In the case of combined slip, the bristles will deflect both in the longitudinal and lateral direction. The evaluation of the combined slip case is similar to the pure side slip case. Next a summary of the equations of the brush model is given, which can handle both pure and combined slip conditions.
sx= 𝜅
1 +𝜅 (65)
sy= tan(𝛼)
1 +𝜅 (66)
scomb=√
s2x+s2y (67)
𝜃= 2kba2
3𝜇Fz (68)
Whenscomb<1∕𝜃then
F= 3𝜇Fz𝜃scomb (
1 −|𝜃scomb|+ 1
3(𝜃scomb)2 )
(69)
tp= a 3
(1 − 3|𝜃scomb|+ 3(𝜃scomb)2−|𝜃scomb|3 1 −|𝜃scomb|+1
3(𝜃scomb)2
)
. (70)
Whenscomb≥1∕𝜃
F=𝜇Fz (71)
tp= 0 (72)
The expressions for the longitudinal force Fx, lateral force Fy and self-aligning momentMzare:
Fx= sx
scombF (73)
Fy= sy
scombF (74)
Mz= −tp⋅Fy (75)
The required model parameters are the bristle stiffness per unit of length kb, the friction coefficient𝜇and half of the tire contact lengtha. The tire contact length2a depends on the vertical tire deflection𝜌, which is dependent again on the vertical forceFz. Assuming the tire to behave as a linear spring in the vertical direction, see e.g. Fig.2, the tire deflection𝜌reads
𝜌= Fz
kz, (76)
wherekzequals the vertical tire stiffness. A first estimate for the dependency of half of the contact lengthaon the tire deflection𝜌can be made by assuming that the wheel is a rigid circular disk that penetrates the road. The next equation can be obtained for half of the contact length,
Tire Characteristics and Modeling 77
0 0.05 0.1 0.15
vertical deflection /r0 [-]
0 0.1 0.2 0.3 0.4
half of contact length a/r 0 [-]
a/r0 = sqrt( 2 /r0 - ( /r0)2 ) a/r0 = sqrt( 0.5 /r0 + 3( /r0)2 ) measurements
Fig. 21 Experimental results and equations to describe half of the contact lengtha
a=r0
√ 2
(𝜌 r0
)
− (𝜌
r0 )2
, (77)
wherer0equals the undeformed tire radius. A comparison with experimental results shows that this is not a very accurate representation, since in reality the contact length is much shorter, as shown in Fig.21. By adapting the contact length equation in a pragmatic way, a better match with the measurement results can be obtained. The empirical equation forathen becomes
a=r0
√ 0.5
(𝜌 r0
) + 3
(𝜌 r0
)2
. (78)
Implementation: divisions by zeroWhen programming the brush tire model equa- tions, care has to be taken to avoid divisions by zero. This can occur at several instances:
∙ when both 𝜅 and 𝛼 are zero, the combined theoretical slipscomb (67) will be zero, and a division by zero will occur in (73) and (74). A solution is modify the denominator in expressions (73) and (74). Sincescomb≥0it can be replaced bymax(scomb, 𝜖), where𝜖 is a small positive number and the functionmax(a,b) returnsaifa≥borbwhenb>a.
∙ The brush tire model equations are only valid whenFz≥0. When a wheel lifts of the ground the vertical tire forceFzwill become zero and the calculation of𝜃 (68) becomes problematic. A solution is to evaluate𝜃for a small positive vertical force, e.g.max(Fz, 𝜖). With this modification the forcesFx,Fyand self-aligning momentMzwill still be zero whenFz= 0as a result of the multiplication withFz in expression (69).
∙ When the wheel locks up, the angular velocityΩwill be zero and the longitudinal slip𝜅will become−1. A division by zero will occur in the calculation of the the- oretical slipsx(65) andsy(66). In can be noted that the brush tire model equations have been derived under the assumption that the wheel is rotating in the forward direction (Ω>0) and that the bristles spend a certain time in the contact zone.
When the wheel is not rotating we basically get the brush model, as discussed in Sect.3.1. The resulting force will be equals to𝜇Fzand in the opposite direction of the sliding velocity
Fx= −𝜇Fz Vsx
√
Vsx2 +Vsy2
, Fy= −𝜇Fz Vsy
√
Vsx2 +Vsy2
. (79)
Assuming thatΩandVxcannot become negative, then𝜅≥−1. The division by zero can be eliminated by modifying the denominator of expressions (65) and (66) tomax(1 −𝜅, 𝜖). The results obtained then are still in agreement with (79).
Implementation: forward and backward drivingThe brush tire model equations have been derived for specific conditions, for example Vx>0 and Ω≥0. When implementing the brush tire model in a vehicle model also other conditions may appear, e.g. driving backwards, vehicle sliding backwards while the driven wheels still rotate in the forward direction, vehicle standing still, etc. To handle these condi- tions, it is more convenient to use the sliding velocitiesVsx,Vsyand rolling velocity Vrto calculate the tire forces and moments, instead of𝜅and𝛼. The theoretical slip equations have to be adapted. The expressions become
sx= − Vsx
max(|Vr|,Vr,min) (80)
sy= − Vsy
max(|Vr|,Vr,min) (81) where Vr,min is a lower boundary for the rolling speed. This value should not be selected too small (e.g. 1 m/s) as it will make the resulting differential equations numerically stiff, leading to long simulation times. It can also be verified that when the rolling speedVris equal to zero that (79) will be obtained. Furthermore the pneu- matic trailtpwill change sign when switching from forward to backward driving, and it will be zero when the rolling speedVr(or equivalentlyΩ) is equal to zero. This can be incorporated in a continuous way by modifying Eq. (75) to
Mz= −tpFytanh(10Vr). (82)