A sign convention is adopted and a contact point is defined to describe the forces and moments acting at the tire-road interface. A commonly accepted definition is the ISO sign convention as shown in Fig.1, see ISO (1991). The wheel is considered to be an infinitely thin disk and the road is assumed to be locally flat. The distributed forces that are present between tire and road across the contact patch are lumped into three forces and moments that act at the contact centerC. The location of this contact centerCis defined by the intersection of three planes:
1. the road plane
2. wheel center plane through the plane of symmetry of the wheel 3. the plane through the wheel spin axisy′and normal to the road.
Three axis are defined at the contact centerC. Thex-axis is defined by the intersection of plane 1 and 2 and positive in the forward driving direction. They-axis is defined by the intersection of plane 1 and 3 and positive to the left. Thez-axis is normal to the road plane and positive upwards. The forces and moments that are generated between tire and road are assumed to act at the contact centerCand are expressed in thex,yandz-axis of the wheel as defined here.
Tire Characteristics and Modeling 49
y
Ω
rl re
r0
y
x z
Vx
A
C S
Fx,Mx Fy,My
Fz,Mz
Vsx
Vsy
γ
1 road plane
2 wheel center plane 3 wheel spin axis plane
Fig. 1 ISO sign convention with adapted inclination angle𝛾
Three forces and three moments can be distinguished:
∙ longitudinal forceFx
∙ lateral forceFy
∙ vertical force (or normal force)Fz
∙ overturning momentMx
∙ rolling resistance momentMy
∙ self-aligning momentMz.
It should be noted that in practice the forces and moments are measured at the wheel center Ausing a measuring hub or wheel equipped with strain gauges. In order to calculate the forces at contact centerC, both the location of the contact center and inclination angle𝛾are required.
Different tire radii may be distinguished. The free tire radiusr0equals the radius of the undeformed tire. The loaded radiusrlequals the distance between the wheel centerAand contact pointC. The effective rolling radiusrerelates the angular veloc- ity of the wheelΩabout the wheel spin axisy′with the forward velocityVxfor a freely rolling tire according to
re= Vx
Ω. (1)
Both the loaded radiusrland effective rolling radiusreare dependent on the vertical forceFz, as is illustrated in Fig.2. The loaded radiusrldecreases almost linearly with increasing vertical forceFz, whereas the effective rolling radiusreshows a smaller dependency onFz. A pointSmay be defined, which corresponds to the instant center of zero velocity for the freely rolling tire. The distance between pointSand the wheel centerAequals the effective rolling radiusre.
0 1000 2000 3000 4000 5000 6000 vertical force Fz [N]
0.24 0.25 0.26 0.27 0.28
tire radius [m]
r0 re rl
Fig. 2 Dependency of the effective and loaded tire radius,reandrl, on the vertical forceFz
The longitudinal slip𝜅of the tire is defined as 𝜅 = −Vsx
|Vx| = −Vx− Ωre
|Vx| , (2)
whereVsxequals the longitudinal sliding velocity of pointS,Vxequals the longitu- dinal velocity of the wheel centerAandΩequals the angular velocity of the wheel.
Note that for a freely rolling wheel𝜅andVsxare equal to zero by definition. When the wheel is locked and does not rotate (Ω = 0)𝜅will be equal to−1. The tire behav- ior can be considered to be linear for small values of longitudinal slip 𝜅, i.e. the longitudinal forceFxincreases linearly with the longitudinal slip𝜅,
Fx=CF𝜅𝜅 (3)
whereCF𝜅equals the longitudinal slip stiffness.
The side slip angle (or drift angle)𝛼of the tire is defined as tan(𝛼) = −Vsy
|Vx|, (4)
where Vsyequals the equals the lateral sliding velocity of pointS. The inclination angle𝛾is the angle between the normal to the road plane and the wheel center plane.
In the sign convention adopted here a positive inclination angle 𝛾 corresponds to a rotation about thex-axis in the negative direction, as shown in Fig.1. For small angles, up to a few degrees, the lateral forceFyand self-aligning momentMzdepend linearly on the side slip angle𝛼and inclination angle𝛾according to
Fy=CF𝛼𝛼+CF𝛾𝛾
Mz= −CM𝛼𝛼+CM𝛾𝛾, (5) where CF𝛼 equals the cornering stiffness,CM𝛼 the self-aligning stiffness,CF𝛾 the camber stiffness andCM𝛾the camber torque stiffness. With the adopted sign conven- tion of Fig.1 and (5) these stiffness are all positive for regular tires. An exception
Tire Characteristics and Modeling 51
Fig. 3 Lateral forceFyand self-aligning momentMz resulting from a side slip angle𝛼(a) and inclination angle𝛾(b)
(a) sideslip angleα- top view V
Fy C
−Mz
α V
C Fy
tp
α
(b) inclination angleγ - rear view/top view
Fy C
Mz
γ
Ω A
Fy C Mz
A V Ω
is the camber stiffnessCF𝛾, which may be zero or even negative force some truck tires. The pneumatic trailtprelates the self-aligning momentMzto the lateral force Fy, according to
tp = −Mz
Fy. (6)
The pneumatic trailtp can be interpreted as the moment arm of the lateral forceFy to produce a self-aligning momentMz, this is illustrated in Fig.3. This figure also shows the effect of an inclination angle 𝛾 on the lateral forceFyand self-aligning momentMz. Note that for a positive side slip angle𝛼and positive inclination angle 𝛾 the contributions to the self-aligning momentMzhave opposite sign.
Based on the preceding definitions, it is clear that the tire forces and moments are a function of various slip quantities and inclination angle. In a way the tire can be considered as a non-linear function with multiple inputs and outputs, as illustrated by Fig.4. In the next sections the focus will be on the relation between the inputs 𝜅,𝛼,𝛾,Fzand outputsFx,FyandMz. As already indicated in Fig.4there are many additional factors that have an influence on the tire forces and moments, but they all will assumed to be constant. In Sects.2,3and4the steady-state relations between the inputs and outputs will be considered. Thereafter the dynamic behavior of the tire will be discussed in Sect.5.
Longitudinal slip definitionIn some literature a different definition for the longitu- dinal slip is used. Brake slip is defined as:
longitudinal slipκ side slip angleα inclination angleγ
vertical forceFz turn slipϕ forward velocityV
longitudinal forceFx
lateral forceFy overturning momentMx
rolling resistance momentMy self-aligning momentMz
road surface (dry/wet/ice) inflation pressure temperature wear
tirefunction
input: output:
Fig. 4 The tire considered as a non-linear function
sxb= Vx− Ωre
Vx ⋅100%, (7)
and drive slip is defined as:
sxd= Ωre−Vx
Ωre ⋅100%. (8)
Both brake and drive slip have a value between 0 and 100%. For modeling purposes a continuous descriptions as given by (2) is more convenient. The effective rolling radiusremay also be referred to as the dynamic tire radius in some literature. Note that all definitions of longitudinal slip, (2), (7) and (8), will result in a division by zero, when the vehicle is standing still (Vx= 0,Ω = 0), which is undesirable from a computational point of view. In Sect.5this will be discussed further.
Sign conventionsOver time various sign conventions and axis systems have been used and standardized. The SAE and ISO sign convention use the same definition for thex-axis, but have opposite definitions of theyandz-axis. Both SAE and ISO define the tire side slip angle as
tan(𝛼) = Vsy
Vx. (9)
Furthermore SAE and ISO define a positive inclination angle𝛾corresponding to a rotation about the positivex-axis. Though this may seem intuitive at first, a positive side slip or inclination angle will then result in a lateral force in the negative y- direction. This makes the definition and interpretation of the cornering and camber stiffness cumbersome, as they will be negative for a regular tire. In the SAE sign convention the vertical forceFzwill be negative when the tire is loaded, which is also not very intuitive.
Tire Characteristics and Modeling 53
SAE Pacejka ISO Adapted ISO
longitudinal slip κ=−VVsxx, all agree side slip angle tan(α) =VVsy
x tan(α) =−VVsyx tan(α) =VVsy
x tan(α) =−VVsyx
positive side slip angle (top view)
Fx V
Fy
Mz
α V Fx
Fy
Mz
α V Fx
Fy
Mz
α Fx V
Fy
Mz
α
positive camber angle (rear view)
Fy
Fz
γ
Fy Fz
γ
Fy Fz
γ
Fy Fz γ
vertical forceFz negative positive (non-right handed system)
positive positive
longitudinal forceFx
Fx
κ
lateral forceFy γ= 0 γ >0
Fy
α Fy
α Fy
α Fy
α
overturning momentMx
Mx
α Mx
α Mx
α Mx
α
rolling resistance
momentMy positive positive negative negative
self-aligning momentMz
Mz
α Mz
α Mz
α Mz
α
Fig. 5 Comparison of tire sign conventions and slip definitions
Adopting the ISO sign convention for expressing the tire forces and moments and adding a minus sign to the ISO definitions for side slip and inclination angle results in an intuitive set of tire characteristics, as illustrated by Fig.5. This figure was first published by Besselink (2000). Note that with this sign convention a positive slip value (𝜅,𝛼) generally results in a positive force or moment and that a positive inclination angle𝛾results in an upward shift of the force or moment curve.