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(1998): The dynamic response of tyres to brake torque variations and road unevennesses. List of Symbols adistance front axle to c.g.; half of contact length axlongitudinal acceleration aylateral acceleration aMslip velocity dependency coefficient for friction Arrolling resistance coefficient bdistance rear axle to c.g.; half contact width Bstiffness factor in ’Magic Formula' B1brake force of rolling wheel cstiffness; factor cclateral carcass stiffness per unit length c gyrnon-dimensional gyroscopic coefficient c px,ytread element stiffness per unit length of circumference C’pxtread element longitudinal stiffness per unit area Ccornering stiffness ; sum front and rear Ccornering stiffness, sum left and right Ccontact centre (point of intersection) Cshape factor in 'Magic Formula' CdAair drag coefficient Cfxlongitudinal stiffness of standing tyre Cflateral stiffness of standing tyre Cfzstiffness of tyre normal to the road CFacornering stiffness Cfklongitudinal slip stiffness Cf7camber stiffness for side force CF(pspin stiffness for side force C gyrtyre gyroscopic coefficient CMaaligning torque stiffness C Mcamber stiffness for aligning torque C Mspin stiffness for aligning torque C Mtorsional yaw stiffness of standing tyre C Mxoverturning couple stiffness against camber C K-'cx,ycarcass horizontal stiffness of standing tyre C gyrgyroscopic coefficient dfznormalised change in normal load, Eq.(4.E2) dttread depth Dpeak factor in 'Magic Formula'; dissipation function Ecurvature factor in ’Magic Formula’ ecaster length; tread element deflection ftrail of c.g.
frrolling resistance coefficient Faxforce for forward acceleration Fdair drag force Fx,totsum of longitudinal tyre forces Fxlongitudinal tyre force Fylateral tyre force Fzvertical (normal) tyre force (load) (0), in Chap.9: Fz 0 Frrolling resistance force (0) FNtyre normal force (0) FNoreference vertical load, nominal load (= Fzo) FVtyre vertical force FHtyre longitudinal horizontal force gacceleration due to gravity; feedback rider control gain Gweighting factor hheight Hheight; sharpness factor in ’Magic Formula’ Htransform; Hurwitz determinant i1 izradius of inertia Imoment of inertia Iwwheel polar moment of inertia Ipwheel polar moment of inertia J1 kradius of inertia; viscous damping coefficient Kcentrifugal force; force acting in belt, wheel centre Iwheel base lsshift; two-point follower length lblength of basic curve lfoffset lunit vector along line of intersection mmass; fraction of 2a where adhesion occurs mccontact patch mass (dummy) mttyre mass mmmass of mainframe (including lower part of rider) mmrmass of mainframe plus rider mrmass of upper torso MBDbrake, drive torque 608LIST OF SYMBOLS Mxoverturning couple Myrolling resistance moment Mz(self) aligning torque M’(self) aligning torque due to lateral deflections M*aligning torque due to longitudinal deflections M -L -Lz,gyrgyroscopic couple Msteer torque nnumber of elements; frequency [Hz] nunit vector normal to the road =(0,0, 1)T nststeer system ratio pLaplace variable [1/m] ptinflation pressure qaverage vehicle yaw resistance arm; generalised coordinate qcontact force per unit length of circumference, vector Qgeneralised force ryaw rate; tyre (loaded) radius r f cradius of carcass (belt), unloaded; cross section crown radius r yofree tyre radius varying along cross section contour, ryo = ryo(yco) reeffective rolling radius of freely rolling wheel rffree unloaded tyre radius rlloaded radius rofree unloaded tyre radius (= Ro) Rradius of curvature Rofree unloaded tyre radius (= ro) sforward position of neutral steer point; half track width sLaplace variable; travelled distance ssx(practical slip component) ssytan (practical slip component) sunit vector along wheel spin axis Swheel slip point; impulse; string tension force Svhvertical, horizontal shift tpneumatic trail; time tccaster length trrise time tunit vector in road plane perpendicular to line of intersection l Tkinetic energy; moment acting around belt, wheel centre uforward velocity of c.g.; longitudinal deflection Upotential energy vlateral velocity of c.g.; lateral deflection Vspeed of travel of c.g.
(with x, y components) Vspeed of travel of wheel centre (with x, y components) Vcspeed of contact centre C (with x, y components) Vspeed of sliding (with x, y components) Voreference velocity =VgRo Vrwheel linear speed of rolling (= Vcx- Vsx) Vswheel slip velocity of slip point S (with x, y components) Vxlongitudinal speed component of wheel centre V* svelocity of contact patch mass (with x, y components) wvertical road (effective) profile (positive downwards) Wwork x.y.zlongitudinal, lateral, vertical displacement x.y.zcoordinates with respect to moving axes system, z axis vertical xo,yo,zoglobal coordinates x,y,zglobal coordinates Xlongitudinal horizontal tyre force X.Y.Zglobal coordinates ycodistance from wheel centre plane ymrlateral offset of mmr c.g.
awheel (side) slip angle; axle (side) slip angle aroad transverse slope angle a'transient tyre slip angle avirtual axle slip angle Pvehicle side slip angle; tyre yaw torsion angle Px.yroad transverse, forward (effective) slope angle Pgyrgyroscopic wheel coupling coefficient, Eq.(6.35) ycamber (wheel inclination) angle y'transient tyre camber angle runit step response function ssteer angle of front wheels ollR, steer angle at V- 0 Aincrement eroll steer coefficient; rake angle of steering axis estring length ratio, Eq.(5.153); eff. radius gradient -drjddt sicamber stiffness reduction factor NLnon-lagging part esmall quantity to avoid singularity cdamping ratio; spin factor (=1 if spin influence is disregarded) hheight ratio, Eq.(6.36) Cacornering stiffness load transfer coefficient r7camber stiffness load transfer coefficient nundersteer coefficient; effective rolling radius gradient - drjdp. offset steer coefficient etyre model parameter, Eqs.(3.6,3.24,3.46) eangular displacement about axis; pitch angle cstring model composite parameter, Eq.(5.160) Klongitudinal wheel slip K'transient longitudinal tyre slip * Kdamping coefficient due to tread width Xwavelength; root characteristic equation Xfraction of 2a where adhesion occurs; user scaling factor ncoefficient of friction ptyre radial (vertical) deflection x,y,ztyre longitudinal, lateral, normal deflection arelaxation length; load transfer coefficient atheoretical slip, vector, Eq.(3.34) * aintersection length in string model with tread elements cstring model length parameter, Eq.(5.153) ccontact patch relaxation length Troll camber coefficient (Pbody roll angle; spin slip P'transient spin slip tturn slip 0phase angle Vyaw angle; steer angle ^c1compliance steer angle Wiotoe angle COfrequency [rad/s] (O0undamped natural frequency «1,2natural frequencies ®ndamped natural frequency (Ospath frequency [rad/m] Qwheel speed of revolution ,,moving axes system, axis along spin axis, horizontal Subscripts and superscripts aaxle; from belt to wheel rim centre bbelt; from belt centre to rim ccompliance (steer) ccontact patch; from contact patch centre to belt; crown; contour Ddrag eeffective effeffective (cornering stiffness) eqequivalent ffree, unloaded; of front frame gglobal i1: front, 2: rear L,Rleft, right mof mainframe mrof mainframe plus rider NLnon-lagging ooriginal; initial; average ounloaded; nominal; at vanishing speed; natural rroll; rolling; rolling resistance; of residual spring; of rider sslip; from road surface tocontact patch; of front sub-frame slat verge of total sliding sfside force (steer) sssteady state ststatic stwsteering wheel ttransition from adhesion to sliding wwheel x,y,zforward (longitudinal), lateral (to the right), downward zrresidual (torque) , ,along, around , , axes 1,2front, rear; leading, trailing edge?
1sincos Ay2 = Ay-bAy(11.83) **4= -hdvAy-hdA6 x(hy ) h kxr = (hrv + sr(pr+yr)Ay -hrAd Axf = -(hfP + efS )Ayz -h^AQ Axs = (hsp + eS -sJ5)Ay/-hsAd wherecan be expressed in the generalised coordinates by taking the variation of 6 (11.63). Now, we may compare (11.82), after having substituted herein ...
Beyond that value the model should operate as usual. In Chapter 8, Sec.8.6 an application will be given. A similar equation may be employed for the lateral transient slip. Another extreme situation is the condition at wheel lock. At steady state Eq.(7.9) reduces to: ^VJu= -Vx(7.27) K which ...
This also appears to hold for a limited number of conducted combined slip tests and the response to vertical axle motions at side slip and braking carried out with the measuring tower. Apparently, the rigid ring model provided with the short wavelength transient slip model is very well capable of ...