Thermo-Mechanical Model for Wheel Rail Contact using Coupled. Point Contact Elements

Transcription

1 IM th July, ambidge, England hemo-mechanical Model fo heel Rail ontact using oupled Point ontact Elements *J. Neuhaus¹ and. Sexto 1 1 hai of Mechatonics and Dynamics, Univesity of Padebon, Pohlweg 47-49, 3398 Padebon, Gemany *oesponding autho: jan.neuhaus@upb.de Abstact A model to calculate the locally esolved tangential contact foces of the wheel ail contact with espect to contact kinematics, mateial and suface popeties as well as tempeatue is intoduced. he elasticity of wheel and ail is modeled as an elastic laye consisting of point contact elements connected by spings to each othe and to the wheel. Each element has two degees of feedom in tangential diections. he esulting total stiffness matix is educed to calculate only the position of the elements in contact. Fiction foces as well as contact stiffnesses ae incopoated by a nonlinea foce-displacement chaacteistic, which oiginates fom a detailed contact model. he contact elements ae tanspoted though the contact zone in discete time steps. Afte each time step an equilibium is calculated. Fo all elements, thei tempeatue and its influence on local fiction ae egaded by calculating fiction powe and tempeatue each time step. Keywods: Rolling ontact, Discete Elements, ontact Stiffness, empeatue Intoduction o calculate the tangential contact foces in wheel ail systems fo e.g. vehicle dynamics, a pofound knowledge of the ceep foce chaacteistic is necessay. o achieve this, the ceep dependent distibution of tangential foces inside the contact zone is necessay. Especially at highe slip the tempeatue in the contact zone becomes high enough to effect significantly on the tangential foces. Many appoaches have been made to calculate the tangential foce distibution. Kalke [Kalke (1967); Kalke (199)] developed the pogam ONA to calculate contact foces. his pogam assumes Hetzian contact and halfspace assumption to calculate the taction foces. Due to halfspace assumption the computation times ae vey high. Appoaches have been made to educe computation time compaed to ONA [Kalke (1982); Polach (2)], howeve these do not include tempeatue effects. In [Sexto (27)] a numeical model fo the wheel ail contact is developed. he contact zone is discetized and each patial aea is descibed by a point contact element. A single point contact element includes diffeential contact stiffness as well as a nonlinea patial fiction foce. By applying defomations on the point contact elements, which ae deduced fom the kinematics of wheel and ail, the tangential foce distibution in the contact zone can be calculated. Also tempeatue and its influence on the fiction coefficient ae egaded in this appoach. In [ombege et. al (211)] a model fo the wheel ail contact compising oughness intefacial fluids and tempeatue is shown. he tangential contact is discetized using independent contact stiffnesses as suggested in [Sexto (27)]. he modeling method is based on FASSIM, but allows vaying coefficients of fiction. In a mico contact model the effects of intefacial fluids and tempeatue on the fiction coefficient ae calculated with espect to oughness, contact kinematics and mateial popeties. 1

2 Also, the Finite Element Method has been used to calculate foces in the olling contact. Nackenhost [Nackenhost (24)] used an abitaian lagangian euleian fomulation fo calculating contact foces using the Finite Element Method fo stationay olling contact. By splitting the motion into igid body motion and elastic defomation and solving the tanspotation poblem of wheel and ail elements though the contact zone simila to tanspot poblems in fluid dynamics. he contact foces can be calculated by this method etaining a fine mesh in the contact zone and a lage mesh outside of it. In [en et al. 211] and [Zhao and Li 211] the tangential foces in wheel ail contact whee calculated using Finite Elements models. Howeve, tempeatue effects wee not taken into account and the computation time fo these models is vey high. he model pesented hee is based on the appoach by Sexto [Sexto (27)]. Instead of using independent point contact elements, the elasticity of wheel and ail is modeled by an elastic laye, to bette model the elastic defomation inside the contact zone. he algoithm used in this model computes the tangential contact foces at stationay olling by computing equilibium of the foces caused by the elastic laye and the tangential contact foces. Additionally, the influence of oughness and tempeatue can be investigated with this model. ontact Kinematics he kinematics of a igid wheel olling ove a igid ail unde slip angle α and angula velocity ω ae depicted in Fig. 1. y Ay a) I ω v z y x α Iz Az Iy A Ay Ix Ax b) Figue 1, heel and Rail Schematic v α x Ax he wheel shown in Fig. 1 a) is moving elative to the inetia fame I in I x-diection with constant velocity v. he contact patch shown in Fig. 1b) is descibed by the efeence system A, which is also moving with the wheel cente in I x-diection with velocity v. A point on the wheel s suface in contact is descibed by the wheel fixed coodinate system, which is otated elative to the inetia system aound the I z-axis by a constant angle α. he angula velocity of the wheel is defined elative to the wheel fixed coodinate system with [ ω ] I ω =. (1) 2

3 he inetia system I is fixed on the ail. he tangential velocity of the point on the wheel in the contact is dependent on the angula velocity and the slip angle α and can be calculated by he longitudinal slippage s can be calculated with I I [ v ωr cosα ωrsin α ] v =. (2) v ωr s =. (3) v In the following, all vectos will be given in coodinates of the -system, unless noted othewise. ontact Model Modelling appoach he ceep foce chaacteistic of wheel and ail is highly dependent on the elastic defomations inside the contact aea. he elasticity of wheel and ail is modeled as an elastic laye consisting of discete elements having an elasticity equivalent to the combined elasticity of wheel and ail. A detail of the elastic laye is shown in Fig. 2 a). he laye consists of coupled massless point contacts P connected by spings with sping stiffnesses c k,x and c k,y in tangential diection to each othe. he laye is also coupled by spings with stiffnesses c x and c y to the igid wheel in x- and y- diection espectively. v c x A total y c y A contact c k,x c k,y P c k,y c k,x x c k,x c k,y c k,y F ( µ, F N, ) c k,x ailing Edge x y Leading Edge a) b) Figue 2, Elastic Laye and ontact Aea If the point P is inside the contact aea, a nonlinea tangential foce F ( µ, FN, ) applies, which is descibed in detail late in this pape. is a point which is fixed on the ail, so is the sliding distance of which is also the displacement of P elative to the ail. he tangential foces cause defomations which influence the neighbohood aound the contact aea and vice vesa. heefoe, not only the contact aea but also the suounding aea A total is modeled with the elastic laye. he size of the total aea A total is not as lage as the total wheel s suface but lage enough, that defomations at the edge of the total aea ae diminishing small compaed to the defomations inside the contact aea. he discetised elliptical contact aea with the suounding total aea is shown in 3

4 Fig. 2 b). he total aea is discetised in ectangula patial aeas with the dimensions x and y. he position of the contact aea bode in dive diection is efeed in the following as leading edge. he elastic laye can be descibed by a set of linea coupled equations. he elation between foce and displacement with espect to the wheel coodinate system is: F = ii oi io oo i o. (4) he foce and displacement vectos in this equation have been soted fo point elements inside the contact aea with index i and outside the contact aea with index o. he displacement vectos can be witten as P [ ], i = P1, x, i P1, y, i P 2, x, i P 2, y, i and [ ] P o P1, x, o P1, y, o P 2, x, o P 2, y, o, =. (5) P, x and P, y ae the displacements of the point P in x- and y-diection espectively. In ode to educe computational effot, the equation system is educed to calculate only the foces and displacements inside the contact aea. he displacements of the points outside the contact aea can be calculated by the second ow of the block matix equation Eq. (4): 1 o oi oo i =. (6) Inseting Eq. (6) in the fist ow of Eq. (4) leads to F 1 ( ii oi oo ) P i =., ed (7) So, the elation between tangential foces and tangential displacements in contact can be descibed by a linea equation system with educed stiffness matix ed. Nonlinea angential Foce he elasticity of wheel and ail is modeled as descibed above. Due to the oughness of sufaces in contact a nomal pessue dependent contact stiffness develops. Futhemoe, the tangential foce fo sticking and sliding conditions has to be applied at point P. his is egaded by a nonlinea foce displacement chaacteistic, deived fom a detailed mico contact model [Neuhaus and Sexto (213)]. Using this model, the tangential foce including the tansition fom sticking to sliding can be computed using measued ough sufaces fo diffeent nominal nomal pessues. he cuves achieved fom this simulation can be appoximated by adequate analytical functions fo efficient use in the olling contact model. Figue 3 shows a esult using this mico contact model. he development of the nomalized foce while moving two sufaces tangentially against each othe is shown. he slope at nomalized tangential displacement of zeo can be intepeted as contact stiffness while sticking. Full sliding is indicated by a gadient of zeo of the foce displacement cuve. Pue sticking only exists when no tangential foce applies; aftewads the cuve ises nonlinea and continuously goes ove to sliding. his cuve can be appoximated well by an exponential function. In case of monotone elative movement in constant diection this foce displacement chaacteistic can be used to model sticking and sliding, because the displacement vecto always points in the same diection as the velocity vecto. 4

5 Figue 3, angential Foce he tangential foce on a single point contact P is dependent on the sliding distance of P and thus of the magnitude of and its diection. As mentioned above, it can be calculated by an exponential law which appoximates the foce displacement chaacteistic seen in Fig. 3 with F = μ F N 1 e k c (8) whee is the unit vecto pointing opposite to the sliding diection. he coesponding diffeential nomal foce is calculated accoding to a given nomal pessue distibution. he nomal pessue distibution can be achieved fo example by using the Hetzian heoy, Finite element calculations o measuement esults. Simulation pocedue he olling contact is simulated by tanspoting the elastic laye as descibed above though the contact aea in discete time steps t. he tanspoting velocity elative to the contact coodinate system A given in coodinates of the wheel system is A t [ ωr ] v =. (9) In ode to maintain a constant gid, the value of t is chosen in a way that an element is tanspoted the distance x in one time step. So, the position of point is equal to the position of its successo in negative x -diection afte one time step if x t = ωr. (1) Because P is fixed to the ail a elative diffeential displacement, P between and applies afte each time step t. It is the displacement between a point on the igid wheel and a point on the 5

6 igid ail afte one time step and can be expessed elative to the contact coodinate system A assuming small angles α by [ v ωr ωrα ] t = A, (11) o with espect to the wheel coodinate system by [ v ωr vα ] t =,. (12) his diffeential elative displacement, P is added to the position vecto of,, afte each time step. Fo new elements enteing the contact aea at the leading edge, the position vecto is set to thei position vecto P immediately befoe enteing the contact, such that =. Afte applying the elative diffeential defomation fo one time step, an equilibium between the nonlinea tangential foce defined in Eq. 8 and the foces fom the elastic laye is found accoding to Eq. 7 by with i F = (13) ed [ F F F ] F R R, 1 R,1 R, nk = (14) whee n k is the numbe of elements in contact. he equilibium fo Eq. 13 is found by using the Newton Raphson method. Fistly, the gadient of the tangential foce F i has to be calculated, which can be done due to the analytical desciption of the foce displacement chaacteistic. Secondly, the Jacobian J is calculated by adding the foce gadient of the nonlinea tangential foce to the constant gadient emeging fom the educed stiffness matix shown in Eq. 15: J F = ed + (15) i Using the Jacobian fom Eq. 15, a diffeential displacement can be calculated and added to the displacement vecto i iteatively until the equilibium condition of Eq. 13 is fulfilled within an elative eo ε el. ed P i (16), F < ε el μf N Afte finding the equilibium fo a defined toleable eo ε a new time step is calculated until a steady state is eached in the contact aea. his is usually the case, when an element enteing the contact aea at simulation stat has cossed and left it at the tailing edge. 6

7 empeatue Model aused by high contact pessue in the wheel ail contact and consequently high fiction powe, the tempeatue in the contact zone plays an impotant ole to descibe the fiction behavio coectly. heefoe, this effect is consideed in this model. he model used hee is adapted fom the tempeatue model descibed in [Sexto (26)]. he fiction powe fo a single element can be calculated by he heat souce qh P = F R v. (17) P R can be calculated elating the fiction powe to a patial aea q H PR =, A = x y. (18) A Assuming a high Peclet numbe, Knothe and Liebelt [Knothe and Liebelt (199)] showed, that the thee dimensional heat tansfe poblem can be educed to a two dimensional poblem fo a stip in x-diection using the appoximated heat tansfe equation v x 2 = κ (19) 2 z with the themal diffusivity defined as λ κ =, (2) ρ c whee λ denotes the heat conductivity, ρ the density and c the specific heat capacity of the wheel [Sexto (26)]. Assuming a constant heat souce, the tempeatue distibution in x-diection, the tempeatue at time step j can then be calculated using the tempeatue at the element at time step j-1 by j = j 1 +. (21) he tempeatue diffeence is computed as = 2 κ v π α q λ H x (22) with α as the heat patitioning facto between wheel and ail. Fo low slippage the facto can be assumed to be.5, this means heat is equally distibuted between wheel and ail. Othewise it can be calculated fom the velocities of the contact elative to wheel and ail. Fo details see [Sexto (26)]. Using this appoach, the maximum tempeatue occus at the tailing edge due to the assumption of constant heat distibution, but in fact the tempeatue eaches its maximum shotly befoe the tailing edge due to the not-constant heat distibution. Howeve this pocedue can be used to egad tempeatue influence because the tempeatue distibution in the main contact aea whee most of the foces ae tansmitted is appoximated well. 7

8 In pinciple, due to the mutual inteaction between fiction powe, tempeatue and fiction coefficient, an iteative loop inside the computation of a single time step is necessay. his can be skipped, because the tempeatue and thus the tempeatue dependent fiction coefficient convege within the time step simulation. he tempeatue dependent fiction coefficient at time step j+1 is assumed to be appoximate equal to the tempeatue dependent fiction coefficient at time step j. Using this pocedue usually a steady state is eached afte an element is tanspoted though the complete contact aea. hus the tempeatue dependent fiction coefficient of time step j+1 is calculated using a linea elationship between the fiction coefficient µ and tempeatue at time step j j 1 μ j = μ 1 (23) E μ j+ whee E defines the slope of the tempeatue dependency. Results he equations above have been implemented in MALAB to model the olling contact. he esults shown in the following ae nomalized and theefoe have qualitative chaacte. his will be done in futue by compaing the defomations of the elastic laye with the defomation computed by a Finite Element model and adapting the sping stiffnesses to minimize the defomation diffeence. Nonetheless the esults show that the effects in olling contact can be modeled plausibly. Fo the simulations, the nomal pessue has been calculated accoding to Hetzian theoy. he nomalized nomal pessue distibution * p p N p = (24) N max is shown in Fig. 4. he x* and y* coodinates have been nomalized to the ellipsis half axes. Figue 4 Nomal Pessue Distibution Fig. 5 shows the tangential foce in x-diection nomalized to the maximal tansmittable tangential foce with 8

9 * F = μ F p N max A. (25) he simulation has been pefomed at pue longitudinal slip s of.8. he leading edge is seen font ight. A zone with sticking fiction, behind the leading edge can be identified with a linea ising tangential foce towads the tailing edge shown in Fig. 5a) whee almost no sliding velocity exists, as shown in Fig. 5b). Also a egion with sliding fiction can be seen towads the tailing edge. Hee, the shape of the tangential foce distibution equals the nomal pessue distibution and the sliding velocities ise towads the tailing edge. he step in the shown tangential foce at the leading edge and at the tansition fom sticking to sliding is caused by the coupling stiffnesses. Qualitatively the shape of the tangential foce distibution matches well with Kalke s theoy and Finite element calculations. a) b) Figue 5 angential Foce Distibution and Sliding Velocity In Fig. 6 the tangential foces in x- and y-diection fo a longitudinal slip s of.8 and slip angle α of.5 degees ae shown. a) b) Figue 6 angential Foce Distibution in x- and y-diection ompaed to pue longitudinal slip, the total tangential foce in x-diection is significantly lowe, because lateal foces in y-diection occu at this slip angle, which is shown in Fig. 6b). Also the aea with sliding fiction is lage compaed to the case with pue longitudinal slip in Fig. 5. he fiction powe fo each patial aea P R and the tempeatue diffeence elative to the suounding ae shown in Fig. 7 a) and b) espectively fo combined slip. he fiction powe eaches its maximum just befoe the tailing edge due to the tangential foce and sliding velocity 9

10 distibution. In the sticking egion, the fiction powe is zeo. As descibed at the tempeatue model, tempeatue eaches its maximum at the tailing edge. a) b) Figue 7 Fiction Powe and empeatue he computing time fo one simulation was 19 seconds on an Intel i7 pocesso using 2 of 4 coes. his is quite low compaed to computationally intensive Finite Element models. onclusions A model fo calculating the tangential foces in the wheel ail contact compising contact stiffness and tempeatue has been developed. he model is based on the olling contact model of Sexto [Sexto (26)]; howeve the point contacts ae coupled with each othe to model an elastic laye which epesents the elasticity of wheel and ail. Also, a nonlinea tangential foce is applied to the coupled massless points, to model contact stiffness as well as sticking and sliding fiction. he set of equations fo descibing contact and suounding aea can be educed to compute only foces and displacements inside the contact aea which educes computation time. he simulation is caied out in discete time steps, in which the elastic laye is moved though the contact aea and an equilibium is calculated afte each time step. empeatue and its influence on the fiction coefficient ae calculated as in the model of Sexto [Sexto (26)] assuming a high Peclet numbe and constant heat souce distibution. his is a satisfying appoximation fo the exact solution of elliptical heat souce distibution. he simulation poduces plausible esults fo tangential foce distibution unde pue longitudinal and combined slip. Fiction powe and tempeatue distibution match qualitatively well compaed to othe modeling methods. 1

11 Refeences Kalke, J. J. (1967). On the olling contact of two elastic bodies in the pesence of dy fiction (Doctoal dissetation, Delft Univesity) Kalke, J. J. (199). hee-dimensional elastic bodies in olling contact (Vol. 2). Spinge. Kalke, J. J. (1982). A fast algoithm fo the simplified theoy of olling contact. Vehicle System Dynamics, 11(1), Polach, O. (2). A fast wheel-ail foces calculation compute code. Vehicle System Dynamics, 33, Sexto,. (27). Dynamical contact poblems with fiction: models, methods, expeiments and applications. Spinge. ombege,., Dietmaie, P., Sexto,., & Six, K. (211). Fiction in wheel ail contact: a model compising intefacial fluids, suface oughness and tempeatue. ea, 271(1), Nackenhost, U. (24). he ALE-fomulation of bodies in olling contact: theoetical foundations and finite element appoach. ompute Methods in Applied Mechanics and Engineeing, 193(39), en, Z., u, L., Li,., Jin, X., & Zhu, M. (211). hee-dimensional elastic plastic stess analysis of wheel ail olling contact. ea, 271(1), Zhao, X., & Li, Z. (211). he solution of fictional wheel ail olling contact with a 3D tansient finite element model: Validation and eo analysis. ea, 271(1), Neuhaus, J.; Sexto,. (213): A Discete 2D Model fo Dynamical ontact of Rough Sufaces. 3d Intenational onfeence on omputational ontact Mechanics 213, 213 Knothe, K., & Liebelt, S. (1995). Detemination of tempeatues fo sliding contact with applications fo wheel-ail systems. ea, 189(1),