ogi and ail, oniding loal ak igulaiy, hav ignifian vaiaion. Th ond xampl dmona ha vial diplamn and alaion of a ody, vial diplamn and alaion of h midpoin of ail, vial diplamn and alaion of h midpoin of idg, oniding loal ak uidn, hav alo ignifian vaiaion. Wha mo, h dynami pon oniding aov wo a xiing a h am im ina lvanly. I how ha h a of val diffn ak igulaii xiing a h am im play mo ignifian on h dynami pon han h a of on ak igulaiy. So mainaining a good ak uu in ailway ngining i vy impoan. ACKNOWLEDGMENT Th auho gafully aknowldg h uppo fom h Naional Naual Sin Foundaion of China und h ah gan No and h Impa and Safy of Coaal Engining Iniiaiv, a COE pogam of Zhjiang povinial govnmn a Ningo Univiy (Gan No. zj17). REFERENCES Chng, Y.S., F.T.K. Au and Y.K. Chung, 001. Viaion of ailway idg und a moving ain y uing idg-ak-vhil lmn. Eng. Su., 3(1): Chung, Y.K., F.T.K. Au, D.Y. Zhng and Y.S. Chng, Viaion of muli-pan non-unifom idg und moving vhil and ain y uing modifid am viaion funion. J. Sound Vi., 8(3): Fýa, L., Viaion of Solid and Suu und Moving Load. Thoma Tlfod, London, pp: Huon, S.G. and Y.K. Chung, Dynami pon of ingl pan highway idg. Eahq. Eng. Su. D., 115(1): Li, X., 00. Nw Mhod of Oial Mhani and Engining. China Railway P, China. Lin, Y.H. and M.W. Thwy, Fini lmn analyi of lai am ujd o moving dynami load. J. Sound Vi., 136(): Lou, P., 005a. Vial dynami pon of a imply uppod idg ujd o a moving ain wih wo-whl vhil uing modal analyi mhod. In. J. Num. Mh. Eng., 6(9): Lou, P., 005. A vhil-ak-idg inaion lmn oniding vhil' pihing ff. Fini Elm. Anal. D., 1(): Lou, P., 007. Fini lmn analyi fo ain-akidg inaion ym. Ah. Appl. Mh., 77(): Lou, P. and Q. Zng, 005. Fomulaion of quaion of moion of fini lmn fom fo vhil-akidg inaion ym wih wo yp of vhil modl. In. J. Num. Mh. Eng., 6(3): Nwmak, N.M., A mhod of ompuaion fo uual dynami. Po. Am. So. Civil Eng., 85(3): Yang, F. and G.A. Fond, An iaiv oluion mhod fo dynami pon of idg-vhil ym. Eahq. Eng. Su. D., 5(): Yang, Y.B. and Y.S. Wu, 001. A vail lmn fo analyzing vhil-idg inaion pon. Eng. Su., 3(5): Yang, Y.B., C.H. Chang and J.D. Yau, An lmn fo analyzing vhil-idg ym oniding vhil' pihing ff. In. J. Num. Mh. Eng., 6(7): Zhai, W.M., Two impl fa ingaion mhod fo lag-al dynami polm in ngining. In. J. Num. Mh. Eng., 39():