SWITCHING PROCESS IN LIMITED SLIP DIFFERENTIAL

Transcription

1 6th Internatonal DAAAM Baltc Conference INDUSTRIAL ENGINEERING Aprl 2008, Tallnn, Estona SWITCHING PROCESS IN LIMITED SLIP DIFFERENTIAL Resev, J.; Roosmölder, L.; Stas, M. Abstract: The energy flow modellng n automotve propulson systems amed to provde base data for the choce of the dfferental parameters n consderaton of the mpact on the vehcle lateral dynamcs s presented n ths paper. The characterstcs of the dfferental n the torque transmsson chan are concerned. Descrpton of the characterstcs and operatonal condtons of the lmted slp dfferental gear tran s gven. Algorthm of energy flow has been provded. Key words: modellng, planetary gear tran, lmted slp, torque transmsson, understeerng, oversteerng. 1. INTRODUCTION An essental dsadvantage of the conventonal dfferental has been noted n case where one wheel of the vehcle slps on a surface wth low frcton, t s lkely to brng the vehcle to halt. In the above case, the conventonal dfferental s unable to transmt the necessary torque to the other wheel. The lmted slp dfferental (LSD) can transmt more torque n ths case but some decrease n the steerng qualtes wll follow due an ncreased understeerng. The am of ths work s to nvestgate possbltes to mprove vehcle s steerng characterstcs by energy flow.e. swtchng the dfferental gear ratos. Some specal terms have been handled n ths paper: dfferental - a mechancal system wth two rotatonal degrees of freedom, where the gears are mostly arranged as a planetary system; lmted slp dfferental - a dfferental, where the nternal relatve moton s subjected to torque losses; swtchng process n lmted slp dfferental - a general process of the gear rato changng n a dfferental. 2. SWICTHING PROTCESS 2.1. Energy flow swtchng Energy flow swtchng s nterpreted here as a smooth change of torque rato or gear rato at the dfferental. The lateral dynamcs of a vehcle equpped wth a dfferental of the knd undergoes the followng processes. When vehcle s drvng strat forward nto the curve and the steerng (front) wheels are startng to turn we have the stage of the relatve small torque dfference of the drvng rear wheels where the dfferental s stll locked, see ponts 1-2, Fg. 1. When the vehcle s movng along a curve and the rotatonal veloctes of ts drvng wheels are equal (ω =ω ο ). The nner wheel of the curve slps back n a relaton to the road surface, whereas the outer wheel slps forward n a drecton of moton. When the vehcle s movng forward nsde the curve ncreasng the angle of the steerng wheels the relatve slp s noted n the frcton elements of the dfferental at ponts 2-3. The unlockng of the dfferental happens here, t s characterzed by the transton from the statc frcton to the dynamc frcton. Dfferental opens at pont 3.

2 Fg. 1. Torque swtchng process on a curvlnear trajectory, where T drvng wheel torque, δ - angle of the steerng wheel, 1-10 characterstc ponts. Lghtly ncreasng the angle of steerng wheels the car drves n understeerg area between ponts 3-4. For approvng the steerng qualty the torque could be swtched opposte by changng the veloctes of the drvng wheels as t has been shown at pont 4. Equatng torques on outer and nner wheel the dfferental locks and starts to contrbute vehcles cornerng from pont 5. Further ncreasng the steerng angle the frcton between elements transts from statc to the dynamc at ponts 6-7. Between ponts 7-8 the dfferental contrbutes cornerng but after some ncrease of the steerng angle t starts to resst the cornerng agan. Dynamc frcton transts to statc frcton (ponts 8-9) and dfferental locks. In stages 1-2, 4-6 and 9-10 frcton elements are stll (locked). Stages 2-3, 6-7 and 8-9 happens transton processes. In stages 3-4 and 7-8 frcton elements are slppng. These stages can be nterpreted to vehcle movement models Steerng process Below the characterstcs of the road vehcle at cornerng at low veloctes have been examned (wth no centrfugal force). At low speeds a lght relaton between the drecton of moton and the steerng wheel angle has been notced. The am for the desgn of the steerng system s a mnmum tyre scrub at cornerng. Therefore all tres should be n pure rollng wthout lateral sldng at cornerng. To satsfy ths requrement wheels should follow the curved path wth dfferent rad orgnatng from a common centre C, Fg. 2. Steer angles δ o and δ should satsfy the relatonshp: B cot ( δo ) cot( δ ) =, (1) L where the subscrpts o and denote outer and nner wheels at cornerng. The steerng geometry that satsfes the above equaton s usually referred as the Ackermann steerng geometry and s vald as a theoretcal reference case where sdeslp of wheels s dsregarded [ 4 ]. Model treatment and nfluence to the steerng characterstcs have been handled n [ 3 ]. In Fg. 2 pont C s also an nstantaneous centre and Ω s the rotatonal velocty relatve to t. The steerng angles of the front wheels δ o and δ have been approxmated to the longtudnal axs of the vehcle used n the calculatons: δ = 0, 5( δ o + δ ). F o F B R C Ω V o V V L δ o F r δ Oversteerng area Understeerng area Fg. 2. Steerng characterstcs of the vehcle, where B s a track of the vehcle, L s a wheel base, F - force, V - velocty, whereas the subscrpts o (outer) and (nner) denote the relatve dsplacement of the nstantaneous centres C. Subscrpt r denotes the resstance and Ω - rotatonal velocty of the vehcle.

3 When sde force s appled to the tre a lateral force wll be developed at the contact patch and the tre wll move along a path at a slp angle wth the wheel plane [ 2 ]. Accordng to the value of the understeer coeffcent or the relatonshp between the slp angles of the front and the rear tres, the steady-state handlng characterstcs may be dvded nto a three categores: neutral steerng, understeerng and oversteerng. In the general desgn of the dfferental t s essental to follow the prncples: The dfferental s expected to provde the vehcle wth the utmost neutral steerng qualtes. The characterstcs that nfluence the steerng of the car should change the steerng for more neutral so that t contrbutes to the steerng qualtes. Over- and understeerng could be used by race cars and mltary vehcles on racetracks or on landscape drve by controllng veloctes of drvng wheels. 3. PERFORMANCE CHARACTERISTICS OF A VEHICLE WITH THE LIMITED SLIP DIFFERENTIAL 3.1. Tractve effort characterstcs At any rad of cornerng the rotatonal velocty of the drvng wheels of a vehcle wth a common dfferental s adjusted to the steerng radus. The applcaton of a planetary gear tran (locked dfferental) leads to resstance at cornerng. As a matter of fact the coeffcent of effcency of a common dfferental s relatvely hgh. Thus n case of the common dfferental s appled the torque of the drvng wheels s approxmately equal. In a loss free case not consderng gear rato at dfferental a certan correlaton s notced: T n = Tout1 + Tout 2, where: T n denotes the nput moment of the dfferental gear tran, however T out1 and T out2 are the output torques on correspondng wheels. BesdesTmn = Gfmnrd, where G s the gravtatonal force on the drvng wheel and f mn denotes the coeffcent of frcton between the drvng wheel (radus r d ) and the road surface calculated on the wheel wth a lower value of frcton on the assumpton that an equal force of gravtaton has an mpact on the drvng wheels. The hghest torque n case of a common dfferental can be gven T n = 2T mn. Here the locked dfferental has taken the form of a planetary gear tran as a result of lockng and enablng the transmsson of a hgher torque on the road: T ( f f ) n Tmn + Tmax = Grd mn + =, (2) max where: T max denotes the torque of the wheel wth a hgher value of frcton and the coeffcent of frcton f max on the road surface. Here the drvng wheels wth equal rotatonal veloctes provde a dsadvantage at cornerng. Dfferental wth a relatvely hgh coeffcent of effcency has been wdely appled n most street and road vehcles. In extreme stuatons (e.g. f mn << f max ) t s essental to ncrease the torque on the drvng wheels. To acheve that we have tred to lmt the relatve mutual rotatonal velocty of the drvng wheels. Theoretcally the torque rato at dfferental can be expressed: T2 k =, (3) T1 where T 2 and T 1 denote the torques of each output shafts of the dfferental n the begnnng of swtch off the brakes. In case of the applcaton of lmted slp dfferental (LSD) the hghest torque value T n = T 1+ k can be expressed: ( ) mn. In case of jeeps (off-road vehcles) that are drven on rough roads, hgher torque

4 values are needed. It s possble to obtan hgher torque values by decreasng the cornerng abltes. Larger torque rato allows applyng more torque on drvng wheels but on the same tme ncreases vehcle s understeerng. The correlaton of the veloctes and the torques of the shafts (rotatng elements) of each transmsson unt can be then automatcally formulated, presented n the form of a matrx and solved by the computer program. Ths wll enable to calculate the values of the torques and veloctes of each shaft for the whole system. The torque and the speed losses can be calculated by the Lagrange multpler technque [ 1 ]. The transmsson systems wth more than one nput or output shaft as the epcyclcal trans can be calculated. Besdes possble over constraned elements of the transmsson system can be detected Dfferental model components A graphcal model of submodules of the vehcle s shown on Fg. 3. Denotaton: Fg. 3. Graphcal (A) and descrpton (B) elements of the model. In descrpton of the model components we need an analyss of the general process of the moton. The vehcle moves along the non constant radus by changng angle of the front wheels. The lateral dynamcs of the LSD can be dvded nto certan stages. The most sgnfcant factor of characterstcs of the dfferental s the constrant on the nner slp. Ths constrant may have a varable or constant mpact on the torque rato. In case t s varable, t can be load- or velocty proportonal. Let us take LSD, whch s a load proportonal. The dfference between the veloctes of the drve wheels: Δ V = V o V. For the modellng of the LSD we need speed- and load characterstcs of the vehcle as the ntal parameters. The general equatons of the LSD for the smulaton of the lateral dynamcs of the vehcle can be correspondngly dvded nto equlbrum, compatblty and consttutve relatons. The equlbrum of the force and torque has been generally shown as ΣT = 0,

5 ΣF = 0. We can overcome the resstance of constant movement by makng use of the tangental force of the drvng wheels: F = F +. (4) r F o As a rule formula can be gven generally as: dv m = F + Fo Fr, (5) dt where m denotes the mass of the vehcle, dv s a dervaton of tme of the velocty dt and F r s a resstance of moton. The compatblty can be gven for the outer and nner drve wheels subsequently: ω r (1 s o ) = V o, ω r (1 s ) = V, (6) where s denotes the slp. Accordng to Fg. 2. the curvlnear moton of the vehcle can be gven as follows: V V o R + 0.5B 1 so = =, (7) R 0.5B 1 s The consttutve relatons of the LSD for the tangental force relatvely of the drve wheels can be expressed as the followng approxmaton: mgμ = ). (8) 2π F, o arctan( 20s, o Moton resstance: F ρa 2 V 2 r = ± krmg + ka, (9) where k r denotes the rollng resstance, k a s a coeffcent of aerodynamc resstance, A s a vehcle front area, μ s a coeffcent of a wheel engagement on the road, g s an acceleraton due a gravty and ρ s an ar densty. Fg. 4. Predcton of the new dfferental 3.3. Predcton Results shown n Fg. 4 characterze the swtchng process n the lmted slp dfferental. Proposed relatons of the motons have been used for modellng of new dfferental concept. Smulatons were realzed n MatLab envronment. 4. CONCLUSION The basc equatons for the descrpton of the module of the dfferental are elaborated. Based on these equatons, the module system of the vehcle model could be derved. The model enables to gude the change of dfferental parameters and the energy flow delverng between the wheels durng the drve for race and mltary vehcles. Energy flow swtchng enables more effectve conerng of vehcle on curve lnear trajectory. 5. REFERENCES 1. Mäg, M., Jacobson, B and Resev, J. Dynamc analyss of planetary gear trans: Proc. SCIENCE 98 (Papstel, J., ed.). Tallnn, 1998, 7-19; 2. Resev, J. Vrtual Dfferental as Torque Dstrbuton Control Unt n Automotve Propulson Systems, Dss. Tallnn Techncal Unversty, Tallnn, 2002; 3. Resev, J., Roosmölder, L. Vrtual Dfferental as Torque Dstrbuton

6 Control Unt n Automotve Propulson Systems. Proc. 4 th Internatonal DAAAM conf. n Estona, 2004, Tallnn, pp 74-77; 4. Resev, J., Roosmölder, L. Torque dstrbuton unt n automotve propulson systems. Proc. Estonan Academy of Scences. Engneerng. Vol. 10, No. 4, 2004, pp ; 5. Wong, J. Y. Theory of ground vehcles. John Wley & Sons, New York, Mäk, R. Wet Clutch Trbology Frcton Characterstcs n Lmted Slp Dfferentals. Dss. Lulea Unversty of Technology, ADDITIONAL DATA ABOUT AUTHORS Jür Resev, Marko Stas Tartu College at Tallnn Unversty of Technology, Puestee 78, Tartu, Estona; jresev@ttu.ee Lembt Roosmölder Department of Product Development at Tallnn Unversty of Technology, Ehtajate tee 5, Tallnn, Estona; lembtr@staff.ttu.ee Phone ADDITIONAL DATA ABOUT AUTHORS