A brush-based thermo-physical tyre model and its

effectiveness in handling simulation of a Formula SAE

vehicle

Ozdemir Ozerem and Denise Morrey∗

Department of Mechanical Engineering and Mathematical Sciences, Oxford BrookesUniversity, United Kingdom

Abstract

The ability to model tyre dynamics precisely is often one of the most critical elements for realistic vehicledynamics control and handling investigations. The industry-standard empirical models are able to predict theimportant tyre forces accurately over a short range of vehicle operating conditions, which is often restricted tothe operating conditions experienced during the tyre testing process. In this paper an alternative and practicalmethod to model Formula SAE tyres has been proposed and studied in a series of possible running scenarios.A simple, analytically-solved brush type tyre model is considered for the physical part with the introductionof a novel approach for defining the contact length formulation that incorporates the influence of inflationpressure, camber angle and velocity while a set of ordinary differential equations are employed to predict thethermal behaviour of the tyre model, which are mostly based on an already-existing method that has not beenexperimentally validated before. The resulting tyre models provide realistic and informative behaviour of thetyre, which has the ability to consider the majority of the typical operating conditions experienced on a FSAEvehicle. The performance of the proposed tyre models are compared against experimental tyre test data, whichshow good agreement and indicates that the tyre models have the ability to give realistic predictions of the tyreforces and thermal behaviour in the case of thermal tyre model. Furthermore, the temperature-dependent tyremodel has been incorporated into a two-track model of Oxford Brookes Racing’s Formula SAE vehicle to studythe effectiveness of the tyre model during transient handling simulation. The resulting simulations suggest thatthe proposed tyre model has the ability to represent realistic operating conditions of tyres, and also that tyretemperatures influence the vehicle dynamic behaviour significantly during on-limit scenarios.

Keywords Tyre, dynamics, modelling, physical, thermal, vehicle, handling, simulation, FSAE

∗Corresponding author: Department of Mechanical Engineering and Mathematical Sciences, Oxford BrookesUniversity, Wheatley Campus, Oxford OX33 1HX, United Kingdom. Email: dmorrey@brookes.ac.uk

Introduction

The only component of a vehicle which provides con-tact with the road surface, and transmits the forcesand moments required for vehicle control are the tyres.Hence, the tyres are one of the most important compo-nents of a vehicle. However, there is still no theory thathas the ability to accurately and faithfully describe tyrebehaviour (Pacejka, 2012). As a result, tyre modellingplays a critical role in vehicle dynamics research, con-trol and simulation studies.

Existing tyre models can be categorised into; em-pirical, semi-empirical, and physical modelling ap-proaches. The empirical type tyre models such as vari-ations of ‘Magic Formula’ (Pacejka, 2012) are oftenregarded as the industry-standard tyre models for han-dling simulations. These curve-fitting type tyre mod-els provide a robust tool to accurately describe thetyre dynamics of passenger type vehicles, where thetyres are not operated at extreme and wide range ofoperating conditions during the laboratory or outdoortesting procedures. However, it is acknowledged thatthese tyre models fail to describe tyre behaviour sat-isfactorily in extreme scenarios, such as motorsport(Grob, Blanco-Hauge and Spetler, 2015), where thetyre temperature and sliding speed characteristics aresignificantly different and often highly changing. Onthe other hand, the physical tyre models with simplemodelling approaches allow easy computation, but alsolack accuracy. In contrast, more advanced and com-plex physical modelling techniques provide very goodaccuracy, but require great computational power anddetailed description of the tyre. In addition, the com-plex physical modelling approach does not easily allowinteraction with real-time applications, as is often re-quired in the current lap time simulation tools and racesimulators. Moreover, published work in literature fo-cussing on the effects of tyre temperatures on vehicledynamics is quite limited and would benefit from fur-ther investigation.

The aim of this paper is to apply basic tyre mod-elling methods to develop a practical, robust and effi-cient physical tyre model which follow analytical solu-tions, and to extend the physical model using thermaltyre modelling methods, which were not previously val-idated using experimental data, with further improve-ments in order to develop a thermo-physical tyre modelsuitable for Formula SAE vehicle simulations and ex-perimental validation of the developed tyre models.

Basic tyre modelling methods

A physical tyre model, such as the Brush model (Pace-jka, 2012), offers a good platform for tyre model devel-

opment as it utilises numerical and analytical meth-ods. The use of rather simplistic assumptions to char-acterise the contact patch dimensions, frictional mech-anism and bristle properties, such as adopting a con-stant value over a range of different operating condi-tions, creates an unrealistic tyre model with poor per-formance. Work by Sorniotti and Velardocchia (2008)and Kelly and Sharp (2012) attempts to model morerealistic brush tyre models. Sorniotti and Velardocchiainvestigated the behaviour of tyre tread elements byusing a brush model with parabolic contact pressuredistribution, which incorporates anisotropic stiffnessproperties, and have outlined the potential improve-ments for a brush model based on bristle and contactpatch characteristics. The model provided good corre-lation with experimental data (Sorniotti and Velardoc-chia, 2008). Kelly and Sharp used an isotropic brushmodel with an arbitrary contact pressure distribution.They incorporated a more realistic frictional mecha-nism between the road surface and the tyre rubber. Fi-nally, they considered simple mechanisms for the tyrethermal behaviour to derive a set of ordinary differ-ential equations to predict the temperature changes inthe tyre. Although they have proposed a great amountof theoretical work with the model, the model has notbeen compared against experimental data (Kelly andSharp, 2012). Sorniotti also proposed a semi-empiricalmethod to predict the tyre temperatures and incorpo-rated the effects in a brush model (Sorniotti, 2009).The model was based on an extension to the earliertyre model developed with Velardocchia (2008).

y

x

x

�xt

v

vmaxFy

C

��

��

bristles

z

a a

x

�u

Fz

x

�Fx

top view

side view

C

road surface

a a

Figure 1: Brush model behaviour in pure lateral(top view) and braking conditions (side view)(Figureadapted from (Pacejka, 2012)).

Figure 1 depicts the brush model concept. In the

1

brush tyre model theory, the tyre is considered to con-sist of bristles that represent the tyre tread, which areattached to the tyre belt. The carcass is responsible forsupporting the tread and consequently, the bristles. Inmost of the traditional forms of brush model, the car-cass is assumed to be a rigid element. The combinationof compliance between these components determinesthe elasticity of the actual tyre (Pacejka, 2012). Asthe tyre rolls, the bristles pass through the tyre contactpatch area, and are allowed to deflect, due to the driv-ing motion which generates adhesive forces at the roadsurface. The bristles are deflected in the contact patchuntil the local friction force can no longer withstandany further shear deformation and transitions into asliding state over the road surface (Kelly and Sharp,2012). The combination of the shear forces arisingfrom the adhesive and sliding regions determines theoverall tyre forces.

The traditional form of Brush model, as proposedby Pacejka (2012), is used as the reference mathemati-cal frame for the physical tyre model described in thispaper. The sign convention used in this work is shownin Figure 2, which follows SAE conventions (Pacejka,2012). A combined slip approach is employed as thetyre is known to operate at its maximum potentialin motorsport, and is expected to be in a combinedstate of simultaneous lateral and longitudinal deflec-tions. The main assumption of the tyre model is theparabolic pressure distribution across the tyre contactpatch. According to Kim and Savkoor (1997), thisis the case when small vertical loads are considered.Hence, as low vertical loads (up to ∼2 kN) are usu-ally experienced with the FSAE vehicles, this seemsto be a reasonable assumption for this application andmakes the solution of the tyre forces simpler by provid-ing analytical solutions (Pacejka, 2012). The bristlesare assumed to respond differently to lateral and longi-tudinal deflections, so that the tyre is assumed to haveanisotropic stiffness and frictional properties. Lastly,the bristles are assumed to have identical characteris-tics along the contact patch length.

Brush model mathematical formulation

The practical quantities of the longitudinal slip, κ, andthe slip angle, α, do not allow for a reliable solution incombined slip conditions. According to Pacejka (2012)these quantities are normalised by representing the pa-rameters σx and σy as the individual theoretical slipquantities for the longitudinal and lateral directions,which are given by (Pacejka, 2012):

σx =κ

1 + κ, σy =

tanα

1 + κ(1)

z

y

x

Fz

Fy

Fx

�

�

C

Figure 2: SAE tyre sign conventions.

Effects of ply-steer are also incorporated in this workas built-in slip angle (Pacejka, 2012).

The resultant theoretical slip, σ, of the tyre in thecombined slip conditions can then be given as:

σ =√σx2 + σy2 (2)

These slip quantities affect the longitudinal and lat-eral deflections of the bristles within the contact patch,based on position of a tread element along the contactpatch length, x, and half the contact patch length, a,as depicted in Figure 1. Within the adhesion region,static frictional forces apply to the bristles which causedeflection in both longitudinal and lateral directions (uand v) depending on tyre slip conditions. These deflec-tions are represented respectively as (Pacejka, 2012):

u = (a− x)σx, v = (a− x)σy (3)

The deflections of the bristles are the main contributorsto the shear forces generated at the adhesion region ofthe contact patch. Therefore, the longitudinal and thelateral contact forces per unit length (qx and qy) withinthe adhesion region can be expressed by assuming thatthe bristles behave as linear springs with anisotropicstiffnesses (kpx and kpy):

qx = kpxu, qy = kpyv (4)

The tyre longitudinal and lateral forces for an as-sumedly rectangular contact patch shape with length2a and width 2b for complete adherence with the road

2

surface can then be given by integrating the contactforces over the contact patch area:

Fx =

∫ b

−b

∫ a

−a

qx dx dy

Fy =

∫ b

−b

∫ a

−a

qy dx dy

(5)

Analytical solution to the equations (5) for null tyreslip conditions then gives the longitudinal stiffness andcornering stiffness expressions of the tyre:

CFκ

(∂Fx∂κ

)κ=0

= 4a2bkpx

CFα

(∂Fy∂α

)α=0

= 4a2bkpy

(6)

In this work, the longitudinal stiffness and corneringstiffness of the tyre follow empirical definitions (Smith,2003), which are given by:

CFκ = CFκ0 · dFze−ccfx(dFz−1)

CFα = CFα0 · dFze−ccfy(dFz−1)(7)

where CFκ0 and CFα0 are the reference longitudinaland cornering stiffnesses of the tyre, respectively. Theuse of an exponential type empirical law for defin-ing the longitudinal and cornering stiffnesses of thetyre eliminates poor extrapolation performance of themodel at higher vertical loads, unlike the bevahivourof the linear fit (Sorniotti and Velardocchia, 2008) orpolynomial type empirical models. The fitted coeffi-cients ccfx and ccfy determine the magnitude of changein longitudinal and cornering stiffnesses in relation tothe vertical load variation acting on the tyre. The ver-tical load ratio dFz is given by:

dFz =FzFz0

(8)

where Fz is the vertical load acting on the tyre, whileFz0 is the reference or nominal vertical load. Themodel behaviour against measured data in the lateraldirection for a range of vertical loads is shown in Fig-ure 3, which shows good agreement. Use of equations(6 to 8) then enables to estimate the bristle stiffnesseskpx and kpy at different vertical loads.

However, the contact patch can no longer be char-acterised by full adhesion in conditions where the tyreis rolling with the presence of slip and limited friction.Hence sliding commences within the contact patch atthe point where the bristle contact forces exceed thelimits of the frictional forces available at the road sur-face. Based on the traditional brush model theory, this

0 500 1000 1500 2000 2500

Vertical load [N]

0

100

200

300

400

500

600

700

800

Cor

nerin

g st

iffne

ss [N

/deg

]

Meas.Model

Figure 3: Cornering stiffness model in comparisonwith measured data (Meas.) at different vertical loads.

occurs when the resultant contact force per unit lengthexceeds the static frictional limits, and in this work itis assumed that the contact patch is characterised bysingular adhesion and sliding regions in order to allowan analytical solution. The transition point, xt, wherecontact transitions into the sliding region is determinedby solving the following condition:√(kpxσx

µsx

)2+(kpyσyµsy

)2(a− xt) =

3Fz8ab

{1−

(xta

)2}(9)

where µsx and µsy are the static friction coefficients inthe longitudinal and lateral directions, respectively.

Given the contact transition point is known, the tyreforces can then be computed by:

Fx =

∫ b

−b

∫ a

xt

qx · dxdy +

∫ b

−b

∫ xt

−a

σxσµkxqz(x) dx dy

Fy =

∫ b

−b

∫ a

xt

qy · dxdy +

∫ b

−b

∫ xt

−a

σyσµkyqz(x) dx dy

(10)where µkx and µky are the kinetic friction coefficientin the longitudinal and lateral directions, respectively,and qz is the contact patch vertical load distribution,which is assumed to have a parabolic shape along thecontact patch length with a uniform profile across thewidth of the contact patch that can be expressed by:

qz =3Fz8ab

{1−

(xa

)2}(11)

Coefficient of friction characterisation

Sharp, Gruber and Fina (2015) highlight the impor-tance of frictional forces at the contact patch by stating

3

that the tyre forces mainly depend on the friction be-tween the tread rubber and the road surface. The phys-ical phenomena which generate the frictional forces are,however, significantly complex. The frictional forcesare generally influenced by factors such as road sur-face and texture (micro and macro roughness), surfacetemperatures of the road and the tread rubber, and thesliding speed of the tread rubber (Sharp, Gruber andFina, 2015). Therefore, the typical approach of assum-ing a Coulomb’s friction law on rubber friction, whichuses constant values of static and kinetic coefficients offriction between the tyre and the road surface, is an in-effective way of accurately characterising the frictionalforces. Moreover, according to Mavros (2009), the slid-ing speed of the tyre is known to have a significant in-fluence on the magnitude and behaviour of the kineticfriction coefficient of the tyre. For instance, Pacejka(2012) uses a linear, velocity-dependent model for sim-ulating the friction behaviour of a bristle element, byapproximating this to the sliding speed of the tyre belt,as the tyre rolls over the ground surface. Nevertheless,this is a relative simplification, as the effects of bristletemperatures are neglected, and due to the fact that inreality the friction decay with increasing sliding speedcan be non-linear. As mentioned earlier, another influ-ence on the tyre rubber and the road surface friction isdirectly related to temperature. Earlier work by Kellyand Sharp (2012) proposed an empirical formulationto describe a master curve for the characterisation ofthe kinetic coefficient of friction between the rubberand the road surface, which was implemented in de-termining the frictional characteristics of the proposedtyre model in this work. The equation is a modifiedSavkoor-type empirical rubber friction model (Mavros,2009), which is a function of both the sliding speed ofthe bristles’ tips over the road surface and the surfacetemperature of the tyre tread. Shape factors cµvs andcµt are included in this exponential decay type model,that is a function of the sliding speed components inlongitudinal or lateral directions, Vsx,y, the differencebetween tread temperature, Tt, and the reference tem-perature, T0, in addition to the pre-defined limits ofminimum and maximum values of the friction coeffi-cient (µ0 and µm). The kinetic friction coefficient inthe longitudinal and lateral directions can be expressedby (Kelly and Sharp, 2012):

µkx = µ0 + (µm − µ0)e−[cµvs(log10(Vsx))−cµt(Tt−T0)]2

µky = µ0 + (µm − µ0)e−[cµvs(log10(Vsy))−cµt(Tt−T0)]2

(12)

In this work, the sliding speed used in determiningthe coefficient of friction is approximated to be equal tothe slip velocity at the tyre contact centre, as a result

10-2 10-1 100 101 102

log10

(Sliding Speed)[log10

(m/s)]

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Coe

ffici

ent o

f fric

tion

[-]

20°C 40°C 60°C 80°C 100°C

Figure 4: Variation of kinetic friction coefficient withrespect to bristle sliding speed for a range of tempera-tures.

of the rigid carcass assumption. However, in reality thesliding speed of the tyre rubber varies along the slidingregion of the contact patch, which may show variationsin local friction coefficient, and potentially influencethe frictional power associated with the sliding. Themodel proposed in this work considers steady-state de-flection of bristles in the sliding region, which considersthe slip speed to be equal to the sliding speed (Pace-jka and Sharp, 1991), an assumption taken in order toallow full analytical solution to the tyre forces givenby equation 10, hence neglecting the localised dynamiceffects. The effect of temperature and sliding speedon the kinetic friction coefficient for a typical FormulaSAE tyre tread compound, solved using equation (12)and generic values is depicted in Figure 4. Increasingthe temperature of the rubber leads to a horizontalshift of the peak friction coefficient. Similarly, Figure5 shows the sliding speed dependency of the kineticfriction coefficient across different tyre operating tem-peratures. The temperature effects are ignored for thesteady-state non-thermal model.

In addition, the reduction in friction coefficient as aresult of increasing contact pressure is accounted forusing a linear relationship (Kelly and Sharp, 2012).The reduction factor, Ccp, is given by:

Ccp = 1− cµcp · dPcp (13)

where dPcp is the global contact patch pressure ratioand the magnitude of reduction is hence controlled bythe coefficient, cµcp. The global contact patch pressureratio is given by:

dPcp =PcpPcp0

(14)

4

0 20 40 60 80 100 120

Temperature [ °C]

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Coe

ffici

ent o

f fric

tion

[-]

Vs = 0.5 m/s

Vs = 1.0 m/s

Vs = 2.5 m/s

Figure 5: Influence of sliding speed on kinetic frictioncoefficient.

where Pcp is the global contact patch pressure and Pcp0is the reference global contact patch pressure value.The reduction factor, Ccp, is then used to determinethe actual static and kinetic friction coefficients in thelongitudinal or lateral directions (µsx,y and µkx,y) us-ing (Kelly and Sharp, 2012):

µsx,y = µsx,y · Ccpµkx,y = µkx,y · Ccp

(15)

Contact patch geometry

The boundaries for the contact patch length are signifi-cant for determining the forces generated at the contactpatch, as indicated by equation (10). Furthermore, thecontact patch length is also responsible for determin-ing the position of the adhesion to sliding transitionpoint, xt. Based on experimental observations of theauthors using a glass rig test bench, in another study,the length of the contact patch was found to be moresensitive to the vertical load intensity than the contactpatch width, which was also found to show a small vari-ation for typical FSAE tyres (Ozerem, 2014). Hence,the contact patch width is assumed to have a constantvalue in the present work for simplicity reasons, that iswcp = 2b, where wcp is the nominal tyre tread width.The contact patch is also assumed to be rectangular.For slick-type racing tyres, there is no need to spec-ify a grooving or cut-out rate of the tread pattern andthe total contact patch area can be assumed to be in acomplete contact with the road surface. However, esti-mation of the tyre contact patch dimensions are fairlycomplex due to a number of reasons, such as the influ-ence of cornering forces, camber angle and conditions of

under-inflation or over-inflation. An experimental pro-cedure for determining the contact patch dimensionswould offer a precise method of empirical contact patchsize determination (Ozerem, 2014). Unfortunately thistype of testing is highly complex and would justify asingle, more detailed study on its own right, and is out-side the scope of this particular paper. Consequently,a simple theoretical method proposed by Gim (1988),that is based on the Pythagorean theorem, is adoptedin this work to determine the contact patch length, lcp,and half contact patch length, a:

lcp = 2a, a =

√R0

2 − (R0 − δz)2 (16)

Hence in this expression the contact patch length isrelated to the unloaded radius, R0, and vertical de-flection of the tyre, δz. The vertical deflection of thetread is directly proportional to the vertical load actingon the tyre and tyre vertical stiffness, Kz. Althoughthe tyre vertical stiffness is usually assumed to have aconstant value, analysis of the behaviour of a rollingtyre during vertical testing shows (Avon Motorsport,2015) that the vertical stiffness is dependent on severalfactors, primarily the inflation pressure, camber angleand angular velocity of the tyre:

δz =FzKz

, Kz = f(Pi, γ, ω) (17)

Assuming that the tyre responds linearly to these pa-rameters, the following new formulation of a tyre ver-tical stiffness equation is derived, based on a linear-fitof empirical data:

(18)Kz =Kz0{[1−(Pi0−Pi)λi] · [1−γλg] · [1−ωλav]}

where Kz0 is the reference vertical stiffness, Pi0 is thereference inflation pressure, Pi is the inflation pressure,ω is the angular velocity and γ is the camber angle ofthe tyre. Lastly, the parameters λi, λg and λav areused to represent the coefficients of a real life tyre’sresponse to the inflation pressure, camber angle andangular velocity respectively, but they require specificexperimental data to be determined. This simple andlinear description of the vertical tyre stiffness, however,does not take into account the dynamic behaviour ortemperature-related effects of the tyre. Nevertheless, itattempts to consider the important parameters whichaffect the vertical stiffness of a tyre, as it rolls on a flatsurface.

5

Model parameter estimation

The model parameters are estimated using a con-strained non-linear optimisation routine that min-imises the least square error between the measured tyreforce data and the simulated tyre behaviour. For thephysical tyre model, which neglects the temperatureeffects, a total number of 20 parameters need to be es-timated that accounts for the pure longitudinal, purelateral and combined slip conditions. This a signifi-cant improvement over the Magic Formula tyre models,which have an excessive amount of parameters (Pace-jka, 2012).

The physical model proposed above allows solutionof tyre force characteristics with constant values of rub-ber temperature and inflation gas pressure. This typeof model offers a reasonable tyre model to be usedin vehicle handling simulations, where the tempera-ture changes are not significant, or where steady-statebehaviour is of interest. The influence of tyre tem-peratures and inflation gas pressure variations are ac-counted with the addition of thermal modelling in thenext section.

Physical model experimental validation

The validation of the tyre model output against ex-perimental data is highly important to prove the ac-curacy of the tyre model estimations. However, accessto the experimental data for tyres is often very chal-lenging due to the limited availability of sophisticatedtyre testing facilities and resources. Furthermore, fora complete handling type tyre model validation, thetyres need to be tested at a wide range of operatingconditions. Avon Motorsport/Cooper Tires 6.2/20.0-13 Formula SAE specification tyres have been mod-elled in this paper, as used on the Oxford BrookesRacing Team’s vehicle in 2014 competitions. Accord-ing to Kasprzak and Gentz (2006), the ‘Formula SAETyre Test Consortium (FSTTC)’ was found to provideteams competing in the Formula SAE competition withhigh quality tyre data, necessary for vehicle handlingsimulations and vehicle development (Kasprzak andGentz, 2006). The experimental data was measuredat the well-known CALSPAN Tire Research Facility inthe USA where the tyres are tested at typical operationconditions experienced in the FSAE competitions. Theexperimental test inputs such as lateral and longitudi-nal slip, forward velocity, camber angle and verticalload acting on the tyres have been applied to the phys-ical tyre model in order to simulate a virtual tyre testbench. For the physical tyre model, the temperature

-15 -10 -5 0 5 10 15

Slip angle [deg]

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

Late

ral f

orce

[N]

Meas.PM

Figure 6: Lateral force behaviour of the physi-cal model (PM) in comparison with measured data(Meas.) as a function of slip angle.

effects are neglected. The 5.2 version of Magic Formulahas also been modelled and parametrised in a simi-lar way with the physical model in order to comparethe proposed models in this paper against a proventyre model. As this work is primarily concerned withthe cornering performance of a vehicle, the compari-son between the models and measured data will onlybe shown for a pure lateral slip case.

Figure 6 shows the comparison of the outputs fromthe physical tyre model and measured data for the purelateral slip test. The physical model is in very goodagreement with the measured data across the testedvertical load and slip angle ranges as the average fittingerror is 6.19%. On the other hand, Figure 7 shows com-parison also with the Magic Formula model, in whichthe difference between the tyre models is almost in-distinguishable. The MF5.2 provides a fitting error ofonly 0.46% less than the physical model. Therefore,even though the tyre temperature effects are not in-corporated, the proposed model performs quite wellwhen compared against both experimental data andthe industry-standard Magic Formula for the pure lat-eral behaviour.

Thermal model

The physical tyre model was extended in order to in-clude the effects of tyre, road surface and surround-ing temperatures with the tyre thermal modelling ap-proach proposed by Kelly and Sharp (2012). Themodel is adopted to a great extent, mainly because

6

0 500 1000 1500 2000 2500 3000 3500

Data point [-]

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

Late

ral f

orce

[N]

Meas. PM MF5.2

Figure 7: Lateral force behaviour of the physi-cal model (PM) in comparison with measured data(Meas.) and Magic Formula model (MF5.2).

of its degree of fit, and suitability with the physicaltyre model proposed in the earlier section.

In this thermal modelling approach, the tyre is as-sumed to consist of only three bodies for simplificationreasons. The main causes of the temperature changesare due to two factors. Firstly, the tyre heating processthrough work done, and secondly, the heat transfer be-tween tyre bodies, road surface and the ambient sur-rounding. The temperature throughout the depth ofthe tread and the carcass are assumed to be uniform.This is also the case across the contact patch area.

Tyre heating process

The primary reasons for tyre heating are based on twofactors:

(i) The tyre deforms due to carcass deflection as itpasses through the contact patch. This processcauses energy dissipation, which contributes toheating of both the tread and the carcass.

(ii) The frictional power acting on the tyre in the slid-ing region of the contact patch, as the tip of thebristles slide over the road surface.

It is assumed that the deflection of the carcass is di-rectly related to the travelling velocity of the tyre, andthe total forces acting on the tyre in the longitudinal,lateral and vertical directions. Efficiency parametersare used in addition to control the amount of heat gen-eration in all directions (Kelly and Sharp, 2012). The

deflection power in the longitudinal, lateral and verti-cal directions are given as:

QDPx = ηxVx|Fx|QDPy = ηyVx|Fy|QDPz = ηzVx|Fz|

(19)

where QDPi, i = x, y, z and ηi, i = x, y, z stands for thedeflection power and efficiency terms in the longitudi-nal, lateral and vertical directions, respectively. Thenet deflection power can be given by:

QDPnet = QDPx +QDPy +QDPz (20)

Simple terms describing the frictional power, QFP ,are used to calculate the tyre heating within the con-tact patch. In contrast to the model proposed by Kellyand Sharp (2012), the model considers the fact thatonly a part of the vertical load across the contact patchis carried by the sliding portion of the contact patch.The heating of the contact patch are hence related tothe frictional forces arising from the sliding portion ofthe contact patch and the sliding speed of the tyretread on the ground surface:

QFP ={|Vsx|·|Fx,slide|+|Vsy|·|Fy,slide|

}RRT (21)

where RRT is a constant which determines the propor-tion of the heat energy that the tyre receives from thisfrictional power.

Heat transfer process

The heat transfer processes between the tyre bodies,road and ambient surroundings have a significant ef-fect on overall temperature across the components oftyre. Constant values for the ambient and the roadsurface temperature are considered here, as the tem-perature changes of these can be considered as negli-gible. Simple principles of thermodynamics are usedto determine the heat transfer around the tyre bodies.For instance, the tread conducts heat to the road sur-face in the adhesion region and convects heat to theambient surroundings, but receives a proportion of theheat generated by the friction power and deflection.

The heat transfer between the road and the ambientsurroundings is related to the heat transfer coefficient,h, and the temperature difference between these bod-ies. The equation is given as:

Q2−5 = h2−5(Tt − Ta) (22)

where Ta indicates the surrounding temperature.Qj , j = 1, 2, 3, 4, 5 and hj , j = 1, 2, 3, 4, 5 are the heat

7

transfer rate and heat transfer coefficient between road,tread, carcass, inflation gas and ambient, respectively.The heat transfer between the tread and the road sur-face is given as:

Q2−1 = h2−1(Tt − Tr) ·Acp,adhere (23)

where Tr indicates the road surface temperature andAcp,adhere indicates the non-sliding portion of the con-tact patch area.

The heat transfer between the tread and the carcassis given as:

Q2−3 = h2−3(Tt − Tc) (24)

where Tc indicates the carcass temperature. The car-cass also convects heat to the ambient air around thetyre sidewall region, which can be given as:

Q3−5 = h3−5(Tc − Ta) (25)

The heat transfer between inflation gas and the carcassis given as:

Q3−4 = h3−4(Tc − Ti) (26)

where Ti indicates the inflation gas temperature.

Finally, the heat transfer around the inflation gas ismuch simpler, compared to the tread and carcass, asit depends only on the heat transfer with the carcass.The actual inflation pressure is then calculated usingideal gas law. Several other factors which influencethe heat transfer around the tyres in motorsport use,such as heat radiation from brakes, engine, exhaust,wheel rims, road surface or sunlight are neglected dueto added complexity in description of accurate thermalsurrounding conditions. However, the main factors areconsidered (Kelly and Sharp, 2012).

Rate of temperature change

The heat generations and heat transfer between thetyre bodies are then used to determine the rate of tem-perature changes for the tyre. The set of ordinary dif-ferential equations to solve the tread, carcass and in-flation gas rate of temperature changes are related tothe specific heat capacity, cp, and mass of the bodies,m, and can be expressed by:

Tt =QFP +QDPt −Q2−1 −Q2−3 −Q2−5

cp,tmt

Tc =QDPc +Q2−3 −Q3−4 −Q3−5

cp,cmc

Ti =Q3−4

cp,imi

(27)

where the subscripts are: tread, t; carcass, c; inflationgas, i. Additionally, the deflection power is distributedbetween the tread and the carcass using a constant,RCT , which determines the amount of power the treador the carcass receives:

QDPt = QDPnet ·RCTQDPc = QDPnet · (1−RCT )

(28)

Additionally, the influence of tyre temperature on thelinear working range of the tyre is accounted for usinga linear drop of cornering stiffness with increasing tem-perature, similar to the model proposed by Mizuno etal. (2005). As the proposed model is a time-dependentmodel, the tyre transient properties are accounted forusing a first-order approach based on tyre relaxationlength (Pacejka, 2012). In this approximation, the lat-eral relaxation length, τy, is defined as:

τy =CFαKy

(29)

where Ky is the lateral stiffness of the tyre.

Thermo-physical model experimentalvalidation

The extended physical model with the addition of thethermal model or in other terms, the ‘thermo-physicalmodel’, has been simulated against the experimentaldata in an identical virtual tyre test bench using theFSTTC tyre testing inputs, in a similar fashion to thephysical model validation described in the earlier sec-tion. The parameter estimation firstly involves in iden-tification of the coefficient of friction and corneringstiffness related parameters by fitting the model pa-rameters using the measured lateral force data, simi-lar to the steady-state physical model parameter iden-tification process. Certain empirical-based thermalmodel parameters are then determined for completetyre model parameter identification. It should also benoted that the experimental data consisted of discon-tinues in time and force data points in between slipangle sweeps and hence the full and continuous tem-perature history of the tyre testing was not available.Therefore, the initial tread temperature of the tyre atthe beginning of each sweep is assumed to be equal tothe measured data.

Figure 8 and Figure 9 show the results of the thermo-physical model from the virtual test bench in compar-ison with the experimental test data and demonstratethat the model is able to provide a good estimation ingeneral for both the lateral force and the tread tem-perature of the tyre. The average fitting error of the

8

-15 -10 -5 0 5 10 15

Slip angle [deg]

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

Late

ral f

orce

[N]

Meas.TPM

Figure 8: Lateral force behaviour of the thermo-physical model (TPM) in comparison with measureddata (Meas.) as a function of slip angle.

thermo-physical model with respect to the measureddata is 6.09% for the lateral force and 8.50% for thetread temperature output. In comparison with thephysical model experimental correlation shown in Fig-ure 6, the model performance improves by 0.1% onaverage fitting error and the thermal model has thecapability of capturing the hysteresis behaviour at thelinear range of the tyre operation at small slip anglesdue to relaxation effects and at higher slip angles dueto frictional changes caused by heating/cooling of thetyre, both of which the physical model is not able toconsider. Furthermore, it is shown in Figure 8 that therelaxation length estimation of the model is of goodaccuracy for different vertical loads and tyre temper-atures. Analysing the tyre model’s tread temperatureestimation (Figure 9), even though the model is basedon simple principles and uses significant assumptions,the general trend is of acceptable accuracy. The mostsignificant flaws in performance of the tyre model pre-diction occur at the lowest and highest vertical loadingcases between 38 and 64 seconds, which suggests thateither the friction or contact patch size estimation ofthe tyre model is not able to produce realistic valuesin such cases. In addition, the cooling behaviour of themodel supports the latter observation, as the model isnot able to provide a sufficient correlation in temper-ature during cooling phases of certain vertical loads,which is most likely to be related with the incorrectsizes of adhesive and sliding regions within the contactpatch.

Figure 10 shows the model behaviour operating ata higher slip angle range, where the advantage of thethermo-physical model over the physical model is high-

0 10 20 30 40 50 60 70

Time [s]

10

15

20

25

30

35

40

45

50

Tre

ad s

urfa

ce te

mpe

ratu

re [

° C

]

Meas.TPM

Figure 9: Tyre tread temperature prediction of thethermo-physical model (TPM) in comparison withmeasured data (Meas.) with respect to time.

lighted. It is shown that the thermo-physical model isable to capture the temperature related lateral forcechange at higher slip angle similar to the the lateralforce trend of the measured data, which is not the casewhen the physical model is simulated. Even thoughthe model is not able to provide a perfect correla-tion against measured data, resulting simulations ofthe thermo-physical tyre model suggests that it couldprovide a useful platform to investigate the tyre tem-perature related effects in vehicle dynamics studies, es-pecially in motorsport, as it is shown to capture impor-tant tyre model behaviour with respect to tyre temper-ature variations.

Vehicle dynamic simulation

Undoubtedly, the main interest of racing teams isto improve vehicle performance. The best practicein achieving improved lap times is usually throughanalysing computational vehicle simulation for a va-riety of manoeuvres. As this paper is aimed at inves-tigating and modelling of tyre temperature effects, thethermo-physical tyre model described earlier is incor-porated into a transient vehicle model using OxfordBrookes Racing’s 2014 FSAE vehicle parameters, andis simulated in a high speed, smooth and low speed,challenging type manoeuvres.

Vehicle model

The vehicle model used in this work is depicted in Fig-ure 11, and is a classical two-track model. The model

9

0 50 100 150 200 250 300

Data point [-]

-3500

-3000

-2500

-2000

Late

ral f

orce

[N]

Meas.Meas. trendPMPM trendTPMTPM trend

Figure 10: Comparison of lateral force estimationof the physical model (PM) and the thermo-physicalmodel (TPM) against measured data (Meas.) at higherslip angle range.

considers the yaw and lateral degrees of freedom asso-ciated with the vehicle movement. A constant forwardvelocity is applied to specifically study the pure steadystate cornering capabilities of the vehicle without con-sidering the longitudinal dynamics involved during ma-noeuvring. As a two-track model, it also considers thelateral weight transfer between all four wheels througha roll balance distributed between front and rear axlesbased on quasi-static assumptions. This enables inves-tigation of individual tyre temperatures and tyre be-haviour around the vehicle. In addition, a simple aero-dynamic model is used to determine the downforce, asa function of vehicle forward velocity, acting on eachwheel through a centre of pressure coefficient and a liftcoefficient. The aerodynamic balance is biased heaviertowards the rear, which provides a higher proportion ofthe aerodynamic downforce on the rear than the front,with the aerodynamic effects being more significant athigher speeds. The vehicle parameters are given in Ta-ble 1. Additionally, the relaxation effects of the tyreare ignored in vehicle simulation in order to generatesmoother results.

The vehicle kinematic equations of motion are givenas:

X = u cosψ − v sinψ

Y = v cosψ + u sinψ

r = ψ

(30)

where X and Y are the displacements of the vehiclecentre of gravity in the longitudinal and the lateral di-rections and ψ is the yaw angle of the vehicle. In addi-

b a

v

u

�

�

�

WF

WR

Figure 11: Representation of a two-track vehiclemodel.

Table 1: Vehicle model parameters.

Parameters Symbol Value Unit

Gravitational acceleration g 9.81 m/s2

Vehicle mass mv 238 kg

CG height hCG 0.28 m

Wheelbase l 1.6 m

CG to front axle distance a 0.83 m

CG to rear axle distance b 0.77 m

Yaw moment of inertia Iz 75 kgm2

Front track width WF 1.21 m

Rear track width WR 1.2 m

CG: centre of gravity.

tion, v and u are the vehicle centre of gravity velocitiesin the lateral and longitudinal directions, respectively.

The vehicle dynamic equations of motion are givenas:

mv(v + ur) = Fy,FL + Fy,FR + Fy,RL + Fy,RR

Iz r = a(Fy,FL + Fy,FR)− b(Fy,RL + Fy,RR)(31)

Vehicle control

As the forward velocity of the vehicle is fixed at a con-stant value, the only control on the vehicle is throughthe steering input acting on the front wheels. A par-allel steering geometry is assumed so that both of thefront wheels steer at the same rate, which ignores theeffects of Ackermann steering. A non-linear steeringcontroller proposed by Sharp, Casanova and Symonds(2000) is used in calculation of the steering input, δ,

10

0 50 100 150 200 250 300 350 400

X [m]

-20

-10

0

10

20

30

40

50

60

Y [m

]

Racing line

Vehicle path

Figure 12: Position of the vehicle during lane changemanoeuvre.

required to follow the desired trajectory line. The con-troller gets sample values of preview path information,ei:n, current lateral off-set error, e1, and angular posi-tion error, eψ and through a set of individual controlgains, K, calculates the required steering angle. Thesteering control input is given as (Sharp, Casanova andSymonds, 2000):

δ = Kψeψ +K1e1 +

n∑i=2

Kiei (32)

Vehicle performance

The vehicle model is simulated for two different sce-narios. The first one is a high speed and smoothlane change manoeuvre (Brayshaw, 2004). The sec-ond event is a typical Formula SAE competition skid-pad event (Formula SAE Online, 2016), which is lowerspeed but consists of challenging cornering events withtight cornering radii. The racing line for both scenar-ios has been generated based on the curvature profilealong the desired trajectory.

The lane change manoeuvre consists of two straightsections, two right corners and a tight left corner sec-tion. The skid-pad event consists of a short straightand two initial turns to the right direction followed bya transition to left turns for two times, before the fi-nal short straight section. The vehicle forward velocitywas set to 28 m/s and 10 m/s for the lane change andthe skid-pad simulations, respectively. The path fol-lowing ability of the vehicle model is shown in Figure12 for the lane change manoeuvre and in Figure 13 forthe skid-pad event. The vehicle is able to follow thedesired racing line with good accuracy as the distancefrom the vehicle centre of gravity to the desired pathdoes not exceed around 1 m in either of these simula-tions.

25 30 35 40 45 50 55 60 65 70 75

X [m]

-20

-15

-10

-5

0

5

10

15

20

Y [m

]

Racing line

Vehicle path

Figure 13: Position of the vehicle during skid-padevent.

The resulting tyre tread temperatures from simula-tion along vehicle trajectory for both events are shownin Figure 14. Although the variation in temperaturesis quantitatively not very high, analysing the temper-ature profile shows that the tyre model follows the ex-pected behaviour. For instance, the temperature dropswhile travelling on the straight sections, mainly due tolack of tyre slip. As the vehicle turns into the firstcorner of the lane change manoeuvre (Figure 14a), thetread temperatures rise at an identical rate, however asthe vehicle negotiates the much tighter corner, the tyretemperatures rise up with different rates in relation toeach other. This is related to the presence of increasedweight transfer at higher lateral accelerations (Figure15a). Consequently, the tyre temperatures are higheron the loaded side of the vehicle due to increased load-ing. Moreover, as the tyre temperatures are relatedto the sliding speeds and generated frictional forces,the tyres heat up significantly more on the tyres ex-periencing higher slip angles. This behaviour is moreevident on the skid-pad simulation (Figure 14b), wherethe front tyres are steered at higher values in order tonegotiate the much tighter corners (Figure 15b). Asa result, the front tyres generate higher slip angles,which improves the front tyre force generation while in-creasing the sliding speeds at the tyre contact patches.In addition, the aerodynamic downforce acting on thetyres are significantly less during this event, comparedto the high speed lane change manoeuvre, which dis-tributes the vertical loads more evenly between frontand rear tyres, as shown in Figure 16. Moreover, thevehicle’s roll balance is slightly biased toward the front,which increases the weight transfer on the front end ofthe vehicle compared to the rear.

During the skid-pad simulation, the vehicle initiallycompletes two right turns which increase the temper-ature of all the tyres. However, as the vehicle goesinto the transition for the left turns and then negoti-ates two complete left turns, the temperature on the

11

(a)

0 50 100 150 200 250 300 350 400

28

29

30

31

Tem

pera

ture

[°C

]

Lane Change

FL

FR

RL

RR

(b)

0 50 100 150 200 250 300

Distance travelled [m]

26

27

28

29

30

31

Tem

pera

ture

[°C

]

Skid-Pad

FL

FR

RL

RR

Figure 14: Simulated tyre tread surface temperaturesfor lane change manoeuvre (a) and skid-pad event (b).

front left tyre drops slightly, while other tyres keep onheating up at different rates. For instance, the frontright experiences the highest temperature rise rate, asa consequence of higher vertical load and slip angle.The simulated slip angles are shown in Figure 17.

Unfortunately, the studied manoeuvres do not im-pose a great amount of tyre temperature variationsdue to assumed simplifications, and hence do not al-low for a quantitative analysis of tyre temperature ef-fects on the vehicle performance. However, the qual-itative analysis is in agreement with the expected be-haviour. Additionally, even though the variation issmall, the changing tyre temperatures and resultinglateral force output of the tyres show that the vehi-cle does not achieve pure steady-state cornering phase,and the driver model needs to make adjustments insteering in order to stay on the desired path. In orderto study this effect in more detail, the steering is ap-plied as a constant step input that generates increasedtyre temperatures and significant variation in vehiclebehaviour. The forward velocity was set to 14 m/s forthis simulation. Figure 18 shows the vehicle trajectorywhere the vehicle initially has a greater turn radiusbut with time it reduces and becomes steadier. Thisis directly related with the tyre temperatures (Figure

(a)

0 50 100 150 200 250 300 350 400-20

-15

-10

-5

0

5

10

[deg

] or

[m/s

2]

Lane Change

δ

ay

(b)

0 50 100 150 200 250 300

Distance travelled [m]

-15

-10

-5

0

5

10

15

[deg

] or

[m/s

2]

Skid-Pad

δ

ay

Figure 15: Simulated steering control (δ) input andlateral acceleration (ay) response for lane change ma-noeuvre (a) and skid-pad event (b).

(a)

0 50 100 150 200 250 300 350 400-1500

-1000

-500

0

Ver

tical

For

ce [N

]

Lane Change

FL

FR

RL

RR

(b)

0 50 100 150 200 250 300

Distance travelled [m]

-1500

-1000

-500

0

Ver

tical

For

ce [N

]

Skid-Pad

FL

FR

RL

RR

Figure 16: Simulated tyre vertical forces for lanechange (a) manoeuvre and skid-pad event (b).

12

(a)

0 50 100 150 200 250 300 350 400-2

-1

0

1

2

3

4

5

Slip

Ang

le [d

eg]

Lane Change

FL

FR

RL

RR

(b)

0 50 100 150 200 250 300

Distance travelled [m]

-5

-4

-3

-2

-1

0

1

2

3

4

5

Slip

Ang

le [d

eg]

Skid-Pad

FL

FR

RL

RR

Figure 17: Simulated tyre slip angles for lane changemanoeuvre (a) and skid-pad event (b).

19), and as the tread temperatures increase the vehi-cle’s lateral acceleration capabilities increase too untilthe tyre temperatures reach their equilibrium point orget a steadier behaviour under the current operatingcondition (Figure 20).

According to the resulting simulations, the point atwhich the vehicle response is expected to become shal-low is after about 20 seconds and the pure steady-statebehaviour requires a significant amount of time or un-til all tyres reach their equilibrium point. However, inreality the effects of the tyre wear are also expectedto influence this behaviour dramatically, therefore itis unlikely that the vehicle will ever experience puresteady-state behaviour.

Moreover, investigating the vehicle cornering capa-bilities with the influence of tyre temperatures, apply-ing a sinusoidal type steering input with constant rateshows that the vehicle is able to corner at higher lat-eral acceleration with the increase in tyre temperaturesexperienced during this test. Figure 21 shows the yawmoment against lateral acceleration capability of thevehicle at different averaged front tyre temperatures.As the front tyre temperatures increase, the vehicleis shown to generate higher lateral acceleration whilemanoeuvring.

-15 -10 -5 0 5 10 15

X [m]

0

5

10

15

20

25

30

Y [m

]

Figure 18: Vehicle trajectory in constant radiussteady-state cornering simulation.

0 20 40 60 80 100 120 140 160 180 200

Time [s]

30

40

50

60

70

Tyr

e T

read

Tem

pera

ture

[°C

]

FL

FR

RL

RR

Figure 19: Tyre tread temperature variation insteady-state cornering simulation.

0 20 40 60 80 100 120 140 160 180 200

Time [s]

0

0.5

1

1.5

[rad

/s] o

r [g

]

Yaw rate

Lateral Acceleration

Figure 20: Yaw rate and lateral acceleration profileof vehicle in steady-state cornering simulation.

13

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Lateral acceleration [g]

-600

-400

-200

0

200

400

600

Yaw

mom

ent [

Nm

]

30

32

34

36

38

40

42

44

46

Ave

rage

d fr

ont t

yre

tem

pera

ture

[° C

]Figure 21: Yaw moment diagram with the influenceof tyre temperatures.

Conclusions

In this paper, a unique application of tyre modellingmethods for a Formula SAE type vehicle has beendemonstrated, which consequently provides a poten-tial tool for improved understanding of the vehicle be-haviour.

The combination of existing elements of the simplebrush tyre model and newly introduced elements re-sulted with the introduction of a steady-state physical-based tyre model and a time-dependent, thermo-physical model. Even though utilising mostly simplemethods and relationships, estimation of both of thetyre models showed reasonably good agreement withan industry-standard tyre model, and more impor-tantly with the experimental tyre test data. For in-stance, the introduction of a new approach to accountfor the load-dependent tread element stiffness provideda good improvement over the existing rigid-carcasstype brush tyre models with constant bristle or carcassstiffness. In addition, the unique vertical stiffness de-scription as introduced in the tyre models improved thebrush model’s ability to respond better to a variety ofoperating conditions that the tyres are expected to ex-perience while manoeuvring. It is also shown that thetemperature primarily effects the frictional behaviourof the tyre and secondarily the stiffness characteristicsof the tyre response. On the other hand the thermalbehaviour of the tyre model requires further investiga-tion, especially with lower and higher vertical loadingconditions, as the thermo-physical model was observedto provide a poor correlation against the experimentaldata under such conditions.

The implementation of the tyre models in tran-sient vehicle simulation was also found to be relativelystraight-forward and highly efficient as the proposedtyre model has shown that it has the ability to re-alistically replicate the thermal behaviour of the tyresduring vehicle manoeuvring. Moreover, both tyre mod-els were also found to require low computational effort.The analysis of vehicle simulation showed that the tyretemperatures significantly and immediately affect thevehicle’s cornering capabilities. Furthermore, it is alsoobserved that the vehicle requires a settling time untilit can achieve steady-state manoeuvring as the tyresare observed to require a certain time until they reachtheir equilibrium temperatures.

Funding

This research received no specific grant from any fund-ing agency in public, commercial or not-for-profit sec-tors.

Declaration of conflict of interest

The authors declare that there is no conflict of interest.

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14

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Appendix I

Notation

a Half contact patch lengtha Distance from front axle to centre of gravityay Vehicle lateral accelerationAcp Contact patch areab Half contact patch widthb Distance from rear axle to centre of gravity

CFα Tyre cornering stiffnessCFκ Tyre longitudinal stiffnessCcp Tyre friction reduction factorcp Specific heat capacityFx Longitudinal tyre forceFy Lateral tyre forceFz Vertical tyre forceh Heat transfer coefficientKz Tyre vertical stiffnesskp Bristle stiffness per unit lengthKy Lateral stiffness of the tyrelcp Contact patch lengthm MassPcp Global contact patch pressurePi Tyre inflation pressureqx Longitudinal contact force per unit lengthqy Lateral contact force per unit lengthqz Contact patch vertical load distributionr Vehicle yaw rateR0 Tyre unloaded radiusu Longitudinal vehicle centre of gravity velocityu Longitudinal bristle deflectionv Lateral vehicle centre of gravity velocityv Lateral bristle deflection

15

V VelocityVs Tyre slip or sliding velocitywcp Contact patch widthxt Contact adhesion to sliding transition pointα Lateral slip angleγ Tyre camber angleδ Wheel steered angleδz Tyre vertical deflectionη Tyre efficiencyκ Longitudinal slipλg Vertical stiffness camber coefficientλi Vertical stiffness inflation pressure coefficientλav Vertical stiffness angular velocity coefficientµk Kinetic friction coefficientµs Static friction coefficientσc Theoretical combined slipσx Theoretical longitudinal slipσy Theoretical lateral slipτy Tyre lateral relaxation lengthψ Vehicle yaw angleω Tyre angular velocity

Subscripts

a Ambientc Carcassi Inflation gask Kineticr Roads Statict Treadv Vehiclex Longitudinal directiony Lateral directionz Vertical direction

Abbreviations

CG Centre of gravityFL Front leftFR Front rightFSAE Formula SAEFSTTC Formula SAE Tire Test ConsortiumMF Magic FormulaPM Physical modelRL Rear leftRR Rear rightSAE Society of Automotive Engineers (formerly)TPM Thermo-physical model

16