Dynamic Performance, System Identification and Sensitivity Analysis of the Ladder Tracks. Ontario, Canada

Transcription

1 Dynamc Pefomance, System Identfcaton and Senstvty Analyss of the adde Tacks D. Younesan 1, S. Mohammadzadeh 1, E. Esmalzadeh 1 School of Ralway Engneeng, Ian Unvesty of Scence and Technology, Tehan, Ian, Faculty of Engneeng and Appled Scence, Unvesty of Ontao Insttute of Technology, Oshawa, Ontao, Canada Abstact Dynamc pefomance of two types of ballasted ladde tacks s nvestgated n ths pape. Two types of tacks wth ladde unts havng 6 and 1 metes length ae studed. The al and ladde unts both ae modeled by Tmoshenko beam. Coupled set of equatons of moton ae then solved usng Galekn technque. The soluton s dected to obtan the thee most mpotant desgn factos n ladde tack systems.e. the al deflecton, the bendng moment geneated n the ladde unt and the ballast pessue. The effects of the ladde unt length and the load speed on the dynamc pefomance of the ladde tack system ae nvestgated. Intoducton Analyss of beams tavesed by movng loads has been of consdeable pactcal mpotance especally n last few decades. Many eseaches studed the vbaton of beams subjected to vaous types of movng loads [1-]. Fyba [3] has also cted moe than efeences n ths aea. The nteest ases fom lots of applcatons n the alway, bdge and tack engneeng. Today's concete cosstes wee developed to eplace wooden cosstes, and natually took the same shape. Snce the cosste system povdes poo load dspesal, howeve, the cosstes gadually snk unde the epettve loads appled onto the tack. ateal movement esstance n ths system s also weak. Fom the 194s to the 196s, epements wee caed out n Fance, the Sovet Unon and Japan wth longtudnal sleepes lad n paallel pas unde the als. The goal was to ceate a alway tack equng a mnmum of mantenance. Howeve, none of the epements wee successful. Based on ths epeence, the Stuctual Engneeng Goup of Japan's Ralway Techncal Reseach Insttute has developed the ladde sleepes n whch paallel longtudnal concete beams ae held togethe by tansvese steel ppes [6]. The lateal steel ppes ae embedded nto the longtudnal concete beams. adde sleepes povde contnuous suppot to the als and act wth them as composte als. When lad n stone ballast, the esultng gound pessue s educed and moe even fo the same oveall weght as fo concete cosstes. The ballasted and non-ballasted ladde tacks esult n ensung tan safety, assung mantenance educton, and envonmental mpovements, pomsng majo pogess n achevng alway effcency and mpovng alway management. The mnmum unt length s 6 metes and ts mamum value s up to 1 metes. Dynamc pefomance of two types of ladde tacks s studed n ths pape. In the fst type tack.e. type A the length of ladde unts s assumed to be 6 metes and the second type.e. type B s assumed to have 1 metes length. The objectve hee s to nvestgate the effect of the ladde unt length on the dynamc pefomance of a ballasted ladde tack. A 4-mete tack s dynamcally smulated fo both types of ladde unts. The al s modeled by a hnged-hnged Tmoshenko beam and the ladde unts ae modeled by a set of fee-fee Tmoshenko beams floatng ove the ballast. The ballast and al-pads ae both modeled by a set of paallel spngs and vscose dampes. The coupled set of equatons of moton s then solved usng Galekn technque. The soluton s dected to obtan the thee most mpotant desgn factos n ladde tack systems.e. the al deflecton, the bendng moment geneated n the ladde unts and the ballast pessue. Mathematcal modelng As shown n Fgue 1, the al s connected to the ladde unts, va al pads at 75 mm ntevals and ladde untes ae contnually suppoted by ballast. Accodng to the model shown n Fgue 1 and usng the Tmoshenko beam theoy, the equatons of moton of the al could be wtten as [4-5]

2 w R (, t) ψ R (, t) w R (, t) ρ R A R + k RA RG R ( ) = p(, t) + F(, t) t (1) F (N) adde unt (6 m) V (km/h) Ral Ral pad Ballast A) Tack type A F (N) V (km/h) adde unt (1 m) Ral Ral pad Ballast B) Tack type B Fgue 1: Two types of the studed ladde tacks. ψ R (,t) w R (,t) ψ R (,t) ERIR k RARG R ( ψ R (,t) ) = ρrir () t n whch, w R and ψ R epesent the vetcal deflecton and otaton of the al. E R, G R and ρ R ae Young s modulus, shea modulus and densty of the al and A R, I R and k R ae the aea, the second moment of aea and the shea facto of the al coss secton. F(,t) s the contact foce between the al and ladde unts pe unt length of the al and s defned as F (,t) = FM δ( M (t)) (3) n whch F M s the magntude of each movng load and M s ts poston. δ s the Dac delta functon. P(,t) s the foundaton foce pe unt length of the al and s defned as P (,t) = F (t) δ( (t (4) P )) n whch F P s the foce between the al and th pad element and s the poston of th pad element. Smlaly fo the equaton of moton of each ladde unt, one can ave to w (,t) ψ (,t) w (,t) ρa + k AG ( ) + K Bw (,t) + C Bw& (,t) t = FP (t) δ( (t)) (5)

3 E I ψ (,t) k A : G ( ψ w (,t) (,t) ) = ρ All paametes n Equaton (5) and (6) have been aleady ntoduced n the al equatons ecept that hee they assume the subscpt to ndcate that they ae paametes of the ladde unts and K B and C B ae the ballast stffness and vscose dampng espectvely. Soluton method In ths study, Galekn s method s used to solve the set of coupled patal dffeental equatons. In ths method the esponses of the al and each ladde element assume to be n the fom of W R (, t) = ϕr ()q R (t) ψ R (, t) = ψ R ()q R (t) (7) fo the al esponse and W (, t) = ϕ ()q (t) (8) ψ (, t) = ψ ()q (t) fo the esponse of each ladde unt. φ R and ψ R ae the mode shapes of the al n vetcal deflecton and otaton and smlaly φ and ψ ae the mode shapes of ladde unts. Usng the pncple of othogonalty of mode shapes, the equatons of moton can be tansfomed nto sets of odnay coupled dffeental equatons. The al s assumed to be hnged-hnged at both ends and ladde unts ae assumed to be feefee beams floatng ove the ballast. adde unts have two gd modes of vetcal and ptch moton as well as fleual mode shapes n vetcal decton.the combnng pocedue of gd and fleual mode shapes s now eplaned fo ladde unts. By mposng fee-fee bounday condtons.e. [4] w ψ =. (9) =., dψ =. d on the equatons of moton of each ladde gven by Equaton (5) and (6), fo chaactestc equaton of natual fequences of each ladde unt, one can get [4] s s a s B s DET ba bb =. s Sn (ba ) s Cos (ba ) s Sn (bb ) s Cos (bb) a s a s B s B s (1) bacos (ba ) basn (ba ) bbcos (bb ) bbsn (ba ) and fo the mode shapes ϕ () = c1 Cos(ba ) + c Sn(ba ) + c3 Cos(bB ) + c4 Sn(ba ) (11) ψ ( ξ) = d1 Cos(ba ) + d Sn(ba ) + d3 Cos(bB ) + d4 Sn(ba ) In above equatons a = + s ( s ) 1 + b I ψ t (,t) (6) B = 4 + s + ( s ) 1 + b ρa ω I b = ; = ; s = E I A k E A I G (1)

4 n whch ω s the natual fequency of the elated fee-fee beam. Usng othogonalty pncple of the mode shapes j ρ A φ () φj() + ρ I ψ () ψ j() = M = j (13) Fo two gd mod shapes of vetcal and ptch moton, one can ave at Fo ω1 =. ϕ1 = / ; ψ1 = 1. Rgd Rotaton ϕ 1 = 1. ; ψ 1 =. Rgd Dsplacement (14) and ~ ~ ( )p(, t)d A ~ F & q (t) F B ~ 1 = (15) 1 = ~ p(, t) d A ~ A ~ & q& (t) F B ~ (16) n whch and A ~ = Numecal esults ϕ1() ϕ1()d ; = ϕ1() ϕ 1 ()d ; F = ϕ 1 () ϕ B ~ ~ 1 () d (17) & q n (t) + ωnqn (t) = ϕn ()P(, t)d n =,3,.. (18) Fo two types of tacks, llustated n Fgue 1, the numecal smulatons ae caed out. The ladde unt length fo tacks A and B s assumed to be 6 and 1 metes, espectvely. The tack paametes ae lsted n Tables (1) and () Item Notaton Value UIC6 al Young s modulus E R 1 GPa Shea modulus G R 77 GPa Mass densty ρ R 785 kg/m 3 Coss sectonal aea A R 7.69X1-3 m Second moment of aea I R 3.55X1-6 m 4 Shea coeffcent k R.4 adde unts Young s modulus E 8. GPa Shea modulus G GPa Coss sectonal aea A 31X1-3 m Second moment of aea I 98.3X1-6 m 4 Mass densty ρ kg/m 3 Shea coeffcent K.43 Table 1 Popetes of the al and ladde unts [4-6]. Item Notaton Value Pad stffness K P MN/m Pad vscous dampng C P 7 kns/m Ballast stffness K B MN/m Ballast vscous dampng C B 173 kns/m Table Popetes of the al foundaton [4-5].

5 Deflecton (m) Tme (s) Fgue : Ral deflecton at ts md-span (V=7 km/h). Bendng moment (N.m) Tme (s) Fgue 3: Bendng moment at the mddle of the ladde unt (V=7 km/h). Ballast pessue (pa) Tme (s) Fgue 4: Ballast pessue at the md-span (V=7 km/h).

6 Deflecton (m) oad speed (km/h) Fgue 5: Ral deflecton at ts md-span. Bendng moment (N.m) oad speed (km/h) Fgue 6: Bendng moment at the mddle of the ladde unt. Ballast pessue (pa) oad speed (km/h) Fgue 7: Ballast pessue at the md-span.

7 Two successve movng loads ae assumed to be tavelng on the tack. The magntude s assumed to be 14 tons and the dstance s 16 metes epesentng a typcal ale load and boge's cente dstance. Fo the pescbed tacks A and B, numecal smulaton s caed out n the tme doman and the esults ae pesented n Fgue to 4. The tme hstoy of the al deflecton at the md-span s llustated n Fgue. As t s shown, the mamum deflecton s bgge fo tack A wth shote ladde unts. The bendng moment as one of the most mpotant desgn paametes n the ladde tack systems s also nvestgated. Fgue 3 llustates the tme hstoy of the bendng moment at the md-pont of a cental ladde unt. It can be seen that the level of the mamum bendng moment s much hghe fo tack B wth longe ladde unts and the mamum value takes place when the second load s passng. The tme hstoy of the ballast pessue at the al md-span s llustated n Fgue 4. It s shown that the pessue s also hghe fo the tack B but the dffeence hee s not as much as the case of bendng moment. Vaaton of the dynamc chaactestcs of these two types of tacks vesus the load speed s llustated n Fgue 5 to 7. As llustated n Fgue 5, the deflecton of the al at ts md-span s an nceasng and deceasng functon of the load speed espectvely fo the tack A and B. Vaaton of the bendng moment aganst the load speed s llustated n Fgue 6. It s seen that fo the tack B thee s a mnmum occus aound the speed of 1 km/h and the bendng moment s an nceasng functon of the load speed fo the tack A. As t s seen fom Fgue 7, the ballast pessue s a monotoncally nceasng functon of the load speed fo both tacks A and B. Conclusons Dynamc pefomance of ballasted ladde tacks was nvestgated n ths pape. Two types of ladde tacks.e. tack A and B wth the ladde unts havng 6 and 1 metes length wee studed. The effects of the ladde unt length on the dynamc pefomance of the ladde tack system wee nvestgated. Coupled set of equatons of moton of the al and ladde unts wee deved usng Tmoshnko beam theoy and then solved usng Galekn technque. It was found that, the al deflecton at the md-span s bgge fo tack A wth shote ladde unts. It was shown that the level of the mamum bendng moment s much hghe fo tack B wth longe ladde unts. Moeove, the ballast pessue s also hghe fo the tack B but the dffeence s not as much as the case of bendng moment. The deflecton of the al at ts md-span s an nceasng and deceasng functon of the load speed espectvely fo the tack A and B. Futhemoe, fo vaaton of the bendng moment at the ladde unts t was found that thee s a mnmum occus aound the load speed of 1 km/h fo the tack B and the bendng moment s an nceasng functon of the load speed fo the tack A. The ballast pessue s a monotoncally nceasng functon of the load speed fo both tacks A and B. Refeences [1] E. Esmalzadeh, M. Ghoash, "Vbaton analyss of a Tmoshenko beam subjected to tavelng mass", Jounal of Sound and Vbaton, Volume 199, pp , (1997). [] E. Esmalzadeh, M. Ghoash, "Vbaton analyss of beams tavesed by unfom patally dstbuted masses", Jounal of Sound and Vbaton, Volume 184, pp. 9-17, (1995). [3]. Fyba, Vbaton of solds and stuctues unde movng loads, Thomas Telfod, ondon, (1999). [4] M. H. Kaganovn, D. Younesan, D. Thompson, C. Jones, "Rde comfot of hgh-speed tans tavelng ove alway bdges", Vehcle System Dynamcs, Volume 43, Numbe 3, pp , (5). [5] M. H. Kaganovn, D. Younesan, D. J. Thompson, C. J. C. Jones, "Non-lnea Vbaton and Comfot Analyss of Hgh-Speed Tans Movng ove the Ralway Bdges", Poceedngs of the ASME Confeence, ESDA 4, Mancheste, UK, (4). [6] H. Waku, N. Matsumoto, H. Inoue. "Technologcal nnovaton n alway stuctue system wth laddetype tack system", Poceedngs of WCRR97, Floence, Italy, (1997).