VEHICLE DYNAMICS MODELING DURING MOVING ALONG A CURVED PATH. MATHEMATICAL MODEL USAGE ON STUDYING THE ROBUST STABILITY

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1 U.P.B. Sc. Bull., Seres C, Vol. 4, Iss. 70, 008 ISSN x VHICL DYNAMICS MODLING DURING MOVING ALONG A CURVD PATH. MATHMATICAL MODL USAG ON STUDYING TH ROBUST STABILITY Oana-Carmen NICULSCU-FAIDA 1, Adran NICULSCU-FAIDA Problematca deplasãr pe traectore curbã a unu automobl reprezntã un subect de studu ntens în încercarea ngnerlor de profl ş automatşt de a realza sstemul de control adecvat. Inante de toate se mpune cunoaşterea rguroasã a dnamc autovehcululu aflat într-o mşcare crcularã, pentru a se putea transpune cât ma exact realtatea fzcã în ecuaţ matematce. Modelul matematc trebue sã cuprndã elementele defntor pentru procesul respectv, în scopul realzãr unu comproms între complextatea ecaţlor ş puterea de calcul necesarã procesãr acestora. În aceastã lucrare este prezentat un model matematc nelnar cu şase grade de lbertate, pe baza cãrua se studazã nfluenţa dferţlor parametr asupra stabltãţ autovehcululu în curbã (ungh de bracare, vteza ş acceleraţa automoblulu etc.). Dn studul realzat cu ajutorul crterulu stabltãţ în sens Lapunov, reese faptul cã, la vteze ma mar de 10 km/h, unghur de bracare mc pot destablza autovehculul. Rezultatele obţnute se pot pune în valoare prntr-un sstem de control. The problem of the moton of an automobl on a curved path represents a subject of hgh nterest, n the strve of the profle engneers and automatsts to desgn the proper control system. Frst of all rgorous knowledge s compulsory, about the vehcle dynamcs on a crcular moton, n order to express as accurate as possble the physcal realty nto mathematcal equatons. The mathematcal model must contan all the defnng elements for the specfed process, so that a compromse can be done, between the complexty of the equatons and the computng power needed. Ths paperwork presents a mathematcal nonlnear model wth sx freedom degrees, based on whch the nfluence of dfferent parameters over the stablty of the vehcle on a curved path s studed (brackage angle, velocty and acceleraton of the vehcle a.s.o.).from the study based on the Lapunov meanng stablty crteron, we conclude that, at speed exceedng 10 km/h, small brakeage angles can destablze the vehcle. The obtaned results can be shown to advantage through a control system. Keywords: brackage angle, speed, robust stablty 1 PhD, Automatc Industral Control and Informatcs Department, Unversty POLITHNICA of Bucharest, Romana Automotve ng., Segula Technologes, Romana

2 50 Oana-Carmen Nculescu-Fada, Adran Nculescu-Fada 1. Introducton Whle movng along the curve, on the contact surface of each wheel, forces appear mantanng the vehcle on the trajectory. Under these forces, the tre s deformed and therefore the velocty of each wheel devates from the tre plane under a certan angle, functon of the lateral stffness of the tre and the force value. The movement on a curved trajectory has been debated n numerous papers of the doman, among whch [4] where a three freedom degrees model s consdered, subjected to the wheel forces gven by the magc formula of H.B. Pacejka, [5] from the same conference where a two freedom degrees model s studed and the tre model proposed by Dugoff modfed by the authors. In the artcle [3] where a three freedom degree model s presented on a self made tre model. In [8] the authors have studed the wheel dynamcs on a curved path and n straght lne and the result obtaned s very usefull to establsh the sdeslp angles. The vehcle stablty s studed n [6] usng coeffcents to correct the forces that act upon each wheel. In [] a common bcycle model s made equvalent to a model n whch nstead of the four forces on each wheel there are consdered two forces wth a sldng pont of applcaton along the symmetry axes of the vehcle. Among the many artcles t s worth mentoned [1] n whch the authors defne a way to control the sdeslp of the vehcle startng from a two freedom degree model. Ths method s used together wth the control of the spnnng speed. The present artcle analyzes the curved moton usng a vehcle model wth sx freedom degrees, whch consders the dsplacement of the vehcle on the transverse drecton, on the longtudnal drecton, also the rotaton movements around the transversal axs, around the longtudnal axs and the vertcal dsplacements for each of the four wheels of the vehcle. The mathematcal model offers better results comparng to the other models, consderng the hgher precson of the estmaton and the wder spectrum of the stablty analyss. The objectve of ths paper s to dentfy the crtcal stuatons that a vehcle may encounter n a curve.. The mathematcal model The model wth sx freedom degrees takes nto account the movement of the vehcle on transverse drecton, on longtudnal drecton, of the rotaton movements around ts transverse axs, around ts longtudnal axs, and of the movements on the vertcal for each of the four wheels of the car. Comparng to the prevously studed model, other fve more movements are consdered: the rotaton around the longtudnal axs (the rollng moton) and the vertcal

3 Vehcle dynamcs modelng durng movng along a curved path. Mathematcal model 51 movements for each wheel, n order to better analyse the stablty of the vehcle wth a hgher precson than the analyzed case based on the bcycle model. l R 3 D α 3 l s R 1 F 1px A F 3p δ β 1 C.G. m F cf β V F 1p F 1py / α 1 R 4 C F 4p α 4 F px R B δ β F p F t F py α C.I.R. Fg. 1. The forces representaton, actng on the wheels of the vehcle In Fg.1. the followng elements appear: m - the mass of the vehcle; l - the length of the vehcle; l s - the dstance between the center of mass and the rear axel axs; - the front wheelbase (the dstance between the front wheels); - the curvature radus; α - the sldng angle of the wheel; δ - the brackage angle; β - the devaton angle; F p - the perpendcular force on the drecton of the wheel, due to the sldng of the wheel, determned by the sldng angle α ;

4 5 Oana-Carmen Nculescu-Fada, Adran Nculescu-Fada F t - the drvng force; F cf - the centrfugal force; mv Fcf = ; R - the resstve force to rollng; R = R1+ R; V - the velocty of the vehcle. As t can be notced n Fg.1., three pars of forces act upon the front wheels, along the wheel ( F t and R ) and perpendcular on t ( F p ). For the front wheels, the force along the wheel s the drvng force ( F t ) generated by the engne mnus the resstve force to rollng ( R ). For the rear wheels the force from the wheel plane represents the resstance to rollng ( R ). In the center of mass the centrfugal force act along the curvature radus ( F cf ). Let us wrte the equaton of the moment around the vertcal axs, crossng through the ntersecton pont between the longtudnal axs and the rear axel: ( F1py + Fpy ) l+ ( Ft R1 R) lsnδ ( F1p Fp ) snδ + ( R4 R3) (1) = Fl cf s cos β The torque gven by the resstances to rollng for the rear wheels ( R4 R3) s much smaller than any other terms and can be neglected. The force perpendcular on the drecton of the wheel, for small deformatons of the tre, s expressed as: Fp = c α α () where: c α - the rgdty coeffcent of the tre adapted to brackage, descrbed by the formulas: cα1 = cα (3) 1 λ H1 4( H1) where: λ the sldng of the wheel; c α the transversal elastcty coeffcent of the tre (expermentally determned).

5 Vehcle dynamcs modelng durng movng along a curved path. Mathematcal model 53 The ntermedate measure H H s descrbed by the formula: cλ λ 1 c α 1 = + tg( α1) μn1 λ 1 λ 1μN1 where: c λ the longtudnal elastcty coeffcent of the tre (expermentally determned); μ the frcton coeffcent between the tre and the the rollng path; N - the sol reacton for each wheel; In order to determne the N reactons from the road, the vertcal movement was consdered equal to the dsplacement of the jonng pont of the body wth the suspenson. The N reacton s consdered proportonal to the movement, whch s almost exact f the tre vertcal deformaton s neglected, compared to the deformaton of suspenson sprng. (4) mv ( N1+ N3) G hg cos β = 0 mv ( N1+ N) l Gls hg sn β = 0 N1+ N + N3 + N4 = G (5) (6) (7) For solvng the equatons system a fourth relaton s necessary, between the 4 unknowns. If we consder: N1+ N4 = N + N3 (8) from the four equatons the forces N can be calculated. For computng the tracton force F t the followng formula wll be used: M η m 0 t Ft = = ct. (9) r where: M m - the engne moment correspondng to the vehcle velocty and to the coupled gear; - the current gear rato; 0 - the man transmsson rato; η - the total mechancal effcency of the transmsson; t

6 54 Oana-Carmen Nculescu-Fada, Adran Nculescu-Fada r - the rollng radus of the wheel. R - the rollng resstance s descrbed by the expresson R = fn (10) where: f the rollng resstance coeffcent. The sldng rear angles, as they result from the fg.1, have the formulas : l s sn ( β) l α3 = arctg ; s sn ( β) α4 = arctg + cos( β ) + cos( β ) The sldng front angles have the formulas: l l α1 = arctg and α = arctg ls ls + respectvely, accordng to the results obtaned n the work [13] where the authors study these parameters startng from the general plane movement equatons of a body, partcularly for the crcular moton. In the same work the result for the sldng angle of the vehcle s presented: l s β = arctg l s From the equaton (1) results the curvature radus of the trajectory,. The radus s calculated usng Mathcad software. These results wll be used to determne the stablty by the help of the robust stablty crteron. In order to determne the numercal value of the Lapunov functon, whch wll be used to study the robust stablty, t s necessary to compute the curvature radus for each case separately. The equaton (1) s graphcally solved. For each par of values of the brakeage angle (δ ) or of the velocty (V ) the functon h( ) s graphcally represented, and the value for whch the functon becomes null, s the soluton of the equaton. The graphcal solvng method s used because of the complexty of the resultng equaton when one unknown s replaced. All the necessary data for the numercal computaton of the curvature radus are to be found n the table below:

7 Vehcle dynamcs modelng durng movng along a curved path. Mathematcal model 55 Table.1 The necessary values for the numercal calculus of the curvature radus Fx parameters Varable parameters Masses [kg] and Dmensons o Velocty Ratos Coeffcents Angle[ ] Momentum[Nm] [m] [ km / h ] m = 1350 l = 1, 4 0 = 4,1 c 8500[ N / rad] f m = 10 l = 1, 6 η = 0,9 r s t c α = λ = M m = 150 l = lf + ls λ = 0,1 μ = 0,9 = 1, 5 = 0, 6 f = 0,015 h = 0, 4 g 10500[ N / rad] r = 0, 91 The results of the numercal calculus are shown n the table below. o The curvature radus for dfferent brackage angles δ [] and veloctes Vkm [ / h ] δ V Table. o δ [] V [km/h] ,1 585,66 396, 303,5 76, ,05 97,4 655, ,7 1431,56 963,8 73, , ,1 131, , ,5 574, ,3 131, , ,74 190, ,7 1454, ,15 408,8 705, 049,6 1759, , ,1 374,8 479,5 093,8 The model s nonlnear and n order to analyze the stablty, the crteron of the robust stablty s used. Ths mples to choose a functon called Lapunov, varable n tme, and ts frst order dervatve s always postve and whch fulfls the condton L(0)=0. The stablty s obtaned only when the Lapunov functon (L) becomes negatve. In order to defne the Lapunov functon, the case of accelerated moton wll be studed. For the Lapunov functon two varables dependent of tme are chosen such that the frst and second dervatves by tme, to be determned. The dervatves derve from two of the equatons of the dynamcal equlbrum of the vehcle, whch are determned from the Fg.:

8 56 Oana-Carmen Nculescu-Fada, Adran Nculescu-Fada R 3 F x3 F y3 D V 3 α3 l s l f R 1 F p1x F x1 F y1 A δ R 4 F 3p F y4 C V 4 α 4 C.G. mv / β β V F px F p1 R F p1y F y B V 1 / F t F x4 F 4p F x F p F py V δ F t C.I.R. Fg.. The forces actng the wheels of the vehcle, n the case of the curved accelerated moton n horzontal plane (x,y) In Fg.. the followng terms are used: F p - the forces perpendcular on the wheel due to the sldng angle of the vehcle; Ft - the tracton force for the front tyres M r Ft = r where: M r - the pullng force on the wheels; r - the tre radus. F x - the nertal forces on x axs for the wheel; F y - the nertal forces on y axs for the wheel; V - the tangent acceleraton of the vehcle;

9 Vehcle dynamcs modelng durng movng along a curved path. Mathematcal model 57 V - the tangent acceleraton for each wheel; ψ - the angular acceleraton of the vehcle around the vertcal axs crossng the center of mass. The equaton of the momentum around the vertcal axs crossng the center of mass s descrbed as: J zψ = Ftlf snδ + ( Fp Fp 1) snδ + ( Fp 1+ Fp) lf cosδ ( Fp3+ Fp4) ls ( Fx 1 Fx) ( Fy1+ Fy) lf ( Fx3 Fx4) (11) ( Fy3 + Fy4) ls ( R1+ R) lf snδ ( R1 R) cosδ ( R3 R4) and The equlbrum force equaton on the transversal drecton s: mv mv sn β + cos β = Ft snδ + ( Fp 1+ Fp) cosδ + ( Fp3+ Fp4) (1) F + F R + R snδ + F + F ( y1 y) ( 1 ) ( y3 y4) The dervatves of the above mentoned varables are n ths case ψ and V, from whch t results that the varables are ψ and V. The dervatves ψ and ψ represent the acceleraton, respectvely the angular velocty for the gyraton moton around the vertcal axs from the center of mass, CG, whle V and V represent the acceleraton and the angular velocty tangent to the trajectory of the vehcle respectvely. The terms contanng the varables ψ and V are grouped together: ψ Jz + mr + 4mr( lf + l ) s = V 4mrsnβ ( lf l ) + s + Ftl f snδ + ( Fp Fp 1) snδ + ( Fp 1+ Fp) l f cosδ (13) ( Fp3+ Fp4) ls f ( N1+ N) lf snδ f ( N1 N) cosδ f ( N3 N4) mv ψ 4mr( lf ls) = Vm snβ cosβ + Ftsnδ (14) + F + F cosδ + F + F f N + N snδ ( p1 p) ( p3 p4) ( 1 ) The terms followng ψ and V n the equatons (13) and (14) are constant.

10 58 Oana-Carmen Nculescu-Fada, Adran Nculescu-Fada The equaton system, from whch the Lapunov functon resdes accordng to the robust stablty theory, s: ax 1 1= bx 1 + c1 (15) ax 1= bx + dx + c whch n ths case, s expressed as: a 1ψ = bv 1 + c1 a ψ = bv + dv + c The chosen functon wll be: L= ax1 bx ct and, accodng to the relatons between velocty and acceleraton from the clascal mechancs, the Lapunov functon becomes: Vt L= a bvt ct (16) It can be easly notced that f t = 0 (the Lapunov functon argument) then L = 0, that s L (0) = 0, whch represents one of the condtons for the system to be stable. In the relaton (16) V s not known, ths varable s measured usng an acceleraton sensor. The second condton for the system to be stable s that the frst order dervatve of the Lapunov functon to be strctly postve: dl ax 1 bx c 0 dt = > (17) In the relaton (17) the second equaton of the system (15) s ntroduced: dl dl = bx + dx + c bx c > 0 = dx > 0( A)( ) x R dt dt The thrd condton of the stablty of the system s that the Lapunov to become negatve. The stablty of the vehcle wll be checked under certan relevant condtons, n order to defne the behavor of the vehcle n curves. The varable elements are the brakeage angle and the ntal velocty n the curve. For ths, fve brakeage angles and seven veloctes wll be chosen. The ends of the angles and veloctes ranges are consdered as the lowest or hghest lmts, whch these elements may have n real condtons. The values of the Lapunov functon, computed usng expresson (16) are shown n the table 3:

11 Vehcle dynamcs modelng durng movng along a curved path. Mathematcal model 59 The Lapunov functon values Table.3 δ( o ) V (km/h) The mathematcal model gves better good results compared to the ones obtaned n [6] where a lnear mathematcal model was used. The mprovement conssts of the hgher precson of the estmaton but also the larger wdth of the spectrum of the stablty analyss. The precson of the stablty estmaton, for ths vehcle, can be notced f small calculus steps are used, for the brackage angle and velocty also. Accordng to the robust stablty crteron, t results that the vehcle s stable as t s shown n the fg. bellow: The stablty lmt 30 The brackage angle (degrees) Unstable Stable system The velocty of the vehcle V(km/h) Fg. 3. The stablty lmt of the vehcle on a curved path The curve from the fg. 3 represents the stablty lmt of the studed vehcle; that means f the vehcle has a small velocty and a brakeage angle of the

12 60 Oana-Carmen Nculescu-Fada, Adran Nculescu-Fada wheel such that the pont correspondng to the respectve coordnates s under the curve, then the moton on the specfed trajectory s stable and f the pont s above the curve then the moton s unstable. 3. Conclusons From the performed analyss, t results that the vehcle s stable untl a 110 km/h velocty when the brakeage angle reaches the value of 10 o, at the value of 15 of the brakeage angle the stablty s at ts lmt for the same velocty, and when the wheels have more than 15 o the nstablty tendency grows more, untl the value 5 o of the brakeage angle when the stablty s lost at veloctes less than 90 km/h. The orgnal mathematcal model wth sx freedom degrees s wthout approxmatons ths contrbutng to the exactness of the results obtaned. The stablty crteron appled s also an orgnal approach regardng ths type of applcaton due to the hgher degree of dffculty of the Lapunov functon fndng. It s mportant to notce that ths crteron shows the tendency of loosng the stablty, ths means that t ndcates exactly the condtons when the vehcle starts loosng adherence or contact wth sol for at least one of the wheels. R F R N C S [1]. M. Abe, Y. Kano, K. Suzuk, Y. Shbahata, Y. Furukawa, Sde-slp control to stablze vehcle lateral moton by drect yaw moment, JSA Revew, pp , 001. []. A. Komatsu, T. Gordon, C. Matthew, 4WS Control of Handlng Dynamcs Usng a Lnear Optmal Reference Model, 5 th Internatonal Symposum on Advanced Vehcle Control, Ann. Arbor, Mchgan, USA, -4 August, 000. [3]. M. Börner, Adaptve Querdynamk-modelle für Personen-kraftfahrzeuge- Fahrzustandserkennung und Sensorfehlertoleranz, Rehe 1, Nr. 563, VDI Verlag, Düsseldorf, ISBN , 003. [4]. M. Lakehal-Ayat, S. Dop,. Fenaux, F. Lamnabh-Lagarrgue, F. Zarka, On global chasss control: combned brakng and cornerng, and yaw rate control, n Proceedngs of Advanced Vehcle Control 000. [5]. W. Mannng, D. Crolla, M. Brown, M. Selby, Co-ordnaton of chasss control systems for vehcle moton control, n Proceedngs Advanced Vehcle Control 000. [6]. N. Noomwongs, H. Yoshda, M. Naga, K. Kobayash, T. Yoko, Study on handlng and stablty usng tre hardware-n-the-loop smulator, JSA Revew 4, , 003. [7]. O. Nculescu-Fada, S. Ilescu, I. Făgăraşan, A. Nculescu-Fada, Vehcle stablty study on curved trajectory, Buletnul Ştnţfc U.P.B., Sera C Ingnere lectrcă, vol. 70, nr., ISSN145434x, 008. [8]. V. Vantsevch, Fundamentals for the Parallel Control of Normal, Longtudnal, and Lateral Wheel Dynamcs, Proceedngs of AVC, 5 th Internatonal Symposum on Advanced Vehcle Control, Ann Arbor, Mchgan, 000.

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