Dynamic Analysis of the Turnout Diverging Track for HSR with Variable Curvature Sections

Transcription

1 Word Journ of Engineering nd Technoogy, 07, 5, ISSN Onine: -449 ISSN Print: -4 Dynmic Anysis of the Turnout Diverging Trck for HSR with Vribe Curvture Sections Wdysw Koc, Ktrzyn Pikowsk Deprtment of Ri Trnsport nd Bridges, Gdnsk University of Technoogy, Gdnsk, Pond How to cite this p: Koc, W. nd Pikowsk, K. (07) Dynmic Anysis of the Turnout Diverging Trck for HSR with Vribe Curvture Sections. Word Journ of Engineering nd Technoogy, 5, Received: December 9, 06 Accepted: Februry 8, 07 Pubished: Februry, 07 Copyright 07 by uthors nd Scientific Reserch Pubishing Inc. This work is icensed under the Cretive Commons Attribution Interntion License (CC BY 4.0). Open Access Abstrct The p presents n nytic method of identifying the curvture of the turnout diverging trck consisting of sections of vrying curvture. Both iner nd noniner (poynomi) curvtures of the turnout diverging trck re identified nd evuted in the p. The presented method is univers one; it enbes to ssume curvture vues t the beginning nd end point of the geometric yout of the turnout. The resuts of dynmics nysis show tht widey used in riwy prctice, cothoid sections with nonzero curvtures t the beginning nd end points of the turnout ed to incresed dynmic interctions in the trck-vehice system. The turnout with noniner curvture reching zero vues t the extreme points of the geometric yout is indicted in the p s the most fvourbe, tking into ccount dynmic interctions occurring in the trck-vehice system. Keywords Riwy Turnouts, Curvture Modeing, Dynmics Anysis. Introduction Typic, used since the beginning of riwy engineering, geometric yout of the turnout diverging trck consists of singe circur rc without trnsition curves. It introduces sudden, brupt chnges of the horizont curvture of the yout t the beginning nd end of the turnout diverging trck, which increses dynmic interctions in the trck-vehice system, prticury unfvourbe in high speed ri (HSR). Investigtion nd evution of geometric youts of the turnout diverging trck re sti current issue. Recenty, iming t smoothing chnges of the curvture t the neurgic regions of the turnout diverging trck, the cothoid sections hve been introduced t both sides of the circur rc [] [] []. The curvture of the ppied cothoid DOI: 0.46/wjet Februry, 07

2 sections in mny cses does not rech zero vue t the extreme points (i.e. t the beginning nd end points of the turnout). The p presents the evution of the seected geometric youts of the turnout diverging trck to indicte the most fvourbe soution for HSR. In the turnout with iner curvture sections, diverging trck is divided into three zones (Figure ): beginning zone of the ength, in which curvture increses inery from k = (or k = 0 ) to k =, R R midde zone of the ength with constnt curvture k =, R n end zone of the ength, in which curvture decreses inery from k =, to k = (or k = 0 ). R R The vrious vues of curvture nd ength of ech section cn be ppied in the designing process. Curvture of the turnout diverging trck is described by n nytic function k( ), where stnds for the ength of the curve. This p presents the identifiction of nytic functions k( ) for iner curvture sections (i.e. cothoid sections) s we s for noniner curvture sections in the poynomi form. The identified curvtures hve been compred using the dynmic mode, described in [4], to find out the most fvorbe soution from the point of view of minimizing the dynmic effects. In this p, the Crtesin coordintes of the turnout diverging trcks re not presented. The method of the identifiction of the Crtesin coordintes from the curvture k( ) is described in [4]. The determintion of prmetric equtions x( ) nd y( ) requires the expnsion of the integrnds into Tyor series [5] using Mxim pckge [6].. Appiction of the Liner Curvture Sections.. Soution for the Beginning Zone In the beginning zone of the turnout the considered issue is identified by boundry Figure. Curvture of the turnout diverging trck (iner curvture sections) R = 6000 m, = 40 m, R = 6000 m, = m, = 45 m, R = 5000 m. 4

3 conditions [] ( 0) k = k k = k () nd differenti eqution ( ) 0 k =. () After determining the constnts, the soution of the differenti probem (), () is s foows: k k k( ) = k +. () The sope of the tngent t the end of the zone, for =, is defined by the formu: θ k + k =. (4) ( ).. Soution for the Midde Zone In the circur rc zone, i.e. for, +, curvture is constnt: k k =. (5) At the end of circur rc the sope of the tngent is defined by the formu:.. Soution for the End Zone k + k θ ( + ) = + k. (6) In the end zone of the turnout the foowing boundry conditions re dopted: k + = k k + + = k for the differenti Eqution (). After determining the constnts, the soution of the differenti probem (), (7) is s foows: k k k k k = k + +. (8) The sope of the tngent t the end of the turnout is defined by the formu: from which the turnout nge k + k k + k θ ( + + ) = + k +, (9) n n = tnθ cn be obtined s ( + + ) (7). (0). Appiction of the Noniner Curvture Sections The curvture of the turnout diverging trck in Figure is not undoubtedy n ide soution. The doption of more gente chnges of the curvture t both 44

4 sides of the circur rc nd the ssumption of zero curvture vue t the extreme points (i.e. the turnout beginning nd end points) of the geometric yout re worth considering... Soution for the Beginning Zone The foowing boundry conditions hve been dopted: to the differenti eqution ( 0) k = k k = k k k k ( 0) = C k ( ) = 0 ( 4 ) () k = 0 () with ssumption, tht coefficient C 0. As resut of soving the differenti probem (), () the foowing curvture hs been obtined: C C C k( ) = k + ( k k) ( k k) + ( k k). () Function k( ) describing the curvture in the considered zone shoud be monotonic nd shoud increse for > 0. In order to obtin fesibe soution the coefficient C shoud be proy djusted. It hs been shown, tht the pproprite C.5;. Tking into ccount the ength of the prmetric curve () nd curve of iner curvture (i.e. generized cothoid) the most fvourbe ssumption seems to be C =.5. Curvture k( ) in this cse is s foows: k = k + k k k k. (4) At the end of the zone, for =, the sope of the tngent is described by the formu:.. Soution for the Midde Zone θ k + 5k =. (5) 8 ( ) Simiry to the midde zone described in the section., i.e. for, +, the curvture is constnt k( ) = k. The sope of the tngent t the end of the circur rc, for = +, is described by the formu: k + 5k θ ( + ) = + k. (6) 8.. Soution for the End Zone Assuming the boundry conditions: k + = k k + + = k 0 k + = k + + = C k k (7) 45

5 for the differenti Eqution () the foowing soution hs been obtined: where k = c + c+ c + c (8) 4 C C c = k k k ( C) ( C) c = k k ( ) C C c = + + k k C c = k k 4 Assuming C =.5 the foowing coefficient formus hve been obtined: c k = k k c = k k c = + + k k c = k k 4 The sope of the tngent t the end of the turnout, for = + +, is defined by the formu: k + 5k k + 5k θ ( + + ) = + k +. (9) 8 8 In Figure the curvture of the turnout diverging trck (for C =.5) with noniner curvture sections hs been shown. The geometric prmeters of the turnouts presented in Figure nd Figure re conform. 4. Seection of the Geometric Lyouts of Turnout Diverging Trcks In order to ensure reibe comprtive nysis of the geometric youts presented in Tbe, the foowing common ssumptions hve been dopted: the turnout nge :n, where n = 50, the curvture vues k, k nd k re common to turnouts, the circur rc rdius R = 6000 m, the ength of the beginning zone nd the end zone re simir nd ensures the fufiment of the kinemtic conditions, the ength of the circur rc ensures reching the ssumed vue of the turnout nge... 46

6 Figure. Curvture of the turnout diverging trck (noniner curvture sections for C =.5) ( R 6000 m, 40 m, R 6000 m, m, 45 m, R 5000 m) = = = = = =. Tbe. Geometric prmeters of the seected turnouts (the turnout nge :50). No Curvture k [rd/m] [m] k [rd/m] [m] [m] k [rd/m] L [m] Constnt / / / Liner / / / Liner / / Liner / Noniner / / / Noniner / / Noniner / The highest veocity on circur rc without sueevtion (i.e. in the midde zone) resuts from the foowing condition: m V =, 6 R whie in the extreme zones the condition is s foows V d mxψ = mx ψ, 6 d, where: V trin veocity [km/h], R circur rc rdius [m], m cceertion on circur rc [m/s ], missibe vue of cceertion on circur rc [m/s ], ( ) function describing ter cceertion in the zones of chnging curvture, ψ rte of cceertion chnges in the zone of chnging curvture [m/s ]. ψ missibe vue of prmeter ψ [m/s ]. It is ssumed tht = 0.6 m s nd ψ = 0.5 m s. 47

7 On circur rc without trnsition curves (turnout ) the cceertion chnges inery from 0 to into ccount m ong the ength of the rigid bse of wgon b. Tking V =, 6 Rb for 0, b with condition the imit of the veocity V ψ = ψ, 6 Rb V mx is obtined s foows V = R ψ. (0) mx, 6 b On circur rc in the midde zone with sections of chnging curvture in the beginning nd end zones the imit of the veocity is described by the formu mx V =, 6 R () In the beginning zone where curvture chnges inery rte of cceertion chnges ψ is constnt. In this zone the foowing condition shoud be fufied: V ψ = ψ, 6 R R from which the minim ength of the beginning zone cn be determined:, V, 6 R R ψ. () Noniner curvture (poynomi) induces chnging rte of cceertion chnges ψ ong the ength of the turnout. The foowing condition shoud be fufied: ψ mx = V mx.5 ψ., 6 R R An increse by 50% of the imit vue ψ is justified by the fct, tht the occurs ony once (for = 0), nd next decreses, reches t the end of vue ψ mx the section (for = ) zero vue. The condition () cn be ppied so for the end zone of the turnout. For the ssumed turnout nge :50 (i.e. n = 50) the foowing sope of the tngent hs been obtined, using Eqution (0): Assuming θ ( + + ) = rctn = rd. n b = 0 m for the turnout diverging trck (Tbe ), using Eqution (0) the mxim veocity Rb, mx V = km h. The ength of the circur rc is obtined s foows: R θ ( ) = + + = m. The veocity imit in turnouts 7 resuts from the Eqution (): V mx = 6 km h (in the comprtive nysis of the turnouts, presented in section 6, it ws ssumed V = 00 km h ). 48

8 The geometric prmeters of the seected seven turnouts re presented in Tbe. The engths of the sections nd resut from condition (), whie the ength resuts from the ssumed sope of the tngent in the Eqution (9) for iner curvture: k + k k + k = θ ( + + ) k nd in the Eqution (9) for noniner curvture: k 8 8 () k + 5k k + 5k = θ ( + + ). (4) The function of ter cceertion ( ) ong the yout, s proved in [4], resuts directy from the yout curvture k( ). The ssumed functions of ter cceertion ( ) for seected turnouts re presented in Tbe. 5. The Dynmic Mode With incresed speed requirements on riwys, the dynmic effects minimiztion is current issue, especiy in HSR. Bsing on the ssumption tht horizont curvture chnges re forcing fctor of the ter oscitions, seected seven geometric youts of the turnout diverging trck re compred in terms on their impct on the dynmic interctions occurring in ri-vehice system. In the presented comprtive nysis of the youts, structur spects of the ri Tbe. Lter cceertion [m/s ] ong the three zones (ength [m]) of the seected turnouts. No Beginning zone Midde zone End zone ( ) = ;9.984 = ; ( ) = ; = ; = ; ( ) = ;0.484 = ; = ;60.00 ( ) = ;9.984 = ; = ; ( ) = ;97.84 = ; = ; ( ) = ;9.859 = ; = ; ( ) = ; = ;

9 vehice re omitted. A dynmic mode with one degree of freedom, consisting of mss with spring nd dm is ppied to compre the dynmic interctions occurring on the vrious turnout diverging trcks. An ddition prmeter ength of the rigid bse of wgon hs been introduced, which resuts in referring to the ter cceertion of the wgon mss center (rithmetic men of cceertions occurring in the front nd rer bogies). The ter cceertion ( t ) occurring ong the turnout diverging trck cn be described by the seprte functions dedicte for different turnout zones. Assuming constnt veocity ong the turnout, s it is done in this p, function ( ) for ech turnout zone is presented in Tbe. Considered cse incudes driven horizont hrmonic oscitions X [7] described by the eqution d X ( t) dx ( t) dt where: D Lehr s dmping coefficient, ω free oscition frequency, ( ω ) + u + + u X t = t (5) dt u = D ω. D Lehr s dmping coefficient D is used s dmping mesure in the riwy engineering. In the presented p D = 0.75 nd ω =.5 π/s re ssumed. The ssumed vue of D hs been obtined in the eximent reserch presented in [8]. As proved in [9] this ssumption hs no impct on concusions from the comprtive nysis of dynmic proties of riwy geometric youts. The function of oscitions X ( t ), describing ter dispcement of the vehice under the force P( t) = m ( t), is the soution to the differenti Eqution (5). The function X ( t ) is the resutnt of the sttic component nd the system oscitions. From the point of view of dynmic effects evution the resutnt cceertion of oscition motion = X ( t) is essenti. The mximum mpitude of the cceertion of osciting motion mx X nd indictor defined s foows 0 + k 0 v w w = X t d (6) where: 0 the point t which curvture of the turnout diverging trck chnges, k the ength of the section on which oscitions re dmped, re ssumed s criteri of the dynmic effects evution presented in Section Resuts of the Dynmics Anysis The ength of the rigid bse (it hs been ssumed b = 0 m ) used in the dynmic mode resuts in more gente chnges of ter cceertion ( ), It is presented in Figure nd Figure 4 s the ine_corr for turnout nd (Tbe 50

10 Figure. Lter cceertion forcing the ter oscitions for turnout. Figure 4. Lter cceertion forcing the ter oscitions for turnout. ). The cceertion of osciting motion X ( ), computed numericy using the dynmic mode described in section 5, for the seected seven geometric youts of turnout diverging trck (Tbe nd Tbe ) is presented in Figures 5-. Aprt from the beginning nd end zones of the turnout diverging trck, the dynmic interctions occur so t the beginning nd end of the midde zone, s shown in Figure 7. The comprtive nysis of the seected youts of the turnouts diverging trck hs been crried out using dynmic indictors: w (6) nd mx X, bsed on cceertion of osciting motion X ( ) from Figures 5-. The computed vues of w nd mx X for the seected seven turnouts re presented in Tbe. As shown evidenty in Tbe nd Figures 5- the gretest vues of cceertion in osciting motion X ( ) occur in the beginning nd the end zone of the turnout diverging trck, wherein the vue of the cceertion is infuenced by the ssumed curvtures k nd k. In geometric youts of the turnout diverging trck, in which k 0 or k 0 (turnout nd 5 in Figure, Figure nd Figure 0), the dynmic interctions re significnty greter (pproximtey 00 times greter) thn in the youts in which k = 0 nd k = 0 (turnout 4 nd 7 in Figure 9 nd Figure ). 5

11 Tbe. Dynmic indictors ( ) X for seected turnouts. w nd mpitude of the cceertion of osciting motion Zones of dynmic effects ong the turnout diverging trck No Beginning zone Beginning of the circur rc End of the circur rc End zone w [m /s ] mx X [m/s ] w [m /s ] mx X [m/s ] w [m /s ] mx X [m/s ] w [m /s ] mx X [m/s ] Figure 5. Acceertion of osciting motion X ( ) for turnout (V = 4 km/h). Figure 6. Acceertion of osciting motion X ( ) for turnout (V = 00 km/h). The ssumption of k = 0 (turnout Figure 8 nd turnout 6 Figure ) resuts in rdic reduction of the cceertion in osciting motion X ( ) in the end zone of the turnout diverging trck. The simutneous doption of both conditions: k = 0 nd k = 0 eds to reduction of the dynmic interctions 5

12 W. Koc, K. Pikowsk Figure 7. Acceertion of osciting motion X ( ) in the midde zone of the turnout (V = 00 km/h). Figure 8. Acceertion of osciting motion X ( ) for turnout (V = 00 km/h). Figure 9. Acceertion of osciting motion X ( ) for turnout 4 (V = 00 km/h). ong the whoe turnout diverging trck; it is concerned youts with sections of iner curvture (turnout 4 Figure 9) s we s youts with sections of noniner curvture (turnout 7 Figure ). The presented resuts eds to concusion tht widey ppied in riwy prctice cothoid sections with curvtures k 0 t the beginning nd end points of the turnout diverging trck re not justified. The dynmic proties of the yout cn be significnty improved by ssuming k = 0 nd k = 0 t the mentioned points, ccepting the fct tht the ength of the resuting turnout wi 5

13 Figure 0. Acceertion of osciting motion X ( ) for turnout 5 (V = 00 km/h). Figure. Acceertion of osciting motion X ( ) for turnout 6 (V = 00 km/h). Figure. Acceertion of osciting motion X ( ) for turnout 7 (V = 00 km/h). sighty increse (Tbe ). The cceertion in osciting motion X ( ) occurring t the beginning nd t the end of the midde zone is not dependent on the curvtures vues k nd k, dopted in the beginning nd the end zone of the turnout, but depends on the curvture chrcteristics. Liner curvtures k nd k (turnouts 4, Figures 6-9) induce greter vues of dynmic indictors (Tbe ) thn noniner ones (turnouts 5 7, Figures 0-). Tking into ccount the dynmic proties nd the ength of the yout, the turnout diverging trck 7 is definitey the most fvourbe. Turnout 7 in comprison with turnout 4 hs better dynmic proties in the midde zone, shorter ength nd insignificnty worse vues of dynmic indictors in the beginning 54

14 nd end zones (Tbe ). 7. The Most Fvourbe Geometric Lyout of the Turnout Diverging Trck As resut of dynmics nysis it hs been proved tht the most fvourbe dynmic proties cn be chieved by ppying noniner curvture in the beginning nd end zones of the turnout diverging trck nd ssuming zero curvture vue t the extreme points of the geometric yout. Assuming k = 0 nd k = 0 the curvture k( ) of the turnout is defined s foows: in the beginning zone, for 0,, bsed on the Eqution (4) the foowing formu is obtined k k k( ) = (7) in the midde zone, i.e. for, +, the curvture is constnt k( ) = k in the end zone, for +, + + the curvture is described by the Eqution (8) with the foowing coefficient vues: c k = c k = c = k + + k c4 =. The sope of the tngent t the end of the turnout, for = + +, is s foows 5k θ ( + + ) = k+ ( + ). (8) 8 In Figure the curvture of the most fvorbe turnout diverging trck 7 is presented. 8. Concusions Typic turnout diverging trck consists of singe circur rc without trnsition curves. It introduces sudden, brupt chnges of the horizont curvture of the yout t the beginning nd end of the turnout diverging trck, which increses dynmic interctions in the trck-vehice system, prticury unfvourbe in HSR. The p presents univers, nytic method of identifying the curvture of the turnout diverging trck. Both iner nd noniner (poynomi) curvtures of the turnout diverging trck re identified nd evuted using dynmic 55

15 Figure. Curvture of the turnout diverging trck 7 (noniner curvture sections). ( k 0, 60 m, k 6000 rd m, m, 60 m, k 0) = = = = = =. mode. The presented method enbes to ssume the curvture vues t the beginning nd end point of the geometric yout of the turnout. The ength of the circur rc is djusted to obtin the ssumed turnout nge. Recenty, iming t smoothing chnges of the curvture t the neurgic regions of the turnout diverging trck, the cothoid sections hve been introduced t both sides of the circur rc. The curvture of the ppied cothoid sections chnges inery but in mny cses does not rech zero vue t the extreme points (i.e. t the beginning nd end points of the turnout). The resuts of dynmics nysis presented in the p show tht cothoid sections with nonzero curvture t the beginning nd end points of the turnout ed to incresed dynmic interctions in the trck-vehice system. Dynmic interctions cn be decresed by ppying curvture reching zero t the extreme points of the turnout. The p presents the evution of the seected seven geometric youts of the turnout diverging trck nd indictes the most fvourbe soution for HSR. The most fvourbe from the dynmic proties point of view is the turnout diverging trck with noniner curvture reching zero vues t the extreme points of the turnout. References [] Fei, W.Z. (009) Mjor Technic Chrcteristics of High-Speed Turnout in Frnce. Journ of Riwy Engineering Society, 9, 8-5. [] Prsons Brinckerhoff for the Ciforni High-Speed Ri Authority (009) Technic Memorndum: Aignment Design Stndrds for High-Speed Trin Otion. [] Wng, P. (05) Design of High-Speed Riwy Turnouts. Theory nd Appictions. Esevier Science & Technoogy, Oxford, United Kingdom. [4] Koc, W. (04) Anytic Method of Modeing the Geometric System of Communiction Route. Mthemtic Probems in Engineering, 04, Artice ID: [5] Korn, G.A. nd Korn, T.M. (968) Mthemtic Hndbook for Scientists nd Engineers. McGrw-Hi Book Compny, New York, USA. [6] Mxim Pckge. [7] Hibbeer, R.C. (05) Engineering Mechnics Dynmics. 4th Edition, PDF Free 56

16 Ebook Downod, Prentice H. [8] Vrg, J., et. (980) Anysis of Admissibe Infuence on Riwy Vehice Moving Aong Trnsition Curves with Liner (Cothoid) nd Cosine Curve Geometry nd Aong Turnouts with Gret Rdii (in Hungrin). The Sci. Works of Riwy Institute, Budpest. [9] Koc, W. nd Pikowsk, K. (999) Inteigent Modeing of the Riwy Trck Lyouts Using Dynmic Criteri. Interntion Conference of Modeing nd Mngement in Trnsporttion, Poznn-Crcow, Pond, 5-6 October 999, Submit or recommend next mnuscript to SCIRP nd we wi provide best service for you: Accepting pre-submission inquiries through Emi, Fcebook, LinkedIn, Twitter, etc. A wide seection of journs (incusive of 9 subjects, more thn 00 journs) Providing 4-hour high-quity service User-friendy onine submission system Fir nd swift peer-review system Efficient typesetting nd proofreding procedure Dispy of the resut of downods nd visits, s we s the number of cited rtices Mximum dissemintion of your reserch work Submit your mnuscript t: Or contct wjet@scirp.org 57