An Adaptive Contact Model for Simulation of Wheel-rail Impact Load due to a Wheel Flat

Transcription

1 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM-7-46 An Adaptve Contact Model for Smulaton of Wheel-ral Impact Load due to a Wheel Flat J. J. Zhu, A. K. W. Ahmed *, S. Rakheja Department of Mechancal and Industral Engneerng, Concorda Unversty, Montreal, Canada * Correspondng author (emal: waz@vax.concorda.ca) Abstract Several reported studes have concluded that wheel-ral mpact load due to a wheel flat predcted by commonly used Hertzan contact model underestmates the mpact force due to a wheel flat between wheel and ral at low speeds and yelds an overestmaton at hgh speeds. In ths study, an adaptve wheel-ral contact model wth radal sprng s developed for predcton of wheel-ral normal contact force. Ths proposed model adapts the contact length, contact depth and ncorporates possble partal and asymmetry of the contact due to defectve wheel profle. A vehcle-track nteracton model ncorporatng a -D rollplane vehcle, 3-D Tmoshenko track system coupled by the adaptve contact model s developed to nvestgate the mpact force between the wheel and ral due to a sngle wheel flat. The smulaton results were compared wth those usng Hertzan contact model and avalable feld test data. The results demonstrate the effectveness of the adaptve contact model n predctng wheel ral mpact load due to a wheel flat n a wde speed range. Keywords: Adaptve Contact, Wheel-ral, Wheel Flat, and Impact Load 1 Introducton Heavy haul freght cars generate sgnfcant force at the wheel-ral nterface, whch s further magnfed when there s a defect n the wheel or ral profle. The rollng contact mechansm s a complex problem and has attracted numerous nvestgatons n the past few decades. In the mathematcal modelng and smulaton of ralway vehcle-track dynamcs, t s the contact model that couples the vehcle system wth track system. An accurate descrpton of contact force between the wheel and the ral s thus a necessary condton to obtan relable smulaton results for the vehcle-track system. The most wdely used vertcal contact model s based on Hertzan non-lnear elastc contact theory, whch descrbes the contact behavor of two cylnders. Such contact model s essentally a pont contact model based on the assumpton that the contact patch s very small [1]. It s further assumed that the contact pont les on the centerlne of the wheel. Ths s a reasonable assumpton n modelng vertcal contact forces between a perfect wheel and ral geometry. However, when there s a defect on the wheel or ral n the contact zone, the nduced mpact force predcted usng Hertzan contact model may not be very relable. A few studes have suggested that the Hertzan non-lnear pont contact model ncorporatng lnear track model consstently underestmates wheel flatnduced mpact loads at low speeds, whle overestmates at hgh speeds []. Multpont contact model has also been proposed to predct contact forces between wheel and ral wth a defect such as a wheel flat [3]. Such a model, however, assumes that the vertcal contact sprngs are dscretely and symmetrcally dstrbuted about the vertcal center lne of wheel regardless of perfect or defectve wheel and ral profle. In the wheel-ral vertcal contact study [3], t has been shown that multpont contact model estmates mpact force very smlar to that of Hertzan contact model. It s not dffcult to vsualze that n the presence of a defect, such as a wheel flat, the radal sprngs wll not be symmetrcally dstrbuted about the vertcal center lne of wheel when the regon wth flat enters or leaves the contact area. For the present nvestgaton of mpact forces due to wheel flats, an adaptve wheel-ral contnuous contact model s developed to overcome the lmtaton of the publshed models descrbed above. The proposed model consders the contact length, contact depth, and possble partal contact due to defectve wheel/ral profle. Ths adaptve contact model s based on the concept of contnuous radal sprngs unformly dstrbuted over an adaptve wheel-ral footprnt, whch has been successfully utlzed n study of ground vehcle contact problems assocated wth pneumatc tres [4]. The adaptve contact model developed for wheel-ral vertcal nteracton neglects the lateral and longtudnal forces. It s also assumed that the contact patch s very small and does not exceed the length of the flat. Due to concal or worn profle of the ralway wheel, wheel radus at the contact pont always vares as the wheel rolls along the track. It s however, assumed that such varatons due to possble lateral moton s small and that there s no slp between the wheel and ral. The adaptve contact mode s used n ths nvestgaton to couple a 6 DOF -D roll plane vehcle model wth 3-D ral system modeled as contnuous Tmoshenko beam. The smulaton results n term of wheel-ral mpact force due to a wheel flat are obtaned usng central fnte dfference method for dfferent forward 157

2 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM-7-46 velocty. The results are compared wth reported analytcal and feld test data over a wde speed range. Development of an Adaptve Wheel-ral Contact Model.1 Contact Force for Perfect Wheel Profle The wheel-ral vertcal nteracton s represented by contnuously dstrbuted radal sprngs that take nto account the stffness of the wheel and the ral. As shown n Fgure 1, the contact patch s desgnated by contact length l e. The radally dstrbuted sprngs are assumed lnear, whle the constant radal sprng stffness K w, s defned as the magntude of force requred to produce unt angular deformaton of the sprng. The contact force s developed by radal nterpenetraton of the wheel nto the ral. The elemental radal deflecton, δ, at an angle α, leads to radal sprng force df as shown n Fgure 1, such that: df = ( K w dα ) δ (1) df n = ( K w δ cos α )dα df t = ( K w δ sn α )dα (4) Upon combnng equatons () to (4) and ntegratng over the entre contact patch (-α w, α w ), the resultant wheelral normal contact force wth perfect contact profle can be expressed as: P = Fn = K wrw(sn α w α w cos α w ) (5) where α w s wheel-ral contact patch angle, whch s defned as half of the angle formed by a lne connectng wheel center to the front contact pont, and a lne connectng the wheel center to the rear contact pont. A symmetrc contact about the wheel center lne s assumed for a defect free wheel. The contact angle can thus be expressed as: 1 Rw δc α w = cos ( ) (6) Rw The contact overlap δ c can be determned from wheel center dsplacement z w and correspondng ral dsplacement z r, such that: zr zw zr zw> δ c = { zr zw (7) From the above relatonshps, when the overlap δ c and the contact force are equal to zero, the wheel contact angle s also zero. In ths study, t s assumed that the ral wheel vertcal dsplacements (z r, z w ) are equal to zero when the wheel and the ral are just about to come nto contact..contact Force for Defectve Wheel Profle Z w Z r α δ Ral Wheel df n α w df df t l e α w δ c h The contact model s extended to nclude a defect n the wheel profle n the form of a flat. As shown n Fgure, the presence of a wheel flat could yeld a contact patch that s asymmetrc about the wheel centerlne. For the contact patch defned by (α f, α r ), combnaton of equatons () to (4) and ntegraton over the contact patch (α f, α r ), yelds resultant normal contact force as: P f = F n = α α r K w R w δ c ( R cos α ) cos α dα (8) Fgure 1: Radal contact representaton of wheelral nteractons where α s the angle between the vertcal centerlne of wheel plane and arbtrary contact pont wthn the contact length. For an nstantaneous wheel radus R, the elemental radal deflecton δ can be expressed as: h δ = R () cos α where h s dstance between the wheel center and the contact patch center, gven by: h = R w δc (3) where R w s nomnal wheel radus and δ c s the wheel-ral deflecton at contact patch center, or the wheel ral contact overlap. The normal and tangental components of the contact force, shown n Fgure 1, can be expressed as: 158 Fgure : Deflecton of an element of wheel-ral contact patch wth defectve wheel where δ c can be calculated from equaton (7); R s the nstantaneous radus of the wheel at a poston α ; α f and α r are postons of the extreme front and rear contact pont. Form equaton (8), t can be seen that computaton of wheel-ral contact force requres: () determnaton of nstantaneous wheel radus R at any pont of contact zone; () determnaton of the contact patch, or front and rear contact angles (α f and α r ) at every nstant; and () establshment of radal sprng constant K w.

3 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM Determnaton of the Radus R at Arbtrary Pont of Wheel Rm wth a Flat The profle of a wheel wth a flat can be descrbed by ts radus and correspondng angle β between a reference lne and the radus, as shown n Fgure 3. The flat s desgnated by lne BB 1 and the reference s chosen as the vertcal lne through wheel center. The ntal poston of the flat s descrbed by the angle between the reference lne and the wheel flat center lne (β o ). The length of the flat s determned by arc angle φ, whch represents half of the chord angle between OB and OB 1. For a small flat length, ths arc angle can be expressed as: l f ϕ = (9) Rw where l f s the length of flat and R w s the nomnal wheel radus. The nstantaneous radus of wheel at arbtrary poston can now be smply expressed by: R < β β ϕ β + ϕ < β π = OR R w (1) Rw f β ϕ < β β + ϕ where D f s the depth of flat and can be estmated usng followng equaton f t s unknown [5]: l f D f = (14) 16 R w The term x n equaton (13) s the dstance between an arbtrary pont on the flat and the front end of the flat (desgnated by x n Fgure 3), and can be expressed n terms of angle β, such that: lf x = + (Rw Df )tan( β βo ), β ϕ<β β +ϕ (15) The nstantaneous wheel radus can be then obtaned as a functon of poston β by substtutng for f nto equaton (1). In ths study, only the haversne flat model s used to represent a flat wth a gven length and depth snce ths model s more commonly used and such type of flat s more commonly observed n practce. The poston of the wheel flat wth reference to vertcal centerlne of wheel would vary as the wheel rotates. Fgure 4 shows the flat poston at an nstant t, where the flat has shfted by an angle γ from the ntal poston. For a constant forward speed V, the angle γ can be expressed as: V γ = ω t = t (16) R W The correspondng new poston of flat s then gven by β o γ. By substtutng (β o γ) for β o n equatons (15), (11) and (1), the nstantaneous wheel radus R at any pont ncludng contact zone can be readly obtaned for a rotatng wheel. For a haversne type flat, f can be determned from equaton (13) by substtutng for x(t), gven by: x(t ) lf = + (Rw Df )tan )) [ β(t ) ( βo γ(t ))]; ( βo γ(t )) ϕ<β(t ) ( βo γ(t + ϕ (17) Fgure 3: Wheel profle wth a flat where f s the varaton n the wheel radus due to wheel flat, and s dependent upon the type of flat beng consdered. When the wheel flat s just formed, t takes the form of a chord. It s often referred to as fresh flat or chord flat. For fresh or chord type flat, f can be obtaned by the geometry of the profle, such that: Rw D f (11) f β ϕ < β β = Rw cos( β βo ) l f f R w R w, D = (1) 4 In practce, the two ends of chord become rounded due to contnuous runnng on the ral. Ths type flat s usually called rounded flat, or worn flat, or haversne flat. The varatons n the radus due to such a flat, f, can be descrbed by [5]: π x f =. 5 D f 1 cos( ) l f + ϕ (13) Fgure 4: Scheme of a rotatng wheel wth a flat Fgure 5 shows the varaton n wheel radus at the contact patch center when rollng at a speed of 5 km/h. For ths case the wheel and the flat are defned by: R w =.457 m, l f = 1 mm, D f = 1.5 mm and β o = 9º. Fgure 5: Changes n effectve rollng radus of a rotatng wheel wth a flat 159

4 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM Determnaton of the Contact Patch (α f, α r ) As llustrated earler n Fgure, a wheel flat would yeld asymmetrc contact geometry about the vertcal wheel centerlne. The analyss of ths geometry nvolves dentfcaton of the wheel-ral contact patch, or the extreme front and rear contact angles (α r, α f ) n the presence of a flat, as shown n Fgure 6. The fgure also shows the contact geometry for perfect wheel, whch s descrbed by angle ±α w. The contact geometry s derved upon consderaton of the dsplacements of wheel and ral, whch yeld overlap δ c usng equaton (7). Furthermore, the nstantaneous radus of wheel rm R s derved from equaton (1). The postons of the front and rear contact pont poston (α f, α r ) are derved from the correspondng rad R(α f ) and R(α r ), such that: R( α f ) cos( α f ) = R w δ c (18) R( α r ) cos( α r ) = R w δ c Wthn the contact patch: R( α ) cos( α ) > R w δ c (19) And outsde the contact patch: R( α ) cos( α ) < δ () -α w R w α w α r α f h Reference lne (α=) Fgure 6: Identfcaton of wheel ral contact patch (α f, α r ) n specfed co-ordnate system Above formulatons suggest that an teratve approach can be adopted to verfy the contact patch coordnates. Assumng a defect free wheel profle, contact patch angles are ntally taken as α w, α w. The wheel rad R at every pont wthn α w are computed untl the extreme contact ponts are establshed usng the above requrements...3 Establshment of Wheel-Ral Contact Radal Sprng Stffness c δ c V The effectveness of the proposed adaptve contact model s largely dependent on the dentfcaton of relable value for the radal sprng constant K w. Although expermental characterzaton would be desrable, an estmate of K w may be obtaned from analyss of wheel-ral nterpenetraton under a statc load P o. Rewrtng equaton (5) yelds an expresson for the stffness K w : p o K w = R w (sn α w α w cos α w ) (1) where α w can be determned from equaton (6). The statc wheel ral overlap δ o should be deally determned by expermentally measurng the statc dsplacement of ral and wheel center. Alternatvely t can be derved from the Hertzan non-lnear contact theory by assumng that the contact patches calculated from the two dfferent models s same under the same statc load. Hertzan contact theory provdes followng relatonshp between appled load and wheel-ral overlap [1]: 3 P = C H δ () where δ s the overlap between wheel and ral; and C H s the Hertzan contact coeffcent. The above relatonshp can also be appled to determne statc overlap of wheel and ral, δ o, by assumng the contact force as the statc load P o. 3 Vehcle-track Interacton model The magntudes of mpact forces caused by wheel defects are strongly dependent upon vertcal dynamcs of the coupled vehcle-track system. The analyss of wheel-ral contact force response thus necesstates development of a representatve vehcle-track system model ncorporatng the contact model. In ths work, the adaptve wheel ral contact model s appled to a three dmensonal model of multple layers track system n conjuncton wth roll plane model of the vehcle system. Such formulaton permts for analyss of the nfluence of a wheel defect on the wheel-ral nteracton of not only the defectve wheel but also the other wheel wthn the same axle. The smplfed vehcle model conssts of half bolster coupled to two half-sdeframes through the secondary suspenson and a complete wheelset as shown n Fgure 7. The varous degrees of freedom nclude: the bounce (z b ) and roll (φ b ) motons of the bolster; bounce motons of the left and rght sdeframes (z sfl, z sfr ); and the bounce (z w ) and roll (φ w ) motons of the wheelset. The elements of prmary suspenson represented by K 1, C 1 and secondary suspenson represented by K, C are assumed to be lnear. Load W stands for a quarter of car body weght, and acts on the center of the bolster. P 1 and P r are vertcal forces at the wheel-ral contacts at the left and rght wheels, respectvely. The model neglects nteractons between the leadng and tralng wheelsets wthn a boge. The contrbutons of the car body dynamcs are also consdered to be relatvely small due to ts low natural frequency. The equatons of motons of roll-plane vehcle model are expressed n the matrx form as follows: [ M ]{ d & } + [ C]{ d& } + [ K]{ d} = { F} (3) Where [M], [C] and [K] are mass, dampng and stffness matrx, respectvely; and vector {d} s the dsplacement vector; { F } s the generalzed external force vector. For a roll plane vehcle model, t s necessary to consder a par of ral system n three- dmenson (3-D). For ths nvestgaton, two-layer track system model consstng of left and rght rals, pads, sleepers and ballast elastcty s developed, as shown n Fgure 8. Two rals are modeled as Tmoshenko beams supported on the sleepers modeled as lumped masses, through the ral-pads and fasteners that are represented by sprngs and dampers. The ballast s modeled as sprngs and dampers nserted between each dscrete sleeper and subgrade. 16

5 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM-7-46 Bolster K, C z Rght Sdeframe K, C sfl Left Sdeframe z sfr z w φ w K 1, C 1 K 1, C 1 Wheelset A ral of length L, modeled as Tmoshenko beam supported dscretely by n sleepers s shown n Fgure 9. Both ends of the ral are assumed fxed, where a denotes the spacng between two adjacent sleepers. P c and X w represent the vertcal contact load and ts locaton along the ral length. For the ral model wth a movng pont load P c and dscrete support forces F (Fgure 9), the governng equatons for vertcal and bendng motons can be expressed as [6]: Z Rl( r )( x,t ) Z Rl( r )( x,t ) kag θl( r )( x,t ) m = P cl( x x t ( t ) δ( x X ) r ) W θl( r )( x,t ) Z Rl( r )( x,t ) θ ( x,t ) l( r ) [ EI ] + kag θl( r )( x,t ) mr + P θ a x x x t l( Z K p, C p a X w P l L z b W Fgure 7: Sx-DOF roll-plane vehcle model P c K b, C b Fgure 8: Schematc of the two-layer 3D track system = 1 r N F δ( x a ) ( x,t ) = F 1 F F 3 F - F F n-1 F n Fgure 9: Model of ral supported by sleepers (fxed end) ) l( r ) (4) where Z R and θ represent ral vertcal and rotatonal motons; k s the shear coeffcent of ral; A s the crosssectonal area of the ral and G s the shear modulus of the ral materal; EI represents the flexural rgdty of the ral. δ functon represents the poston of vertcal forces on the ral and converts the concentrated forces nto dstrbuted forces. Subscrpt refers to th sleeper, and notaton l(r) refers to left (rght) sde ral. m s ral mass of unt length. r s the radus of gyraton of ral cross-secton. P a s longtudnal force appled on the ral whch s neglected n ths study. L b L w φ b P r X As shown n Fgure 8, the sleepers are model as lumped masses. A set of sprngs and dampers represent the complance of the ballast. The lateral dstance between two ral supports s L s. The vertcal dsplacement Z s and roll dsplacement φ s of th sleeper due to ral pad force {F s } are derved from the followng dfferental equaton: [ M ]{ } [ ]{ } [ ]{ } { } T s d& s + Cs d& s + Ks ds = Fs (5) where [M s ], [C s ] and [K s ] are the mass, dampng and stffness matrces; {d s } s the dsplacement vector. The detaled dervaton of the equatons s presented n [7]. 4 Smulaton Results Coupled vehcle-track system model developed n the prevous secton consdered the ral as contnuous system, whle the vehcle and sleeper components are modeled by dscrete or lumped parameter systems. Mathematcally, the model s represented by a set of coupled ordnary (ODE) and partal dfferental equatons (PDE). It s essental to explore an effectve method for analyss of coupled partal and ordnary dfferental equatons wth suffcent accuracy and stablty of the soluton. In ths study, a central fnte dfference method (CFDM) s appled to solve for partal and ordnary dfferental equatons [7]. In ths study, a nomnal wheel flat s defned as 1 mm long and 1.5 mm deep. Ths flat s ntroduced only to the left wheel, whle the rght wheel s assumed to have a perfect profle. The vehcle and track parameters lsted n Table 1 and the above flat sze are selected to facltate a drect comparson of results wth those avalable n the lterature. Smulatons are carred out for track length correspondng to 5 sleepers and results are presented for a tme segment where any effect of boundary condtons s absent. Fgure 1 llustrates a segment of the tme hstory of wheel-ral contact force at the left and rght wheel-ral nterface for a forward speed of 7 km/h. When the flat on left wheel comes nto contact, the contact force reduces due to loss of contact as the left wheel suddenly drops whle the left ral moves up untl the wheel hts the ral. Thus an mpact force s produced. As shown n Fgure 1, there s a total loss of contact at the left wheel for the speed and flat sze consdered. After the mpact, the contact force oscllates for about a cycle pror to dsspatng due to the dampng. Fgure 1 further shows that the contact force at the rght wheel-ral nterface also vares n a smlar manner. The magntude n ths case, however, s much smaller and takes places wth a tme delay n relaton to the left wheel. For the speed and flat sze consdered, the mpact force at the left wheel wth flat s found to be more than 3.5 tmes the statc load, whereas the peak force at the other sde s about 1.5 tmes the statc wheel load. In order to examne the mpact sequence n terms of wheel and ral motons, the tme hstory of wheel and ral dsplacements s presented n Fgure 11. The results show the change of wheel and ral movement at the left and rght track as the vehcle runs along the track. When the flat on left wheel comes nto the contact regon, the left ral moves up and wheel drops down pror to mpact between wheel and ral. After mpact, the left wheel and ral oscl- 161

6 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM-7-46 late around ther ndvdual statc poston for a short duraton. Meanwhle, at the rght sde, both ral and wheel move up at frst to compensate for the roll moton of the axle as left wheel drops due to the flat. The resultng oscllatng motons of the ral and wheel at the rght sde yeld the varaton n the contact force at that wheel. Table 1: Parameters of vehcle and track system [8] Vehcle System Car body mass (quarter of vehcle) Bolster mass (half) Mass moment of nerta of bolster about centerlne of track (half) Mass of half sde frame (half) Mass of wheelsets Mass moment of nerta of wheelset about centerlne of track Prmary suspenson stffness Prmary suspenson dampng coeffcent Secondary suspenson stffness Secondary suspenson dampng coeffcent Dstance between left and rght secondary suspenson n boge Dstance between left and rght wheel bearng center Wheel radus Track System 15 kg 3.5 kg 87.5 kg.m 3.75 kg 11 kg 4.1 kg.m 6.5 MN/m 1 kn.s/m.55 MN/m 44.4 kn.s/m 1.6 m 1.6 m.475 m Shear coeffcent.34 Ral cross secton area m Shear modulus of ral 81GN/m Elastc modulus of ral Second moment of area of ral about Y axs Ral mass per meter Pad stffness Pad dampng coeffcent Ballast stffness (*) Ballast dampng coeffcent (*) Sleeper mass N/m m 4 6 kg/m 14 MN/m 45 kn/m 4 MN/m 5 kn.s/m 7 kg Mass moment of nerta of sleeper 9 kg.m Sleeper spacng.685 m Ral support dstance (*) 1.55 m Radal sprng stffness N/m/radan Note: The parameters wth(*) are not gven by reference [8] and assumed accordng to typcal freght car[3] 16 Fgure 1: Contact force tme hstory n the proxmty of wheel flat contact Fgure 11: Tme hstory of wheel-ral dsplacement 4.1 Comparson wth Hertzan Contact Model The most wdely used contact theory for vertcal wheel-ral contact force smulaton s the Hertzan contact model. One of the motvatons for the present nvestgaton was to ntroduce a new radal sprng adaptve contact model that can accommodate asymmetrc contact regon for defectve wheel profle. In order to compare the developed model n applcaton to wheel flat, the vehcle-track system model ncorporatng Hertzan contact model and the developed adaptve contact model are smulated for dentcal parameters. A common value n lterature used for Hertzan contact coeffcent s c H =.85x1 11 for Hertzan contact model, whle the radal sprng stffness of the adaptve model s selected such that both models yeld dentcal statc contact force. The smulaton s carred out for a 1 mm long and 1.5 mm deep flat at a speed of 7 km/h. The contact force result for one steady-state mpact cycle s shown n Fgure 1. As the results show, the mpact force from Hertzan contact model (558. kn) s sgnfcantly larger than that from adaptve model (44.7 kn). The mpact force predcted by adaptve contact model s therefore 7.5% less than that from Hertzan contact model at 7 km/h for gven parameters. The results n Fgure 1 further show that the

7 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM-7-46 Hertzan contact model predcts the loss of contact of wheel-ral second tme after the mpact, and the duraton of the second loss of contact s even larger than the frst one, whch s unlkely n practce. In order to examne the effect of speed, the peak mpact load due to the same flat wthn the speed range ~ 18 km/h s plotted n Fgure 13. As the results show, both models predct smlar trend for change n speed. However, n comparson to adaptve contact model, Hertzan contact model underestmates wheel-ral mpact load at low speeds, and overestmates the mpact load at hgh speeds. Such a trend for the performance of adaptve contact model s very encouragng snce t s well known n the lterature that for smulaton of mpact force due to wheel flat, Hertzan contact model may underestmate at low speeds whle may overestmate at hgh speeds. Another notceable dfference between the two contact models as shown n Fgure 13 s the fact that beyond 9km/h, the peak mpact force predcted by Hertzan contact model reduces consderably as speed s ncreased, whereas the adaptve contact model exhbts very small reducton wth ncreasng speed. extensve feld measurements n Svealandsbanan, Sweden on the man lne between Esklstuna and Sodertalje n cooperaton wth Chalmers Unversty of Technology n [9]. The response of wheel-ral contact force due to a 1 mm long,.9 mm deep wheel flat was measured at dfferent travelng speed. The wheel ral contact force was measured utlzng an nstrumented sleeper bay over whch a boge wth a flat wheel was moved at dfferent speeds. A sample of expermental tme hstory of mpact force at 5 km/h s shown n Fgure 14. As the result shows, the wheel-ral contact force recorded s zero except when the wheel s on the nstrumented sleeper bay. The mpact sequence can be clearly dentfed from the result. As the wheel flat approaches contact regon, there s an ntal drop n the contact force from the statc value. Ths s followed by a relatvely sharp peak force referred to as the mpact force. Fnally the mpact force s followed by a damped oscllaton of the contact force as the wheel travels away from the nstrumented bay. The second ncrease n Fgure 14 corresponds to rear wheel wth perfect profle enterng the nstrumented bay. The expermental vehcle and track parameters and descrpton of wheel flat (1 mm long and.9 mm deep) are also smulated usng the developed model for ths nvestgaton. Majorty of the parameters are obtaned from reported studes by Anderson and Oscarsson [1] and by Nelsen and Oscarsson []. They also carred out numercal smulaton of wheel-ral mpact force response due to such wheel flat to compare wth expermental results n [9]. The tme hstory of the contact force from the current study s presented n Fgure 15. The result for 5 km/h shows very good agreement wth the feld test result presented n Fgure 14. The peak contact force (15 kn) and the trend predcted from ths smulaton s very smlar to the peak (11 kn) and trend observed n the feld test. Fgure 1: Comparson of contact force from adaptve and Hertzan contact model 6 Impact Load (kn) Hertzan Contact Adaptve Contact Speed (km/h) Fgure 13: Comparson of wheel-ral mpact load at dfferent forward speeds 4. Comparson wth Publshed Feld Test Data In order to nvestgate the nfluence of wheel flat on the dynamc wheel-ral contact force, the Centre of Excellence CHARMEC (CHAlmers Ralway MEChancs) carred out Fgure 14: Tme hstory of wheel ral contact force measured n one nstrumented sleeper bay at 5 km/h [16] The smulated peak contact or the mpact load for the same parameters s evaluated for speed n the range of 5 to 1 km/h. These results are presented n Fgure 16 along wth expermental result from [9]. Fgure 16 further presents the numercal results obtaned n [, 1] for Hertzan nonlnear contact model along wth sngle wheel vehcle model, and lnear and nonlnear track model. Although some expermental results are scattered, t s easy to see that the present vehcle-track model wth adaptve contact model 163

8 IISc, Bangalore, Inda, December 1-13, 7 NaCoMM-7-46 shows closest trend to the experment among all the results presented n Fgure 16. Partcularly, the effectveness of the adaptve contact model as opposed to Hertzan model at hgh speed s hghly sgnfcant. The response of the vehcle-track system n terms of contact force s examned for two dfferent flat szes at varous forward speeds. Results are compared wth reported expermental and computatonal data n tme doman and n terms of peak mpact force at dfferent speeds. The results demonstrate that adaptve contact model s more realstc for accurate representaton of the contact between wheel and ral. The comparatve study shows that the proposed model, although smplfed n terms of vehcle and track systems, can predct wheel-ral mpact load better than nonlnear Hertzan pont contact model. Ths study also shows that wheel flat can cause wheel-ral mpact load not only between the defectve wheel and ral, but also between the ral and cross wheel. Further studes are proposed wth nonlnear track model, and for establshment of accurate radal sprng stffness for adaptve contact model. Fgure 15: Tme hstory of wheel ral contact force calculated n present smulaton at 5 km/h It s also apparent that the smulated results obtaned n ths nvestgaton tend to predct the upper bounds of expermentally observed mpact loads for speeds beyond 4 km/h. From the nfluence of nonlnearty n the track system [] as shown n Fgure 16, t s possble that the present model wth track system nonlnearty may produce even a better agreement wth the expermental values over the entre speed range. 16: 5 Conclusons Unlke the sngle or multple pont contact models, the proposed adaptve contact model s based on contnuous wheel-ral contact n the contact patch, and accounts for asymmetry of the patch and partal contact as the flat enters and leaves the contact regon. References [1] Z. Q. Ca, Modelng of ral track dynamcs and wheel/ral nteracton, Ph.D. Thess, Queen s Unversty 199. [] J.C.O. Nelsen and J. Oscarsson, Smulaton of dynamc tran-track nteracton wth state-dependent track propertes, Journal of sound and vbraton, 75(4), [3] R. G. Dong, Vertcal Dynamcs of Ralway Vehcletrack System, Ph.D. Thess, Concorda Unversty [4] K. Wang, Dynamc analyss of a tracked snow plowng vehcle and assessment of rde qualty, MAsc. Thess, Concorda Unversty [5] D. Lyon, The calculaton of track forces due to dpped ral jonts, wheel flats and ral welds. Paper presented at the Second ORE Colloquum on Techncal Computer Programs, May 197. [6] S. W. Weaver, et al., Vbraton problems n engneerng, 5 th ed. Wley, New York c199. [7] J. J. Zhu, Development of an adaptve contact model for analyss of wheel-ral mpact load due to wheel flats, MAsc. Thess, Concorda Unversty 6. [8] M. Fermer and J. C. O Nelsen, Vertcal nteracton between tran and track wth soft and stff ralpads---full scale experments and theory. Proc. Instn Mech.Engrs, Part F: J. Ral and Rapd Transt, 1995, 9(F1), 39~47. [9] A. Johansson and J. C. O. Nelsen, Out-of-round ralway wheels - Wheel-ral contact forces and track response derved from feld tests and numercal smulatons, Proceedngs of the Insttuton of Mechancal Engneers, Part F: Journal of Ral and Rapd Transt, 3, Vol: 17 Iss: Page: 135. [1] C. Andersson and J. Oscarsson, Dynamc tran/track nteracton ncludng state-depent track propertes and flexble vehcle components, Vehcle system dynamcs 33, 47-58,