Keywords: multibody simulation, wheel-rail contact, real time simulations

Transcription

1 Detemination of wheel/ail contact points in the simulation of a ailwa vehicle dnamics Ju Auciello, Stefano Falomi, Monica Malvei, Enico Meli, Paolo Toni Depatment of Enegetics S. Stecco, Univesit of Floence, Ital Astact Numeical simulation of sstem dnamics is toda a standad in the design of ailwa vehicles thei tpical applications ae the suspension kinematics, handling pefomance and ide comfot as well as the geneation of load data fo lifetime pediction. One of the ke points in this tpe of simulations is the model of the wheel/ail inteaction, in othe tems the definition of the foces exchanged etween the wheels and the ail in the contact points. The diection and the magnitude of the contact foces depends on the nume and the location of the contact points. The pocedue that allows to define the geomet of the contact has then a significant effect on the eliailit of the simulation. The component of the contact foce nomal to the contact sufaces can e defined as a function of the elative indentation etween the sufaces. The component of the contact foce tangent to the contact sufaces depends on the elative speeds etween the sufaces in the contact aea (wheel sliding). The authos have een woking on the definition of efficient and eliale models of the inteactions etween the wheels and the ails and in paticula fo the definition of the contact points. Diffeent algoithms have een analed and compaed, the ae ased on semi analtical appoaches and on neual netwoks. The pape will summaie the poposed methods and the esults otained fom the simulation of two diffeent sceneies. Two diffeent models have een used in this test: the fist one was ealied with a commecial softwae, while the second one was developed and implemented the authos. Thee is a gloal ageement etween the models, some diffeences can e seen duing the tansients, due to the diffeent methods fo the detemination of the contact points fo the integation. Kewods: multiod simulation, wheel-ail contact, eal time simulations 1 Intoduction Numeical simulation of sstem dnamics is toda a standad in the design of ailwa vehicles. The multiod analsis is applied fo example to suspension kinematics and compliant kinematics, handling pefomance and ide comfot as well as to the geneation of load data fo lifetime pediction. These simulation tasks ae usuall caied out as off-line simulations pefomed means of commecial o customied softwae.

2 In the last eas Hadwae in- the-loop (HIL) simulation has extensivel een intoduced in the design of vehicle contol sstems and the test of electonic contol units, oth in the automotive and in the ailwa field. The eliailit of this tpe of tests is stictl elated to the popeties of the numeical model that simulates the condition that the tested device would meet in eal opeative conditions. The numeical model is a epesentation of the dnamics of the sstem in which the tested unit is inseted (single vehicle, entie tain etc.). In ode to otain on this tpe of test igs eliale esults the availailit of a ealistic eal time model of the sstem dnamics is then necessa. Ove the last eas the MBS (Multiod Simulation) method has then een estalished in the eal-time simulation domain, tpicall fo the design of vehicle contol sstems and the test of electonic contol units [1,2]. The numeical model descied in this pape epoduces the complete thee dimensional dnamics of a ailwa vehicle unning on a geneic tack, it has een developed with the ojective of a eal time implementation, in ode use its esults to contol the actuatos of Hadwae In the Loop test igs. One of the ke point in this stud was the definition of a eliale and efficient model desciing the foces aising in the wheel/ail inteaction aeas. The contact foce vecto is composed a nomal component, esulting fom the weight of the vehicle, and a tangential component that esults fom the fiction etween the wheel and ail in the contact aea. This component, named ceep foce, aises in the contact aea when a taction o a aking toque is applied to the axle. These foces pla an impotant ole in the wheelset dnamics and thei value depend on a nume of paametes, including the position of the contact point etween the odies. The intoduction of a well defined full thee dimensional wheel/ail inteaction module in a standad multiod model is not eas and ma sensil incease the complexit and the computational uden of the numeical pocedue [3-10]. In the model descied in the following sections the wheelset is assumed to have six degees of feedom with espect to the ails. The local defomation of the contact suface at the contact point is allowed and the nomal contact foces ae defined using Het s contact theo o in tems of assumed stiffness and damping coefficients. This tpe of appoach allows the sepaation etween the wheel and the ail and to manage multiple contact points. One of the main polems coelated with this appoach is the definition of the contact point location on line. In most elastic foce models, the thee-dimensional contact polem is educed, fo the sake of efficienc, to a two-dimensional polem when the location of the contact points is seached fo. In ode to find a solution fo the polem of contact mechanics, detailed desciptions of the sufaces in contact, as well as the kinematics of the odies, ae equied. Because wheel and the ail pofiled sufaces, the pediction of the location of the contact point on line is a not eas task, especiall when the most geneal thee-dimensional motion of a wheelset with espect to the ails is consideed. In man cases, some simplifications ae made in the geometic and/o the kinematic desciption. A common method used in man existing compute algoithms to find the location of the contact point is to intepolate

3 some pecalculated tale enties. The location of the contact point is given as function of some coodinates that measue the elative position of the wheelset with espect to the ails. The degee of accuac of such algoithms depends on the nume of coodinates that ae used to define this elative position. The developed numeical pocedue can e schematied as shown in Figue 2. The dnamics of each wheelset depends on the extenal actions (taction/aking toques) and on the contact foces. The solution of the diffeential equations desciing the wheelset dnamics allows to calculate the kinematical paametes (displacement and otation) fom which the position of the contact points can e identified. The ceep foces ae then calculated taking into account the position of the contact points and the wheel ceepages, otained fom the wheelset kinematics. MULTIBODY MODEL Initial conditions Phsical paametes (Inetial popeties of the odies, constaints, phsical constants) Geometical paametes (geometical popeties of the odies and the constaints, tack pofile, contact od sufaces) Contact points Contact foces Kinematic vaiales Kinematic vaiales Inteaction etween odies Contact points Contact foces CONTACT MODEL Figue 1: numeical pocedue lock diagam. 2 Definition of Wheel/Rail contact points In Figue 2 the efeence fames used to descie the wheel and ail pofile geometies ae shown. Concening the ail coodinate sstem, x is paallel to the tack diection, is on the plane defined the ails and nomal to x, is consequentl defined. The oigin O is fixed on the tack axis, in othe tems it is on the plane identified the ails and is equall distant fom oth of them. The simulated vehicle can move on a geneic tack, defined γ () s, epesenting a geneic thee dimensional cuve defined means of its paamete s. A second auxilia coodinate sstem elative to the tack is defined, its oigin O ' is on the median ailwa cuve γ ( s (t ) ), it is not fixed, and depends on the wheelset motion on the tack. The axis x ' is tangent to the tack cuve, ' axis is nomal to the ail plane and ' is consequentl defined. The oigin O of the wheelset coodinate sstem is on the plane ', then: '

4 [ O ( t) O' ( s( t) )] γ '( s( t) ) = 0 (1) It is then evident that fo an t value, s(t) and then O can e calculated fom the position of the wheelset cente of mass. ' ' x O ' ' γ (s) Wheel-set G o sala γ (s) ~ s tack γ ( ) ail γ ( s( t) ) ' O ' G ' ' x O x x ' a) ) Figue 2: a) ail coodinate sstem, ) wheel coodinate sstem. The wheel is a evolution suface, the wheel adius depends on the coodinate, the function s = s( ) descies the wheel pofile in the plane, the coodinates of a geneic point on the wheel suface can e expessed, in the wheelset coodinate sstem, as follows: ( P O ) s( )sinu = s ( u, ) = s( )cosu (2) whee u is the angle etween the adius that identif the geneic point on the wheel and diection. The ail suface is an extusion suface, the function = ( ') descies the ail pofile in the plane, then the coodinates of a geneic point on the ail suface can e expessed as: x P ' = ' (3) ( ) ( O ) the coodinates of a geneic point on the wheel can then e expessed in the auxilia coodinate sstem:

5 ( P O ) = O O + [ ] O ( R (4) ' ') 2 The displacement ( O O ' ) and the otation matix [ 2 ] R etween the wheelset and the tack efeence sstem depend on the elative displacement etween the wheelset and the ail, then, fo each wheelset configuation the wheel suface can e expessed with espect to the auxilia efeence fame as a function of the paametes u and : The point P coesponding to ( u ) P = s, (5) P on the ail suface is then given : ( u, ) ( u, ) x P = ( ) (6) ( ) n P P ( ) t P c ( ) n P c P c P O O a) ) Figue 1: a) ail suface, nomal and tangent diections, ) definition of the diffeence suface The diffeence suface is then defined as: u, = P P k (7) ( ) ( ) D The contact points ae located in the points in which this suface assumes a local minimum. Then, in ode to find the contact points the local minima of this suface ae seached. c c (, ) ( u, ) min D u i i, u, (8) i = 1,2,.., n 4

6 The minima ae calculated a pocedue that analticall educes the polem dimension to a single equation that is numeicall solved. This method is allows a good pecision with low computational loads. 3 Contact foces etween wheel and ail Once the local minima have een identified, the indentation etween the sufaces is conventionall defined in the diection nomal to the ail: c c c ( P P ) ( P ) p = n (9) Whee n' is the unita vecto nomal to the ail in the contact point. The indentation value is used to calculate the nomal component of the contact foce, accoding to the elasto-viscous appoach as sum of a tem that depends on the nomal penetation etween the od (the elastic component), and a tem popotional to the suface elative velocities in the contact point (the viscous tem). The magnitude of the tangential component of the contact foces is calculated on the asis of Kalke and Het theo [12,13]. The Het s theo is used to define the contact aea dimensions and shape, that depend on the nomal foce magnitude, the mateial popeties and the local pofile geomet. The Kalke linea theo esults ae used to define the components of the ceep foces as follows: Tx = f11ξ T = f22η f23ϕ M sp = f23η f33ϕ (10) whee f 11, f 22, f 33, f 23 ae the linea ceep coefficients, depending on the contact ellipse semi-axis and on the comined modulus of igidit (thei values ae taulated)[10]; ξ, η and φ ae the values of the ceepage components. 4 Vehicle multiod model The multiod model is composed of seven igid odies (the ca od, two ogies, fou wheelsets) connected thee-dimensional non linea elasticviscous foce elements modelling the connection elements etween the odies (fo example the vehicle suspensions, dampes etc.). On each wheelset the following foce act: the ceep foces in the contact aea; the foces due to the inteaction with the oogie; the extenal applied aking o taction toque; the weight.

7 The numeical model was ealied in the Matla/Simulink envionment, that allows to otain a numeicall efficient model, to test diffeent tpes of integation algoithms, to manage singulaities. The stuctue of the model is modula so diffeent susstems can e easil modeled and sustituted in the main pocedue. In this tpe of design envionment diffeent and complex sstems can e modeled (electical, pneumatic etc.), while a toolox devoted to the multiod simulation allows to easil model mechanical sstems with an high nume of odies and constaints. The esults of the simulations pefomed with the developed model wee compaed with those otained with models of the same vehicle ealied with Adams Rail and Simpack. The enchmak case is the Mancheste wagon, whose featues ae known in the liteatue and whose Adams Rail model is availale [14]. The validation pocess is necessa to evaluate the accuac of the model and to veif the pefomance of the model in tems of computational uden. even if the ae otained using diffeent softwae envionments and diffeent modeling appoaches. The diffeences ae paticulal evident in the definition of the contact points and of the contact foces. 5 Results In this section some the esults otained fom the compaison with the Adams Rail model ae iefl descied. In tests the vehicle tavels along a cuve with a constant speed. The paametes that ae vaied in the fist set of tests wee: the cuve adius values (500, 1200 and 2400 m); the ail angle was 1/40 and 1/20; the cant angles wee 40/1435 ad and 90/1435 ad; the speed values wee 15, 30 and 45 m/s. The wheel/ail adhesion coefficient was 0.2 in all the tests. In this set of tests the displacements of the wheelset centes and the contact foces wee analed and compaed. Figue 2 shows the paametes elative to one of the pefomed tests: the tack cuvatue and the cant angle. Figue 3 shows the compaison etween the esults otained the Adams and Simulink multiod models, the time-histo of the wheelset cente of mass displacements is analed in the pesented example. The esults show a good ageement etween the diffeent models, a diffeence etween the simulations can e noted in the tansient, it is due to the diffeences in the contact foce calculation. Tale 1 summaies the hunting fequencies calculated though simulations pefomed imposing diffeent vehicle speeds, a quite good ageement etween the esults otained with the Simulink and Adams models can e oseved. The calculated citical speeds ae 71 m/s with the Simulink model and 73 m/s with the Adams Rail model.

8 a) ) Figue 2: R = 1200 m, α = 1/ 40ad, β = 40/1435ad, V = 30m/s Test featues: p a) tack cuvatue, ) cant angle. a) ) c) d) Figue 3: wheelsets cente of mass displacement: compaison etween esults otained with Adams and Simulink models: a) Fist wheelset (font ogie, font wheelset), ) Second wheelset (font ogie, ea wheelset). Speed (m/s) Hunting Fequencies (Het) ADAMS SIMULINK Tale 1: hunting fequencies, compaison etween the esults otained though Adams and Simulink models The simulation esults allow to conclude that the Simulink and Adams multiod models give sustantiall compaale esults. The time histo analsis shows

9 that the Simulink model is geneall chaacteied an highe tansient damping, due to the diffeent method fo the calculation of the nomal component of the contact foce (in the Simulink model a non linea elastoviscous appoach is implemented). 6 Conclusions The ojective of the wok pesented in the pape is the development of a numeicall efficient model desciing the thee-dimensional dnamics of a ailwa vehicle. The challenge is the ealiation of a eal time implementation, to e used fo example to contol an HIL test ig. In this tpe of test ig the eliailit of the ealied tests depends on the accuac of the simulation of the vitual envionment, on the othe hand the eal time constaint imposes limitations on the complexit of the numeical models and on thei integation times. The development of such models has then to find a compomise etween accuac and efficinenc. This pape summaies the main featues of a ailwa vehicle multiod model ealied in the Matla-Simulink envionment. The model was tuned in ode to epoduce the dnamical ehavio of a enchmak ailwa vehicle. The esults of the simulations caied out with this model ae compaed with those otained with the Adams Rail multiod model of the same vehicle. The compaison has shown a sustantiall good ageement etween the models and the elative eos ae acceptale. It has to e highlighted that the models evaluate the local defomation of the wheel and the ail in the contact one means of two diffeent appoaches. With espect to the existing and availale ailwa multiod models, its featues ae then a moe detailed modeling of the wheel/ail contact polems and the possiilit to easil otain an executale implementation that could un in eal time conditions. In the developed model the aspects elative to the contact point definition and contact foce calculation has een caefull investigated since the have a significant effect on the vehicle dnamics. Refeences [1] Wolfgang Rulka, Eli Pankiewic, MBS Appoach to Geneate Equations of Motions fo HiL-Simulations in Vehicle Dnamics Multiod Sstem Dnamics (2005) n.14, pp [2] L. Pugi, M. Malvei, A. Taasconi, A. Palaolo, G. Cocci, M. Violani, HIL Simulation of WSP Sstems on MI-6 Test Rig, Poceedings of the XIX IAVSD Intenational Vehicle Sstem Association, Milano 29 agosto-2 setteme Vehicle sstem Dnamics, Vol. 44, Supplement 2006 pp ISBN

10 [3] A. A. Shaana, M. Toaa, H. Sugiama, K. E. Zaaaa, On the Compute Fomulation of the Wheel/Rail Contact Polem, Nonlinea Dnamics, Spinge 2005, vol. 40, pp [4] Pomo, J. Amosio, Dnamics analsis of a ailwa vehicle in eal opeation conditions using a new wheel-ail contact detection model, Int. J. of Vehicle Sstems Modelling and Testing, 1, 2005 [5] De Pate, A. D., The geometic contact etween tack and wheelset, Vehicle Sstem Dnamics 17, 1988, [6] L. Baea, A. Roda, J. Caalleia, E. Gine, Railwa Tain-Tack Dnamics fo Wheelflats with Impoved Contact Models, Nonlinea Dnamics, Spinge 2006, vol. 45, pp [7] T. G. Kolda, R. M. Lewis, V. Tocon, Optimiation diect seach: new pepspectives on some classical and moden methods, SIAM Review, Vol. 45, N. 3 (2003), pp [8] M. W. J. C. Lagaias, J. A. eeds, P.E. Wight, convegence popeties of the Nelde Mead simplex method in low dimensions, SIAM Jounal on Optimiation, No. 9 (1998), pp [9] R. V. Dukkipati, J. R. Amot, Compute aided simulation in ailwa dnamics, Macel Dekke, 1988W. Schiehlen. Multiod sstem dnamics: Roots and pespectives. Multiod Sstem Dnamics, 1, , [10] J.J.Kalke, Thee-Dimensional Elastic Bodies in Rolling Contact, Kluwet Academic Pulishes, [11] E.A.H. Volleegt, J.J. Kalke, G. Wang, " CONTACT 93 Uses Manual, VORtech Computing, Industial and Scientific Computing, Jul 1992, evised Mach [12] K.L. Johnson, Contact mechanics, Camidge Univesit Pess, [13] P.J. Vemeulen., K.L. Johnson, Contact of Nonspheical Elastic Bodies Tansmitting Tangential Foces, Jounal of Applied Mechanics, Tansactions of the ASME, June, [14] Iwnicki S., The Mancheste Benchmaks fo Rail Vehicle Simulatos' Swets & Zeitlinge B.V. Lisse 1999, (ISBN )